COMENIUS UNIVERSITY BRATISLAVA FACULTY OF MATHEMATICS AND PHYSICS

DEPARTMENT OF DIDACTIC MATHEMATICS

THE ROLE OF EMOTIONS AS TRANSMITTED BY THE USE OF

CARTOONS AND GAMES IN CONSTRUCTING MILIEU IN

NEGOTIATING MATHEMATICAL KNOWLEDGE IN

PRIMARY AND LOWER SECONDARY SCHOOLS

Doctoral Thesis by

CLAUDIA SORTINO

ADVISOR: PROF. FILIPPO SPAGNOLO

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INDEX

Presentation Body, thought and language:

emotion as the reason for changes in information…..………….pag.6

Aim of the thesis

- Thought and language in contextualising experience………………………………...pag.10

- Research questions and hypothesis………………………………………….………..pag.12

Chapter 1 Theoretical frame of reference: the relationship between the body’s emotional state

and the sensory experience of contextualising experience

Abstract…………………………………………………………………………………………pag.14

1.1 Introduction……………………………………………………………………………..pag.17

1.1.1 Theoretical frame of reference (Clamat’s cartoon).

Context in a cartoon:

space-time relationships and the phenomenon of clousure……....pag.18

1.1.2 Theoretical frame of reference (‘Guess the number’).

The use of games for negotiating meaning in passing from

a natural to a pre-algebraic language………………………..pag.21

1.2 The role of affect in learning mathematics and historical enquiry: the relevance of the history

of mathematics in teaching…………………………………………………………….pag.22

1.3 Methodology: tools and the teacher’s role…………………………………………….pag.24

1.3.1 The structure of the a-didactic situation in cartoons and games…………….....pag.25

1.3.2 Validation of the two experimentations…………………………….……….....pag.27

Chapter 2 Affective learning: the emotional state of the body and sensory experience in a

two-fold alternating of perceptive and recalled image.

Abstract………………………………………………………………………………………..pag.30

2.1 History………………………………………………………………………………....pag.36

2.2 Let us see what is inside the box………………………………………………………pag.37

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2.3 Memory and learning: the role of the hippocampus and the amygdala………………..pag.43

2.4 Perception and perceptive images……………………………………………………..pag.48

2.5 Typical elements of affective learning from the neuro-scientific point of view………pag.49

2.6 …. but can learning become appetitus noscendi (the wish to learn)…………………..pag.52

2.7 Motivation: what are its origins?....................................................................................pag.53

2.7.1 Motivation and reward………………………………………………………….pag.56

2.8 Cultural and environmental experience: neuronal Darwinism and the plasticity

of the brain……………………………………………………………………………...pag.59

2.9 Knowledge and social life: language and inferred communication…………………….pag.61

2.9.1 The triadic model of sign……………………………………………………….pag.61

2.9.2 Syntax and understanding………………………………………………………pag.62

2.9.3 Sharing knowledge:

- inferred communication…………….………...………………………………pag.62

- gestures in contextualisation…………………………………………………..pag.64

- Mirror neurons and reciprocity in inferred communication…………………..pag.64

2.10 The Theory of Embodiment.…………….………...……………………………………pag.64

Chapter 3: Cartoons as environments of mediation

Abstract ……………………………………………………………………………………….pag.76

3.1 Introduction: explaining the choice of cartoons as tools of mediation………….……..pag.80

3.2 Cartoons: general and specific objectives………….……………………………….….pag.81

3.2.1 Explaining the choice of cartoons as a substitute for a written text…….……...pag.82

3.3 The five senses of the cartoon………….………………………………………………pag.83

3.3.1 Perceiving images: seeing ? understanding…………………………………….pag.84

3.3.2 Seeing sounds and hearing silence……………………………………………..pag.87

3.3.3 The sensations of taste, smell and… touching images………………………....pag.88

3.4 The visual code of cartoons: structure and educational characteristics…………….......pag.89

3.5 History: the origins of comics…………………………………………………………..pag.93

3.6 The iconic role of images: Invisible Art and Scott McCloud’s point of view………….pag.96

3.7 The language of cartoons as an ‘environment’ for understanding written text……….pag.100

3.8 First Experimentation: Clamat’s cartoon

3.8.1 The experimental context: a sample…………………………………………..pag.102

3.8.2 Methodology: instructions and organisation of cartoons……………………..pag.102

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3.8.3 Structural analysis of each single cartoon and

the role of the tools of mediation…………………………………………...…pag.104

3.8.4 Conclusion of the experimentation:

- qualitative analysis of the experimental work, the teacher’s role and the tools used

in describing the problem……………………………………………………...pag.110

- results of the experimentation………………………………………………..pag.111

Chapter 4: Second Experimentation: introduction to pre-algebraic language in primary and

lower secondary schools. Experimental analysis of a a-didactic situation:

Guess the number’

Abstract………………………………………………………………………………………pag.112

4.1 Introduction…………………………………………………………………………..pag.115

4.2 What is meant by the term ‘language’………………………………………………..pag.116

4.3 References to various works on algebraic language:

- the use of a ‘symbol’ to indicate a number………………………………………….pag.117

- discussions in class………………………………………………………………….pag.117

- the aim of the teaching contract: teaching aim ………………………………….….pag.117

- learning by discovery……………………………………………………………….pag.118

4.4 Explanation of the activity and experimentations results…………………………….pag.118

4.5 The experimental context: the sample…………………………………………….…..pag.120

4.6 Phases in the game and a qualitative description of the experience…………………..pag.120

4.7 Instructions and teacher’s strategies…………………………………………………..pag.122

4.8 Qualitative analysis: results of teaching phase………………………………………..pag.123

4.9 Conclusions……………………………………………………………………………pag.128

Chapter 5: Conclusion

Abstract………………………………………………………………………………………..pag.129

5.1 The role of visual images and symbols in establishing a teaching context, which is easily

recognisable by the learner……………………………………………………………pag.129

5.2 Considerations on the importance of the social-cultural context in a teaching-learning

context, which permit the use of cartoons or arithmetical games…………………….pag.132

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Chapter 6: Information for constructing the milieu

Abstract…………………………………………………………………………………….. pag.136

6.1 Mathematical descriptors for interpreting a mathematical cartoon: a possible grammar for a

mathematical cartoon…………………………………………………………………pag.137

6.2 Textual analysis of a mathematical cartoon:

Donald Duck in the land of mathemagics…………………………………………….pag.143

6.3 Components of affective learning from a didactic-neurophysiological

point of view………………………………………………………………………….pag.149

6.3.1 The teaching aim: the importance of contextualisation in teaching………......pag.152

6.4 A teaching activity for introducing the use of cartoons in geometry: Math Maps……pag.154

Appendix

1 The discrete and the continuous:

aspects and methods for a philosophy of mathematics…………………………….…pag.160

2 How the brain constructs the visual image……………………………………………pag.165

3 Perceiving shapes and movement……………………………………………………..pag.174

4 An interview with Claudio Stassi (designer-advertising cartoonist)……………….....pag.181

5 Teaching units for Clamat’s cartoons…………………………………………………pag.188

6 An a-priori analysis of Clamat’s cartoons…………………………………………….pag.196

7 Learner-secretary’s protocol for the ‘Guess the number’ game……………………....pag.200

Bibliography………………………………………………………………………………..pag.203

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PRESENTATION

Body, thought and language: emotion as the cause of changes in information

Introduction

In this research I have studied the teaching/learning processes in lower secondary schools, with the

aim of constructing particular teaching environments (milieu) and refining particular ‘tools or

attitudes’, which the teacher can use in order to improve the understanding of mathematical topics.

Considering that ‘school learning operates by transforming conceptual thought’ (Vigotskij 1966,

Chapter 4), the teacher’s activity is connected to a logical perception, which is characteristic for

each and every learner. Indeed, every pupil is endowed with ‘simple’ and ‘elementary’ ideas which

characterise the initial phase of learning. When the teacher teaches, it must always be remembered

that, in any given moment, something is changing in the learners’ minds.

Guy Brousseau (1989a) defined the teaching concept of mathematics as science, ‘a science which is

concerned with the production and communication of mathematical knowledge and in what way

this production and communication are specific’; this science has specific study aims:

• the essential operation of disseminating knowledge, the conditions of this dissemination and

the transformations which it produces, both for knowledge and its users;

• the institutions and activities whose aim is to facilitate these operations.

My ideas about teaching emphasizes places attention not only the teaching phase but also the

possible repercussions in the learning phase.

The specific use of a game and particular tools or teachers’ attitudes in our experimentation are

favourable in mediating and negotiating mathematical meaning in class. It is important to underline

that the word ‘attitude’ indicates the totality of all those gestures, words and fictitious strategies

which the teacher can knowingly use in introducing various mathematical aims into a particular

atmosphere, one which emotionally involves the learner; in most cases, the teacher’s attitude is

declared and made explicit throughout the teaching phase. An important aspect in communicating a

mathematical aim will be to encourage the learner to take into consideration the problem proposed

as their personal problem (devolution), to be able to identify mathematical tools in resolving

problems. Devolution appeals to the motivation of learner (Spagnolo, 1998), who does not only

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have to accept the game proposed (a synonym for a situation) but they must research the best

strategies which will let the learner to win.

In my experimentation, I have created cartoons and the ‘Guess the Number’ game. One of the

theoretical reference models in both cases is Guy Brousseau’s Situation Theory, which is present in

the choice of methodology in creating the situation/problem. The theoretical reference paradigm for

this research is the systematic ‘knowledge-learner-teacher’ approach. Studies on possible

relationships between these three regard the theory of learning research including content, methods

and a paradigm. Knowing the research paradigm means possessing:

• a suitable language

• a suitable methodological tool and

• a suitable statistical tool

From the teacher’s point of view, checking the tools for learning research means:

• having an idea about the positive role of mistakes

• constructing a research paradigm with which to resolve teaching problems and

• having autonomous ideas with which to communicate the process and result of the research

The research theory of learning is an important element for teachers and it includes: knowing about

the research phases, constructing an a-didactic situation for the problem, formulating research

hypotheses which are appropriate to the a-didactic situation, and producing documentation for those

who do not know this activity.

A theoretical reference for analysing the teaching situation and learners’ behaviour during the game

(in addition to those of semiotic, epistemological, psychological and historical enquiry) is that of

neuroscience (Jean Pierre Changuex, Antonio Damasio and Edelman). In both experimentations

the a-didactic situation is based on a game and it was very interesting to study the parts of the brain,

which are concerned with analysing effective learning with a game-like activity. Analysing the a-

didactic situation with reference to mathematical cartoons takes into consideration the use of

specific graphic tools, used in creating cartoons, which seem to facilitate mathematical

communication. These tools are used in attempting to recreate an environment which is familiar

and easily recognisable by the learners.

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The guessing game ‘Guess the number’ was analysed and subdivided into specific phases when

used as a teaching tool for negotiating between arithmetic and pre-algebraic meanings, and the

iconic role of symbols. With this activity the passing from arithmetic and pre-algebraic language

was analysed by starting with the problem, which uses a purely natural language.

In our experimentation the use of cartoons is to verify and check the notions acquired or present in a

class but their use is also provided for when introducing new, mathematical concepts. It was

stimulating to analyse how a verbal language gradually develops and how its syntax and semantics

become ever richer and more complex; this is also the case with gradually-developing graphic

language. Commencing with a function which is predominantly narrative and pictorial, graphic

language is integrated ever more deeply with its cognitive and communicative functions, becoming

a most suitable ‘environment’ for the communication and learning of notions. Within a neuro-

physiological and Vygotskian perspective, the teacher’s role is fundamental as the mediator of this

process. By creating or choosing appropriate designs and suitable contexts, they must try to

stimulate interest and encourage social interaction. Indeed through these activities, children record

many experiences of movement in which their ‘cognitive unconscious’ (Lakoff & Nùñez, 2000) is

recalled, updated and made explicit. Regarding the multi-sensorial use of cartoons as linguistic

mediators, the approach to reality is fundamentally far from being a simple connection between

scholastic learning and everyday experience. In accordance with neuro-physiological studies, the

recent theory of Embodied Mathematics (Lakoff & Nùñez, 2000) and the results obtained from my

experimentation, I have sought to understand how certain scientific knowledge within the learners’

minds is constituted, considering that mathematical concepts, logical structures and tools used are

the basis of a real bodily experience (in the case of cartoons, it is ‘virtual’).

The study of teaching activities has been dealt with from many points of view:

• from the dynamics of interaction between problem situations and the learner, teacher-learner,

or between the learners themselves

• from the use of the teacher’s attitudes and precise tools with which to mediate and negotiate

meaning

• the learner’s behaviour and

• the teacher’s teaching strategies regarding the twofold role of teacher-tutor.

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Bearing in mind three important, theoretical strands regarding education:

• cognitive interaction (J Piaget, J S Bruner and D P Ausubel), as characterised by the idea

that education is first and foremost a passing of information. The aim, therefore, is to ensure

that the most possibly correct information reaches the learner.

• social interaction (Lev Semionovich Vygotskij [1896-1934] and Alberto Bandura) who

consider education as the product of the interaction between learner and teacher.

• contextual interaction (B F Skinner, R M Gagnè and L J Cronbach) who not only highlight

the importance of subjects but also the context in which interaction takes place,

this thesis falls within the field of neuro-physiological studies as regards cognitive processes related

to affective and motivational learning.

“…… We must study the structure as much as the function but we must study them inside the black

box and not only from the outside” (F J Crick).

For this reason, as part of my activity as teacher-researcher, I believe it important to inquire, within

the limits of scientific knowledge, how certain learning processes occur from a biological point of

view. I also wondered what effectively the role of emotions in learning is and if it would be

possible to use emotions not by ‘imposing’ knowledge but by encouraging learning to become a

‘wish to learn’ (apetitus noscendi, Changeux 2003).

“Some think that the most difficult problem is to explain the relationship between the mind and brain. I, on the other hand, do not find anything

difficult. The mind is simply what the brain does! The important question is: “What does the brain do?”…” (Marvin Minsky, 1989, Society of the

Mind)

In conclusion, we can say that the tools used for analysing and constructing these environments are:

neuroscience (Jean Pierre Changeux, Antonion Damasio, Edelman), epistemology, Situation Theory

(G Brousseau), Semiotics (McCloud), psychology (Lev Semionovich Vygotskij) and Embodiment

Mathematics (Lakoff & Nùnez). The components of this inquiry in the teacher-learner-knowledge

environment are: culture, sociality, images, symbols, sensory-motor experience, motivation and

affect in learning mathematics, the teacher’s role and the teaching environment (milieu). We can

consider the following:

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AIM OF THE THESIS

Thought and language in contextualising experience

The context or environment has an essential function in negotiating meaning as means of

communication in the teaching system. According to Chevallard and Johsua (1982), the context

comprises three components: the teacher, learner and the knowledge to be taught. Subsequently,

Cornu and Vergnioux (1992) introduced the concept of the noosphere, thereby considering the

social context which involves and interacts with the teaching system. The use of various linguistic

registers and personal, cultural experience are important elements in encouraging communication.

Vygotskij has drawn on Piaget’s assertion: “Being aware of an operation indeed means passing

from an action to a linguistic plan; it means, therefore, inventing it in your imagination, to be able to

express it in words” (Vygotskij 1990, p.227). Clearly, in order to pass from one to the other, the

action must be contextualised and then recognised and accepted buy the learner.

In order for understand and encourage the process of passing from an antion to a linguistic plan in

the teaching process, the connection between the ‘word and its meaning’ must be taken into

consideration; this is not stable but subject to a process of evolution (Vygotskij, Chapter VIII).

Observing various experiments regarding the evolution of the brain, it has been seen that in general

culture - sociability

representations, symbols, language

sensory-motor

experience

teacher’s attitude

affective factors and motivation

Epistemology Neuroscience

Historical enquiry Psychology

Semiotics Neuroscience

Neuroscience Psychology Embodiment

Situation theory

Neuroscience Semiotics

milieu or teaching environment

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this evolution is strictly personal since it is tied to a process of cultural, spatial-temporal

contextualisation of the experience, which, in certain cases, draws on an individual, emotional state

(Changeux 2003). The basis of this contextualisation recalls the concept of a medium or

environment (French, as used in English ‘milieu’), like that of a subsystem which interacts directly

with the learner (materials, games, etc). This milieu can be initially defined as the totality of that

which acts on the learner or that which the learner acts on (Brousseau 1977). One can think about

the interaction between learner and milieu, devoid of the substantial involvement of the teacher

(with the stated meaning of a teaching contract but the teacher can take on the role of tutor or

supervisor), as that which defines an a-didactic situation. Whilst, if one also considers an explicit

educational system (for example, the figure of the teacher), then one can speak of a teaching

situation.

“In a teaching situation, which has been prepared and created by a teacher, the learner generally has the task of interpreting the questions asked,

information given and obligations imposed, which are constant in the teacher’s teaching method. These (specific) habits of the teacher, awaited by

the learner and the learner’s behaviour awaited by the teacher, make up the ‘teaching contract’.” (Brousseau, 1980a p.127)

The a-didactic situation has been described by Brousseau (1986, p.50) as:

“The definitive and referential a-didactic situation, which typifies knowledge, can be studied in a theoretical way, but regarding the teaching situation,

as much for the teacher as the learner, there exists a type of convergent idea: the teacher must ceaselessly help the learner to remove all their teaching

strategies from the situation so that personal and objective knowledge is left”.

Sometimes milieu are defined on the basis of real concrete arguments, sometimes the intention for

choosing these arguments is added, and sometimes as something stable, on other occasions as

something which is developed and modified in the learner. In general, the function of the milieu is:

in the teaching system, to define that part connected to specific a-didactic uses, planned by the teacher and teaching aims but without the necessary

and constant presence of such aims (for example, without the direct participation of the teacher).

Brousseau (2000) has emphasised that

“the learner learns by adapting to a milieu which is constitutes a factor of contradictions, difficulty, disequilibria, a little like that which takes place in

human society. As the result of the learner’s adaptation, this knowledge reveals itself with new answers, which are the proof of learning”.

My main research problem is the study of the conditions in which knowledge is constructed, the

aim of which is its optimisation, checking and reproduction in schools. The teaching situations

which I considered are specific to the knowledge, which I wanted to be inculcated. By ‘teaching

situations’, I mean:

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a totality of relationships established in an explicit or implicit way between the teacher, learner (or group of learners) and surrounding elements

(instrumental or material), with the aim of students learning, that is, they construct a certain consciousness, which has been previously established.

In order that the learner construct their own knowledge, they must personally concern themselves

with solving the problem, which has been set in the teaching situation, that is, they must involve

themselves in the activity. It can, therefore, be said that the learner has now reached ‘the

devolution of the problem’. Originally (Brousseau, 1986), devolution was defined as:

“the action through which the teacher has the learner accept responsibility for a learning situation (a-didactic) or a problem and personally accept the

consequences of this devolution”.

My objective is to study the acceptance of this devolution, connecting it to affective learning in

order to emphasise the tie between the word and image and their meaning to the learner.

With the aim of reconciling the rigidity of mathematical language with the evolutionary nature of

the meaning of terms, Vygotskij has stated that:

“a complete elimination of discordances in favour of general and unconditionally correct expression

can be reached beyond language and its mathematical skill. We can only say one thing: our spoken

language, by virtue of its own fluctuations and discordances between grammatical and

psychological features, is habitually found in a state of equilibrium between the ideal of

mathematical harmony and fantasy in a never-ending movement, which we call ‘evolution”

(Vygotskij, 1990 p.339). In this direction an important perspective for teaching mathematics has

been opened up which also encompasses my work: theoretical research into neuro-physiology

regarding learning and the experimental analysis of interference, which moulds the learner.

This takes place between the meaning of the term in its internal language, common language (that

is, everyday extra-curricular experience) and the specific meaning of mathematical terms.

Research Questions and Hypotheses

The choice of carrying out experimental research which includes games or cartoons does not arise

only from the necessity of investigating concepts about learners, rather from the need to suggest a

new way of ‘doing’ mathematics, which appeals to a motivational state as regards personal needs.

Various objectives have guided me in the selection of teaching tools which are to be used in

structuring my experimentation. Some of these are:

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• studying multi-sensorial aspects in teaching and learning activities for mathematics

• analysing the game-like characteristics of mathematics in relation to the motivation and

interest of doing this type of mathematical activity (appetitus noscendi, J Changuex, 2003)

• developing a real sensitivity in the learners in interpreting and comprehending symbolic

images

• organising a grammar which is the most characteristic possible in creating and interpreting

a mathematical cartoon

• analysing, from a neuro-physiological point of view, the use of parallel and serial thought

by means of diagrams

• analysing the role and meaning of the graphic tools, used in creating cartoons, for students

which are recognised by the cartoon’s iconic code or those which have been introduced ad

hoc by the teacher (the teacher’s implicit tools)

• analysing the problem of mathematical communication in multi-cultural environments.

Regarding the ‘Guess the number’ game, my objectives are

• thoroughly analysing the relationship between natural and symbolic language

• analysing how the constructing of patterns intervenes in the process of anticipation

We can, therefore, outline the following research hypotheses:

H1 constructing teaching situations, involving a conscious use of cartoons to facilitate

devolution

H2 constructing teaching situations, involving a conscious use of arithmetic games to

facilitate devolution

H3 constructing learning/teaching milieu which encourage an instrumental use of

functional emotions as regards mathematical knowledge (from the learner’s point of

view)

H4 constructing learning/teaching milieu which encourage an instrumental use of

functional emotions as regards mathematical knowledge (from the teacher’s point of

view).

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Chapter 1

Theoretical frame of reference:

relationships between the body’s emotional state and sensory experience

in contextualising experience

Abstract

Regarding research hypotheses H3 and H4, in the first chapter of my thesis I have described and

justified the reference paradigms used in analysing and structuring my experimentation: G

Brousseau’s Situational theory, for methodology and neuroscience, and Vygotskian theory and

Embodiment theory for analysing the use of tools of mediation in the learner-teacher-knowing how

to teach system.

The choice of making use of cartoons and an arithmetical game is linked to using social contexts,

which are easily recognisable by the learner. One of the fundamental elements on which this

research has been based is a diligent enquiry into common knowledge as a starting point for

introducing mathematical concepts. As G Brousseau maintains (2002), “Learners’ behaviour

operates like a detector in the environment”. The interest in the aim of this research was motivated

by the hypothesis that topical and schematic processes located in conscious play-like activities can

facilitate the awareness of knowledge and contribute to developing competence in mathematics.

In specifying by topic hypothesis H1 (paragraph 1.1.1), my study regarding the use of cartoons in

teaching has its origins in an analysis, from a neuro-scientific point of view, in the word ‘game-like

tool’ and recent studies in Embodiment mathematics. Whilst watching a cartoon whose context is

recognised, we can be enchanted by a bodily, multi-sensorial and virtual experience; a cartoon’s

multi-sensuality is tied to its iconic code. By means of their perceptive nature, graphic descriptions

encourage the concepts to be constructed in the dynamic process of action/ understanding/

development. Various studies (Arzarello 2004) have recognised that gestures allow perception and

image to be superimposed, thereby fostering a processes of anticipation and contributing to the

construction of knowledge. Can there be the same effect if we observe gestures? In my research I

have attempted to explain how recognising gestures and the fundamental role (in this phase and the

anticipatory) of space, time and that magic phenomenon which McCloud calls ‘clousure’

(observation of the parts and perceiving the whole) is possible. The most important conclusions are

15

that the presence of the space-time context and clousure make the cartoon a powerful teaching,

multi-sensorial tool since it reproduces contexts which are easily recognisable in the same social

context. I believe that, in the absence of a familiar context, learning cannot take place.

One of the questions that I had set myself about the hypothesis H2 (paragraph 1.1.2) regarded the

close relationship between abstract thought and concrete experience in mathematical studies. I

wondered whether the leap towards abstraction could be an autonomous process or, if personal, it

could be connected to the teacher’s attitude (H3 and H4) and the teaching environment, in which

the situational problem is developed. Starting with a concrete situation (the ‘Guess the number’

game), I introduced specific considerations about the use of language and the meaning of actions.

Thus I attempted to motivate the learner in the use of specific, mathematical objectives. E

Fischbein’s observations (1981, p.23) about the importance of using concrete material is

particularly important, especially that which produces questions based on thinking about that

material. The second experimentation deployed Situation Theory and various studies from the field

of neuro-science and much effort was spent in defining the specific phases required for developing

the experimental activity. This greatly encouraged the creating of an environment in which the

student could actively participate by elaborating thought processes and motivating the need for

seeking solution formulae, which would allow the learner to win and have fun. The idea of solution

formula arises as a need which is tied to an ‘internal’ condition of necessity and the pursuit of well-

being in accordance with Changuex (2003), who defines ‘motivational states’ as conditions which

regulate and guide voluntary behaviour.

In paragraph 1.2 I have included a brief reflection on the influence of affective factors (hypothesis

H3 and H4), which are connected to mathematical activity. These factors constitute one of the

variables to consider in interpreting learners’ behaviour. As tools of research, I have dipped into

the fields of neuroscience, Vygotskian theory (regarding the importance of social interaction) and

recent studies in Embodiment mathematics. The use of a game-like tool combined with an

epistemologically-correct presentation and historical research into mathematical content have most

decisively contributed to the understanding and acceptance of a mathematical concept. Knowledge

about the historical use of tools, such as cartoons and arithmetic games, can also serve to make the

teacher aware of their use (hypothesis H4).

Paragraph 1.3 describes methodology (hypotheses H3 and H4). In organising the activity, I have

deployed Situational theory and semiotics and aspects of neuro-science, the latter to analyse the

16

activity’s game-like structure. With reference to Vygotskian analysis (regarding the nature of

‘common and theoretical knowledge’), the methodology used was to suggest the establishment of a

relationship between perceptions and concepts, which are common to learners and theoretical

knowledge. The teacher’s role is fundamental and it has been specifically described in both the

experimentations, referring to available tools. Referring to the methodology used in the second

experimentation, the following statements are fundamental: motivating the translation, triggering

the translation, explaining the translation phase and inducing meta-cognitive reflections, thereby

encouraging a process of generalisation. However, in experimentations involving cartoons, the

teacher must appear as neutral as possible, leaving the learner to interact only with the cartoon and

the guided questionnaire. In paragraph 1.3.1, I have summarised the four phases of the a-didactic

situation in cartoons and in the game, with passing reference to Situational theory: the instructions,

the action phase, formulation and validating phase. Using the notion of ‘games’ as a model. as

defined by G Brousseau (Spagnolo, 1998, p.102), I have described the strategies used in the

experimentations and how they can be checked from the learner’s point of view of the learner (H3)

and the teacher (H4).

Much attention has been placed on the validating phase, which has been described in paragraph

1.3.2. With cartoons, this takes place at the end of the activity when the teacher lists the various

answers to the questionnaire, thereby leading learners to falsifying their answers. With the ‘Guess

the number’ game, validating takes place at the end of each instruction. Thus, it can be deduced

that the research process for individual solution formulae tends to highlight each learner’s

conceptual errors; this then becomes an opportunity for revising their reasoning and mathematical

skill in trying to reformulate correct strategies.

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1.1 Introduction

By arguing my research hypotheses H3 and H4, this thesis can be located within various neuro-

phsysiological areas of research, concerning interaction and the development of the more important

cerebral activities, with reference to learning and the role of emotions. Some of these areas of

research are still at an experimental stage but my work has only made use of notions which have

been scientifically demonstrated. Moreover, the content of my thesis falls within the realms of G

Brousseau’s Situational theory and this has been much described in other publications (F Spagnolo

1999, D’Amore 1999). The theoretical construct has, however, been modified by references to

Vigotskian theory (Vygotskij 1990) (language and socialisation) and the use of particular tools for

mediating meaning, which have been analysed from neuro-scientific and Embodiment points of

view (Lakoff & Nùnez, 2000).

The starting point of my work is the study of neurophysiology in highlighting the merit of particular

attitudes and teaching tools which are relevant in encouraging the teaching and learning of

mathematical concepts. These particular tools are found in activities which recreate easily-

recognisable environments and the learners find the work easier and work better. The choice of

using cartoons and a game depends on making use of contexts which can be easily recognised by

the learner. With cartoons, the stimulus is wholly constructed and organised by their visual code

(and in the case of cartoons for teaching, by the teacher) or as the result of an enrichment which

takes place over time. The meaning in this case is derived from knowledge, in turn derived from

the tool used.

One of the fundamental components of the basis of my thesis is careful research into common

knowledge as a starting point for introducing mathematical concepts. In structuring

teaching/learning by modules, it is important to conduct an effective a priori analysis. The teacher

must be fully aware of these tasks: defining teaching aims, the skill necessary to fulfil them, content

and measurable results. The start point is not only how much is known about the subject but the

learner’s learning style, even with a structure hypothesis, supposing that the learner may construct

their mathematical knowledge engaging in this activity (M Ajello 2000).

With the twofold role of researcher-teacher, I have attempted to construe teaching/learning where

produced teaching material is available to others; the structure of teaching/learning emphasises the

entire procedure which has brought it into existence. My interest in this subject matter was

18

motivated by the hypothesis that the topical and schematic processes found in game-like activities

and appropriate teaching/learning situations can facilitate the awareness of knowledge and, in

general, contribute to developing competence in mathematical concepts:

The use of games and mathematical cartoons can encourage such a process since their multi-sensorial

involvement (from an affective point of view) includes games and cartoons in a motivational framework,

which binds them to a state of pleasure and, therefore, well-being.

1.1.1 The theoretical frame of reference (Clamat’s cartoon.)

Context in a cartoon: space-time relationships and the phenomenon of ‘clousure’

My studies regarding the use of cartoons in teaching have their origin in neuro-physiology and

related studies. My interest in carrying out this type of experimental research began with questions

I had set myself in reply to the following: “like a cartoon, a game is a game-like tool and children

can therefore have fun……”. But what does the expression ‘game-like tool’ mean? Do children

only have fun or can they learn something specific? Is it possible to use game-like tools as tools for

optimising the devolution of knowledge in learning/teaching? Looking at an image, what is it

exactly that captures the learner’s attention and why? Can cartoons be effectively used in

mathematics?

From a bodily experience point of view, when children see a cartoon they see an immediate image

of a known context, which is recalled by the spatial and temporal contexts of the creating of the

cartoon. By virtue of their perceptive nature, graphic representations encourage concepts to be

constructed in the dynamic process of action/understanding/development, precisely in the same way

as the functioning of a language. Various papers have discussed cartoons as ‘vectors of the

personal learning’ of mathematics and they supplement school teaching as teaching software and

exercise books for the holidays (Frederic Pourbaix 2001).

However, I believe that cartoons possess an extra quality when compared with these tools and this

is due to their game-like and multi-sensuality nature. What precisely makes a cartoon interesting

and enjoyable? There exists a paper (R Tortora and D Iannece 2003) in which, with the aim of

suggesting a constructivist approach along the lines of the Cartesian model, the learners are asked to

describe a story (The Seven Messengers, Buzzzati, 1958) with drawings. This work reveals that,

19

only by using graphic language, the learners can consciously express not only the flow of time but

also the space covered by the messengers.

Regarding the use of real situations in which the learner can ‘find themselves’, we can quote from

Vygotskij (1990, p.236):

“ the general rule of development can be found in the fact that the becoming aware of and developing competence really only belong to a higher stage

of the development of any function. These faculties appear late. They must be necessarily preceded by an unconscious and unintentional functioning

state of a given type of conscious activity. To be aware, it is necessary to possess that which one must be aware of. To achieve competence, it is

necessary to have that which must be subjected to our will”.

Applying the Vygotskian model of the development of the highest mental functions to the cognitive

functions of graphic language, we can discover, by analysing children’s Protocols, the choice and

organisation of data and the answer to the problem in the cartoon:

a) a natural cognitive function in their spontaneous interpretation of drawings for organising

the solution and describing the data;

b) their vast and conscious logical and communicative functions, which can be developed by a

conscious use of particular tools for constructing problems in a cartoon.

However, the innovative part of my analysis is the neuro-physical point of view: the child utilises

certain tools and pictures since s/he recognises the context in the cartoon and the space and time

relationships contained therein; s/he, therefore, knows what to do (and is motivated to do so!) with

the problem. The pattern of graphic representation recalls a context which has been organised

according to an easy-to-recognise relationship of order. One element which makes the use of

cartoons particularly important and magic as an mediators of meaning is the phenomenon of

‘clousure’ (Scott McCloud 1996a): “the observation of parts and the perception of the whole”. The

understanding of a text, represented and described by means of combinations of squares in which

part of a story is ‘drawn’, begins with the most critical and careful interpretation of each single

cartoon. From a strand of research, developed by a researchers in Parma, Italy regarding the

exploration of ‘primary intuition’ (the concepts of infinity, the infinitesimal part and continuity),

there is a paper by N Alberti (N Alberti, 2002) about the revelation of ‘intuitive’ continuity in

children from lower secondary school and his analysis was performed with cartoons. But where

does the intuitive continuity in children come from?

From a neuro-phsyiological point of view, the phenomenon of continuity is bound to the same

human nature which is present when looking at images and in letting oneself be involved by the

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skill in recalling experiences from one’s own background. Continuity will be said to exist for those

who recognise that particular context whilst, for those who have never had that experience, they

will see only the images in series, devoid of any semantic or historical significance. Knowledge

cannot exist without a context and a context cannot exist without a spatial and temporal

organisation of the events. I would like to suggest through my research the teaching potential of

cartoons, stating with neuro-physiological considerations in explaining bodily experience which the

reader has when reading a cartoon (to use M Pellitteri’s term, 1998) by means of our five senses:

reading or better still understanding a comic strip requires that the reader has some type of personal

experience, intuition of a context tied to the intuitive shapes of space and time and a continuity of

action, which is subjected to the discrete description in the cartoon.

Reading a cartoon requires a careful and detailed observation of the graphic elements which are

present according to: appropriate rules of the visual code, the number of cartoons, their type and the

time taken to narrate the story, which is usually different to the personal reading time of the comic

strip. A context is, therefore, created which typifies an initial semantic domain in which various

objects are read and interpreted. The next cartoon recalls another semantic context which, if

integrated into the preceding domain, allows us to interpret the event, which is narrated in the comic

strip. It is the important phenomenon of clousure which has permitted McCloud (Scott McCloud,

1996 a) to ‘relate’ the power of the cartoon as a medium through the sole use of cartoons. The fact

of clousure and the spatial and temporal contexts make the cartoon a powerful teaching tool because

it reproduces contexts which the learner can easily recognise. Indeed, the latter is encouraged to

solve the problem because they know the context and how to get about inside it: without a known

context, there could be no learning.

Notwithstanding the clear necessity of depicting the structure of universal meaning in the

educational field, it is necessary to recognise the importance of the socio-cultural environment in

interpreting this universality. As G Brousseau (2001) maintains:

“Learners’ behaviour operates like a detector in the environment”.

To this end, it is important to make a careful a priori analysis of the tools, used in creating cartoons,

and the language used in the game. Both must try to mediate the same structure in a group of

learners (even if with different nuances) in constructing unique knowledge.

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1.1.2 Theoretical frame of reference (‘Guess the number’).

The use of games for negotiating meaning in passing from

a natural to a pre-algebraic language

One of the fundamental questions in teaching mathematics is the researching of the close

relationships between abstract thought and concrete experience. Not wishing that teaching become

uninteresting (!), it is important to analyse the link between knowledge and experience in

mathematics. One of the questions that had I set myself was if the leap towards abstraction was an

autonomous process in children or, the leap being personal, if concerned the teacher’s attitude and

role. Given my experience in using the ‘Guess the number’ game and in order to construct a

mathematical thought, it is useful to start with a concrete activity in introducing specific

considerations regarding the use of language and the meaning of actions. In this way, an attempt is

made to supply the learner with effective and credible motivation is using specific objects when

‘doing’ mathematics. With the ‘Guess the number’ game, I tried to highlight the conditions

(inherent in the teacher’s positive and functional attitude) which are favourable to the complete

opening up of potentialities, which induce the passing from an arithmetic language to a pre-

algebraic one.

Generative models

With reference to the concrete/abstract dynamic, I believe that E Fischbein’s observations (E

Fischbein, 1981, p.23) are very important:

“Concrete models must be conceived in such as way that, as well as providing the child with the

necessary intuitive support, the child must also be able to free themselves of this support. The

material used must be capable of provoking questions to which the child may reply, having thought

imaginatively about the matter and its logical patterns. Teaching material which does not do this,

includes the answers or which answers questions which the child can answer are of little use. Worst

still is the risk of blocking or suffocating the mathematical thinking processes of the child”.

Fischbein has clarified that, when Piaget spoke of “a period of concrete operations”, the former did

not mean that the child should not break away from reality, only that concepts and operations

should (for him) directly correspond to concrete reality (E Fischbein, 1981, p.22). Moreover,

Fischbein has insisted on the importance of questions, the necessity of encouraging a child to use

their imagination and learn to logically justify their answers. He has also suggested not using any

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sensory subjects but generative models, the latter which capable of facilitating productive thought.

However, in order that conceptual constructions may conform to the teacher’s expectations, it is

necessary for the teacher to propose experiences which are tightly bound to the concept in hand.

When the teacher suggests that, during the game, the learner writes their own instructions and to

then play amongst themselves, my experimentation is trying to recall recent a recent experience.

Leaving their imagination to run free, the teacher waits for the moment when the need to calculate

their own solution formula arises; thereafter, they begin to reflect and logically analyse their

choices. The importance of the formula arises as a need and its generality makes it particularly

prominent. This experience proves itself as a tool for facilitating the learner’s productive thought in

inventing instructions and, therefore, in calculating formula which can be logically justified and

motivated.

From a neuro-physiological point of view, the motivation for calculating a formula comes from the need to have fun by

playing, a need which is connected with an ‘internal’ condition of necessity. Indeed, hedonistic factors regulate

motivational states, defined as internal conditions, which direct and regulate voluntary behaviour (Changeux, 2003).

Specific motivational states, known as impulses, imply a need and satisfying it is a source of pleasure. With the aim of

achieving an aim, impulses increase the level of supervision and, therefore, by diminishing the behavioural threshold,

the impulses strengthen the capacity to act.

1.2 The role of affect in learning mathematics and historical enquiry: the relevance of the

history of mathematics in teaching.

Much research has demonstrated that affective factors can influence decision-making in

mathematics. Whilst I was unable to dwell upon this research, I restricted myself to emphasizing

that the affective factor is one variable to consider when interpreting learners’ behaviour. We

should study their moments of difficulty so that we can intervene effectively, attempting to

understand the origin of their problems and interpreting their thought and behaviour. In my

opinion, neuro-physiological studies are an excellent tool of enquiry. Learners’ attitudes is one of

the main factors in motivation and a demonstration of the results of learning. I also believe that a

positive attitude can be encouraged by a presenting of the discipline which is most tied to the

learner’s accurately-recognised preferences (for example, the use of attitudes and game-like tools).

Subsequently, there will follow an introduction and a view of the discipline which is

epistemologically correct. A favourable situation linked to epistemology and the historical enquiry

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into mathematics would also favour the emergence of negative emotional reactions, fundamental in

understanding and accepting a concept.

Regarding affective learning and the use of hedonistic factors, we can recall the chapter on neuro-

physiology, with which I deal in this thesis. With reference to the importance of the historical

enquiry into the origins of specific mathematical topics, we can consider three fundamental factors

(B D’Amore, 1999; M Ajello 2000):

• knowing about the evolution of ideas is for the teacher of particular relevance regarding

epistemological descriptions and, therefore, didactic choices and, moreover, it is

indispensable for perceiving learning times and all those processes which require personal

re-elaboration (for example, the current definition for the limit of a function in scholastic

texts);

• telling learners about the origins of ideas, problems and theories which have forged the

mathematics which is taught today, making it more concrete, nearer to human needs and

more fascinating a method of inquiry;

• in addition to being imprinted into the memory like fables, stories about people and

anecdotes serve to demythologize mathematics and mathematicians, thereby creating a new

relationship with the discipline. This relationship is no longer mysterious but curious and

you can discovers that stories about people have their own real and personal story, which is

not only scientific. Studying mathematics, therefore, takes on human connotations which are

more acceptable to the learner. (For example, Carlo Federico Gauss’ (1777-1855) famous

mathematical principle of brilliantly and unexpectedly solving at the age of 8 the arithmetic

problem of calculating the sum formed by the addend 100:

1+2+3+4+5+………..+98+99+100).

I believe that historical enquiry can also be important regarding the use of particular mathematical

teaching tools (for example, mathematical games and cartoons), meaning that………

• knowledge about the historical use of tools, such as cartoons and games, for transmitting

information can contribute to making their use a conscious one by the teacher. In function of

the limits and capabilities of teaching tools, the teacher can make an informed choice about

the characteristics of the topic in hand.

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1.3 Methodology: tools and the teacher’s role

The methodology used takes account of neuro-physiological research into learning styles and the

role of emotions and, therefore, the use of tools and the teacher’s attitude (implicit tools); if studied

a priori, they activate in learners ‘an area of proximal development’. Thus these tools of mediation,

skilfully organised in mathematical cartoons or during teaching (various phases of the ‘Guess the

number’ game) transmit mathematical content or the main characteristics of theoretical knowledge.

With reference to the nature of ‘common and theoretical knowledge’, the Vigotskian analysis of the

relationships between common and scientific concepts emphasizes differences between learners’

common knowledge and theoretical knowledge, thereby offering a coherent perspective. This

analysis regards the systematic nature of scientific concepts and the proceeding of scientific

concepts from discourse to facts, the opposite of common concepts which precede facts in the

discourse. And it is in this perspective that the methodology which I will initially describe is

suggested in establishing a relationship between perceptions, learner’s common ideas and

theoretical knowledge. For example, when learners familiarise themselves with a cartoon, they

provide their own interpretation of the situation-problem in relation to their own background and

subsequently they suggest (often with excellent results) a correct solution to the problem.

The teacher should not be present during the activity but encourage learners to freely communicate

their observations and questions without being judge. I believe that (Spagnolo 1999):

a) the teacher should not intervene in the communication phase between learners either by

listening to an interesting proposition or a false statement. It is the situation which is

created which must be the validating source and choice of one strategy or another.

b) the teacher should convince the learners as to their neutrality but not indifference and

provide stimulation, their wish to succeed and ensure that the rules are respected.

Moreover,

c) the teacher should not refuse what is different from the teacher’s model and expectations,

s/he should not give importance only to the results but above all the processes (which can be

interesting), whatever the result.

d) the teacher should leave sufficient time for the learners to come up with personal answers by

means of trial and error and to teach them the necessity of verifying such answers.

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Regarding teaching tools, I believe the following to be important:

a) the teacher should correctly identify various meaningful aspects of mathematical thought to

make them become subjects of experience, discussion, explanation and, therefore, education.

b) the teacher should use various tools to facilitate the passing from experience to concept,

such as: bodily movement (or through an experience, like in the cartoon), games, curiosity,

imagination, discussion, description, symbolisation and the use of appropriate language in

the relevant semantic field.

In practical teaching, I believe the following four teaching statements to be important (this attitude

is more explicit in my experience than ‘Guess the number’). Having created a problem using the

most possible language, the teacher should

- justify the literal linguistic translation, in formulating the problem passing from natural language

to mathematical

- trigger the translation, motivating each learner to model the problem, according to their way of

reading and understanding the problem in ma thematical terms

- explain the fact that a natural language is being translated into a mathematical one, thus keeping

separate the natural world from the mathematical one and thereby assisting the learners to

understand where they are

- infer meta-cognitive reflections about the problem, trying to encourage generalisations.

1.3.1 The structure of the a-didactic situation in cartoons and games

The notion of ‘a a-didactic situation’ in Situational theory is fundamental. In order to understand its

importance, it is necessary to analyse the various phases which constitute the a-didactic situation

and, in my opinion, the teacher’s role therein. In a teaching situation, this situation acquires a

particular meaning since the logic in its structure is the only characteristic which motivates the

learner. But how is it possible to create such a situation without such a teacher?

In structuring the a-didactic situations in both my experimentations, I reflected a great deal about

the teacher’s role and appropriate attitudes so that his/her presence did not influence the learners’

personal reflections. In my experience, the teacher (as tutor) should implicitly intervene during a a-

didactic activity when the learners’ reflections and ideas are effectively acting as a break to the

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logical and productive stimulus. These interventions should not follow a logic, which is inherent in

the role of teacher, but an emotional strategy which does not alter the learners’ emotional state,

thereby projecting them from a game-like situation to one of teaching.

I do not think it by chance that G Brousseau attributes the notion of games (Spagnolo, 1998, p.102)

to teaching and, therefore, a-didactic situations, to modelling. Brousseau has also listed the possible

meanings for the term ‘game’ in defining possible strategies and ways of checking them. However,

Barra (1987, 52) has traced meaningful connections between games and the teaching of

mathematics, outlining a series of arguments with which to define the functionality of games with a

teaching aim. He confirms that “one of the most general characteristics of games is reducing or

neutralising difficulties and, consequently, a directed action, that is the tensions which develop in

the positive achievement of an action”. With this value, games allow for various stages: the

definition of the instructions, the action situation, formulation and validation, each of which

possesses a structure and specific characteristics, not the subject of improvisation.

First phase: instructions

A a-didactic situation always begins with specific instructions during which the teacher can play

with a learner to explain the rules of the game then have two learners repeat the activity. This not

only reduces any ambiguity in verbal language but it acts like feedback: one the one hand the

learner can repeat the situation to check and modify the action and, on the other, the teacher can

understand the feedback process, which has been activated by the learner.

Second phase: action

The action situation involves learner formulating their strategies or learning how to solve the

problem in hand. The learners act on the problem by formulating hypotheses and strategies (the

dialectic of action), which are put to the test by further experience. By means of a dialogue with the

action, the learner intuitively or rationally adopts a strategy, which had been previously rejected,

putting acceptance or refusal to the test, using experience. S/he subconsciously formulates an

implicit model, a set of relationships or rules on the basis of his/her decisions, without formulating

them.

Third phase: formulating phase

Formulating knowledge is confirmed by the capability of the subject to start again, recognise,

identify, separate and reconstruct such knowledge in a linguistic system. In this phase the learner is

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motivated by the situation to formulate their own implicit model, to verbalise their own strategies,

to discuss and defend them so that they are taken up by other learners. The starting point for each

learner is the progressive working out of a negotiated language between the learners, which should

lead them to formulating the strategy in a sort of formulating dialect. Both action plans and

formulation require empirical or cultural corrective processes, which are relevant to the knowledge

being learnt.

Fourth phase: validation

The cognitive models formed in the third phase can be accepted or rejected by the class; this is

metaphorically understood as a type of scientific community which discusses each strategy. The

hypotheses accepted by everyone become theorems whilst the mistakes encourage individual

reasoning to be reviewed to correctly reformulate the strategies. Thus the making of a mistake is an

indispensable stage in the process of building knowledge. With this phase, we can begin to

formalise mathematical concepts which, in traditional teaching methods do not represent a starting

point but a point of departure.

1.3.2 Validation of the two experimentations

My experimentation includes both cartoons and the ‘Guess the number’ game; in the former the

first three phases taken place in a personal analysis of each learner when s/he receives the cartoon

and guided questionnaire. The formulation dialect becomes more important if analysing the cartoon

with the questionnaire is not done by each learner but a pair of learners. In either case, this phase is

subsequently picked up again by the teacher who, once the task has been completed, lists all the

learners’ answers to each question in the questionnaire. Thereafter, a fourth phase follows:

validation.

The sequence of the four phases in the second experimentation is more explicit. The first concerns

the instructions: the action phase commences and, therefore, the game between learners and teacher.

We can observe that whether the teacher plays against the learners or vice versa, a phase of

symbolic transcriptions is written on the board which outlines various moments in the game and the

various strategies suggested by the learners. There follows the formulating and validation phases.

Below is a diagram of the activity:

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1st PHASE: action situation – 1st instruction

2nd PHASE: individual teacher-secretary T’s 1st implicit teaching

3rd PHASE: action situation – 2nd instruction

4th PHASE: individual teacher-secretary T’s 2nd implicit teaching

5th PHASE: game against the teacher – 3rd instruction

6th PHASE: individual teacher – secretary 7th PHASE: Discussion on strategies

teacher writes 1st instruction learners note

down result and the first guess

learners left to reflect

teacher writes strategies on board

1st validation phase

T begins 1st implicit teaching by translating into symbols

teacher writes 2nd instruction

learners note down result and first guess

learners are left to reflect

T transcribes other possibile strategies

2nd validation phase

T begins 2nd implicit teaching by translating into symbols

T writes the 3° instruction

teacher writes down strategies

teacher notes down results

learners left to reflect

3rd validation phase

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8th PHASE: teacher’s intervention is aimed at the functional recall of

the implicit interventions to encourage effective understanding of the

pre-fixed knowledge

9th PHASE: creation of an instruction – 4th instruction teacher-tutor

10th PHASE: individual: verification of

the development of acquired knowledge

It is important to observe that the validation phase is evaluated empirically by subsequent

instructions (by the teacher to the learners) to invent their own instruction and experiment with

them by playing against each other or the teacher.

As we will subsequently see, the experimentation confirms that research into solution formulae

originate as a necessity, a need by the learner to know how to have fun; thus, the activity is takes

place in a personal motivational state. Moreover, the research processes into individual solution

formulae tend to highlight each learner’s conceptual errors, which subsequently encourage learners

to review their own reasoning and mathematical skill, then to correctly reformulate the strategies.

learners write their own instructions

learners calculate the solution formula

learners left together to play

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Chapter 2

Affective Learning: the emotional state of the body and sensory experience in the

alternating between perceived and recalled image

Abstract

In arguing my two hypotheses H3 and H4 in this chapter, I studied affective learning by analyzing

recent studies in neuroscience. An understanding of psychology is essential in teaching but to

obtain results, the psychologist must combine behavioral studies with the parallel study of what

happens inside the brain. Learning, as an activity essential to survival which is tied to emotions, is

reinforced learning but what I will attempt to prove is if knowledge, as a primary impulse, is also

tied to a motivational state, which is located in a structure of self-gratification.

The first two paragraphs of this chapter are specific in that they treat various parts of the brain: how

it produces thought, functioning by electrical impulses, producing and receiving chemical signals in

transforming molecules. It was very interesting to discover that the various sizes of images of the

cortex in parts of the body (homunculus distorto) are similar not in size but the precision with which

cortex is controlled. Moreover, M Gazzaniga’s modular theory of the brain confirms that the

human brain is organized in a modular way, that is, relatively independent functioning units which

behave in parallel.

In paragraph 2.3 I analyzed the specific roles of the hypocampus and amygdala in learning and

memory. As part of the prosencephalon, the limbic system is composed of three structures: the

amygdala, the hypocampus and the hypothalamus; the latter is concerned with survival and it

controls vegetative functions and emotions. After briefly mentioniong two nineteenth-century

works by S Freud and H Bergson regarding memory and the importance of emotions and context in

memory, I have analyzed how a memory is imagined, recalled and connected to other experiences.

A determining role in this process involves the amydala and hypocampus; the latter is rich in

endorphin receivers and participates in the processing of emotions. Of particular interest is Teyler

and Discenna’s theory (1985), according to which the hypocampus probably has codified into its

neuron circuits a complete map of all the cortex circuits which can be activated after perceptive

experiences. In the short term, the hypocampus probably memorizes a trace or spatial-temporal

index of cortical activations. The repetition of signals probably reinforces the trace, thereby

facilitating its recognition. The hypocampus performs the functions of imagination and the recall

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of information. The amygdala is tied to the loss of global and anterograde memory. The

hypophyseal-hypothalamus axis regulates the production of hormones following a physical-mental

experience and bodily variations are performed by means of chemical signals, composed of

hormones and nerves. Hormonal variations in the body produced by a positive or negative emotion

are recorded by the amygdala.

With reference to my research hypotheses, paragraph 2.4 includes a definition of perception as an

experience involving the senses and perceptive images, ie all that which is filtered by our senses.

The latter create dispositional neuron frameworks (Damasco 1995) which, together with a

topographically-organized map, are shaped after a visual experience. The most important results in

this field concern the way in which experience is associated with semantic concepts (which become

conscious) and the procedure with which the ima ge is recalled. The hypocampus indexes the

experience, providing us with a space/time connotation with reference to mental images and the

images of mental states (of the body).

Elements characterizing learning and emotion from a neuro-scientific view are highlighted in

paragraph 2.5. If endorphins are released into our body, it is indicative of a gratifying action, which

takes place in a system of gratification and reinforcement. But recent research has shown that

(paragraph 2.6) circuit-producing dopamine in the Accumbens nucleus leads us to think that

learning can be perceived to be essential to survival if learning can also be tied to a state of not only

well-being but pleasure and self-gratification. This would thus create a dependence on the pleasure

derived from repeating the action (appetitus noscendi).

In paragraph 2.7, I have analyzed the concept of motivation, emphasizing what can explain changes

in behaviour. What is observed is that reaching specific objectives raises the behavioural threshold,

it determines the disappearance of one or more components of behavioural sequences, reducing

motivation and leading to the total elimination of the answer.

The distinction between motivation and reward in paragraph 2.8 is interesting. Reward

mechanisms are activated by external signals which stimulate the internal reactions of desire and

aversion. Reward comes in the form of dopamine. An area in the Accumbens nucleus, the so-

called shell, has an essential role in producing emotions due to its connections with the amygdala

and limbic system whilst its nucleus (core) is directly involved in motor control. This serves as an

interface between motivation and action, identifying the dopamine which has been released. These

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neuronal systems are implicated in creating dependence; will the same not happen with research

into the relationship between pleasure and knowledge and scientific research in general

(Changeux)? Using electro-physiological methods on the level of dopamine neurons, it has been

observed that in certain cases their activation no longer coincides with reward but anticipation

following learning (Schultz and Dickinson 2000).

The hypothesis proposed by Changeux states that acquisition of knowledge may be indirect and the

result of the selection of pre-representations, the so-called preliminary models. Pre-representation

is located within a hierarchical organization and is parallel to cerebral networks so as to be subject

to the constraints of the context, in which it appears. Proceeding by trial and error, children tend to

recognize, identify and ‘categorize’ objects and phenomenon in their surrounding world. According

to Changeux, a pre-representation can be stabilized (at least) in function of the signal received from

the external world. The number of pre-representations should diminish with experience; in other

words, learning is eliminating (Changeux, 1993).

As a supporter of neuronal Darwinism, Gerald Edelman formulated a theory about the selective

functioning of the brain and this is described in paragraph 2.8. Changeux takes up where he left off,

putting forward the idea of which knowledge to choose and which to eliminate, ie, information

when compared to reality and found to be unsuitable is eliminated. From a teaching point of view,

this idea confirms the fact that being born with the same possible logical connections, ability in

some subjects (including mathematics) is in relation to one’s own experiences. It is important that

the sensory experience leaves a strong imprint on a neuronal level, which is why it is also important

to involve culturally-disadvantaged learners on an emotional level.

In arguing my research hypotheses regarding creating a context for encouraging negotiation of

meaning between knowledge, learners and the teacher, paragraph 2.9 analyzes the use of spoken

language for communicating descriptions, meaningful content and/or thoughts. On a social level

we can observe that if spoken language is a characteristic of the human species, sound structures

used in communication can be different.

At the beginning of the 20th century, the American philosopher Charles Sanders Peirce and Swiss

linguist Ferdinand De Saussure re-examined the classical distinction introduced by Aristotle

between an external object, its mental image and its communication in words. For Aristotle, the

relationship between things and concepts was ‘natural’ and based on imitation whilst the

relationship between sound and concept was ‘arbitrary’, possibly the result of tradition. Peirce

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constructed a general theory of signs, latter known as semiotics, in attempting to understand how a

person can create in their ‘mind’ the same (if not more in a more developed state) of what is found

in another person’s mind. Peirce postulated that there exists a threefold relationship between: an

object in the external world, the interpreter (the person’s interpreted thought) and the acoustic

image or representamen, which is formed in one’s head. He maintained that the sign is a mental

representation which joins the interpreter and the representamen. However, for Saussure the sign is

a physical entity with two faces: signified is the obverse and the signifier is the reverse. According

to Changeux, significance refers that the subject has an object: the distribution of neurons of the

corresponding paths and its role is probably that of transmitting knowledge between individuals.

Moreover, Pierce confirmed that effective communication takes place when an acoustic image

produces a modification in the listener’s conscious state, action plans and effective behaviour. The

language of a given community, therefore, corresponds to the total of stored traces in the brains of

people belonging to the same community (Changeux, 2003). Language is not simply a collection of

names and meanings. Syntax is not limited to chaining together isolated words with appropriate

and complementary senses but it reflects the way in which propositions or complete phrases can

take on different meanings in function of the context. There exist rules (morphological rules) which

govern the meaning of words in a context of a sentence and their complexity varies with each

language. The understanding of a story, therefore, does not merely originate from the accumulation

of words and sentences without a line of definitive thought. Rather anticipations are presumed

which allow for the ‘gathering together’ of numerous cerebral territories. And the conscious

construction of learning environments, including that of cartoons, can be included in this

consideration. Images and words possess a strong evocative power, including themselves thus in

anticipatory activities, which facilitate the comprehension of the problem-in-a-story.

But communicating thoughts via language and understanding them are not reduced to decoding a

linguistic signal. As Vygotskij (1962) wrote, “the rational and intentional communication of

experience demands a mediatory system” and a “propagation of affect”. Sperber and Wilson (1986)

go even further maintaining that when one communicates, one has the intention of modifying the

“cognitive environment” of the interlocutor’s brain, that is, “the totality of hypotheses which the

interlocutor is capable of mentally representing and accepting as true”. Knowing how to recognise

another’s intentions and to attribute intentions to events makes communication possible, even in the

absence of a code. the cognitive aim of each individual in conversation is to maximise the

relevance of information elaborated. As we can infer from recent research (Arzarello, 2004),

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gestures also play a fundamental role in verbal production since they intervene in the process of

conceptualisation. Gestures can possess explorative, anticipatory and explanatory functions, social

aspects or contribute to the dialectical construction of knowledge in class, if suitably encouraged by

the teacher.

With reference to my experimentation involving cartoons, I wondered if observing someone

gesticulating could help in some way to contextualise experience. The recent discovery by the

neuro-physiologist Giacomo Rizzolatti of mirror neurones (Rizzolatti et al, 1990; Rizzolatti and

Arbib, 1998) helped me to conceive of a plausible neuron mechanism for the “reciprocal intention

to communicate” within a common intentional framework. By experimentation, researchers

observed that mirror neurons are activated when a monkey looks at the experimenter making the

same gesture. These neurons might intervene in the processes involved in recognising another

person’s actions, differentiating them from other actions and using them to adapt behaviour in a

suitable way. It is, therefore, plausible to suppose that mirror neurons participate in the inferential

communication of intention (Changeux, 2003).

In paragraph 2.10 I have written about the Theory of Embodiment, as described by Lakoff and

Núñez (2000). In their book they ask how normal human cognitive mechanisms are employed in

creating and understanding mathematical ideas. In its known form, mathematics is limited and

structured by the human brain and mental capacities. Therefore, the only mathematics we know or

can know is a brain-and mind-based mathematics.

35

Introduction

This is a "reading" attempt of the use of a game or of a comic strip to favourite the

comprehension and storage of didactic experiences during the activities of learning-teaching

mathematics. The recreational activity may be analysed under several points of view and in

particular in this chapter I will try to analyse it from the neurophysiological point of view.

The psychology is surely essential for the didactic activity, but to reach some results it has to

combine the study of the behaviour with the parallel study of what happens inside the brain and

look into the box. By treating the brain like a black box the experimenter studies entrances and exits

and, on the basis of the results, he tries to deduct the structure and the functioning of what is inside

the box.

"We have to study both structure and function, but we have to study them

from inside the black box and not only from outside" (F.H. Crick).

For this reason, for my activity of teacher-researcher, I reputed important to ask to myself within the

limits of the scientific knowledge how do certain learning processes occur from a biological point

of view (Ferreri, 1992,1993). I asked to myself what effectively the role of the emotions in the

learning process is and if it's possible to use them not to "impose" the knowledge but to favourite

that the learning activity becomes a "wish to learn".

"Some people think that the most difficult problem is that of explaining the relation between

mind and brain. I instead don't find it difficult: the mind is simply what the brain does!

The important question is: "What does the brain?"…"

(Marvin Minsky, 1989, The society of the mind)

Konorsky maintains that the plasticity of the brain is one of the principal properties of the nervous

system. It allows the realisation of enduring functional changes; the other fundamental property is

the excitability which refers to rapid modifications which leave no traces in the nervous system. The

human intelligence does not only consist in the continuous augmentation of the knowledge, but also

in rework, recategorize and then generalize the information in new ways. Our starting hypothesis is

thus the following:

the mind is a function of the brain,

36

which works exclusively concretely for the activation and turning off of the neurons. I find

Changeux's position (2003) towards the science very interesting, because of his attempt of building

a philosophy of the science of the brain.

The learning as essential activity for survival related to the emotions is a reinforced learning but

what I'm going to try to prove is whether the knowledge, as primary impulse, may also be

connected to a motivational state inserted in a scheme of autogratifying (appetitus noscendi). In

this sense the recreational learning is not only connected to a wellness condition, but we can also

hope to connect it to a pleasure state.

The didactic activity and particularly the didactic result will depend from the teacher's

consciousness that during the didactic activity "something is changing". The consciousness of

modifying schemes of neurons will concretely characterise the didactic activity, which will have to

aim to construct the great building of the knowledge trying to present some activities or a language

recalling some cognitive maps already existing in the person who is hearing to learn.

2.1 History

"The most difficult thing to understand is why we are able to understand"

(Albert Einstein)

Our brain considered as fabric of the thought operates through electrical impulses, through

production and reception of chemical signals, through transformations of molecules .

The first scientific interests are dated of the III century b.C.; in that age Erasistrate, Aristotele's

nephew, for the first time sectioned a human brain. Five centuries later Galeno of Pergamo

attributed both intellectual activity and memory to the brain, "prince of the viscera". But we have to

wait for Leonardo da Vinci (1452-1519) and then for Andrea Vesalio (1514-1564), with his work

De umani corporis fabbrica libri septem, in order to affirm that the anatomical studies have become

a scientific discipline. In 1664 Thomas Willis publishes in London his celebrated Cerebri anatome,

thus revealing himself as the "discoverer" of the nervous system. With Franz Gall, Claude Bernard,

Parkinson and Broca the neurological studies have great developments, but a very important result

is connected to a discover dated 1875 due to Camillo Golgi. He ideated a methodology which

apparently in casual way colours only some cells of a zone. With this method only some neurons

are coloured and it's then possible, through some serial sections, to identify all the cells of a

37

determined zone. Santiago Ramon y Cajal, in the same period by using the Golgi's colouring

analysed almost all the structures of the nervous system and described them in a huge work whose

title was "Textura del sistema nervoso del Hombre y de Los Vertebrados".

Cajal defined the notion of nervous system constituted by separated cells, well defined and

communicating among themselves by synapse; he moreover acquired numerous proves

demonstrating that the interconnections existing among the neurons are not casual, and that on the

contrary they are highly structured and specific.

The principles of the neuronal function are very similar also in organisms having great differences

such as the sea snail and the man; the majority of our knowledge is due to the study of the squid.

Even the brain structures are so much similar, for example in a cat and in a man, that for the study

of the largest part of the functions it seems almost indifferent on which brain we are working.

But a real quality change has been given by the possibility of the human brain functional study by

means of the PET (tomography by emission of positrons). This technique allows to register from

outside the presence of marked substances emitting positrons. We know that the glucose is the

neuron's oil and that the cells consume more glucose when they are active rather than when they are

still. The quantity of radioactivity measured in particular areas shows how much they are active.

This technique permits to visualise the cerebral most active zones, on the basis of some sense

stimuli or of emotional/pathological states.

2.2 Let us see what is inside the box…

The brain is a tissue of our body and therefore is composed by highly specialised cells. Their

electrical and chemical signals may be made out, recorded and interpreted and their chemical

behaviours may be identified. The number of the nervous cells (neurons) forming the more or less

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1400 grams of the human brain is in the order of 1011 (a hundred milliards) with the approximation

of a factor 10 in more or less.

A neuron is composed by a cellular body having a diameter between 5 and 100 microns, from

which a principal fibre runs off, which is the axon, and a great number of fibrous ramifications

called dentriti. The dentriti receive the incoming signals, the cellular body analyses them and emits

an answer through the axon which establishes connections with the dentriti and the cellular bodies

of other neurons. Between these connections there are some connecting pieces called synapse. The

signalising system is double: chemical and electrical. A neuron usually establishes connections with

a number of other neurons going from 10 to 10.000, then the possible estimation of the number of

the synapse is around 1015. The brain macroscopically represented has the shape of a walnut kernel,

divided into two hemispheres and occupies the skull; by making a section we can see a darker zone,

the grey matter constituted by the neurons's cellular bodies, and lighter zone, white matter

constituted by the extensions of the axons.

The "external rind", the cortex is constituted by a lamina made of nervous tissue having the

thickness of about 2mm, densely folded upon and under the cerebral hemispheres with a total area

of 14dm2. Observed at the microscope it shows a high cellular density, of about 105 neurons/mm2.

The most "ancient" part is the hind brain, including both cerebral trunk and cerebellum. The second

one is the mesencephalon, which in the human being is limited to the upper part of the cerebral

trunk. The third one is the prosencefalo, divided into archicorteccia (hippocampus and amygdule),

paleocorteccia (olfactory ways) and neo-cortex (that is the really cortex) picture 2.

In the nineteenth century, thanks to Broca's, Jackson's and Trush's studies, which were based on the

correlation between functional damage and anatomical lesion, some areas of the cortex highlighted,

which had particular specialisation, such as for example the visual, acoustical and motory area, as

well the area responsible for the somatic sensitivity. The area extending about between the ears

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constitutes the primary motory cortex, which exerts the voluntary control of the muscles. In parallel

position and just behind to it there's the area of the primary somatic sensitivity, which receives the

information coming from the cutis, the bones, the articulations and the muscles.

Let's observe the different dimensions of representation upon the cortex of the various parts of the

body in the image of the distorted homunculus (picture 4). We can see that the representation areas

are proportional not to the measures of the part but to the precision with which this one has to be

controlled.

For example in the human being, the motory regions and the areas of the somatic sensitivity

dedicated to the face and to the hands are the most developed.

The primary visual cortex is situated behind the brain and precisely on the interior surface of the

occipital lobes. The primary acoustical areas are positioned in the temporal lobes, the olfactory

bulbs in a region situated in lower part of the frontal lobes. The areas interposed between the sense

zones and the front of the motory ones are called association areas, because they have no direct

connections with the lower structures but just only with the other cortical areas. They operate on the

basis of already elaborated information.

40

The symmetry of the two hemispheres is only illusory: each hemisphere, for what concerns the

motory and sense activities, is interested in the opposite side of the body. On the contrary the

distribution of the most specialised functions is totally different and asymmetric. These functional

asymmetries reflect anatomical asymmetries.

The linguistic faculties principally depend from certain areas of the link hemisphere, whilst the

perception of the melodies, the analysis of the visual not linguistic structures, such as the

perspective pictures, depend from the right hemisphere. This last is also the most important for what

concerns the expression faculty and the recognising of the emotions.

The two hemispheres are reciprocally connected and the information do freely pass from one

hemisphere to the other one especially thanks to the callous body. It's interesting observing that in

case of lesions some functions may be deputed to the other hemisphere; moreover many functions

are carried out by both hemispheres together.

Michael Gazzaniga, by studying the behaviour of subjects having the brain divided (that is patients

having had for therapeutic reasons a section of the callous body, thus having no more connection

between the two hemispheres) developed a modular theory of the brain:

"…many current theories of the human thought affirm that the solution of problems happens

only at the level of conscious experience and is a product of our linguistic system in itself.

One of the principal principles of a great part of the psychological research is that the

elements of our thinking processes proceed serially in our "conscience" to build then into

cognitive materials. I really think that this idea of a linear and unitary conscious experience

is totally wrong. My thesis is that the human brain has a modular organisation. By using

the word modularity I mean that it's organised in unity having a relatively independent

working and acting in parallel… These modular activities often act independently from our

verbal consciousness…" "understanding that the mind has a modular organisation means

that certain behaviours we have must be accepted in all their being capricious, and that a

particular behaviour may well have no origin in our conscious processes of thinking."

Gazzaniga identifies in our brain a special component which he calls "the interpreter" which creates

the theory tending to give an explanation of a possible behaviour originating during the waking

state by one of the independent modules. It's seat is the link hemisphere dominating in the right-

hand subjects and it's it self which proposes the theories tending to explain the actions promoted by

the other cognitive not verbal systems.

41

A group of structures called limbic system and formed by the hippocampus, the amygdule and the

hypothalamus constitutes a part of the prosencefalo. Hippocampus and amygdule have a basic role

in the elaboration of the emotive behaviour and in the storing processes.

The hypothalamus dedicates to the survival, it has the control of the vegetative functions, it

controls the emotions. Moreover it also controls hunger, thirst, body temperature, sexual impulse,

pleasure and sleep. These structures have connections with many other parts of the brain, among

which the thalamus and the cortex, and these connections make it possible that our emotions may

effectively change the cognitive levels. The hypothalamus-hypophysis axis regulates the

production of the hormones following to a physical-psychical experience-stimulus. In fact, the

variations of the body are connected to the mind through chemical signals constituted by the nerves

and by the hormones produced. The hormonal variations of the body are recorded by the amygdule

and then an "either positive or negative emotion" originates.

The thalamus is another cerebral structure.

It exerts the last control onto the messages coming from all sense-organs, with the only exception of

the olfactory organ, before they reach the cortical areas.

Inside the thalamus a real elaboration of the signal occurs, which is then transferred into a different

code. The presence of reciprocal thalamus-cortical projections means with no doubts that the

functional state of the cortex may influence the way in which the sensory stations of the thalamus

examine the stream of information directed towards the cortex.

Near the thalamus there is the ‘corpo striato’. It consists of three big cellular groups: the ‘nucleo

caudato, putamen e globo pallido’. These structures have a determined role in the coordinate the

movements and in the processes of learning. The ‘corpo striato’ receives information from many

parts of the cortex and the sensorial systems included. It send fibres to the parts of the brain that

checks the movements. It checks the link between the stimulus and the action implicit in the habit.

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The ‘formazione reticolare’ is another structure linked to the thalamus. It consists of a neuronal

network that let to receive messages from also the acoustical areas. The ‘formazione reticolare’ has

also the function of to produce a generic state of surveillance of the Central Nervous System and a

mechanism of report of the condition of activity of the System as that connected to the sleep-watch

condition: attention and no.

The neuron and the synaptic communication

A neuron is the principal unit of the Central Nervous System and it has the property of own shelter

and own support itself. Most of neurons are composed by three parts: the ‘cellular body’, the

dentriti and the axon. Neurons have unique characteristics: an outer membrane that are capable of

produce and of transmit the electrical impulse and of keep the ‘ionic gradient’ the capacity of make

and of liberate some chemical messengers called neurotransmitters. Finally it has an unique

structure connected to the transfer of the information from the neuron to another in succession: the

synapse.

During the evolution the first system of communication among the cells could be a model like to the

endocrine system. The cell communicates with the others with a production of a specific chemical

substance that through the circulatory system arrives at the target cells and changes their behaviour

(to contract a muscle, ‘secretion’…). This system needs of specialized cells: some of these produce

the chemical messenger (hormone) and others receives information. Together with this system there

is a second more speed and precise: the Nervous System. There is a hypothesis: the appeared of

classes of cells that makes chemical neurotransmitters. This cells head for the target organs to get

into direct contact with them. In this way the neurotransmitters must cover only the minimum space

among the cells (the synapses).

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The synapse is a structure that let neuron to communicate to the other target cells that can be or

another neuron or another muscle or a gland.

The synapse is composed by the pre-synaptic neuron (in general it is in the terminal part of its

axon) of the post-synaptic neuron (in general it is near the cellular body or the dentriti) and of the

synaptic space that divides the two cellular portions.

The synaptic communication is based on the transformation of an electrical impulse into a chemical

signal (neurotransmitter) which in its turn is transformed into an electrical signal or into molecular

modifications. The response of the target cell depends from some variables: from the kind of

neurotransmitter and from the kind of receptor to which it's connected. If the neurotransmitter if of

excitatory kind, it will cause the opening of canals which will bring to the breaking out of the

potential of action. Vice versa, in case of an inhibitory neurotransmitter it occurs a state of hyper

polarisation which will make the post-synaptic membrane not excitable and the cell inhibited. The

post-synaptic membrane may in the end transmit the information to a second messenger which will

induce a range of intercellular chemical modifications destined to last.

2.3 Memory and learning: the role of the hippocampus and the amygdule

"He who has only once understood what the remembrance is, will remain prisoner for all

the eternity of a lonely and same remembrance. Who thus has got only one remembrance

is richer than who would have got the whole world" (Kierkegaard - In vino veritas)

Keeper of our deepest individuality, warrantor of our acquisitions and of our past, the memory is

considered the most indispensable activity of our brain (Ferreri, 1992).

In ancient Greece, in the VI century b.C., Parmenide was one of the first scholars interested in the

memory. He considers the memory uniquely connected to the relation between heat and light, and

cold and darkness inside the organism; in this we can distinguish the importance of the sensory

perceptions in the memorising (storing) process.

In 1881 Theodule Ribot in his work "The diseases of the memory" defines the memory as a

biological fact, "a general function of the nervous system". He assimilates it to the habitude, that is

to the acquisition of automatism thanks to the repetition. As example of these secondary automatic

actions he cites two examples, the learning of the locomotion and of writing. The peculiarity of

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acquiring automatism is connected to a particular faculty of the nerve cell which undergoes an

impression and returns it back in form of image or sensation. The remembrance is conserved as a

transformation of elements that the perception receipts. But Ribot considers too much static the

ancient notion of "trace":

"The organic memory does not only suppose a modification of nervous elements, but the formation

among themselves of associations determined for each particular element and the stabilisation of

some dynamical associations which, for their continuous repeating, will become as much stable as

the primitive anatomical connections".

In the second half of the nineteenth century, at only three years of distance one from another, two

men were born whose works, being really so much divergent, have strongly influenced the

contemporary way of thinking: Sigmund Freud (1856) and Henry Bergson (1859).

Henry Bergson in "Matter and Memory" (1925) tries to explain the relation between spirit and

matter, and through an analysis of the memory he affirms that the memory is "the proper interaction

of spirit and matter". Our memory is like a big personal library where each book is a remembrance

and therefore the past still lives in the present. Nevertheless, the memory is not only a collection of

remembrances; there are two kind of totally different memories: the memory-practice and the pure

memory. The first one is the technical memory, practice allowing us to adapt the present in function

of the past; it's the one permitting the storage of the motion learning; it's "the memory of what we

are able to do".

The second one is spontaneous, independent from our will and is constituted by images-

remembrances. He defines the relationship between memories and perception in this manner: "the

memory has to be at the beginning a power/faculty which is absolutely independent from the matter.

If then the spirit is a reality, it's here, in the phenomenon of the memory, that we can experimentally

come into contact with it". The memory, practically inseparable from the perception, interpenetrates

the past into the present, thus making in an unique intuition a synthesis of present and future. The

pure memory retains faithfully the images of the remembrance together with their chronological

order.

Sigmund Freud on the contrary will pay his attention to everything expressing the disorder at the

level of the conscious personality. Observing some amnesiac subjects he highlighted the importance

of the emotion in the context of the memory. The emotions are indispensable for the creation of a

remembrance because they organize it into a sequence of events. In this way they establish the

importance of it. The sense of time and of order are essential to make it possible that a

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remembrance is considered as such and not as a mere thought or vision without any connection with

the past events.

If we don't recognize or apparently forget the ideas having no emotional content, the remembrance

continues to persist nevertheless, unobserved and unrecognised in part - so affirms Freud - because

of its emotionally neutral character and in part for having become a fragmentary and deformed

image, not different from the deformed and fragmentary images which constitutes the essence of the

dreams. A remembrance becomes a remembrance only in the context of the present: and it's in such

a context that it's organised and receives a sense. Freud noticed the fragmentary character, the

ambiguity of the images, of memory and attributed their lack of sense to the lack of a context. The

remembrances are not still, they on the contrary constantly develop generalisations, new creations

of the past which, inside a system in dynamical evolution, give us a sense of continuity between

present, past and future. In the end, a remembrance without context doesn't exist; and, considering

that the context must necessarily continuously change, no fix or absolute remembrances can exist.

"the memory without the present cannot exist".

In 1890 the American William James, on the basis of Hermann Ebbinghaus's studies, in his work

"The principles of psychology" introduced the idea that the memory might be distinguished in

primary memory (short-term memory) and secondary memory (long-term memory). The short-term

memory allows us to remember for short time periods (only for seconds) limited quantity of

information. The long-term memory will assure the remembering of bigger quantity of information

for very long periods without well defined limits.

Mortimer Minshkin has tried to determine how a remembrance is stored, recalled or connected to

other experiences. The problem has not been totally solved yet, but the ways which have a

determining function in this process are connected to two structures situated on the internal surface

of the temporal lobe in both cerebral hemispheres: the hippocampus (from the Greek: seahorse) and

the amygdule (almond).

In the formation of the hippocampus we can individuate three structures which are positioned

folded all around the hippocampus slit: the giro dentato, the hippocampus properly named and the

subiculum. The information coming all the sensory and associative regions of the neocortex

converge in a region called area endorinale, and from hereon transmitted to the three successive

populations of neurons of the hippocampus. The signal arrives first to the granular cells* of the giro

dentato, then to the pyramidal cells CA3 and finally to the pyramidal cells CA1:

46

After being elaborated from these cells, it is retransmitted to the area endorinale and then to the

neocortex. The hippocampus contains a very big number of endorphin receptors and therefore takes

part to the processes connected to the emotion.

Teyler and Discenna's (1985) theory is particularly interesting; it affirms that the hippocampus has

codified inside its neuronal circuits a complete map of all cortical circuits which may be activated

consequently to perceptive experiences. The activation at the cortical level would therefore be

continuously monitored by the hippocampus which would at short-term memorize a trace or index

of the space-temporal order of the cortical activation. Should the same sequence of cortical

activation occur, the hippocampus would then be able to recognise the event as already happened.

The repetition of the signals would consolidate the trace, thus making the recognition easier. The

hippocampus contains the proceedings for storing and recalling the information.

Recent studies have revealed the importance of the hippocampus, particularly of the CA1 cells in

establishing the spatial context of the memory. John O'Keefe and J. Dostrovsky of the London

University College have discovered that the single CA1 neurons in the rat's hippocampus produce a

discharge when the awake animal moves towards a certain point, which corresponds to the

localisation field of the neuron. This discover implies that the discharge of the CA1 neurons has the

aim of making a map of the habitat/environment by entrusting the information to the memory.

For what concerns the amygdule in the animals it had been seen that to obtain a total and

anterograde loss of the memory, the simple distancing of the hippocampus wasn't enough; it was

necessary damaging the amygdule at the same time. The amygdule receives afferences from the

anterior hypothalamus and sends back further fibres to the same hypothalamus. It's moreover united

to the middle dorsal nucleus of the thalamus, which is connected to the giro cingolare and to the

frontal cortex. Finally, the cingolare cortex and the orbito frontal cortex do project into the complex

of the amygdule, thus closing a fourth limbic circuit. The description of the fourth limbic circuit

shows that in our five senses the olfaction and the taste are the only ones having target-structures

iill ggiirroo ddee nnttaattoo,, hhii ppppooccaamm ppuuss ssuubbiiccuulluumm

47

of the peripheral receptors belonging to the limbic lobes; for the other senses the information arrives

at the limbic region after multiple synaptic connections in the neocortical cerebral cortex. From here

comes the explanation of the terribly evocative power of the smells in our memory.

The amygdule, moreover, is connected to the rest of the body (see the violet part of the Doctor

Ferreri's graph) and therefore records every variation of the body's state.

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2.4 Perception and perceptive images

Our definition of the perception is the experience of our senses (internal and external inputs)

The perception creates some perceptive images, which are nothing more than what is receipted by

our sensors and organizes topographically at a neural level in the primary sensory cortex

(particularly for the visual one it’s the V1).

The perceptive images creates some disposition neural representations which we call Disposition

Neural Scheme (DNS).

The disposition representation is a latent excitement potentiality which is waken up

whenever the neurons discharge at a particular proceeding, at certain rhythms for a certain

period and towards a determined target (another group of neurons) (Damasio, 1995).

The DNS and the topographically organized maps do originate when the information reach the

primary visual cortex. Starting from the geniculate lateral body (a section of the thalamus) the

visual experience, in order to become aware, has to pass throughout the primary cortex to the visual

associative areas, to reach then by means of the “giro del cingolo” (this is our supposition) the pre-

frontal areas where it is associated to other semantic concepts.

In the visual associative areas we find the real image. Each time we recall the image the opposite

process occurs; that is, when we recall an image we put in activity a “map” and consequently also

the semantic content associated to it by means of the effect that the experience has caused to us.

Recalling an image does not only put in action a visual map belonging to the primary cortex but by

means of the “giro del cingolo” (probably) will recall all that other sensory maps connected to them,

following a temporal scanning (semantic maps, auditory maps, gustatory maps…).

The DNS are the indexes which put in action and allow to recall these topographically organized

maps; in other words, they are a scheme of neurons putting themselves in action in synchronous

manner:

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These indexes, as already seen, are kept in the entorinale area or in the areas near the hippocampus,

whilst the various traces and schemes are preserved in the primary areas.

The hippocampus makes an index of the experiences giving us a space/temporal description either

referring to the mental images or to the images of emotional states, that is with reference to the

state/condition of the body. This hypothesis sounds reasonable considering that we observed that in

case of lesion of the hippocampus the process occurs no more.

The experience is then stored not only spatially but also temporarily, and the groups of DNSn

distributed on all the primary sensory cortices constitute the remembrance (smell – vision – sound –

touch – taste).

2.5 Typical elements of affective learning from the neuro-science point of view

“If the brain is really the factory of the thoughts, meaning the place where the thought

itself is made, it’s necessary to admit that for a person the construction

of his brain and that one of his thought are a whole” (A. Pronchiants)

Before introducing in this part a “reading” attempt of the use of a comic strip or of a game to

favourite the comprehension and the storing of mathematical didactical experiences, I think it will

be useful to recall and highlight the elements characterising the emotional learning from a

neurophysiological point of view.

First of all we can observe that if inside our body a release of ENDORPHIN occurs, that means that

some satisfying “action” is occurring. This event is part of a system of gratifying and reinforcing.

All the systems discharging endorphin are connected to well-being situations and are inserted into a

system of this kind.

The amygdule is rich of neurons producing endorphin and it’s commonly thought that this fact rules

the transmission of the signals to other part of the nervous system. In reply also to emotional states

generated by the hypothalamus, the amygdule may make it happen that the emotions influence what

is perceived and learned. The reciprocal effect of the amygdule onto the cortex may explain why the

events having emotional strength might remain deeply impressed. The amygdule with its faculty of

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mediating between the senses and the emotions is a structure which could be at the basis of this

selective attention (Mishkin).

The motivational behaviours may be regulated by factors which are different from those one

determining the needs, and then we can talk of:

Hedonistic Factors – Ecological Limitations – Forestalling Mechanisms (accumbens-dopamine)

Enlarged Motivational System (because it considers also the production and therefore the state

of endorphin)

The emotions originates feelings…

Damasio (1995) in his book defines the emotion as a variation of the bodily state and the totality of

the alterations defines a profile of shifting from a range of middle states (homeostasis). All the

emotions generate feelings, feelings of the emotions, and reasons and feeling proceed constantly

together.

During the course of the emotions (variations of the “landscape of the body”) it occurs a stream of

information towards the periphery and from the periphery, depending on the changes caused

(viscera, muscles) there is a chemical reply (neurotransmitters, neuromodulators, hormones) turning

back of the signals to the brain. All this shows the changes brought by the emotion.

A feeling depends from the superposition of the body image onto “something else” of our integrated

knowledge. The feelings are as cognitive as any other perceptive image; they give us the cognition

of our muscular and visceral state.

“The observation process continues, this experience that your body is being living while thoughts

regarding specific contents run, is the essence of what I call feeling” (Damasio, 1995)

The hypothalamus with the body and the brain: hypothesis of the somatic marker

The temporal scanning is what connects the different sensory experiences to an image and makes

the remembrance possible. The absence of temporal synchronisation causes hallucinations, arising

of images recalled without being inserted in a precise context.

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The disposition representation constitute our store of the knowledge, both INNATE (control of the

metabolism, pulsations and instincts) and ACQUIRED.

Then we have:

Let's remember that the hypothalamus is the part deputed to the survival.

It's directly connected to the limbic system, which is connected to the neurocortex.

Whenever an hormonal variation occurs in our body, it influences both limbic system and

neurocortex (the hypophysis understands if there's a variation of the quantity of hormones, and if

necessary it produces them).

Only integrate systems do exist and then interconnections between the hypothalamus and the limbic

system throughout which it informs the brain about what happens in the body.

When an emotion causes a perturbation of our homeostasis, it's perceived from the limbic system

through the hypothalamus and this changes the situation experience-object and creates a secondary

experience to which possibly a well defined marker (a particular feeling) might associate.

But how are the somatic markers acquired?

When we recall a map it's important observing that we can also recall the Somatic Marker of the

experience, that is some special examples of feelings generated starting from the secondary

emotions.

PPrriimmaarryy EEmmoottiioonnss

SSeeccoonnddaarryy EEmmoottiioonnss

categories of objects, situations

Interweaving of the emotions with creation of one’s proper history of interaction with the habitat/environment

Experience

They are defined as a bodily state connected to an innate scheme (fear, pleasure),they depend from the limbic circuits, particularly from the amygdule and from the cingolato anteriore (anterior part of the giro del cingolo )

52

EXPERIENCE

control

Influence

The Somatic Marker is a "status (positive or negative)" of our body photographed in a determined

instant. It doesn't decide for us but gives a "sign" to the experience.

The majority of the somatic markers has probably been created in our brain during the process of

instruction and socialisation through the connection of specific categories of stimuli with specific

categories of somatic status, and are based on the process of the secondary emotions. All the limbic

system (amygdule) operates referring to the survival and therefore our somatic marker will take part

to the choice we make of this one or that one neural scheme.

This fact, from a didactical point of view, may be useful in order to evaluate the kind of mistake a

student could do or the difficulties which a student might have in understanding an argument which

generalizes or for example recalls a previous concept. The somatic marker which a student might

associate to a didactical experience may in fact depend, so I think, not only from the language

(often really too much artful and complicated for the maturity of a classroom) which could be used

by the teacher but also from the kind of examples or instruments used.

2.6 …but can learning become "appetitus noscendi" (the wish to learn)?

Jean Pierre Changeux (2003) referring to the motivational learning talks about a certain appetitus

noscendi. What he wants to show is that the knowledge as primary impulse may be connected not

only to a necessity status but also to a motivational status inserted in a scheme of auto-gratifying .

System of Internal preference (innate regulating dispositions posed to

assure the survival)

Somatic Markers

External group of circumstances which includes not only entities and events with which the organism has to interact, but also social

conventions and ethical rules

53

The game activity with no doubt is inserted in a system of gratification and reinforcing in which the

status of necessity is connected to a status of well-being through the production of endorphin. But

the existence of a circuit connected to the production of dopamine in the Accumbens nucleus lets us

think that the learning may be considered essential for the survival when it's possible to connect it to

a status of well-being, but of pleasure and auto-gratifying too: in this case the pleasure pushes to

repeat the action!

The circuit of the motivation, in this sense, inserts inside a principle of survival and if consciously

"exploited" may lead the possibility that the didactical activity might insert inside a circuit of total

"dependence" from the knowledge (appetitus noscendi). It's certainly not easy to succeed in

inserting it inside a circuit of such dependence but the consciousness of its existence may let us

think about the use of particular instruments which might favourite and induce the development of

it.

2.7 Motivation: what are its origins?

Our brain is continuously engaged in the exchange of energy and information with the external

world. Beside the "opening" of the nervous system, the brain is characterized by what is called its

"motivation". The brain doesn't act like a mere machine which passively elaborates the information

coming from the external world. The spontaneous activity of specialised groups of neurons leads

the organism to explore and experiment continuously the physical, social and cultural environment,

to catch the answers and to compare them to what there's in its memory. Consequently, the brain

develops really surprising faculties of "auto-activation " (Lappane and Dubois, 2001) and of auto-

organisation then. It's just in this sense that I use the term "motivation" (J.P. Changeux, 2003). The

human being and the animal show behaviours of spontaneous exploration of their habitat destined to

satisfy some elementary vegetative needs necessary to the survival.

We have to consider the brain as a motivated system provided with innate dispositions to the

acquisition of knowledge, to the exploration of the world and classification of it into categories

(appetitus noscendi).

The behaviour depends also from needs and desires, and the motivation is an interior condition of

the organism, the existence of which has been solicited to explain the variability of the behaviour

answers.

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The motivational status are defined as the internal conditions which lead and regulate the volunteer

behaviour.

The specific motivational status, called impulses, are controlled by basic homeostatic regulating

processes which are essential for the survival (hunger, thirst, breathing, temperature).

The interior needs give motivations to our behaviour and guide it towards particular directions in

order to reach or avoid some specific objectives…

These activities are a source of pleasure and therefore there's a great natural solicitation to repeat

these behaviours.

With the term threshold of the behaviour we mean the possibility of "doing" some actions.

Obviously, if it decreases we are more disposable to do some actions and to the movement then.

The interior needs requires the organisation of the single components of behaviour into specific

sequences organized for the achievement of determined aims. The need implies a sequence of

already codified actions which in certain conditions may be also connected to automatism.

The achievement of an object, specifically, :

- elevates the threshold of the behaviour

- determinates the disappearance of one or more components of the behaviour sequence

- reduces the motivational status

leading to the total elimination of the answer.

The motivational status have three particular functions:

Level of surveillance

Threshold of the behaviour

Ability of acting

Motivational Status

increase

reduce

Make more powerful

55

- guide function: they direct the behaviour towards determined objects

- activating function: they increase the surveillance/watch condition and push the individual to

acting

- organising function: by blending the single components of behaviour into coherent

sequences directed to the achievement of the aim

Specifically, the concept of motivation may be reduced to a complex reflex regulated by the

excitatory and inhibitory control mechanisms which can be recalled by numerous stimuli.

It's possible talking about homeostatic mechanisms as servomechanisms: a control system and a

feed-back detector which records the signals or alterations of the reference value.

There are also other more complex conditions (curiosity and sexual lusts) which are defined

impulses; at their basis there are no well defined conditions of physiological privation, but

nevertheless they imply a need and the satisfaction of it.

There are three factors regulating the motivational behaviours:

1) the evolution selection: it has fixed some motivational behaviours so that the answers could

be appropriate to the ecological demands of the animal.

+ System to

be controlled -

Internal incentive stimuli (smell, sight)

Feed-back detector

Elements of control

(Endocrini, of the behaviour, autonomi)

Sign of the reference value

+,- + -

+ - outers

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2) The homeostatic regulation: it often has an anticipatory nature, the biological mechanisms

activate or disable some behaviour answers before a deficit might occur or a need might be

felt ("circadian" rhythm).

3) The hedonistic factors: the pleasure is one of the factors contributing to the origin of the

motivated behaviours. Some rewarding systems are those one at the basis of which there's a

dopaminergetic* system ( which uses the dopamine as neurotransmi tter). This system starts

from the cerebral trunk and from the mesencephalon and projects to the shell of the

Accumbens nucleus and towards the pre-frontal region. This nucleus serves as interface

between the motivation and the action. It has a central role, because it individuates the

discharged dopamine thus regulating the selective contact of the brain with the external

world.

2.7.1 Motivation and reward

We have to distinguish the motivation facing to the environment and the reward coming from this

last one. The rewarding mechanisms are activated by signals of the external world which stimulate

internal reactions of desire or of aversion.

Some neuronal circuits determinate almost automatically a positive or negative sensation depending

on the kind of stimulus received from the environment.

The ways of reward…

The ways of the reward are the dopaminergetic* ways, which start from the trunk and the

mesencephalon and project then to the Accumbens nucleus and to the pre-frontal cortex. They

receive afferences from the diencephalon (tween-brain) and the telencephalon (end brain).

The region of the Accumbens nucleus in particular, called "shell", has an essential role concerning

the emotions for its connections with the amygdule and the limbic system (learning) whilst the

"core" enfolding is more directly engaged in the motory control.

The Accumbens nucleus/core serves as interface between the motivation and the action. It has a

central role because it individuates the discharged dopamine thus regulating the selective contact

of the brain with the external world.

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These neuronal systems are implied in creating the dependence.

Nowadays it's admitted that the drug addiction is connected to the faculty proper to these substances

to stimulate the discharge of dopamine through the projections of the dopaminergetic* neurons of

the cerebral trunk towards the Accumbens nucleus. Adaptive changes are inducted in the cerebral

circuits, so that they can regularly work only in case of their presence.

In conditions of absence of drug the neuronal trace persists and the abrupt interruption of its

assumption causes the so-called "abstinence syndrome " (sickness, suffering, anxiety…).

Experimental studies demonstrate that these symptoms have a "motivational" origin more than a

"hedonistic" one. The biochemical dysfunction (decrease of the transmission of dopamine into the

Accumbens nucleus) of the motivational system causes a loss of control in drug assumption, the

forced use of it and addiction.

The list of the behaviours the control of which we may lose is long: the game, the bulimia, the sport,

the sexual activity, the need for money… Dysfunction which strike the dopaminergetic neurons do

probably contribute to these forced behaviours.

Won't the same thing happen with the pursuit of pleasure connected to the

knowledge and with the scientific research in general?

J.P. Changeux

The anticipation of the reward

The human being shares with the animal the pleasure for the reward and consequently he learns to

foresee it and to look for the conditions which surely allow to obtain it. This fact represents a

fundamental resource of the predisposition to acquire knowledge. This anticipation of the reward

has been recorded with electrophysiological methods at the level of the dominergetic* neurons, and

it has been observed that in some cases the activation of the dopaminergetic* neurons doesn't clash

with the reward anymore but anticipates it following to the learning (Schultz and Dickinson 2000).

The pre-representations

The action of the environment on the brain does not consist only in giving "instructions" to the

brain itself in a passive and direct manner. The hypothesis proposed by Changeux is that the

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acquirement of the knowledge is indirect and it results from the selection of pre-representations

which are also called preliminary schemes. The spontaneous activity has then a central role,

contributing to a some kind of "generator of diversity" of Darwinian type (Changeux 1983,

Edelman 1987). The pre-representations would correspond to dynamical spontaneous transitory

status of activity of neuron populations able to form multiple combinations.

The pre-representations would activate in combined way innate structures and groups, neuronal

distributions resulting from previous experiences (Tsodykis et al., 1999).

The pre-representation is inserted inside the hierarchic organisation and in parallel with the cerebral

nets, so to undergo the bonds of the particular context in which it appears.

The genesis of the pre-representations constitutes thus a simple neuronal realisation of the

"productivity" (Fodor and Pylyshyn, 1998) or , if it may be so defined, of the "creativity" of the

cerebral cognitive processes. This conception has a decisive consequence: the pre-representations

may create multiple "distributions of functional relations" activating different territories

functionally distinguished of our brain or come out from "different sources of sensory-motor

information" (All Port, 1985; McCarthy and Warington 1990). They are at the basis of the

numerous hypothesis of sense produced by our brain.

The cognitive games and the comparison to the reality of the world

Following to several repeated attempts, proves and errors, the child since its birth and the adult

later, "projects" pre-representations in the world surrounding it in an explicit manner at first,

through motory actions, then in silent manner, "mentally". It then dedicates to cognitive games

(Changeux 2003). By simply proceeding throughout attempts and errors, the child tries to

recognize, to identify, to "classify" objects and phenomena of the world surrounding it. But how

does the external world react on the transitory neuronal status which cause these behaviours?

The schema proposed by Changeux is that a pre-representation may be established or not depending

from the signal received from the external world. It constitutes a test of the agreement, of the

conformity or also of the adaptation of the pre-representations towards the environment. This test

permits to discover whether it "gives a sense" or not and it might be based on two possible

mechanisms.

The first one is the "selection per reward" preferably used to evaluate the actions. The signals

received from the environment put in action some neuronal ways which interfere in the motivation

and/or in the pleasure of the reward.

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The clashing in the time of the pre-representations generated internally and of the positive answer

evoked externally would originate the stabilization of the fit "hypothesis".

The second one is the "selection per resonance" which is founded on the correspondence between

the perceptive activity caused by sensory stimuli and the pre-representation existing in the moment

of the sensory experience.

These two selection modalities would entail the stabilization of meanings or of knowledge in form

of representation "maps".

A reduced neuronal model of the external reality would then be selected and saved in memory in

the brain. As a consequence of the selection, the number of the pre-representations should decrease

with the passing of the experience in the world. In other words, to cite the formula of the neuronal

Human Being (Changeux 1983):

"learning is to eliminate".

Democritus said: "We catch nothing absolutely true of the reality, but just what casually happens,

according to the temporary dispositions of our body and to the influences which fall on us or annoy

us".

2.8 Cultural and environmental experience: the neuronal Darwinism and the plasticity of the

brain

Gerald Edelman (1987), supporter of the neuronal Darwinism, highlights a fundamental principle

which is at the basis of the biological brain and that is the individual variability connected to genetic

and of the environment factors. He affirms that the brain might work as a selective system and what

we call learning mi ght really be a form of selection.

His theory is based on three essential theses:

1) during the growing of the brain in the embryo, among the cerebral cells (neurons) a model of

connections highly variable forms, which is different in each individual.

2) After the birth, in each individual a scheme of neuronal connections is fixed, but some

combinations of connections are chosen preferably than to other ones as consequence of the

stimuli received by the brain through the senses.

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3) This selection would particularly happen in groups of cerebral cells reciprocally connected in

index-cards or maps, and these maps dialogue among themselves to create categories of thing or

of events.

Changeux (2003) resumes Edelmann's concept and in his book talks about a learning per selection

and elimination, that is that the information which compared to the reality are not adequate are

eliminated. The way of forming of the connection maps which he calls pre-representations would

make us equal at the moment of the birth from the point of view of the temporal scanning with

which we may connect. The cultural and environmental experience will later on characterize these

connections.

From a didactical point of view this argument make us think that potentially we cannot talk about

"not fit" schoolboys referring to the study of mathematics, because we all born with the same

opportunities of logical connections. It's only in relation to one's proper experiences that we can talk

of preference for a certain type of studies or another one. Before that the brain during the period of

growth might acquire the full faculty of elaborating and analysing the sensory experience, the

experience itself will have to leave a trace of it in the cerebral tissues. In this sense, we may always

hope to succeed in making also the most culturally disadvantaged schoolboys understand the

mathematics, trying to involve them emotionally, because in this way the importance and the

responsibility towards the teacher for succeeding in reasoning in a correct and "mathematical"

manner are less felt.

The typical behaviour of a species, that is its instincts, is also influenced from the experience, the

learning and the memory. It develops and is influenced by the relations with the environment. With

the passing of time the visual, olfactory, taste, auditory, tactile and of balance experience puts in

action and favourites the development of specific nerve ways, whilst some other ones are no more

used.

In fact, if we observe a child whose attention is focused in taking possess of a game we become

aware of how the faculties of perception and of movement of an adult are not innate and of how on

the contrary they have to develop with the passing of time through the training.

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2.9 Knowledge and social life: language and inferred communication

In the wide context of biological evolution the use of spoken language, to communicate

representations, meaningful contents and thoughts in order to verify their social truth, has generated

a discussed question: the difference between languages.

Undoubtedly the spoken language is proper to human beings but the sound structures involved in

communication can deeply change from a group to another.

2.9.1 The triadic model of sign

It was at the beginning of the XIXth century that two great and strong characters thoroughly

changed our comprehension of knowledge communication through spoken language. They were the

American philosopher Charles Sanders Peirce and the Swiss linguist Ferdinand De Saussure.

They both approached Aristotle’s classic distinction between the external object, the immaterial

mental image, and its communication through words.

Aristotle would think that the link between the “thing”- the concrete object- and the concept was

“natural”, so to say, based on imitation; while the link between the sound and the concept was

purely arbitrary and coming from tradition.

Peirce has moved forward and came to a new general theory of signs: semiotics, focused on the

various ways of acquiring and communicating knowledge. He tried to understand how someone

could create into someone else’s mind something equivalent, if not even better than what was

previously in his mind (no matter whether the thing was “real” or “abstract”).

Peirce assumed a triadic relation between an object of the external world, an interpreting thinking

called “interpretant” (we will talk of “referent”) and the acoustic or visual image called

“representamen” (that refers to the form which the sign takes, not necessary material).

Peirce thinks that the sign is a mental representation that joins a concept and an acoustic image

rather than a link between a thing and a word. The sign would be the basis of his triangle and links

the interpretant to the representamen.

Saussure moves further than Peirce, he sees the sign as a psychic entity with two sides, like a sheet

of paper: the thought- signified- is the recto while the sound- signifier- is the verso.

According to Changeux the signified is the knowledge that a subject has of an object: the geography

of the correspondent neural routes. The only purpose of the signifier would be making knowledge

pass through people.

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Peirce gives some explanations on the effectiveness of the communication through the signifier;

there are three categories of interpretants: affective -“feeling”- focused on people’s understanding of

the specific role of the sign, energetic which requires a physical effort, logic -mental- that causes a

change in habits.

An effective communication can only happen when the acoustic image modifies the status of

conscience of the listener and more over his-her action plans and real actions.

Saussure adds some more important tips; the linguistic sign –which produces changes in the

conscience balance- is not motivated, so the connection between signifier and signified is arbitrary.

There is not any relation between the concept and the sound used to represent it. It is a convention

socially shared and accepted which ends in a language as a result of all evidences left by every

group member.

Finally it is necessary to talk of a human skill that let neural nets get out of the brain: the writing

skill. There are some neural nets involved in reading and writing, but the skill to develop reading

and writing ability yet is into newly born children brains. In fact populations who do not use

writing still have the ability and use it to different purposes.

2.9.2. Syntax and understanding

Peirce and Saussure’s theories leave space to some critics, like the ones from Changeaux according

to whom language is not only a collection of names and signified stored into memory. The ability

the brain has to create a syntactic order by compounding words and sentences gives language a new

dimension unknown to human ancestors.

Syntax is not a mere way of arranging words in sentences in their necessary relations but also a

mean to understand how words and sentences can have different sense depending on the context.

Morphology studies the patterns by which words are formed from other words, including inflection,

compounding, and derivation and its complexity changes together with different languages.

Neuropsychological clinical studies provide important information on neural nets involved in

syntactic elaborations.

Some detailed imaging studies show disunion between syntax and semantic in sentences

comprehension (Dapretto e Bookheimer, 1999). The word comprehension (comprensione) is made

of two Latin words cum and prehendere: “take” “with”. Comprehension is not a pure accumulation

of words but involves forecasting and anticipating in order to bring together different brain’s areas.

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2.9.3 Sharing knowledge

The huge variety of sentences that people produce has to take in account two problems: their “truth”

and their interpersonal variety. The combinatorial explosion of possibilities requires proper

perception and elaborating ways in order to allow an effective and shared comprehension inside a

social group.

Inferred communication

Claude Shannon and Warren (1949) have produced a codified model of communication, now

widely accepted: there is a source that sends a message that travels through a channel, as acoustic or

electric impulses, decoded by the receiver. If the message is a sequence of letters it will be encoded

by a sequence of electric impulses each one corresponding to a letter.

An effective communication needs that sender and receiver share the same code, more over there is

a possible trouble source that is the background noise. This scheme suits telecommunication but is

not worth enough to fully explain the complex verbal and non-verbal human communication,

especially the communication of “thoughts” (Changeux and Ricoeur, 1998).

Dan Sperber and Deirdre Wilson (1986) have showed how sender and receiver are not engines, and

human communication is much more than encoding-decoding a signal. Vygotskij (1962) says that:

“the rational and intended communication of experience needs a mediating system, and especially

an expansion of feeling”. Sperber and Wilson assume that when communicating the aim is to

modify the cognitive environment of the listener’s brain, otherwise the “set of hypothesis he has

and he can figure out as true”.

According to the hypothesis that the brain works with context-related pre-representations, they

suggest that every sender would try to transfer her-his ideas into listeners’ brains.

A communication is possible even without a code, as long as people involved recognise each other

purposes and cooperate aiming to a common result; this is how, according to Sperber and Wilson

conversation and cooperation are born.

But how are words able to produce such a switch on long-term memory structures? Sperber and

Wilson in their book “La pertinenza” described how during a conversation a certain amount of

information are old and related to the ideas the person has of the world, while other information are

new and no related to the old ones. When these two sets of data, the old and the new ones, get in

touch - inference process - new information are born. If there is also a multiplying effect Sperber

and Wilson talk of pertinent information.

The more the multiplying effect the more the pertinence. During a conversation every one aims to

magnify the pertinence of the information.

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Gestures in contextualisation

As Arzarello (2004) says, gestures are important in verbal processes linked to conceptualisation.

Action are organised on the basis of images and gestures; they can be a tool of thinking, can

anticipate or explain things, can have social aspects (in a conversation they own to the individuals

involved in) and, if properly used, in a classroom they have a role in the knowledge building

process, finally manipulation helps realising mental experiments.

It appears that gestures allow the link between perception and mental images helping so the process

of knowledge building.

I finally asked myself if only the observation of gestures could help giving context to an experience.

Mirror neurons and reciprocity in inferred communication

The fresh discovery by neurophysiologist Giacomo Rizzolati of mirror neurons (Rizzolatti et al.,

1990; Rizzolatti and Arbib, 1998) helps understanding the neural basis of the wish to communicate.

These neurons are in the cortical area that is in charge of motion: the ventral pre-motor cortex, part

of the pre-frontal one.

Rizzolatti’s team was studying the role of this area when, recording neurons activity in monkeys

awake, they registered some impulses corresponding to voluntary movements of hand and mouth.

The interesting point was that these neurons pulsed also when monkeys simply looked at

researchers doing the same gesture; just like if they were at the same time motor neurons and

sensorial neurons!

Rizzolatti’s team hypothesis was that these neurons - called mirror - were involved in performing

specific actions as well as their representation; so they would have had a role in imitating actions

and understanding them.

Mirror neurons would play a role in recognising other people actions and adjusting reactions to

them as well as in differentiating one action from another. It is possible to assume that mirror

neurons share a role in inferential communication of intentions (Changeux, 2003).

What has been said above leads to my experiment and especially to the importance of having an

environment easy to recognise for pupils into cartoons. Pupils look at the cartoons and recognise the

context, they become part of the story telling. It is so possible to say that they “virtually” live the

mathematical problem! They feel it, experience it and become acquainted to it; and when you know

something you are no longer scared of it, but ready to deal with it. And after all it is only a game!

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2.10 The Theory of Embodiment.

Mathematics as we know it is limited and structured by the human brain and human mental

capacities. The only mathematics we know or can know is a brain-and-mind-based mathematics.

As cognitive science and neuroscience have learned more about the human brain and mind, it has

become clear that the brain and the body co-evolved so that the brain could make the body function

optimally. Lakoff & Núñez (2000) in their book will be asking how normal human cognitive

mechanism are employed in the creation and understanding of mathematical ideas. The basic form

of the argument is this:

1. The question of the existence of a Platonic mathematics cannot be addressed scientifically.

Platonic mathematics cannot in itself be perceived or comprehended via the human body,

brain and mind.

2. All that is possible for human beings is an understanding of mathematics in terms of what

the human brain and mind afford. The only conceptualization that we can have of

mathematics is a human conceptualization. Therefore, mathematics as we know it and teach

it can only humanly created and humanly conceptualized mathematics.

3. What human mathematics is, is an empirical scientific question, not a mathematical or a

priori philosophical question.

4. It is only through cognitive science – the interdisciplinary study of mind, brain and their

relation- that we can answer the question: What is the nature of the only mathematics that

human beings know or can know?

5. If you view the nature of mathematics as a scientific question, then mathematics is

mathematics as conceptualized by human beings using the brain’s cognitive mechanisms.

6. However, you may view the nature of mathematics itself not as a scientific question but as a

philosophical or religious one.

Throughout their book they will give the argument rests on analyses to the effect that human

mathematics makes fundamental use of conceptual metaphor in characterizing mathematical

concepts. Conceptual metaphor is limited to the mind of living beings. Therefore, human

mathematics (which is constituted in significant part by conceptual metaphor) cannot be a part of

Platonic mathematics, which –if it existed- would be purely literal.

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RECENT DISCOVERIES ABOUT THE NATURE OF MIND

In recent years, there have been revolutionary advances in cognitive science-advances that have an

important bearing on our understanding of mathematics. The most profound of these are:

The embodiment of mind. The detailed nature of our bodies, our brains, and our everyday

functioning in the world structures human concepts and human reason.

The cognitive unconscious. Most thought is unconscious-not repressed simply inaccessible

to direct conscious introspection.

Metaphorical thought. The mechanism by which the abstract is comprehended in terms of

the concrete is called conceptual metaphor. Mathematical thought also makes use of

conceptual metaphor, as when we conceptualize numbers as point on a line.

Many of the confusions, enigmas and seeming paradoxes of mathematics arise because conceptual

metaphors that are part of mathematics are not recognized as metaphors but are taken as literal.

Metaphors are an essential part of mathematical thought, not just auxiliary mechanism used for

visualization or ease of understanding. Consider the metaphor that Number Are points on a Line

(Number don’t have to be conceptualized as points on a line; there are conceptions of number that

are not geometric!) or Numbers Are Sets (we don’t have to conceptualize numbers as sets).

Conceptual metaphor is a cognitive mechanism for allowing us to reason about one kind of thing as

if it were another. Metaphor is not simply a linguistic phenomenon but rather, it is a cognitive

mechanism that belongs to the realm of thought. “Conceptual metaphor” has a technical meaning: it

is a grounded, inference - preserving cross-domain mapping – a neural mechanism that allows us to

use the inferential structure of one conceptual domain (say geometry) to reason about another (say

arithmetic). Lakoff & Núñez (2000) believe that revealing the cognitive structure of mathematics

makes mathematics much more accessible and comprehensible. Because the metaphors are based

on common experiences, the mathematical ideas that use them can be understood for the most part

in everyday terms.

Mathematics as we know it is human mathematics, a product of the human mind. Where does

mathematics come from? It comes from us! We create it, but it is not arbitrary, not a historically

contingent social construction.

Mathematic is a product of the neural capacities of our brains, the nature of our bodies,

our evolution, our environment and our long social and cultural history.

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THE THEORY OF EMBODIED MATHEMATICS

As a cognitive scientists, Lakoff & Núñez (2000) ask certain basic questions:

1. What are mathematical ideas from the perspective of cognitive science? What commonplace

cognitive mechanisms do they use?

2. Given that innate mathematics is minuscule – consisting of subitizing and a tiny bit of basic

arithmetic – what cognitive mechanisms allows this tiny innate basis to be extended to

generate all of advanced mathematics?

3. How are mathematical ideas grounded in our experience?

4. Which mathematical ideas are metaphorical and which are conceptual blends?

Encounters with the Romance

In doing the research for their book Lakoff & Núñez (2000) kept bumping up against a mythology –

what they have called Romance of Mathematics. As their research progressed, it became clear that

their findings contradicted this mythology. What they will do is state here what the romance is and

how their results contradict it, so that at least the points at issue will be clear.

The Romance of Mathematics

• Mathematics is an objective feature of the universe; mathematical objects are real;

mathematical truth is universal, absolute and certain.

• What human beings believe about mathematics therefore has no effect on what mathematics

really is. Mathematics would be the same even if there were no human beings, or beings of

any sort. Though mathematics is abstract and disembodied, it is real.

• Mathematicians are the ultimate scientists, discovering absolute truths not just about this

physical universe but about any universe.

• Since logic itself can be formalized as mathematical logic, mathematics characterizes the

very nature of rationality.

• Since rationality defines what is uniquely human and since mathematics is the highest form

of rationality, mathematical ability is the apex of human intellectual capacities.

Mathematicians are therefore the ultimate experts on the nature of rationality itself.

• The mathematics of physics resides in physical phenomena themselves – there are ellipses in

the elliptical orbits of the planets, fractals in the fractal shapes of leaves and branches,

logarithms in the logarithmic spirals of snails. This means that “the book of nature is written

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in mathematics”, which implies that the language of mathematics is the language of nature

and that only those who know mathematics can truly understand nature.

• Mathematics is the queen of the sciences. It defines what precision is. The ability to make

mathematical models and do mathematical calculations is what makes science what it is. As

the highest science, mathematics applies to and takes precedence over all other sciences.

Only mathematics itself can characterize the ultimate nature of mathematics.

Of course, not everyone who believes some of these statements believes them all. These theme are

commonplace in popular writings about mathematics and many of them are taken for granted in

textbooks and mathematics courses, as well as in everyday discussion about mathematics.

The Romance of Mathematics is not a story with a wholly positive effect. The Romance serves the

purposes of the mathematical community. It is part of a culture that rewards incomprehensibility, in

which it is the norm to write only for an audience of the initiated – to write in symbols rather than

clear exposition and in maximally accessible language. The Romance of Mathematics is not an

entirely harmless myth – that at least indirectly it is contributing to the social and economic

stratification of society. Lakoff & Núñez (2000) goal is to give a more realistic picture of the nature

of the mathematical cognition and in the process to make mathematics more accessible as well.

A Question of Faith: Does Mathematics Exist Outside Us?

The central claim of the Romance is that there is a transcendent mathematics – one that has an

objective existence, external to human beings or any other beings.

Proof plays a central role in the romance. It is through proof that human mathematicians transcend

the limitations of their humanity. Proofs link human mathematicians to truths of the universe. In the

romance, proofs are discoveries of those truths. But there is no any scientific evidence that this is

true (Lakoff & Núñez 2000). There is no way to tell empirically whether proofs proved by human

mathematicians are objectively true, external to the existence of human beings or any other beings.

The Inherent Implausibility of Transcendent Mathematics

Lakoff & Núnez point out that even on the basis of mathematics itself – without any scientific

evidence – the claim that transcendent mathematics exists appears to be untenable. One important

reason is that mathematical entities such as numbers are characterized in mathematics in

ontologically inconsistent ways. Here are some of the ways that natural numbers are characterized

within modern mathematics:

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1. On the number line, all numbers are points on a line. On the number line, points are zero-

dimensional geometric objects. If numbers are literally and objectively points on a line, then

they, too – as abstract transcendent entities – must be zero-dimensional geometric objects!

2. In set of theory, numbers are sets. Zero is the empty set. One is the set containing the empty

set. Two is the set containing (a) the empty set and (b) the set that contains the empty set.

And so on. In set theory numbers are therefore not zero-dimensional geometric entities.

3. In combinatorial game theory (Berlekamp, Conway, & Guy, 1982), numbers are values of

positions in combinatorial games. Combinatorial game theory is a normal, respectable

(though not widely known) branch of mathematics. On the transcendent mathematics

position, the entities of combinatorial game theory and their properties should be real.

Numbers, therefore, should really be values of positions in combinatorial games and not

points on a line or sets.

None of these branches of mathematics has a branch-neutral account of numbers.

Again according to transcendent mathematics, there should be a single kind of thing that numbers

are; that is, should be a unique ontology of numbers. Numbers should either be zero-dimensional

geometric entities lined up in an order, or they should be sets of the appropriate stricture, or they

should be values of positions in games, or something else but only one kind! That is an inherent

problem in the philosophical paradigm of transcendent mathematics.

Why the Only Mathematics Is Embodied Mathematics

The study of mathematical cognition may tell us about how we human beings conceptualize and

understand mathematics, how mathematics might be realized in the human mind and brain, how it

might be learned, or how we make mathematical discoveries, but it cannot tell us anything about

mathematics itself!

After all if mathematics exists independent of human beings, studying the cognitive processes and

neural structure of the human mind and brain won’t get us any closer to the true nature of

mathematics.

But the only access that human beings have to any mathematics at all, either transcendent or

otherwise, is through concepts in our neural systems. For human beings mathematics is embodied

mathematics. The only mathematics we can know is the mathematics that our bodies and brains

allows us to know (Lakoff & Núñez, 2000) .

Because the embodied mathematics is an empirical theory about the embodied mind, the theory of

embodied mathematics is framed within the study of embodied cognition. The elements of

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embodied cognition are not axioms and proofs but image schemas, aspectual concepts, basic-level

concepts, semantic frames, conceptual metaphors, conceptual blends and so on.

Weak and Strong Requirements for a Theory of Embodied Mathematics

In the weak sense, embodiment means that every concept we have must somehow be characterized

in the neural structure of our brains. Everything we learn can be learned only through a neural

learning mechanism capable, by virtue of its structure, of learning that kind of thing.

Everything we know either must be learned or must be built into the innate neural writing of our

brains. When someone presents you with an idea, the appropriate brain mechanism must be in place

for you to understand it and learn it. Every bit of the meaning of language must also be accounted

for by neural and cognitive mechanisms.

But there are even stronger requirements (for a discussion, see Núñez, 1999). Cognitive science

must explain how abstract reason is possible and how it is possible to have abstract concepts and to

understand them.

The reason is that abstract concepts cannot be perceived by the senses. You cannot see or hear or

smell or touch the concepts of justice, responsibility and honour much less the concepts of

ecological danger, evolution, due process or entropy. It is a challenge for cognitive science to

explain – in terms of our bodies and brains – exactly how we can comprehend such concepts and

think using them. The same is true of every concept in advanced mathematics: continuity, limits,

fractals, open sets, infinite interactions, transfinite numbers, hypersets, logarithms, infinitesimals.

The answer to questions about mathematical cognition must be given in terms of the kind of

cognitive and neural mechanisms found in the automatic, unconscious, human conceptual system.

Strong Constraints

Mathematics has changed enormously over time and forms of mathematics often vary from

community to community across the mathematical world. Mathematicians differ in their

interpretations of mathematical results.

Today there is much more interest in studying specific details of specific subject matters: the

relationship between modular forms and elliptic curves, the topology of four-dimensional spaces,

non-linear dynamics and complexity, and questions of computability relevant to computer science.

A cognitively adequate account of mathematical cognition must be flexible enough to characterize

historical change, different forms of mathematics (e.g. standard and non-standard analysis),

alternative understandings of results and the characterization of all the fashions.

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Next, a theory of embodied cognition applied to mathematics must answer the question of how,

with pretty much the same kinds of brains and bodies, we are able to do all this.

The Challenge for an Embodied Mathematics

The properties of mathema tics are, in many ways, properties that one would expect from our folk

theories of external objects. The reason is that they are metaphorically based on our experience of

external objects and experiences.

SOME PROPERTIES OF EXTERNAL OBJECTS

THAT ARE CHARACTERISTIC OF MATHEMATICS

Universality ( two plus two is always four, regardless of culture)

Precision: in the physical world, two objects are two objects, not three or one.

Consistency for any given subject matter.

Stability: basic physical facts – that is, particular occurrences at a given time and place –

don’t change.

Generalizability

Discoverability: facts about object in the world can be discovered.

Thus, mathematics, too, is universal, precise, consistent within each subject matter, stable over

time, generalizable and discoverable.

The Properties of Embodied Mathematics

Recent research in neuroscience, cognitive science and the history of mathematics points in the

direction of an embodied mathematics (Lakoff & Núñez, 2000). The theory of embodied

mathematics makes the following claims.

1. Mathematics is a product of human beings. It is shaped by the nature of our brains, our

bodies, our conceptual systems and the concerns of human societies and cultures.

2. The parts of human cognition that generate advanced mathematics as an enterprise are

normal adult cognitive capacities – for example, the capacity for conceptual metaphor.

3. Simple numeration is built into human brains. Human beings can “subitize” – that is

instantly and accurately perceive the numbers of entities in a very small collection.

4. mathematics is fundamentally a human enterprise arising from basic human activities, for

example, counting and measuring, architecture, gambling, motion and other change,

grouping, manipulating written symbols, playing games, stretching and bending objects.

5. the mathematical aspect of these concerns is precision.

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6. Precision is greatly enhanced by the human capacity to symbolize. Symbols also permit

precise and repeatable calculation.

7. Conceptual metaphor is a neurally embodied fundamental cognitive mechanism that allows

us to use the inferential structure of one domain to reason about another.

8. The stability of embodied mathematics is a consequence of the fact that normal human

beings all share the same relevant aspects of brain and body structure and the same relevant

relations to their environment that enter into mathematics.

9. Mathematics is not monolithic in its general subject matter. Rather there are mutually

inconsistent versions of geometry, set theory, logic and so on. Each version forms a distinct

and internally consistent subject matter.

10. Mathematics is effective in characterizing and making predictions about certain aspects of

the real world.

Several of these proprieties are not obvious and need further clarification. Exactly what make

inference and calculation stable in a view of mathematics as embodied? If mathematics can be

shaped by the concerns of culture, how can mathematical results be stable across culture?

The stability of Inference and the

Appearance of Timelessness

Mathematical inference tends to be stable across individuals, time and culture because it uses basic

cognitive mechanism – mechanism like category formation, spatial-relations concepts, conceptual

metaphors, subitizing and so on.

Important consequences for mathematics follow from the fact that certain aspects of human

conceptual systems are universal and that they have inference-preserving mechanisms such as

conceptual metaphor.

1. Inferential stability. The first important consequence is the mathematical proof and

computations are cognitively stable. Proofs made using inference-preserving cognitive

mechanisms remain valid. Correct computations remain correct.

2. The possibility of discovery. Once mathematical concepts and assumptions are discoveries

within a mathematical community, it is possible to make discoveries by reasoning alone –

that is, without recourse to empirical evidence.

3. Abstraction.

4. Stable, natural connections among the branches of mathematics.

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5. The systematic evolution of mathematics over time. Generation after generation of

mathematicians draw out the consequences of assumptions and models established by

previous mathematicians. Once results are established, they are stable and take on a

seemingly “timeless” quality.

How culture Gives Rise to Alternative Versions of mathematics

In the Romance of Mathematics, culture is assumed to be irrelevant. If mathematics is an objective

feature of this or any other universe, mere culture could not have any effect on it.

• Mathematics is independent of culture in the following very important sense: once

mathematical ideas are established in a worldwide mathematical community, their

consequences are the same for everyone regardless of culture. (However, their establishment

in a worldwide community in the first place may very well a matter of culture.)

• Mathematical is culture-dependent in another very important sense, a sense recognizable by

mathematicians such as Wilder (1952) and Hersh (1997). Historically important, culture

specific ideas from outside mathematics often find their way into the very fabric of

mathematics itself. Culturally specific ideas can permanently change the actual content of

mathematics forever.

Some examples where the permanent content of mathematics has been shaped by culture are:

1. The idea of essence (everything in the universe has an essence that defines what it is and is

the casual source of its natural behaviour).

2. The idea that all human reason is a form of mathematical calculation called logic.

3. The idea of foundations for a subject matter (every respectable subject matter is to be

conceptualized metaphorically as if it were a physical structure like a building,

which has to have secure, solid, permanent foundations if it is not to “collapse” or

“fall apart”.

The Foundations of Mathematics movement of the early twentieth century brought all tree of

these ideas together:

1. The essence of a subject matter is to be given by a small set of axioms.

2. Mathematical reasoning is a form of mathematical calculation, which allows all

mathematical truths to be calculated (using mathematical logic).

3. All respectable subject matters can and must have secure foundations on which

everything in the subject matter is built.

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The Foundation movement itself collapsed but they all shaped the structure of mathematics itself.

One cannot even imagine contemporary mathematics without these ideas. They are products of

human culture and human history. Thus, the very idea that there must be absolute “foundations” for

mathematics is itself a culture-dependent feature of mathematics. Mathematics is grounded in the

human body and brain, in human cognitive capacities and in common human activities and

concerns. Since mathematical ideas are products of human beings with normal human cognitive

capacities living in a culture, it is perfectly natural that general cultural ideas should be applied to

many special cases, including mathematics (Lakoff & Núñez 2000).

THE PHILOSOPHY OF EMBODIED MATHEMATICS

The Embodied Nature of Mathematics

There are the properties of mathema tics that any philosophy of mathematics must account for. One

immediate consequence is the disconfirmation of the Romance of mathematics.

• From a scientific perspective, there is no way to know whether there are objectively

existing, external, mathematical entities or mathematical truths.

• Human mathematics is embodied, it is grounded in bodily experience in the world.

• Human mathematics is not about objectively existing, external mathematical entities or

mathematical truths.

• Human mathematics is primarily a matter of mathematical ideas, which are significantly

metaphorical in nature.

• Mathematics is not purely literal; it is an imaginative, profoundly metaphorical enterprise.

• There is no mathematics out there in the physical world that mathematical scientific theories

describe.

A second immediate consequence is a disconfirmation of the radical social constructivist theory of

mathematics.

Mathematics, being embodied, uses general mechanism of embodied cognition and is grounded in

experience in the world. Therefore, it is not arbitrary! That means:

• Mathematics is no purely subjective.

• Mathematics is not a matter of mere social agreement.

• Mathematics is not purely historically and culturally contingent.

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Ontology and truth

Consequently, two of the central questions of the philosophy of mathematics have been:

• What are mathematical objects?

• What is mathematical truth?

The most common answers are those of the adherents of both the Romance and the Foundations

movement:

• Mathematical objects are real, objectively existent entities.

• Mathematical truth are objective truth of the universe.

But there are such new answers by the cognitive science:

• Mathematical objects are embodied concepts – ideas are ultimately grounded in human

experience and put together via normal human conceptual mechanism, such as image

schemas, conceptual metaphors, and conceptual blends.

• Mathematical truth is like any other truth. A statement is true if our embodied understanding

of the statement accords with our embodied understanding of the subject matter and the

situation at hand. Truth dependent on embodied human cognition (Lakoff & Johnson, 1999,

chs. 6-8; Núñez, 1995).

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Chapter 3

Cartoons as environments of mediation

Abstract

The third chaper is dedicated to cartoons. In the first paragraph I have explained my choice for

using a cartoon as a teaching tool. From my experimental analysis, cartoons, as linguistic

mediators, have an ‘iconic’ function which restricts the semantic field of reference in relation to the

number of fields of reference in a written text; and cognitive involvement takes place on different

linguistic planes. The idea of using cartoons as a tool for mathematical trasposition originated from

the need to be able to directly communicate mathematical instructions, thereby reducing as much as

possible the difficulties inherent in understanding written language. With this aim I created

cartoons in which situations are depicted through the use of universal1 symbols and images.

Involving play, these images have tried to encourage, from an emotional point of view, an effective

transfer of the problem, in agreement with my hypotheses H1, H2 and H3.

In paragraph 3.2 I have summarised the general and specific objectives of an activity of this type.

As used in my experimentation, Clamat’s cartoon is a collection of cartoons which have been

organised by teaching theme and level. This use of cartoons proves to be a excellent tool for:

communicating, mediation, reflecting on the most appropriate reasoning, conjecturing, interpreting

a problematic situation, developing one’s own intuitive capabilities, critical capacity and personal

re-elaboration. In my opinion, the activity offers a starting point for reflecting about the

relationships between problematic situations, the need for information in resolving them, selecting

information and the relationship between solutions and restraints (the ‘acceptability’ of a solution).

Paragraph 3.3 describes the paradox at the heart of reading a cartoon: it is mono-sensorial since it is

only involved in the physical sense of sight but contemporaneously it is multi-sensorial as to what

the story narrates.

Sub-paragraphs 3.3.1, 3.3.2 and 3.3.3 describe graphic language, taken to mean a system of

conventions, an iconic code with balloons, signs for movement, visible examples of onomatapoeia

and a variable caligraphy. An iconic code is not natural and innate to man but rather a product of

convention, a result of a cultural transformation of reality with representational aims. Perceptive

learning processes are, therefore, structured so as to continually establish associations by analogy, 1 The term ‘universal’ involves not only the territories in which the experimentation takes place but also those which the knowledge about, at least the type of cartoon is ascertained.

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resemblance, past experience and similarity, according to classes to which they belong. The sense

which is most involved in these classificatory frameworks is that of sight.

The ‘line’ is the basic design element in cartoons (the visual element). It is used as an expressive

and linguistic element: it does not communicate only shapes and proportions to the reader but it also

provides shadow, ‘materiality’ and the substance from which the subject is made. With reference to

American cartoons (and recently, Italian science fiction), a new device has been introduced:

fuzziness (another example of co-language). The prudent use of shadow in cartoons is a powerful

device for introducing a particular emotion to the reader and the most visual element providing

sensations of movement and rhythm are the film lines (additional, invisible signs). It is precisely

thanks to the co-language of impaginated sequences of images that the reader is able to deduce (by

intuituion, knowledge of this co-language) everything that takes place from the beginning to the end

of the story.

Designing sounds is a very tricky part of cartoon design. From a sensorial point of view, the

paradox lies in the contrast between the visibility of descriptive onomatapoeia and hearing its

meaning, ie, the interior sound which the reader constructs in their mind. ‘Seeing sounds’,

therefore, is an inalienable and creative process by the reader and they originate on their own

because they are referred to by the designer and consequently recalled by the reader. Eye-ear

interactivity is inevitably ‘virtual’, depending on the reader’s concentration.

Regarding the senses of taste and smell, three references can be found in design literature:

throughout the passage of history, by descriptive and/or visual elements and by the smell of the

paper. The descriptive indicators of smell are rarely present, leaving narration and design with

which the reader infers and imagines the characters’ suspected mood. The allusion-delusion game

is valid for suggesting the sensorial perceptions of smell and taste regarding our experiences and

situations which, whether they happened or not, we can imagine. The aim of graphic signs, as is the

case with narration, is precisely not that of demonstrating a given experience, rather to suggest it,

and if possible, influence it.

Paragraph 3.4 describes one the most critical points of this analysis: the cartoon’s ‘auxiliary’

graphic co-language is expressed for recounting the events in the story. The visual code is made up

of everything which concerns the techniques of the results of signs and all the rules of impagination;

that is, the totality of all the graphic and composite conventions which in some way ‘create’ or

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enrich an event in a story and which includes at least two elements: in the cartoon (visual and

auditory elements, captions, balloons, visual and graphic onomatapoeia) and structural and/or

surrounding the story (impaginated, graphico-editorial elements and paper-technical elements).

Another element which characterises cartoons is the presence of characters.

Cartoons can be used for educational aims and, as a means of communication, they possess great

potential: the emotional involvement from a sensorial point of view, ease of reading, potential for

synthesis, a personal time for interpreting an experience which is objectively told and the conscious

amd instrumental use of images. Nor is the educational aspect a strange and unusual phenomenon

in W Disney cartoons.We can recall the famous project entitled the ‘Great Parodies’: Micky

Mouse’s Hell (1949), Paperon Bisbeticus Domato (1998), the Iliad became Paperiade in 1959 and

the Ulysses’ Paperodissea in 1961. Of the Promessi sposi, there were two parodies containing two

important Disney families of characters: I promessi paperi (1976) and I promessi topi (1989). Nor

are parodies from the great operas missing: Micky Mouse and the Magic Flute, Paper-Dames and

Celest’Aida, Micky Mouse and Reno’s Gold, Micky Mouse and the Barber of Seville. Being able to

reflect and think about an action performed by a character leads us to express concretely the action

and analyse it critically.

Paragraph 3.5 is purely historical as it describes the history of the origins of cartoons. It was

interesting to discover that cartoons did not begin as a product for children and young people but

they were invented at the end of the nineteenth century as a form of journalistic and humouristic

expression.

Paragraph 3.6 relates Scott McCloud’s point of view regarding the iconic role of images. He

spectacularly discussed cartoons through cartoons, talking about the espressive power of cartoons

by means of the very same graphic art; in his book he defines cartoons as sequential art. He

observes that when we abstract an image by using a cartoon, we concentrate on specific details

more than removing details. Thus cartoons have a specific power of concentrating our attention on

an idea through the cartoon’s imaginary universality. The more a face is cartoon-like, the more

people can describe it. McCloud’s ideas about the plane of the figure is the domain of the artistic

subject. The area described by these three vertices, ‘reality, language and the figure’, represent the

entire figurative vocabulary in the cartoon. If the visual iconography constitutes the cartoon’s

vocabulary, then clousure is its grammar and since the definition of a cartoon is hinges on the

arrangement of its elements, it is possible to say that a cartoon is clousure.

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Paragraph 3.7 introduces languages as different aspects in a global environment of communication.

Consequently, these aspect are strongly interconnected and interwoven in continuous and reciprocal

interaction. The most generally widespread idea regarding the language of cartoons is that it is a

juxtaposition of a language of words with a language of images. The fact is that, even if we were

dealing with a similar and simple juxtaposition, the global effect would probably not be the words

taken on their own but also their relationships (D Barberi, p203). The mental processes and

perceptive mechanisms that are necessary for gathering information and the meaningful connections

in an illustration (in selecting and organising visual stimuli) are not at all spontaneous and natural

but they must be decodable; educational and cultural variables influence this.

Paragraph 3.8 is very specific and it describes my first experimentation. In sub-paragraph 3.8.1 the

sample is described the sample whilst the methodology is described in paragraph 3.8.2. There are 4

cartoons which present the problem of the pots, ladders, roads and house. One of the key aims of

Clamat’s Cartoons is to provide a structure or framwork that allows learners to see relationships and

make meaning. Symbols help to encode information, imprinting this on the visual memory and

facilitating recall and Clamat’s Cartoons can be easily integrated into lesson planning. The main

character is a lovely mouse (Clamat) who lives out and talks about everyday problems. The

learners receive two pieces of paper, one with the drawing and other with a guided questionnaire to

assist in reading the cartoon.

In sub-paragraph 3.8.3 I describe each single cartoon and the role of the tools used in making the

cartoons. The connection to a mathematical problem arises from a real context and this enhances

the action and the attempt to resolve the problem. In all Clamat’s cartoons, I have attempted to

represent an environment which is simple and easy to recognise by learners who possess similar

cultural information. Moreover, I have emphasised the action of the main character by designing

specific tools such as a brush or ladder. The thesis of the problem is bound up with the reader’s

skill and creativity in interpreting the images and various objects in ballons, omomatopoeia and

mediatory mathematical tools.

In sub-paragraph 3.8.4 I have listed the results of my experimentation with reference to my research

hypotheses H1, H3 and H4. In the qualitative analysis of this experimentation, I have described

how some learners have interpreted the tools used in mediating the situation-problem in

encouraging transfer. The learners were interested and very curious throughout the activity. One of

the important features of iconic images is its capacity to interest and facilitate those learners who

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have lexical and decoding weaknesses regarding the letters of a written text. It is highlighted by

the Protocols that the class understood the iconic-textual language of the situation problem and the

learners created reasoned models, which liberally used ‘personalised icons’ to make their proof

more flowing and linear.

3.1 Introduction: explaining the choice of cartoons as tools of mediation

The aim of this thesis is to analysis the use of cartoons are linguistic mediators, highlighting what

an associative memory for images can influence in understanding a text and its ‘anticipation’. The

analysis of teaching situations has been carried out by referring to the strategic use of specific tools

of mediation, which are present when creating cartoons.

The term ‘implicit’ (as in ‘implicit tools’) indicates the use of particular and fundamental tool-

symbols in understanding a problem described in a cartoon. Some of these tools can be recognised

by the grammar in the cartoon whilst others are introduced on an ad hoc basis by the teacher. Our

experimentation was performed in three classes in a secondary school in Palermo and Ficarazzi: two

Class 1s, three Class 2s and two Class 3s. The learners were aged from beween 10 and 13 years and

four cartoons were created for the experimentation, being organised by teaching theme and level.

Experimental analysis tells us that, as linguistic mediators, cartoons possess iconic functions which

allows the semantic field of reference to be restricted in relation to the number of fields of reference

in a written text; cognitive involvement often occurs on different linguistic planes. Unlike a written

text, cartoons make it possible for a mathematical problem to be presented in a context which is

easy to recognise by the learners. They recognise the context in which a mimed action is developed

by the main character and communicating the problem is more direct and involving. In order to

create the most favourable environment for teaching, I introduced specific tools into the designs

and, in recalling a tool, I did not only refer to the use of certain symbols and descriptions of

particular situations but also to the teacher’s specific attitude in using that tool. The idea of using

cartoons as a tool for mathematical transposition originated from the need to be able to directly

communicate a mathematical instruction, for example, geometric and, therefore, a certain logic. I

thus aspired to reducing to a minimum the clear difficulty in understanding spoken language and so

the logic implied in that language.

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Bearing in mind these aims, I created cartoons in which situations use ‘universal2’ symbols and

images, attempting to:

1. encourage from an emotional point of view effective transfer of the problem, thereby

motivating the learner to find a solving strategy by suggesting that the main character

solve the problem;

2. show the translation phase in mathematical symbols which enables the passing from a

natural context (even fantastic!) to a mathematical one, in which solving strategies can

be found;

3. communicate the role of defining simple mathematical objects and symbols for

expressing concepts;

4. identify the resources required in achieving an aim, selecting information from the

context;

5. encourage joining up resources to the aim, leading the learner to efficiently link together

actions with the aim of solving the problem;

6. encourage conjecture;

7. lead to the solving of a problematic situation by recognising and using the geometric

properties of the graphics.

3.2 Cartoons: general and specific objectives

Clamat’s cartoons are a collection of cartoons organised by teaching theme and level so that they

can be used as part of a syllabus, bearing in mind a cross curriculum. As mediators of meaning,

cartoons have a twofold objective regarding the presence or otherwise of non-Italian learners in the

class.

The use of universal symbols in a specific context an excellent tool for:

• communication/mediation/knowledge of the meaning of symbols and mathematical activity;

• reflection, understanding one’s own way of discussing, conjecturing, interpreting and

representing a problem;

• developing one’s own intuitive, critical capacities and personal re-elaboration.

2 The term ‘universal’ involves not only the territories in which the experimentation was conducted but also that in which knowledge was ? of at least one type of cartoon.

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This tool encourages the fulfilment of the following objectives:

General objectives for non-Italian learners

• constructing a shared context in which mathematical activity makes sense

• a common language for talking about mathematics

• mediation and communication of constructions with typical mathematical meaning, dictated

by another culture

General objectives for an all Italian class

• constructing a shared context in which mathematical activity makes sense

• knowledge and understanding mathematical symbols and a closer meaning regarding a

situational problem

Specific objectives for my experimentation are

• reflecting and improving one’s own way of interpreting, discussing and explaining a

problem, mediated by a visual experience

• analysing, from a neuro-physiological point of view, one’s own sensibility and intuitive skill

in understanding images in combination

• a comparison between different strategies and their validation;

• a deeper understanding and/or strengthening of specific mathematical arguments;

• analysing the role and meaning of graphic tools,as used in creating cartoons for learners.

These include those recognised by the cartoon’s iconic code and those introduced on an ad

hoc basis (the teacher’s implicit instruments) by the teacher.

3.2.1 Explaining the choice of cartoons as substitute for a written text

We believe that cartoons as teaching tools which substitute or analyse in greater depth the solving

of a classic problem (which would otherwise be treated conventially with a written text) is an

excellent way to diffuse notions and encourage an understanding and analysis of one’s own

abilities. Thanks to their structural organisation, cartoons operate in a play-like way on a cognitive

level in the learner (this definition is decidely more pertinent than that of the reader’s, Pellitteri,

1998). Cartoons involve many more sensorial points of view and making it possible to check how

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the scene develops and therefore the story. Cartoons not only involve characters (to which the

reader should feel some attachment) and their actions but irreplaceable and essential balloons!

Cartoons tell a story! The deal with a discourse about concepts: cartoons express the simplest way

of saying what you want to say, using a system of conventions, an iconic code which includes

amongst its balloon: signs indicating movement, onomatopoeia (which in semiotic language

indicates a term for an imitating harmony of sounds) and variable handwriting. The triangular

relationship between the main character who is doing the action, their thought translated into

symbols and described in a balloon, and described in words in the second balloon provide a perfect

tool for encouraging communication and the transfer of the problem. Without realising it, the

reader is immersed and involved in a not unpleasant problem (personal pleasure and teaching

contract!) with one (initial!) visual experience. The same reader will then try to solve the problem

with their own tools, critical and logical skills.

The balloons are excellent tools of communication for the transmission of knowledge, thought and

intention. Their presence catapults the reader into the character’s thought who at that moment is

looking, listening and alive. As magical realities and by virtue of their close and logical structure,

balloons allow us to enter into someone’s thought, to read about their volition and even listen to

their voice, a voice which involves us, calls us, asking us to participate. This is a magic door which

makes use of the false union between images and words. Whilst crossing not only two languages

but two communicative strategies (Pellitteri, 1998), we can approach a line of reasoning, which

makes us become an active part of that thought. All this seems to activate our critical and logical

capabilities, thereby developing our sensibility in understanding images.

3.3 The five senses of the cartoon

Cartoons are really a multi-media instrument, since, in addition to design and text, an entire series

of co-languages can be added, treated by other parts in the narrative and/or representation: the

cinema, theatre and painting. Cartoons are the language of graphics, taken to mean a system of

conventions like an iconic code that includes, signs indicating movement, visual onomatopoeia and

variable handwriting. They are a tightly woven web of languages, one placed on the other and

fused together, thereby creating a sort of super-language. The paradox lies in the fact that cartoons

are mono-sensorial and contemporarily multi-sensorial, the latter quality because they only involve

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one physical sense: sight. You look at a story, a cartoon only. You read captions and balloons, you

appreciate designs, you understand the whole narrated episode in its finest nuances, thanks to the

sign co-language. Yet cartoons are at the same time multi-sensorial because reading an album

means picking up a book or magazine containing cartoons and leafing through its pages (tactile

sense) and smelling the paper (olfactory sense). But these are objective sensorial references: rather

multi-sensoriality is found the story and this will be dealt with below.

3.3.1 Perceiving images: seeing ? understanding

Rudolf Arnheim (in a well-known text on the relationship between arts and the science of

perception (Art and Visual Perception)) and Wassily Kandinsky (in his famous book Dots and Plain

Surfaces) discuss the main mental processes which enable someone to recognise an object.

Arnheim puts much emphasis on a concept which is based on visual perception and derived from

classical Plutonic forms: the pattern. This term in English means a shape, a modulus, a profile

based on identifying an object’s geometric structure. It is on the basis of a pattern (in addition to

the perceived rule of passed experience, which facilitate most of man’s cognitive processes) that we

can recognise the letters of the alphabet in any typographic character. This is due to their

similarities being more than their differences with the spaces filled in by our capacities to abstract

and completing. The iconic code is not natural and innate in man although it is the product of

convention: the ways of designed representation have varied throughout history and are, therefore,

the result of a cultural transformation of reality with representational aims.

Umberto Eco (1975, pp.270-74) discussed this with his ‘recognising codes’ and a ‘code of iconic

representation’. He defines the notion of an iconic code as the “system which corresponds to a

system of graphic vehicles, perceptive unities and codified codes….[…..] which depend on a

previous codification of perceived experience”. The processes of perceived learning are, therefore,

structured so as to continuously establish associations by analogy, resemblance, passed experience,

similarity and class to which they belong; the domain in which they are predominantly involved in

classificatory frameworks is that of sight.

Lines do not exist in physical reality, which is composed of plane surfaces, lights and colours.

Even the most immediate graphic for representing an object has always been the line. One of the

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main differences between cartoons and illustrations is that in the cartoon (a medium which can be

used quickly) the recognisability of the object should be immediate, render the idea with the

smallest possible number of strokes, that is it should assume a graphic quality and be concise. What

is it that makes for a comprehensible design, irrespective of the number of strokes, style, realistic

(or not) nature and colours? According to D Barberi (p.31), it is possible to define perception as a

physiological and intellectual process with which a perceiving subject selects a representation of the

world, full of meaning. The eye decides the visual hierarchy and the decision, about what to make

an object and what to leave in the background, is due to the rules of relative size, where smaller

figures always stand out. Lines which have not been designed are virtually traced out by the

reader’s eye in accordance with the ‘law of amodal completion’, whereby an ‘incomplete’ design is

reconstructed and mentally defined.

The line is the cornerstone of cartoon designs and it is used as an expressive and linguistic element.

The line is not only the dividing element between an object and its surrounding space, it does not

only communicate shape and proportion to the reader, but it is the means of providing shadow,

materiality, the substance of which the object is made. What really makes the lines in a cartoon

design three-dimensional is their careful modulation. Additional signs, the so-called ‘intuitive

signs’ are not of negligible importance in the final inked result which do not appear on the paper in

the first draft. It is also interesting to note that in American serial cartoons (and recently in Italian

science fiction cartoons), a new device – somehow related to perspective - has been introduced: the

use of fuzziness. By means of computerised touches, parts of the image can noe be out of focus.

This is yet another example of a co-language which creates a type of ‘contamination’ with the

photograph whilst giving a pleasant three-dimensional and realistic effect. The shrewd use of

shadow in the cartoon is a powerful devide for introducing a particular emotion into the reading: for

example, a lamp casting a shadow on a bedside cabinet makes us think that the lamp is switched off.

As with many other elements in cartoons, shadow can also be realistically and humorously

reproduced in a detailed way and hinted at with an infinite number of variations. Shadow in

representation and visual communication are not only vehicles for understanding volume and

brightness in a particular scene but also dramatic intensity, dynamism and expression.

It is important to observe the result of graphic techniques, that is, the type of strokes, the

background, the style by which an object is represented or an ‘event’ (the crash of lightening, rain,

the sea, a tree). The way in which an artist creates the ‘style’, which makes his designs

recognisable, is known as idiolect, a code of iconic representation which the reader can distinguish

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from other artists’ representative codes. The result of a sign is a characteristic basis of cartoon

expressivism and it is important to notice how the recognisability of an object derives not only from

structural similarity to the real object (pattern) but also the sign substrate which it characterises: it

is exactly the type of stroke (uniform or modulate) and how it is managed (shadow, broken lines,

halftone screen, background).

By cartoon sequence, we do not mean the very quick succession of images giving the effect of

movement due to the optic phenomenon of flickering just like in the cinema but alternating between

designs showing various parts of the scene. The uniqueness of the sequence concept in cartoons

can be found in the concluded and separate scenes, where the sequence lives right in the spaces

between the cartoons. It is thanks to this co-language of impaginated sequences of images that the

reader infers (partly by intuition and partly by being acquainted with this co-language) what

happens from the beginning to the of the scene. However, it is necessary to note that the

understanding of a cartoon is tightly bound to a traditional reading (speculating on what many

people know that you go from right to left in eastern thinking) and that is may depend on factors

which are apparently commonplace but fundamentally coherent for both the narrative and the

graphic: graphic rules, if followed, allow the impagination to breathe. The spaces between the

cartoons must be less for those placed side by side and greater between one strip and another so that

the reader is certain at first glance to understand the correct sequence on the page. Innovative by

genre, subsequent captions are the real protagonists in science fiction or better still the unburst

balloon (that is, the dots which the speaker has designed) of quotes in discursive form. The reader’s

impression is that of an excited discussion of many voices, as performed for example on the radio

by the various characters; what is surprising is is that simulatenous events are represented in still

more cartoons.

In a scene which appears dynamic, the visual element which more directly provides us with the

sensation of movement and rhythm are the film lines. In that they are vectorial paths (with a

direction, a reverse and dynamic intensity), film lines give velocity and force to the narrated and

represented action. An image without balloons and film lines could be equal to a photograph in that

the former would capture a single instant in an event. Instead, additional signs make a narrative of

the scene, giving it body and time: the disentangled lines imply that the scene is lasting (or has

lasted) for about 1/10 of a second. These lines represent velocity, the wind, the going forward and

reversing of a car and a character’s decision.

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3.3.2 Seeing sounds and hearing silence

Lines, modulations, perspectives and the use of textures are visual elements, that is, those which

delineate the story as well as how it would be visible with obvious and appropriate differences.

Instead, film lines are additional signs which are invisible in reality and which serve to signal

movement and temporality. Indeed, there exists a fundamental difference between the nature and

origin of signs which narrate and describe visual events in cartoons (movement, the wind, speed….)

and signs which refer to auditory phenomena, such as noises (words, explosions….) and music.

Sound is simply invisible in the cinema and in photography; its respresentation was introduced (to

the exclusion of Futurist experiences in painting and poetry) by cartoons with a variety and

complexity of shapes and styles, more so than for movement. From a sensorial point of view, the

paradox lies in the contrast between the visuality of an onomatopoeic graphic and the hearing of its

meaning, the interior noise which the reader constructs in their minds. ‘Seeing sounds’, therefore,

is an inalienable and creative process by the reader.

When we think, we create sentences which are articulated with thoughts. When we read a cartoon,

we fall into a dimension of its own in which noises in our imagination seem real. The exclusive

task of graphic operations is, therefore, to suggest to the reader not only so that they may

understand the explosions (boom) in a cartoon, it means that ‘something is exploding’ and the

graphic must make it clear to the reader as to what type of explosion is happening. It is interesting

to make clear the importance of the visual equivalent of onomatopoeia: by convention logos exists

which, with an infinite number of designs, once read evoke the same noise in the cartoon. For

example, the term ‘screeching’, commonly used to represent the friction of rubber on the road, is

represented by the logo ‘skree! This conventional relationship between sound and graphic

onomatopoeia is valid for many noises: ‘crying’ in cartoons becomes ‘sigh!’; ‘sniff’; ‘burp’; the

previously-mentioned ‘boom’ becomes ‘sboom’ which imitates the word ‘bang’; ‘ka-boom’

simulates an extended bang; and ‘bang’ is associated with gun shots and short and brief crashes or

the bang of combustible powder etc.

Tackling the problem from a purely feasible point of view, designing and not writing, elaborating a

logo, appropriate lettering which is faithful to the precise sound to be represented is not simple.

Designing sounds is a part of cartoon design which is particularly tricky and it is necessary to

dedicate much care and attention as a poorly-designed logo can ruin the effect of a cartoon, which

otherwise was well conceived. There exists another paradox, typical of cartoon narration, where

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differences between cartoons and illustrations come to light. If the scene in the illustration is

particularly quiet, in that the design describes only what is visible, the cartoon will have many

textual appendices, sounds, film strips and colours, which thereby make it noisy and lively.

Sounds in illustrations belong to a framework as much as colours and objects; they are not a

narrative element but a descriptive one. The case is reverse with cartoons: the representation of

people who engage in a dialogue or who find themselves at sea in a storm, in the absence of

balloons and onomatopoeia, comprise not only a descriptive device but also one which is purely

narrative: sounds hide themselves because they are hinted at by the designer and consequently

evoked by the reader. For example, music in cartoons cannot be included without designing

musical notes or texts which are impossible to identify as songs: eye-ear interactivity is inevitably

‘virtual’, depending on the reader’s concentration. This seems the right moment to quote D Barberi

and the relationship between words and images in cartoons and in the presence-absence game of the

very same words:

“All cartoons are constructed as relationships between images and words, even those where there are no words because their absence in a context, in

which they are normally present, is particularly meaningful…..[ ] It is important not to forget that quality is found in a cartoon in the relationship

between words and images and not with one or the other individually. And this relationship reveals itself in uncountable ways, not by a simple

catalogisation.” [D Baberi, The Alchemy between Images and Words]

3.3.3 Sensations of taste and smell and… touching images

The senses of taste and smell are two topics which may appear inappropriate to cartoons but they do

have much potential. In the literature about design there are at least three references to the senses of

taste and smell:

1. a reference through the story

2. a reference through graphic elements and/or visual ones

3. the real smell or paper

Rarely are the graphics of smell present in cartoons but only the story and designs, from which the

reader infers and imagines the characters’ mood. The comment “…what a stink!” is a narrative

element, yet thanks to a detailed reconstruction and graphic onomatopoeia which, the smell seems

to emanate in some way from the page and – depending on the reader’s fantasy – one can smell a

repulsive smell. The allusion-illusion game is a particularly valid way for suggesting the sensorial

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perceptions of smell and taste of a part of our experience and conversely situations which, whether

they have happened or not, we can imagine. The determining factors for the results of the sense of

taste and chewing food are on the one hand elements of graphic onomatopoeia scattered throughout

cartoons (gnam, glut) and on the other little ‘balls’ which represent breadcrumbs or the speed and

greed with which a meal is consumed. More than with sight and hearing and regarding taste and

hearing, it is obvious to think that a well-designed cartoon cannot really provide the character’s

phsyiological sensations to the reader. The aim of graphic signs and narration is precisely not that

to verify a given experience but to suggest it and, if possible, influence it: to infer what the reader

hears, even if very slightly, ie a sort of sensorial reminiscence associated on the one side to one’s

own passed experience and on the other hand the imagination of perceptions, which are not yet

experimented upon but hypothetically constructable in our mind.

3.4 The visual code of cartoons: structure and educational characteristics

One of the most important points of this analysis is the auxiliary graphic co-language which is

expressed by the cartoon when recounting events in the story. Put simply, it is possible to consider

the visual code as everything which concerns techniques relating to the results of signs (the way in

which each actor expressly visualises the objects in the story with one’s own graphic-illustrative

idiolect,) and on the other hand the totality of the rules of impagination. However, a visual code is

something more: it is a collection of all those graphic and composite conventions which in some

way ‘create’ or enrich an event in a story. Whilst accepted by the reader, signs constitute symbols

in a visual code, which signify a movement, a noise, brightness etc. Moreover, the intimate fusion

of co-languages is indispensable in creating and understanding a cartoon story: it is thanks to this

fact that it is natural for the reader to pass from one cartoon to the other.

The visual code of a cartoon includes at least two groups of elements:

A. in the cartoon

1. visual elements

2. auditory elements

2a. captions

2b. balloons

2c. visual onomatopeia or graphics

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B. structural and/or surrounding the story

1. impaginatory elements

2. graphic-editorial elements

3. paper-technical elements

Wanting to extend my analysis of group A, visual elements, we have already seen that they regard:

film lines (in a dynamic scene, they provide the sensation of movement and rhythm, in addition to

the speed and force of the action described).

stroboscopic effects (an optical phenomenon derived intermittently seeing a moving body. This is

due to mini black-outs between one frame and another, which our eyes cannot fill, and the end

result is to see the people moving in bursts. One important thing to highlight is that the

stroboscopic effect has much to do with time and this is not always respected. Often thoughts and

speech, as found in the balloons, are extended, compared with the few instances required in

depicting the events; the only way out is to have captions in the monologue and implicitly distribute

them in a longer scene).

colours (they can be present two circumstances: in completing a pre-existing Indian ink design or

for constituting the design. However, colour in cartoons suggests shape, the solidity of objects and

the sense of touch through the coloured nuances of metal and wood in describing the human body.

Recently, colour of American cartoons has been used more and more with the aid of computers to

chromatically give the finishing touch to albums, especially those about super-heroes.

Regarding auditory elements we can find:

the lettering (the material writing of texts written in space-texts and it describes the variable

influence of defining a voice and atmosphere. Reading a sound means observing single signs which

come together, giving rise to a given word or sound and, on the basis of the language which they

refer, decode them and providing meaning (M Pellitteri, p94). Generally, block letters manually

hand-written by the copywriter or directly by the author is used in the captions; this, however, can

be impersonal with one character not being differentiated from another. If used coherently in the

story, each character can contribute in strengthening and characterising the story. An effective use

of lettering is also possible with the title on the first page, and American cartoonist have used this to

great effect. It has always been the convention in cartoons for the text in balloons and the captions

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to be graphically incoherent for two reasons: the first is a technical problem with colour cartoons.

If the text was directly written onto plates, a small error during printing would be sufficient to

make the balloons and captions illegible and vice versa; if composing texts on a separate film or

with the computer: the results would be excellent. The second reason for incoherency is a purely

artistic in nature: often and voluntarily, texts are reviewed many times by the author prior to

printing and it is not convenient to write the texts beforehand on the plates, making any alterations

thereafter impossible).

balloons (‘nuvoletta’ in Italian or ‘ectoplasme’ in French. These are the graphic metaphorisation of

breath and air which is exhaled on speaking and they are simply a graphic convention for reading

the audio during the action; ballons include thoughts, punch-lines and various characters’ dialogues.

The balloon is an element which more than any other marks out the reading time of the whole

comic strip but not the time taken for reading the words, given that each reader will have their own

reading time, but the time taken for the action).

onomatopoeia (graphic or visual. Conventional graphic device for representing sound).

Another element which distinguishes cartoons is characters.

With reference to characters, there now follows part of an interview with Claudio Stassi, cartoonist

for various Italian publishers; he highlights fundamental points in creating and portraying characters

in cartoons for educational purposes:

• the reader immediately feels empathy with the main character, deciding to let themselves be

accompanied in the mathematical, historical experience, etc

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• the ideal character is humanised so that the reader more easily accepts their advice and is

interested in their problems. The following comes to mind “….oh, there is a person who

resembles Micky Mouse, who wants to explain something to me…”

• the cartoon must be as colourful as possible as a black and white image would be less

striking

• a first balloon outlines the problem, translated and described only with symbols and adjacent

images, all of which are logical, clear and meaningful

• a second balloon (with supporting role!) contains the problem in a written form

• there is always a main character, who is accompanied by secondary characters which the

former organises and/or uses

In this way cartoons can be used with educational aims. As a means of communication, cartoons

possess huge potential: multi-sensorial, emotional involvement, easy to read, can be synthesized, a

personal time for interpreting an experience (objectively told), care taken in the conscious and

instrumental use of images. For example, adventure cartoons have for a long time popularised the

tight relationship that exists between the historical dimension and geography with a graphic and

visual narrative which, when used in a conscious and congenial way, can fascinate and arouse the

reader’s curiosity and hold it until the end of the story; this suggests a series of functional

information with a teaching aim. On the best examples of the equilibrium between history, cartoons

and teaching is the series of volumes entitled ‘A Man, an Adventure’, published by Cepim in

November 1976. The publisher Sergio Bonelli entrusted the creation of these adventure cartoons to

the best designers of that time: each time their starting point was a historical event, which had really

happened. A freely-constructed narrative event was proposed, one which centred on a character or

heroe located in the historical facts (R Farnè 2002, p.275). Meant for an older audience, Martin

Mystère, born in 1982, had an extraordinary collection of ‘cases’, in which science, art,

anthropology and history provided the material for an adventure, made up of suggestive

interference and a documented recalling.

The educational aspect in W Disney’s cartoons is not a strange and unusual phenomenon but they

have a character which often has a multiplicity of topics and tales, thanks to the extraordinary

versatility of the fantastic world of these cartoons and the willingness of characters to adapt

themselves to various roles and situations. We are reminded of the famous project, the Great

Parodies, a rich collection of cartoons, coming from the best of Disney comics: Micky Mouse’s

Hell, 1949, inspired by the first verse in Dante’s poem, narrated by Guido Martina and designed by

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Angelo Bioletti; in 1998 there was The Taming of Donald Duck, from Shakespeare’s The Taming

of the Shrew by Silvano Mezzavilla and Giorgio Gavazzano; the Iliad became Paperiade in 1959

with Guido Martina and Luciono Bottaio whilst in 1961 Guido Martina, together with Gian

Giacomo Dal Masso and Pier Lorenzo De Vita, made Paperodissea; Giovanni Battista Carpi, the

master of the Italian Disney Academy, produced Hamlet (Micky Mouse, Prince of Denmark, 1960)

and War and Peace (1986), Le Miserable (The Mystery of the Chandelier, 1989). From the

Betrothed Lovers, two parodies, with characters taken from from two great families of Disney

characters, were made: Betrothed Ducks (1976) and Betrothed Mouse (1989). Nor are paradies

from the world of opera missing: Donald Duck and the Magic Flute, Donald-Dames and

Celest’Aida, Donald Duck and the Gold of Reno, Donald Duck the Barber of Seville.

Being able to reflect and think about the personalities or the action of main or secondary character

leads the reader to make the action objective and analyse it critically. Cartoons allow us to break

free from usual reading models, thereby inferring an organisation of images, which is apparently

free and personal, of data provided by the cartoon, always with a teaching aim.

3.5 History: the origins of comics

From a historical point of view, cartoons were not originally invented as a product for children and

young people but they originated at the end of the nineteenth century as a form of journalistic and

humouristic expression. Their aim was to increase the sales of daily newspapers, often in open

competition amongst themselves, particularly the Sunday editions, with inserts of illustrated pages.

A group of proto-cartoonists, headed by Rodolphe Topffer and Wilhelm Busch, used to design

‘little stories in pictures’ for the more important European dailies, more often than not humourisitic

and accompanied with rhyming captions,; this was the beginning of the official origin of cartoons,

which is conventionally dated to 1896. The formal origins of cartoons is said to go back to the end

of the 19th century in American dailies as variations of the typical, illustrated Sunday papers and a

product of the cut-throat competition between the two great powers in American journalism: the

magnates William Randolph Hearst and Joseph Pulitzer. Early on cartoons became a dynamic and

popular past-time, as compared to static traditional illustrations. This mass medium was

surprisingly useful socially as it made an important contribution to the speed and ease with which

large communities of foreign immigrants to America could learn English; they would otherwise

have been linguistically marginalised.

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Cartoons soon transformed themselves into big business, so much so that so-called ‘syndicates’ or

‘sindicates’ were formed, ie, agencies who used to sell (and still sell) single series to American

dailies. However, the authors often directly designed the daily without any intermediaries. Only

starting from the 1920s and 1930s were other types of cartoons developed, with the introduction

first of the comic-book (the classic cartoon album for young people) and after the magazine.

Therefore, even before cartoons were recognised rightly as a graphic-literal form of art, they were

seen and used as a mass medium, a type of popular entertainment and a product which was purely

commercial.

A strong tradition meant that the very first speaking balloons appeared one Sunday in February

1896, including no less than a parrott, amongst countless other characters, from whose beak was a

balloon containing one sentence. It is said that that animal was probably the very first cartoon

character or the first animal to have received the gift of speech from a balloon; however, the event

was controversial. If one really wants to discredit the myth of Down Hogan’s Alley as the first

cartoon with balloons, Romàn Gubern has informed us with conviction that the balloon was a

graphic-phonetic convention, which had already been unknowingly introduced by English

illustrators in the 17th century and adapted thereafter by Americal satirical designers at the end of

the next century. Therefore, the distinction between the two verbal channels in cartoons was clear:

the caption and the balloon.

As was said about proto-cartoonists, captions were the very first carrier of the cartoon voice and

over they years they have developed incessantly and differentiated into ma ny different kinds and

uses. The first type of captions, the little rhyming stanzas, were found at the end of the nineteenth

century and the beginning of the twentieth,the linguistic register and atmosphere in the story was

usually playful and almost always paternally didactic. We can think about the great Italian tradition

of the ‘Corrieri dei Piccoli’, which presented little cartoon stories until the first decade of the 20th

century under the guidance of Antonio Rubino; paradoxically this publication did not include

cartoons and balloons, limiting itself itstead to the use of rhyming caption texts, which would be

recited. The ‘Corrierino’ contained beautiful pages of Italian cartoons with characters such as

Attilio Mussino’s Bilbobul, Sergio Sto Tofano’s Bonaventura and Rubino’s Quadratino, all of

whom were described in an attractive and intelligent way. However and with the benefit of

hindsight, the Corrierino displayed significant limitations in their handling of cartoons: abolishing

the balloon, then thought by many to be particularly unsuitable for those starting to read, to

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substitute it with little rhyming stanzas. This model of rhyming captions, typical of the Italian

tradition, was ousted by a more faithful description of the image.

P Boero and C De Luca (1986 p.208) have recalled that starting in the 1930s, in particular between

1933 and 1935, the market was occupied by a myriad of periodicals: in 1933 Il Monello, Rin-Tin-

Tin, Primarosa (for children) and Tigre-Tino appeared. The Pappagallo and Mastro Remo were

published the year after and they included the first stories of Bibì and Bibò, Laurel and Hardy and

Bombolo, the latter recounting the adventures of Ironfist. In 1935 there was the Grandpa Ebe’s

Comic, Intrepid and the Three Little Pigs. In the meantime, a succession of family characters by

Walt Disney appeared in the weeky Micky Mouse, from which the universally known character

Micky Mouse arose, protagonist of many hilarious events. He successfully fought against gangsters

and villains and he distinguished himself by his courage and optimism. Another character, invented

by Walt Disney in 1934, was Donald Duck with his various defects and weaknesses. Often his feats

concluded with bouts of disaster which were sometimes put right by his nephews: Qui, Quo and

Qua. His rich and greedy uncle Paperon was opposed to Donald Duck’s lack of means and famous

misfortune whilst his cousin Gaston, fortunate in a cheeky kind of way, led a well-to-do and slothful

life. With time Donald Duck became more human, thereby becoming the prototype of the lower

middle-class.

Still in the 1930s, super-heroes like Superman, Batman, Flash (the fastest man in the world),

Spectre, Hawkman and Dick Tracy were born, all characters who fought against crime and

injustice, two important topical themes in a period when America was fighting against the power of

gansters. The, starting in 1934-35, a school of Italian cartoon authors developed in Milan at the

publishers’ Mondadori. A group was formed, bringing together the most important Italian

designers of the time: Walter Molini, Gianni Scolari, Giove Toppi and others who made their way

by inventing stories and narratives in the mould of the European cultural tradition. Yet again in

Milan in the 1930s Cesare Zavattini was active in the field of cartoons and, together with other

Italian authors, he published in ‘Audace’, ‘Adventurous’ and ‘Micky Mouse’.

The widespread diffusion of comics in 1954 is thought to have produced bewilderment amongst

youths and the large agencies were, therefore, forced to down-size their plans, suspending the

production of the more critical cartoons or those likely to arouse opposition. Examples of these

included horror comics and they agencies censored themselves, in a similar way to the cinema. The

new political reality of the 1960s made way for a relaunching of cartoons with the editor Marvel

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publishing an interesting series: The Fantastic Four, Spiderman, Hulk and X-Men. We have not

only seen in Europe the circulation of American comics but the Belgian school has also produced

the hugely successful Tintin. Of all the characters in the post-was period, particular reference can

be made to Goscinny and Uderzo’s Asterix and Obelix with their interpretation of a counter-story

narrating the supposed conquests of the Roman leader Julius Ceasar. Having concluded the war

period with the experience of Corrieri dei Piccoli, the renewal of cartoons in Italy was encouraged

by a new generation of authors who, whilst aspiring to the standards set by American comics, were

preparing a flowering of cartoons, which were typically Italian. The beginning of this process saw

the publication of Diabolik, Hugo Pratt’s Asso di picche, A Gappellini’s Tex Willwe, the latter

being one of the most important designers in the field of Italian cartoons.

If we think of particularly classic examples coming out of the USA - Burne Hogarth’s Tarzan and

Alex Raymond’s Flash Gordon - we can see how the design of each cartoon is a real illustration

with a high degree of formal refinement and accompanied only by a literal text: this is more a

comment about the image than a true caption, as intended nowadays. From these image-descriptive

frameworks, the modern model of captions evolved in a parallel fashion to the origins and

development of balloons. When it was realised that it was allowed and possible to make characters

talk directly from within the design, the captions were gradually transformed into an external voice

with multiple uses. It has been this split between the characters’ voices and the external voice

which has given the captions an interior voice, a narration with properties which we can call

intensive variables: with its own speed, rhythm, timbre and intensity. These properties depend on

other characteristics of the captions and we can call these extensive variables: the shape, dimension,

colour, lettering and positioning of the cartoon.

3.6 The iconic role of images: Invisible Art and Scott McCloud’s point of view

The potential of cartoons is to evoke unlimited emotions. Scott McCloud displays in his book the

need to define the word ‘cartoon’ and to describe it, referring to Will Eisner, the great master of

cartoons who used the expression Sequential Art. McCloud speaks of icons by which he means

almost any image used in representing a person, a place, a thing or an idea. Symbols are only one

category of icons, which includes:

• images for representing concepts, ideas and philosophies

• icons of language, science and communication

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• icons which we call figures and designed images for resembling subjects (obviously by

varying the resemblance, you also vary the level of iconic content).

The meaning in non-figurative icons is fixed and absolute and their appearance has no effect on

their meaning because they represent invisible ideas. However, the meaning of figures is fluid and

variable in relation to the degree of diversity in real life (the level of abstraction from reality varies).

Words are icons which are completely abstract and they bear no resemblance to the original object

(for example, an eye). With reference to figures, we can see that when we abstract an image by

making a cartoon, more than eliminating details we concentrate on specific details so much that a

cartoon has special powers:

• it concentrates our attention on an idea and

• the universality of the imaginary in the cartoon: the more cartoon-like a face is, the more

people can describe it.

When we look at a photo or realistic design of a face, it is seen like the face of another person but

when we enter into the world of cartoons…. ☺, you see yourself! Cartoons are a space into which

identities and awareness are dragged. We do not observe merely the cartoon, we become the

cartoon. The reader makes a little voice of what they read in the balloons in the text, a concept.

The great Marshall McLuhan noticed a similar type of non-visual awareness when people interacted

with inanimate objects. For example, when we look at something we feel much more than our five

senses tell us. The carrier becomes an extension of our body, it absorbs our sense of identity and

we become the machine. Our capability of extending our identity to inanimate objects can mean

that pieces of wood become legs (crutches), pieces of metal become hands (cutlery), pieces of

plastic become ears (telephone receiver) and the constant awareness of oneself influences the

external, including the object of our extended identity. It is just like when the awareness of our

biological ego (☺) is a simplified and conceptualised image…….also our awareness of this

extension is greatly simplified. All that we find in life can be separated into two spheres: concepts

and senses. Our identity permanently belongs to the conceptual world but we can see it, feel it,

smell it, touch it and taste it. It is a simple idea and all the rest initially belongs to the sensorial

world, our external world.

Gradually we expand beyond ourselves, encountering the sight, smell, contact, taste and sound of

our bodies and the surrounding world and we discover that objects in the physical world can cut

across us; these objects possess their own identity or that of our extensions, they begin to glow with

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the life that we give them. Cartoons take their place in the world of concepts. In reducing the

image of a face to two points and two lines, it can seem that our scale of iconic abstraction is

complete but in reality we can still continue to abstract images. Words are definitive abstractions!

Words, figures and other icons are the vocabulary of the language called a cartoon.

The plane of a figure is the realm of an artistic subject. The area described by the three vertices

‘reality, language and the plane of the figure’, represent the complete, illustrated vocabulary in

cartoons or any visual art.

The plane of the figure

1 2 3

FACE

THE PLANE OF THE FIGURE

reality language meaning

1. the margin of the retina

2. the border of language

3. conceptual margin

Cartoons must be able to express the designer’s most intimate ideas so as to mature as a medium.

The phenomenon of closure (observing the parts and perceiving the whole) is, for example, a mental

process in which

becomes a face

There exists a means of communication and expression which makes use of closure like no other, a

means by which the public is a spontaneous and aware participant and closure is the agent of

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change, time and movement: the space between individual cartoons, that is, the margins,

accommodates much of the magic and mystery which are at the very heart of cartoons. In the limbo

of the margin, human imagination takes two separate images and transforms them into one idea.

Closure allows us to connect these moments and mentally construct a continuous and unified

reality. If visual iconography is the vocabulary of cartoons, closure is its grammar and, given that

our definition of cartoons hinges on an arrangement of elements, we can say that a cartoon is

closure. Our mind fills the passing moments between various still moments, which are described in

time between the cartoons, by creating the illusion of time and movement. As, through clousure,

figures and intervals between the figures create the illusion of time, words introduce sound,

representing that which can only exist in time…sound.

Moreover, it is very important to observe how a figure can evoke an emotional or sensorial response

and lines, more than figures, are visual metaphors. Backgrounds are another important tool for

signifying invisible ideas, particularly in the world of emotions. The most used over time, the most

complex and versatile sinesthetic13 icon is the balloon! One of the most important features of the

use of iconic images is its capacity to stimulate people and especially children to reading books,

facilitating those who find it difficult to lexically decode the letters in a written text. From a

pedagogical point of view, the use of images has always been a possibility in enhancing

comprehension and appreciating the contents of a text, when reading is a daunting task. Thanks to

their connection with reality, the use of icons offers many possibilities in: the education of aesthetic

sensibilities, using and developing the imagination, motivating the skill of observation, creativity,

and recalling passed experiences and/or memories in critically analysing experiences with one’s

background and one’s own reality, thereby stimulating curiosity and the desire to know more.

More than with the written word, the use of icons encourages an exchange of impressions and

communication between different readings, enriching them with different kinds of information.

When the integration between verbal text and image is performed with mediated awareness, images

add meaning to their decorative role and the reader is induced, by the creation of the same graphic

creation, to search for meaning, establish connections between the image and the verbal text.

Compared to a written text, images can clarify and make more immediate the understanding of an

difficult text or they can be used in representing, through their own code, the content of a verbal

text. In contrast to what one might think, the use of images encourages comprehension of the text

and makes children more active, thereby stimulating them to researching meanings, characterised

13 Sinesthesia: the art which in some way can unify the senses and various art forms which attract these senses.

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by an authentic reading. In many cases it will be the figures more than the words which are stored

in the reader’s memory, as enduring signs of a reading containing authentic descriptions. The way

by which words and design co-operate in the text is through redundancy, a repetitive meaning when

the words repeat what the design has already described, and returning, when the difference between

design and word is noticeable. The consequences of this relationship are that there is a excess of

information in the text, which reduces ambiguity. For example, for non-readers, images have a pre-

eminent formative and cultural role, which if gradually educated, is not lost in subsequent years

with the acquisition of literacy but these images are transformed and correlated with further reading.

3.7 The language of cartoons as an ‘environment’ for understanding written text

D Barberi (2002) has defined languages as environments more than tools of communication: the

former is lived whilst the latter is used in function of our communicative aims. Languages are

various aspects of the global environment of communication and they are consequently strongly

interconnected, interwoven and in continuous reciprocal interaction. Language is an environment

inside of which we are and inside of which we think. Many times we try to express what we think

in one language with another and every language ends up by being crossed by many if not by all the

others. Ideas are formed within a language. The language in which we are currently thinking will

not only have the characteristics of a tool used for transmitting ideas but it will be the environment

in which we are when when are shaping these ideas. According to D Barberi four types of

relationships between the language of cartoons and other languages can be identified:

• inclusion: the language of cartoons is part of the general language of the narrative, as it is,

for example, for the cinema

• generation: historically, generation is a derivation of language, as is the case with

illustration, caricatures, illustrated literature, with which it shares some characteristics

• convergence: horizontal relationships also exist with languages and cartoons are not

descendents but they are tied by the fact of possessing common ancestors (as in the case of

painting, photography and graphics) or sharing expressive factors (like if the case of poetry

and music, theatre and the cinema)

• adaptation: this happens when the cartoon finds it simpler to mimic and reproduce itself

within another language in using its expressive potentialities rather than attempting to

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construct ‘equivalent’ expressive potentialities. The most important example of the

adaptation of another cartoon language is that of the cinema; if cartoons wants to achieve a

desired effect, they cannot limit themselves to copying the cinema but cartoons must literally

construct the cinema within themselves.

As R Farnè confirms (2002. p.118) beginning to read is the arrival point of a long process of

symbolisation which, by design and reading the design, is transformed into definitive reading.

However, letters have their presuppositions not so much in decoding single letters but in the habit of

deriving meaning from the images and signs in reconstructing the story.

The mental processes and perceptive mechanisms required for gathering the information and

meaningful connections in an illustration, by selecting and organising the visual stimuli according

to determined criteria (resemblance/differences, figure/background etc) are not spontaneous and

natural, but they must be decoded; the competence of this depends on educational and cultural

variables. The problem is not so much tied to the mere recognising of the subject and immediate

global approach with the image as the skill of penetrating the details of figure, pulling together

individual features through question and observation. Effective linguistic competence enables us to

carefully read text because we understand the meaning of what is listened to and because that text

has been adapted to one’s own skill and stimulated by one own’s interests. In the same way,

knowledge about images helps us to look at the figures, dedicating time and attention to images,

where we will find connections and meanings with one’s own internal and external world. For

example, a primary role is carried out by the albums of pre-school age children where images

clearly predominate quantitive and communicative aspects. The content and plot of the story are

exclusively transmitted by iconic language, with its own autonomy and requiring a reading code

and substitutive or complementary interpretations, as regards the verbal language which is adjacent

to the iconic language. In the sphere of linguistic education, illustrated albums are shown

themselves to be useful in improving the verbal expression of children who, when telling and

describing images, learn to construct structurally correct and articulated discourse. By

reformulating and reconstructing the story, learners also acquire specific logical compentences

which regard narrative capabilities and the mechanisms of coherence and cohesion.

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3.8 Presentation of the First Experimentation

3.8.1 Experimental context: the sample

The experimentation was carried out in three classes in two secondary schools in Palermo (Italy)

and two classes in Ficarazzi, a village near Palermo. The students in the sample were from between

10 - 13 years old.

3.8.2 Methodology: instructions and organizing the cartoons

Four cartoons were deployed in the experimentation, being organized according to the various

teaching topics. A teaching unit was prepared for each cartoon thus:

◊ The problem of the pots

◊ The problem of the ladders

◊ The problem of the roads

◊ The problem of the house

What are Clamat’s cartoons?

Structure

One of the key aims of Clamat’s cartoon is to provide a framework within which students can

establish relationships and meaning, similar to an index system in a library. This allows concepts to

be filed away in compartments in the memory, thereby making their retrieval much easier. Much

information can be contained within a small space and the ideas can be arranged in such a way so as

to identify relationships between concepts and encourage thinking from a larger perspective.

Symbols

Symbols help to encode information, imprinting on the visual memory and facilitating recall. A

standard set of symbols has been used throughout Clamat’s cartoons, thereby encouraging

connections between maps to be explored and discovered. Of these, there are various mathematical

CCllaammaatt’’ss CCaarrttoooonnss

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symbols such as arrows, question marks and dots and their usage is very important for encoding

information.

Introduction to the Activities

The activities found in Clamat’s cartoons can be easily integrated into lesson plans and those

described below provide an excellent opportunity for students to have an overview of a topic, in

short, to glimpse the bigger picture. These activities will also help learners to make the all-

important connections, which establish meaning.

The main character to be found in these activities is Clamat, a friendly mouse. He is prone to

everyday problems and he like to talk about these.

Instructions

1 This activity can be introduced by demonstrating one of Clamat’s cartoons to the class. This

simplest and probably most effective activity is to have a class discussion about Clamat’s

problems.

2 The learners receive two photocopies: one with the cartoon strip and the other a guided

questionnaire.

3 The teacher asks the learners to answer the following questions from the questionnaire:

1. What is Clamat’s problem?

2. What are the data in the problem?

3. What strategies could Clamat use in resolving the problem?

4. Why have you chosen these strategies?

5. What does the message in the bubble mean?

6. What do the arrows represent?

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3.8.3 A structural analysis of each cartoon and the role of the tools of mediation used in

creating the cartoons

The connection to the mathematical problem arises from a real context and this enhances the action

in the cartoon and the effort expended in resolving the problem. It is clear that the more evocative

the image, the greater the reading time approximates to the time taken to perform the action, thereby

more directly raising awareness of the experience, which involves the learners’ emotions. Our

emotions are heightened with greater incisiveness by a careful and well-defined use of the image

and a particular colour or object, which is connected with a particular example of onomatopoeia. I

have included a simple environment in all of Clamat’s cartoons, which is easily recognisable by

those learners who possess a similar cultural background. Subsequently, I made sure to highlight

the actions of the main character by designing a specific tool, such as a paintbrush or ladder.

Regarding specific tools used in creating the cartoons, I used arrows and their meaning of

directionality to highlight the relationship between data; question marks indicated the data to look

for. The argument of the problem is connected with creativity and the reader’s skill in interpreting

images and various objects in balloons, onomatopoeia and mediatory mathematical tools. An

observation of Clamat, his facial expression and gestures should evoke the greatest number of past

experiences and increase the emotional state produced by the cartoon. All this should lead us to

understand that:

Clamat does not know what to do..…

Clamat is not sure..…

Clamat is happy…

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Clamat is interested in .…

Clamat is crying ……....

Let us now analyse the first cartoon, relating it to experimental data: the problem of the pots

Looking at the cartoon, the image which immediately attracts

our attention is the character of Clamat, who in turn reminds us

of well-known Diddle. In combination with the ‘question

mark’, Clamat’s facial expression tells us that he has a problem.

Our attention is then transferred around the character, trying to

make sense of the image: the paintbrush, the pots of paint next

to the fence, the fence and paintbrush tip in the same colour and

the pots are all aids to comprehension.

I then drew the same pots in the balloon and included an arrow to indicate the relationship between

one pot and the number of painted lengths of wood in the fence. It was evident, during the

experimentation that the role of this arrow was clear to the learners. In order to repeat the

procedure, I then drew dots and a question mark to indicate the number of lengths of wood to be

calculated, corresponding to a specific number set by the teacher. This context was immediately

recognised by all the students, some of whom spontaneously recognised Clamat’s problem,

referring to it at home. A formulation of this problem could be:

Clamat wants to paint a fence made up of 31 lengths of wood and he needs to know how many pots

of paint he will need, given that one colour will cover 2 lengths, 2 colours will do 4, 3 colours will

do 6 etc.

It is important to remember that the difficulties often encountered in interpreting specific subjects

the cartoons do not depend exclusively on a superficial interpretation of the symbols and

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corresponding symbols nor on a poor memory but mainly on the difficulty in recognising and

reconstructing the context. The conceptualisation of the visual image arises from a personal

interpretation of various information and this acquires semantic value in relation to one’s own

creative skill and background. This comparison-contrast between perceived and recalled images

creates a semantic conflict which can be translated into a type of personal, visual conceptualisation,

which may or may not be sufficient in seeking solutions.

Let us now consider the second cartoon, made up of 4 individual cartoons: the problem of the

ladders:

There are four individual cartoons in this cartoon strip, which contain the story. Clamat’s problem

is obvious: having seen the cheese on the table, which is 10cm from the ground, Clamat thinks

about how he can reach it. He himself is 25cm from the table and he comes up with the idea of

leaning a ladder against it. How long, therefore, does the ladder have to be?

Bearing in mind the aim of involving the most learners in thinking about Clamat’s feelings, I

thought about including Clamat, being interested in the cheese, in the first cartoon and, from this,

the problem of how to reach it. Clamat is thinking but, in that moment, he is not alone as the reader

also provides company. Clamat discovers a strategy but someone must help him to do the

calculations; the story can only have a happy ending in this way:

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The learners told me throughout the exercises how much more user-friendly Pythagoras’ theorem

had become when they used the cartoon and one told me that she never would have thought that

Pythagoras could have helped Clamat! In demonstrating the proof, the learners went back again to

the drawings which they had done in class with their teacher, but this time, some of them drew a

line from the table to the cheese!

Let us now consider the third cartoon, made up of two individual cartoons: the problem of the

roads. The first cartoon precedes the second in that insufficient attention paid to the first cartoon

makes it difficult to understand the second and resolve the problem. The problem is: “Which road

should Clamat take? How can he measure them and know for certain which is shorter?”

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In order to facilitate comprehension, I could have included some form of transport, like a bicycle, in

the first cartoon and, indeed, it was subsequently necessary to make a small adjustment to the

second cartoon. The coloured lines remind us of the roads in the first cartoon; they had been drawn

into lines so that the individual parts could be calculated. This class teacher had asked me to do an

exercise involving direct and inverse formulae for solving flat shapes, such as: squares, rectangles,

triangles and rhomboids. These two cartoons were found to be very effective in encouraging a

spontaneous discussion about the concept of measuring and the properties of each shape. What

struck the teacher was not so much the content of the discussion but the spontaneity and curiosity

which had initiated it; of note was the fact that the learners immediately recognised the context.

The final cartoon was given to the highest classes in the lower secondary school to reinforce the

concept of the expansion of a solid. ‘The problem of the house’ deals with the following: “Clamat

is painting his house and he need to know the extent of the surface to be painted. Do you know how

to calculate the total surface area of a house, knowing the measurements?”

The use of colours was one way of helping learners who did not understood the concept of the

‘expansion’ of a solid very well. Whilst commenting on the images, many of the learners could

help themselves by understanding the significance of the colours used. The nearness of a thoughtful

character who was painting his house made it easier for the learners to interpreting the balloons and

calculate the areas for each single figure. This way of describing the problem reminded them of the

drawings they had made whilst doing their calculations. Below are some examples of the learners’

work:

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3.8.4 Conclusion

A qualitative description of the experimentation, the teacher’s role and the importance of

tools in describing a problem

The teacher’s implicit tools are those which are implicitly included in the creation of a cartoon.

Some of these are recognizable by their iconic code and the grammar of the cartoon while others are

introduced by the teacher in order to facilitate comprehension of the problem by the reader.

Some of the implicit tools used in this experimentation were:

- balloons were used as a mediator between the reader and the thinking processes/intention

of the character. The number of balloons used and their contents were chosen by the teacher who

also organized the content and number of suggestions in relation to the cognitive development of

the class and the teaching aim;

- arrows to show implication, a logical and associative link;

- ? the question mark, used either to show a problem or the aim of a problem to be calculated.

The choice of this tool n the cartoon could not be casual but subject to a particular logic, which was

inherent in the problem;

- dots, indicating a repeated action.

With reference to the implicit use of the semiotic-specific tools in the cartoon, the teacher’s role

was not to teach but to supervise: the teacher distributed the cartoons and a set of questions to the

learners and then supervised each learner. In this phase, the teacher’s role was to encourage the

learners to interpret the problem as a polyphonic intertwining between the story, told in images, and

the story, told in words.

The learners’ answers to the following questions:

1. What do you think the balloons mean?

2. How do you think the arrows are used?

regarding the balloons

were:

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Ø the balloon cannot be removed because it says how many pots of paint are needed to paint

two lengths of wood;

Ø they are necessary in understanding the data for doing calculations;

Ø the balloon helps me to think about the problem;

Ø the balloon summarizes the problem;

Ø the balloon is related to the mind, ie. Clamat’s problem;

Ø the balloon gives us the data for the problem;

Ø the balloon suggests that Clamat will shortly eat the cheese;

Ø the balloon suggests that Clamat should concentrate on taking the cheese;

Ø the balloon shows what Clamat thinks the house is made of.

Ø the arrows show that one pot of paint will cover two lengths of wood;

Ø they mean: ‘correspond to’;

Ø the arrows replace the ‘equal’ sign;

Ø the arrows show how the house is, once it has been opened up in all its parts;

Ø the arrows show the house in pieces;

Ø the arrows indicate the passage between the closed house and the opened up house.

Results of the experimentation

With reference to my research hypotheses H1, H3, H4, the results of my experimentation are as

follows:

-The learners seemed to understand the iconic-textual language of the problem and the meaning of

the semiotic intermediary, which was used in constructing the cartoons. An important observation

is that we obtained similar results with foreign pupils (H1-H3).

- Moreover, various lines of reasoning have been identified which facilitate problem-solving by

making it more linear (H4).

- Finally, it was observed that the least interested and weakest learners participated in the activity.

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Chapter 4:

Introduction to pre-algebraic language in primary and lower

secondary schools. Experimental analysis of an a-didactic situation:

Guess the number

Abstract

In discussing hypotheses H2, H3 and H4, the aim of the ‘Guess the number’ activity is to analyse

the teacher’s role in an a-didactic situation, attempting to clarify the teaching processes and

teacher’s role. I thought it important in my experimentation to bear in mind the learners’

spontaneous concepts and natural language (words, gestures, ability, culture etc) as a starting point

for organising an activity, the aim of which is translating a symbol from natural language into a

mathematical one.

I constructed a teaching situation which included a game whose role was to encourage cognitive

continuity in passing from arithmetic to pre-algebraic thought. The aim of my work was to use

affective learning and a game to produce the need for using a symbol and researching a formula as a

winning strategy. I attempted to analyse the most suitable conditions for encouraging

communication and the understanding of a mathematical context, trying to reduce to a minimum the

objective limits of natural communication between people. Within this parallelism I analysed the

teacher’s attitude, which is constant throughout the activity; the latter uses a game which is

malleable and easily adaptable to logical phases which are essential in performing the activity. The

teacher’s implicit strategies can be studied and organised on the basis of an appropriate a priori

analysis of the problem and activity. From a Vygotskian point of view, these tools in my

experimentation have contributed to the development and understanding of specific topics, like, for

example, multi-tasking and synthesis.

In discussing my research hypotheses H3 and H4, in paragraph 4.2 I will describe the concept of

language. Language is an environment with us ‘inside’ and ‘inside’ which we think. Often we try

to express what we think in one language with another. Therefore, each language ends up by being

crossed by many if not all languages.

In paragraph 4.3 I will describe various studies (Frielander & al, 1988; Kieran, 1989: MacGregor,

1991, Malara, 1999, Radford 2001, Malara N A and G Navarra, 2002, Malara 2003) which have

proved that often algebra is not constructed as a tool and object of thought but that falsifying

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mechanisms and computational aspects are enhanced. Consequently, the language of algebra does

not develop in importance as a language adapted to describing reality, like a tool for reasoning and

anticipating. We observed throughout the various activities that the learners made different

suggestions for describing the numbers to be worked out and this depends a great deal on their

cultural and environmental influences. The choice of symbol is freely made by collective

negotiation. The teacher should leave the learners to freely talk out their hypotheses with their

peers so that they feel protagonists both as regards translating the symbol and subsequent

falsification. The teacher’s aim is to educate and develop the learners’ sensibility in creating a new

and complex language, thereby encouraging critical reflections. Discursive processes, under the

prudent mediation of the teacher in an atmosphere which emotionally involves the learner, can

facilitate the process of consciousness-raising (for Vygotskij, an essential feature of ‘scientific

concepts’) and contribute to developing competence mathematical concepts.

In paragraph 4.4 I will describe an interpretative framework for my second experimentation and the

results obtained, with reference to hypotheses H2, H3 and H4. The identified teaching activities

are: the teacher justifies the translation, initiates it and the teacher makes explicit that the translation

is from a natural to a mathematical language; this leads to meta-cognitive reflection of the problem,

which encourages generalisation about the problem. Regarding the results, the sign ‘=’ is

developed, which is a reflection on i) the importance and potential of the concept of a ‘formula’ and

ii) symbol falsification.

In paragraph 4.5 I will describe the context of the experience and field of enquiry.

Paragraph 4.6 lists the phases of the game and a qualitative description of the experience. The aim

of the game is to guess how the teacher is able to work out the number thought of by the learners.

There are 11 phases in the game and the last phase is very important since it is the moment when

the teacher asks the learners to invent their instructions and to play together. Motivated by the logic

of the game, the learners in this phase contextualise the importance and value of the concept of a

formula.

Paragraph 4.7 lists the various instructions used in the experimentation and the teacher’s strategies

in guessing the number thought of by the learners.

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Paragraph 4.8 From a qualitative analysis of the experience it emerges that, with the implicit

justification of needing space on the board, the teacher translates and briefly rewrites the

instructions in mathematical terms. The model of the activity described on the board encourages a

comparison to be made between the learners’ results and reflections about the existence of a

common strategy for all the learners.

During the experimentation, the teacher (with the explicit role of secretary) carefully describes the

model which will enable the experience to be summarised and generalised in the final phases. In

order to occupy less space on the board, the teacher should suggests using brackets and then

eliminate them, thus leading to reflection on the use of brackets, of an unknown when applying the

distributive properties of multiplication and the use of the ‘=’ symbol. Leaving the model on the

board helps the learners in the validation phase to analyse the possible strategies under

consideration and to eliminate those without general validity. By its very nature, the square on the

board is an effective tool for mediation since it means that the activity can be checked and verified.

When the teacher suggests that the learners play against the teacher, an exchange of mathematical

roles takes place: from a arithmetical to a symbolic logic (a supporter in the game). At this point,

the teacher’s choice of numbers (0 or a negative number) is an excellent tool for reflecting on the

iconic nature of an unknown. Moreover, the final phase of the game enables the learners, by means

of symbol falsification, to discover the importance of calculating a winning solution formula. The

validating phases are alternated with the instructions and the learners discover on their own how to

check their strategies, with the teacher performing the role of secretary. During the falsification

phases, the teacher should emphasise the answering time (the quickest possible) which in turn leads

to reflection on the time problem for arithmetic calculation, clearly very different to the teacher’s

answer in the first instruction.

Paragraph 4.9 includes the conclusions of the second experimentation with reference to my research

hypotheses H2, H3 and H4. The teacher uses tools (attitudes, words, gestures, models on the

board….) so that symbols become tools of mediation for meaning, ie, particular signs with an

objective and specific meaning. The teacher brings these into the class as external objects to be

studied, beginning with their close relationship to a ‘sign’, which lives and has meaning in natural

language.

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4.1 Introduction

The analysis of this aspect of teaching has been carried out from many points of view: the

theoretical reference for the methodology is G Brousseau’s Situation theory whilst I used the field

of neuroscience for analysing the effect of affective learning in a game. Regarding social and

human aspect, I used the theory of Embodiment mathematics and Vygotskij’s theories.

I thought it important in my experimentation to bear in mind the learners’ spontaneous conceptions

and their natural language (words, gestures, skill and culture) as a starting point for organising an

activity, which is aimed at translating a symbol from a natural to a mathematic language. I

constructed a teaching situation involving a game for encouraging cognitive continuity in passing

from arithmetic to pre-algebraic thought. The aim of my work was to use affective learning and a

game to produce the need for using a symbol as a winning strategy. Following a process of natural

growth inherent in the logic of the game, the learners use a language which is richer and more

suitable for their new demands. By using the ‘Guess the number’ game, I have attempted to analyse

the most suitable conditions for encouraging communication and the understanding of mathematical

content without the objective limits of natural communication between people. Studying the limits

of this parallelism, which is located in motivation (a key element for the learner), I have attempted

to observe one of the teacher’s functions during the a-didactic situation on the basis of ‘attitudes’,

the latter which are tools for encouraging the fulfilment of the teaching aim. These implicit

strategies are suitably studied and prepared before class, thanks to an a-priori analysis of the

problem. The teacher’s activities are made explicit during teaching. Thanks to the teacher’s two-

fold role – teaching and a-didactic – the learners effectively work in an field of meaning which

becomes richer and more varied, assimilating a language which is more conscious and unnatural.

The identified teaching activities are: the teacher justifies the translation, initiates it and the teacher

makes explicit that the translation is from a natural to a mathematical language; this leads to meta-

cognitive reflection of the problem, which encourages generalisation about the problem.

Considering such activities and the sub-division of the game into specific phases, I attempted to

perform a detailed analysis of the experimentation from a Vygotskian point of view. These tools

contribute to the development and understanding of various specific concepts concerning the use of

an algebraic language, for example, multi-tasking and synthesis. In this way symbols are tools of

mediation, particular signs with a specific meaning of their own which the teacher brings into the

classroom as external objects to be studied, beginning with their close relationship with the ‘sign’,

which exists and has meaning in natural language.

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4.2 What is meant by the term ‘language’

As a consequence of my hypotheses H3 and H4 in the twofold activity of teacher-researcher, I

believe it is important to take as a stating point learners’ spontaneous conceptions, their natural

language (words, gestures, skill, culture…) and focus attention on the phase of translating the

symbol, which in practical teaching, binds natural language to mathematical language and,

therefore, mathematics. I agree with D Barberi (2002) in believing that languages are various

aspects of the global environment of communication. Consequently, they are tightly

interconnected, interwoven and in continuous and reciprocal interaction with each other. Language

is an environment with us ‘inside’ and ‘inside’ which we think. Often, as temporarily and

efficacious solutions, we try to explain what we think in one language with another. Therefore,

each language ends up by being crossed by many if not all languages. It is important to study the

most suitable conditions when using a highly communicative language so that the communication

of a mathematical subject can be understood to the best of its possibilities, within the objective

limits of communication between people.

4.3 References to various works on algebraic language

Numerous studies (Frielander & al, 1988; Kieran, 1989; MacGregor, 1991, Malara, 1999, Malara

2003, Radford 2001) have documented that even if learners can easily transform symbolic

expressions, they have difficulty in translating relationships into algebraic language and developing

their own formal reasoning. I have also verified this in my experience, particularly when learners

have to invent an instruction and calculate a solution formula. This probably occurs because

algebra has not been constructed as a tool and object of thought and falsifying mechanisms and

computational aspects are enhanced. Consequently the language of algebra does not develop in

importance as a language adapted to describing reality, like a tool for reasoning and anticipating a

‘translation’ of knowledge into ‘formulas’. The language of algebra is a derivation of new

knowledge by means of symbolic falsification, permitted by algebraic formulism. It is important

that learners distinguish between solving and describing a problematic situation and that they learn

to negotiate similar meanings, distinguishing natural from algebraic language, improving the

potential of working with classes of problems.

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The use of a ‘symbol’ to indicate a number

It is interesting to observe that throughout the activity, in translating the instructions into results,

learners make various suggestion about the number to be guessed. These are not generally made

spontaneously but it is the teacher who asks them to suggest a symbol. It is important that the

symbol is freely chosen by collective negotiation. Subsequent to my experimentation, I was able to

verify in the literature (the ArAl-Arithmetic project which encourages pre-algebraic thought,

directed by N A Malara and co-ordinated by G Navarra) that the choice of letters depends on the

environmental conditions in the classroom. In general an empty space is most requested to mean

that the learner does not know what to put in that space.

Other suggestions include ‘?’ or ? .

Discussions in class

It is important that learners compare their choices in such as way that their discussions highlight

differences and analogies of their writing. The teacher should make the learners be aware that they

are protagonists in translating the symbol and falsification. It is important that the learners reflect

on the various languages, knowledge and processes used in negotiating their hypotheses and

strategies with those of their peers without worrying about being judged by the teacher (that is why

the teacher’s role changes throughout the game; here the teacher is a secretary). Thus the learners

can assess the validity of their observations and convictions in making conscious and correct

choices. Indeed, the learners barely perceive their contribution to a collective construction of

knowledge, whilst elaborating strategies and analysing a new language in which they made

numerous suggestions. The teacher’s role is here pivotal in that s/he should not be excessively

rigorous but be able to identify (and have identified) and transmit the most correct ‘actions-

translations’, commenting on erroneous, improbable or misleading translations.

The aim of the teaching contract: teaching aim

The teacher’s aim should be that of educating and developing the learners’ sensibility in creating a

new and complex language, thereby encouraging critical reflection on the differences and equalities

of meanings in two mathematical examples. The use of a game enables the learners to discover

which letters go in the place of numbers whilst being able to solve more problems in less time.

Through play learners reflect about the meaningful application of the various properties of

mathematical objects (for example, the use of brackets, distributive properties and understanding of

various meanings of ‘=’).

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Learning by discovering

Learning by discovering has had strong affective reverberations on understanding processes. These

have become elements of history (mathematical), which have been experienced by individual

learners. At the moment of having to create their own instruction, a need is produced (not to please

the teacher) to calculate the solution formula if learners want to win. I believe that discursive

processes, under the prudent mediation of the teacher in an atmosphere which emotionally involves

the learner, can facilitate the process of consciousness-raising (for Vygotskij, an essential feature of

‘scientific concepts’) and contribute to developing competence mathematical concepts.

4.4 Explanation of the activity and experimentations results

With the aim of making a model of researcher-teacher activity, the activity is organised in the

following way:

• initial aim: to solve the problem

• condition which justifies modelling: not knowing how to solve the problem

• discursive phase: the teacher listens to individual learners’ comments and ideas

• translation phase: the teacher translates the problem described in natural language into

mathematical symbols and then begins to reread it and reflect on the above

• a mathematical model is created and the problem of this model is reread: in this way the

use of mathematical tools for solving the problem is highlighted

• the mathematical problem in the model is solved

• go back to comment on the solution of the real problem: the relationship between the

model and reality can be exploited in interpreting the mathematically-obtained result

Directing the translation of the problem formulation from natural to mathematical language (L1),

together with the fact that there is a shift in mathematics, the learners have a mathema tical problem

(1) which they can solve in the mathematical sphere and then return to the initial problem (2).

Thus, from time to time when analysing various problems, the learners effectively work in an field

of meaning which becomes richer and more varied, assimilating a language which is more

conscious and unnatural.

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(2) (1)

I have considered the following four teaching points: after having created a problem, using the

most natural language possible, the teacher should

• justify the linguistic and literal translation in the formulation of a problem, passing from a

natural to mathematical language

• initiate the translation, motivating each single learner to model the problem in their own

way in mathematical terms

• explain the fact that the translation is from a natural to a mathematical language, thus

keeping separate the natural from the mathematical world and thereby helping the learners to

understand when they are a part of one or the other

• lead the learners to a meta-cognitive reflection about the problem, encouraging its

generalisation.

In discussing my hypotheses H2, H3 and H4, the following objectives of pre-algebraic language

have been encourage:

• increasing the importance of the sign ‘=’

• reflecting about the importance and potential of a ‘formula’ concept

• reflecting about the falsification of symbols as a tool which enables

Class of problem problem L1

class of problem

2nd problem

L2

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1) the passing from a more articulated and difficult problem to an easier one

2) more problems to be solved in less time

• increasing the importance of the translation phase into mathematical symbols and passing

from a natural to a mathematical language

• the importance in using symbols for abstraction

• increasing the importance of using mathematics for solving problems

• reflecting on the use of brackets

• reflecting on the operation of multiplication

4.5 The experimental context: the sample

Our experimentation regards the first, second and third classes in a lower secondary school. The

learners are 11-13 years old and on average there were 22 learners in each class. In the first and

second classes, the aim of the experimentation was to introduce symbolic language, starting with

natural language; in the third class, the aim was to lead to reflecting about the use of pre-algebraic

language which the teacher consciously used after having introduced the concept of identity and

equations. Whilst constantly reminding the learners of the empirical experience of the ‘Guess the

number’ game, the teacher presented these topics and the learners were left to explain the meaning

of the symbols.

4.6 Phases in the game and a qualitative description of the experience

After having subdivided the learners into groups, the teacher explained that the aim of the game was

to work out how the teacher could guess a number thought of by the learners, who then received the

following information:

• 1st phase: the teacher addresses each group of learners, with pen and paper, and asks them to

think of a number and remember it, without communicating it either to the teacher or to the

other groups

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• 2nd phase: the teacher addresses all the groups, asking them to apply the instructions (on the

board) to their number and remember the result (1st instruction of the game)

• 3rd phase: the teacher asks the first group to communicate their result and writes it on the

board under the first instruction*. Then, the teacher asks for the results from all the groups

and these are written next to previous result*.

• 4th phase: the teacher writes (quickly) the numbers which each group had thought of under

each result and asks the learners to work out how it was done

• 5th phase: the teacher listens and asks the groups to talk amongst themselves; each group

tries to be the first to solve the problem (20 minutes)

o Once this communicative phase has finished, each group discusses their own ideas

and solving strategies, which are then written on the board, leading the learners to

comment on their validity

• 6th phase: before writing the 2nd instruction, the teacher will begin a literal translation phase

into the symbols of the instructions in the text, necessitating this due to a lack of board space

• 7th phase: at this point the teacher will write the 2nd instruction under the 1st (the latter

written in symbols) and invite the learners to think of another number and apply the

instructions, communicating the result

-the teacher will quickly write the number thought of by the learners under each result

-the communication phase with groups of learners (15 mins) follows then writing on the

board possible new strategies and/or falsifying the previous ones

• 8th phase: needing more space on the board to write the 3rd instruction, the teacher begins

translating the symbols of the 2nd instruction, which will be written under the 1st instruction

• 9th phase: the teacher gives the 3rd instruction to the learners, this time having the whole

class play in groups against the teacher. Then the teacher thinks of more numbers (4 or 5)

and communicates the results.

o There is a time limit of 15 minutes after which the person, who guesses the numbers

thought of by the teacher and who can explain their strategy, is the winner

• 10th phase: in the final phase, the teacher will compare the possible strategies and, after some

discussion phase for analysing the validity of the strategies, will communicate the teacher’s

own solving strategies (unless a group has not already understood and used the strategies).

• 11th phase: the learners are asked to invent an instruction and play amongst themselves.

Once the game has finished, the teacher can hand out the following questionnaire:

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• Describe each step of the procedures, followed in arriving at the solution

• Comment on the meaning of the formula which you used in the 1st phase

• Comment on your solving strategies: what is different about the strategies used in solving

the game?

• Does changing the positions of the symbols in the result change anything?

4.7 Instructions and teacher’s strategies

Instruction no.1

• multiply the number thought of by 5

• add 6

• multiply this sum by 4

• add 9 to the total

• multiply the final number by 5

• write down the result

Teacher’s strategy

Subtract 165 from the result above and divide it by 100 to obtain the number thought of.

Operations performed on a number n give the following results:

1. 5n

2. (5n) + 6

3. (5n + 6)*4 = 20n + 24

4. (20n + 24) + 9 = 20n + 24 + 9 = 20n + 33

5. (20n + 33)*5 = 100n + 165

Instruction no. 2

• add 3 to the number thought of

• multiply this sum by 5

• add 7

• multiply by 4 the final number

• note the result

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Teacher’s strategy

Operations performed on a number n give the following results:

1. n + 3

2. (n + 3)*5

3. (n + 3)*5= 5n + 15

4. (5n + 15) + 7 = 5n + 15 + 7 = 5n + 22

5. (5n + 22)*4 = 20n + 88

Instruction no. 3

• multiply the number thought of by 3

• add 4

• multiply this number by 5

• add 9 to this number

• multiply the total by 4

• write down the result

Teacher’s strategy

Operations performed on a number n give the following results:

1. 3n

2. 3n + 4

3. (3n + 4)*5 = 15n + 20

4. (15n + 20) + 9 = 15n + 20 + 9 = 15n + 29

5. (15n + 29)*4 = 60n + 116

4.8 Qualitative analysis: results of the teaching phase

From a qualitative analysis of the experience it emerges that, with the implicit justification of

needing space on the board, the teacher translates and briefly rewrites the instructions for the first

instruction in mathematical terms. In this transcription the teacher should encourage the learners to

suggest the steps to take and which mathematical objects to use. Thus the learners are induced to

make critical reflections about the use of brackets, equality and the distributive properties of

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multiplication. During the experimentation it can be observed that various learners compared the

results and numbers, which they had thought of during the game. The modelling of the activity

described on the board encourages a comparison to be made between the results and the learners’

reflections about the existence of a common strategy for all the learners.

The teacher writes the second instruction. After the game, with the same justification as before and

in the same way, the teacher commences the transcription into mathematical symbols under the

previous instruction. Once again the teacher has voluntarily left the results with their number

underneath to encourage comparison and prepare the framework which will enable the learners to

summarise and generalise the experience in the final phase.

There would, therefore be on the board:

1st instruction: { [ ( n * 5 + 6 ) * 4 ] + 9 } * 5 = n * 100 + 165

665 1065 1265 965 265

5 9 11 8 1

2nd instruction: [ ( n + 3 ) * 5 + 7 ] * 4 = n * 20 + 88

108 168 468 248 188

1 4 19 8 5

3rd instruction: 1. multiply the number thought of by 3

2. add 4 to the total

3. multiply this sum by 5

4. add 9 to the total

5. multiply this amount by 4

236 416 596 96

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During the activity (where the teacher recalls learners’ attention to writing the text more concisely),

the teacher will suggest the use of brackets and then remove them; learners can then suggest

possible mathematical operations. Thus the learners will spontaneously reflect on:

1. the use of brackets

2. on the use of an unknown

3. on applying the distributive properties of multiplication

With the second result, the teacher will highlight the use of the ‘=’ symbol as:

1. the equality of two quantities which are the same, written in a different way

2. how an element can connect two expressions, one difficult and the other easy

Leaving the model of the experience on the board, the teacher makes sure that the learners make

very good use of the time for reflection in comparing various results and in the checking phase; they

will find it easier to analyse possible strategies and disregard those which are not valid. By

suggesting that the learners play against each other, the teacher will facilitate an exchange of

mathematical roles. Knowing the number, the learner will initially work with arithmetic logic

whilst the supporter of the game (having to use symbols including unknowns) will obviously think

in symbolic logic. The teacher’s choice of number could be an excellent tool for reflection. The

careful choice of a negative number or zero leads to greater awareness of the iconic nature of

unknowns. Indeed, we can remember the typical answer to learners when they solve an equation

for which the answer is zero or a negative number: “There isn’t a solution because it would be

negative” or “because we can’t leave –x = 3, the solution is 3!”

After the final validation phase the teacher reminds learners of what was written on the board and

the reflections that have been gradually written down and converging towards a pre-determined

teaching aim. In this moment the board is a very powerful tool of mediation since, if well-analysed,

it leads to conscious and critical reflection on the use of mathematical tools and the generalising of

problems, which can be analysed with a formula. Due to model-like nature of the board, it means

that the activity can be checked and verified visually. As a final instruction, the teacher can suggest

that the learners invent a new instruction and play amongst themselves. In this way the teacher lets

the learners discover the need to have (and calculate) a winning solution formula of their own.

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With the first instruction, the possible strategies which the learners have suggested during the

activity are :

1. in relation to how many numbers there are in the result:

if 3, subtract 1 from the first digit (for example, 265 =› 2 - 1 = 1)

if 4, subtract 1 from the first two digits (for example, 1 065 =› 10 – 1 = 9)

2. the teacher makes a note of each group’s result, performing the operations in reverse to

go back to the result and applying to it the reverse operations from that of the instruction

With the 2nd instruction, bearing in mind what is written on the board

3. in relation to how many numbers there are in the result:

if 3, subtract from the first two numbers the third and divide the result by 2 (classes II°E - III°F)

4. the teacher uses the 165 + n*100 result in the first case and n*20 + 88 on the second

case, noted by each group with the opposite operations (class III° F)

With the 3rd instruction, the learners start playing against the teacher. Their strategies for

working out the number thought of by the teacher are:

• the learners have worked arithmetically on the result written down by the teacher,

proceeding with opposite or inverse calculations to instruction

• the learners have proved that calculating a solution formula, using the previous examples and

showing some difficulty in translating the instruction into mathematical symbols due a

practical problem of time (there are only 10 minutes left before the lesson finishes), a

problem to do with the calculation and problems in applying distributive properties

• by trial and effort, the learners thought of a number and seeing if, after the instruction, they

could obtain the same result as the teacher (only some learners in group D followed this

strategy).

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With the 4th instruction the learners enjoyed themselves in inventing an instruction. The

teacher’s request has seemed to enhanced:

• the semantic value of the translation of symbols with specific mathematical symbols

• and the importance of being able to calculate the right solution formulae.

The validation phase

Before introducing the second and third instructions and immediately after the third, the learners

were left alone to check their strategies whilst the teacher had the role of secretary. The results

were:

• there was a problem of time with point 2 in the first instruction. The teacher could not

function as a walking calculator and needed time to perform the calculations. This clashed

with what they had seen and the answers were given in a short period of time, almost

instantaneously.

• Point 1 in the first instruction was no longer valid with the second instruction and it could

not, therefore, be considered.

• Point 3 in the second instruction was longer valid with the third instruction (as was the case

with the first instruction) and it could not, therefore, be considered.

• No comment was made regarding point 4 in the second instruction.

The Problem of Answering Times

During the first falsification phase (or at the very most the second), the teacher emphasis the time

taken to answer (the quickest possible), thereby leading to a reflection on the time taken to perform

arithmetic calculations, which is logically different to the teacher’s answer, which used a pre-

algebraic strategy in the first instruction.

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4.9 Conclusions

Finally, with reference to my research hypotheses H2, H3 and H4, we have obtained the following

results with my experimentation:

• the passing from syncopated algebra to symbolic algebra was highlighted at the 3rd point

of the translation into the mathematical language by the necessity of symbolic falsification.

In this phase the sign ‘=’ and distributive properties operate

• in the first phase of translating into a mathematical language, the need for use of symbols as

icons arises

• in the second phase the meaning of the use of brackets is inferred

• the situation-problem provides a simplification of expressions by consciously falsifying

symbols

• solving more problems in less time implies the conscious acquisition of formulae as a tool

for solving problems in class

• a fundamental situation for passing from arithmetic to algebraic thought: it contributes by

also giving meaning also limited situations (zero and negative solutions).

Regarding hypotheses H3 and H4, the teacher uses tools (attitudes, words, gestures and boardwork)

to enable the symbols to become tools for mediation meaning, particular signs with a specific and

meaningful objectives. The teacher brings them into the classroom as external objects to be studied,

commencing with their close relationships to a ‘sign’, which lives and has the meaning of a natural

language. In this way the teacher can contribute to constructing a milieu which is favourable to the

devolution of mathematical topics. Using a game, learners enjoy themselves and participate in the

activity, being in full control throughout. Moreover, the teacher’s role as secretary enhances

motivation in a game-like activity, making the learners the main actors in this situation-game.

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Chapter 5 Conclusion

Abstract

At the beginning of the 20th century, the philosopher Samuel Alexander (1927) was already asking

himself about the “empirical existence of the mind and its neuronal basis”. Our conscious reflex

developed in and through the conscious of others and this would seem to be connected to our social

life and, more specifically, our ‘social instinct’; thanks to this fact we are aware of others and

ourselves. According to Changeux (2003), the use of language in elaborating knowledge is much

more than the simple connection between sound and senses but it involves in some way the

‘sharing’ of neuronal workspaces and the contextualised division of descriptions. The latter are

developed within language, as well as by the collective selection by members of a social group.

Studying the behaviour of subjects with a divided brain (patients in which the Corpus colossum had

been divided for therapeutic reasons so that the two hemispheres were no longer connected),

Michael Gazzaniga developed a modular theory of the brain:

“… many current theories of human thought confirm that the solution to problems takes place only

on a level of conscious experience and it may be a product of our linguistic system in itself for

itself. It has been one of the principal assumptions of much psychological research that elements of

our thought processes proceed serially in our ‘conscious’ by building themselves out of cognitive

material. I believe that this idea of a linear and unitary conscious experience is irreparably wrong.

Instead, my thesis is that the human brain has a modular organisation. By modularity, I mean that it

is organised into relatively independent functioning units, which act in parallel…….”

5.1 The role of visual images and symbols in creating a teaching context which is easily

recognisable by the learners

The most generally widespread idea about the language of cartoons is that it is a juxtaposition of a

language of ‘words’ with a language of ‘images’. The fact is that, even if we are dealing with a

‘similar’ and simple juxtaposition, the global effect would probably be not of words taken in

themselves but their relationships (D Barberi, p.203). As with reading a written text, learning to

read a cartoon is the arrival point of a long process of symbolisation which, by design and reading

the design, is transformed into definitive reading. Therefore, reading has as its presuppositions not

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so much in the de-coding but in the habit of treating a sense of images and signs in the

reconstruction of a story.

My experimentation has helped me to verify that the mental processes and perceptive mechanisms

are not spontaneous and natural at all but they must be decoded, the competence of which depends

on various educational and cultural variables. These mental processes and perceptive mechanisms

are required in gathering the information and meaningful connections contained in an illustration

(or in a symbolic expression) and in selecting and organising visual stimuli according to determined

criteria (similarity/differences, figures/background etc). It can be inferred from the results of my

experimentation that interpreting symbols is not so much tied to recognising the object described

and the immediate global approach that one can have with that image, rather the skill in penetrating

details in the figure, putting together single features by question and observations.

Learners, attracted by Clamat’s image, found themselves in a story where they attempted to

understand as many nuances as possible. Knowledge about images helps us to look at the figures,

to seek connections and meaning with one’s own internal and external world. We live in a world

which is full of icons and the learners are, therefore, accustomed to decoding images. The results of

the experimentation have been satisfying, the learners de-codified the images which, together with

the balloons, contributed to creating a satisfactory teaching environment, in which to negotiate the

mathematical aims of the activity in hand.

Why are cartoons important?

How we can re-evaluate them from a cognitive point of view?

…and from a learning point of view?

The experimentation produced the following results:

v full control of the sequential understanding of content

( heuristic description)

v categories highlighted by cartoons lead to a concrete expression of the action

(recognition of a context)

v the presence of unreal situations leads to the experimentation of many different strategies

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v the presence of an ICON

creativity (parallel thought)

flashbacks

categories,

classifications

SCHEMA (structure)

Relations of equality and order

v CARTOONS are serial in nature

(narration, telling of a story)

v TEXT + IMAGE (consciously mediated):

The reader re-evaluates the meaning of the message

comprehension by ‘redundancy’

v The phenomenon of CLOUSURE:

“observing the parts and perceiving the whole”

For example, the mental processes which make lines…

…become a face !!!

The following observation can be made:

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It is necessary to bear in mind the appropriate equilibrium between all the elements used in the

cartoon. Too many words and/or images expressing the same information could be confusing. If

the images used are too evocative in certain cases, it could lead to misunderstanding in the text. I,

therefore, believe that these images could erroneously recall a particular somatic marker and give

the sensation of being able to use conceptually pre-conceived maps. In this way it would not be

possible to modify one’s own interpretative model, which would instead be anchored in a semantic

field which is not appropriate to understanding and incorporating new information (a

misconception).

5.2 Some considerations on the importance of the social-cultural context in a learning-teaching

context, which permit the use of cartoons or arithmetic games

Our research has been analysed from numerous points of view: the epistemological, neuro-

scientific, psychological and semiotic. Other research, like that of F Arzarello (2004) has illustrated

the complexity which exists between cultural, biological and cognitive aspects of concrete

expression (Radford 2002), which accompanies the production of mathematical ideas in various

specific contexts. The use of various analytical tools allows us to note different ingredients which

intervene in concretely expressing mathematical objects. With the teacher’s help, learners can

elaborate and create new ideas through a varied language, incorporating not only words and

standard mathematical descriptions but also other non-standard languages, such as that found in

gestures or cartoons. In our experimentation involving the use of cartoons, the variegated language

of mediation between the teacher, learner and teaching situation arises from an approach to

perceptive-motor mathematics, which is tied to a symbolic-reconstructive approach. In a context of

social interaction which is managed by the teacher, this approach produces not only learning based

on doing, touching, moving and seeing (Nemirovsky, 2003) but also on recognising the teaching

environment, using an experience which has been really lived.

Emotions are indispensable for creating a memory because they organise the memory into a

sequence of events; this establishes the importance of emotions. A sense of time and order are

essential because a memory is considered as such and not a thought or a vision without any relation

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to past events. But a memory does not exist without a context. With our experimentation we have

seen the importance of using a game or a cartoon, aware of the fact that cultural and corporal

characteristics of the class may be similar. Thus we can hope that the mathematical message

transmitted by a real situation-problem, described by a cartoon or created by the teacher in a game,

is the same for each learner. Each learner must be able to recognise the context and know how to

move inside of it. In this way and if we know the real expression of mathematical enquiry, learning

can be stimulating.

As a primary impulse, knowledge can be connected not only with necessity by also a motivational

state found in a framework of auto-gratification (J P Changuex, 2003). Our experimentation has

shown that (for example, with the ‘Guess the number’ game) learning can be integrated into the

mathematical subject to be described, thanks to a strong emotional connotation and motivation,

which is turn is bound up with the learner’s will to continue playing and win by having fun.

However, thanks to the iconic code of cartoons and the phenomenon of clousure in the phase of

context recognition, all our five senses intervene providing a corporal connotation to learning,

which is entirely integrated into the mathematical subject.

Undoubtedly our experimentation is located in system of gratification and reinforcement for which

a state of necessity is connected to a state of well-being. Once the experimentation had finished, the

learners and teachers continued to ask us for more cartoons and other mathematical games. This led

us to suppose that learning can be conceived as being essential for survival (Changeux, 2003) if it

can be tied not only a state of well-being but also pleasure and auto-gratification: in this case, it is

the pleasure factor which makes us repeat the action! (appetitus noscendi).

“From reality”, Democritus said, ”we take absolutely nothing which is true,

but only that which sometimes happens in accordance with the momentary inclinations of our body

and the influences which touch or bump into us”

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Below I would like to explain my results with reference to my research hypotheses H1, H2, H3 and

H4:

1. Regarding hypotheses H3 and H4, the teacher used tools (attitudes, words, gestures,

icons, boardwork……..) in the game and cartoons, both of which contributed to

developing and understanding various specific concepts in the teaching aim. In this way,

the symbols became tools of mediation, signs with their own specific meaning, which

the teacher brought into the classroom as external objects to be studied, beginning with

their close relationships with a ‘sign’, which lives and has meaning in a natural language.

2. Regarding the ‘Guess the number’ game, the board is a very powerful tool of mediation

for the teacher since, if well analysed, it leads to a conscious and critical reflection about

the use of mathematical tools and the generalisation of analysable problems by using a

formula. With its model-like nature, the game infers checking and verifying phases of

the activity and a comparison of knowledge acquired with a single (initial) visual

experience.

3. Discursive processes (developed throughout the validation phase, appropriately

organised in the two experimentations under the teacher’s careful mediation in an

atmosphere which emotionally involved the learner) had a consciousness-raising role

(according to Vygotskij, one of the essential characteristics of ‘scientific concepts’) and

they generally contributed to developing competence in mathematical concepts.

4. In discussing my research hypotheses, it was interesting to observe that passing from

common to mathematical language, the learners used a language which was tightly

bound up with their own cultural and social experience. This language not only puts

natural language in relation to mathematical language but also the concepts which can

arise from daily life. These ideas often involve mathematical objects, creating

misconceptions (we can think about the notion of 0 and its daily use in common

language). Through my experimentation, it was possible to notice and analyse various

jumps in logic tied to common experience whether analysing each learner’s personal

interpretation for each cartoon or their active participation during the translation phase of

the ‘Guess the number’ game.

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5. From a multi-cultural point of view, it is important to highlight that the use of cartoons

proved to be an excellent communicative tool. Thanks to their iconic code, cartoons

evoke lived experiences which diminished linguistic differences on entering and exiting

the cartoon. At the end of the exercises, the learners not only understood the problem,

which they attempted to solve, but they produced a purely personal iconic code for

communicating logical procedures, the meanings of some calculations and even the

organisation of some of the same data in the problem. All this is possible could be

inferred by the learner’s Protocols, only a few of which have been reproduce due to a

lack of space.

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Chapter 6

Information for constructing the milieu

Abstract

My experience of using games and cartoons in learning and teaching mathematics has been very

interesting from many points of view. At the conclusion of the experience I pondered as to what

had been the essential elements in creating a context in which a teaching experience could be

developed and measured.

In paragraph 6.1 I dwell upon my experience with cartoons. Reading a cartoon, we can notice that

the content and plot of the story arise from the reciprocal interaction between a graphic and spoken

language. Iconic language takes on its own autonomy and requires a reading code and substitute or

complementary interpretation as regards the adjacent verbal language.

In addition to the fact of temporal/spatial sequencing and clousure (McCloud, 1992), there are other

elements to take into consideration when reading a cartoon: various onomatopoeia and icons used as

mediators of mathematical meaning. Indeed, some of these objects are part of symbolical usage for

each teacher in mathematics. After my experimentation, I sought out elements found in the

interpretation of a story told with images, in which the author explains the art of cartoons through

the exclusive use of a cartoon of itself! With the aim of constructing a possible grammar of

mathematical cartoons, I believe that temporal and spatial contexts represent the syntax whilst our

background represents semiotics. Clousure, the correct use of onomatopoeia and the intermediary

tools used by the teacher have the very difficult role of recalling the most from our background,

with the aim of providing semantic meaning to the sequential images in cartoons.

Paragraph 6.2 is a structural analysis of cartoons: Donald Duck in the land of mathematics. This

cartoon speaks about the queen of sciences in a amusing and absorbing way. The main character is

the adorable but unfortunate Donald Duck. His irony and charm can be seen in his problems and

his expressions, both which are very human. The story begins with a concrete problem about daily

life: Donald Duck is full of debts and he is obliged to give money to the mercenary Uncle Donald.

The icons and words used have the exclusive aim of evoking a familiar context, one that is easily

recognisable. As a narrative sequence, the cartoon is tied to a real context of mathematical history;

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various historical leaps occur always in a real context, which is easy to recognise by the reader with

a logical continuity which derives from the placing side by side of the two images.

In paragraphs 6.3 and subparagraph 6.3.1 I have attempted to explain the features of affective

learning from a didactic-neurophysiological point of view. Of great importance is the

contextualising phase in constructing a teaching activity which encourages the teaching aim.

Paragraph 6.4 introduces a series of exercises in the realm of geometry, which allows for the use of

cartoons: maths maps. These are visual summaries which provide an understanding of sets and

geometry; they both make use of the techniques of mind-mapping. In order to have a visual map

for summarising a topic is very similar to seeing an image of a puzzle, prior to assembling the

pieces. This process allows us to gather a great deal of meaningful information, which will be

represented in a small space. One of the main objectives of maths maps is to provide a structure

which allows learners to see relationships and extract meaning. This structure is very similar to that

of cartoons which can be used educationally by making use of various geometric maps.

6.1 Mathematical descriptors for interpreting a mathematical cartoon: a possible grammar for a

mathematical cartoon

Reading a cartoon, we can notice that the content and plot of the story arise from the reciprocal

interaction between a graphic and spoken language. Iconic language takes on its own autonomy

and requires a reading code and substitute or complementary interpretation as regards the adjacent

verbal language. Analysing how a visual experience is located in temporal and spatial scansion is

not very simple. It is necessary to bear in mind one’s own cultural baggage and stimuli (internal

and external) which have characterised our cultural make-up. My experience of the use of games

and cartoons in learning and teaching mathematics has been very interesting from many points of

view. With this activity, I have realised the important of a context or environment for learning and

knowing. There can be no knowledge without a context and that cannot be created without the use

of objects or images which are organised spatially and temporally, according to a well-defined

logic. Moreover, it is important that this context is familiar to the reader and that it reproduces

internal structures which are common and easily recognisable. At the conclusion of the experience

I pondered as to what had been the essential elements for creating a context in which a teaching

experience could be developed and measured.

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An essential element which must be taken into consideration is the phenomenon of clousure. By

placing side by side two images which are temporally mismatched and located in the same spatial

context, we can narrate a story thanks to the reader’s imagination and intuition. The placing

together of two cartoons or more can be used to transmit a message and communicate an

experience. There are many elements which have assisted me in creating cartoons. In addition to

the element of temporal and spatial sequencing and closure (defined as “the observation of the parts

and intuition of the whole” (McCloud, 1992)), there are many other elements to take into

consideration when reading a cartoon: the various examples of onomatopoeia and icons used as

mediators of mathematical meaning. The latter, as well-know objects to learners, are strategic tools

which the maths teacher can use for a teaching aim, which is expressed by the cartoon. These

objects are part of symbol usage for each teacher of mathematics. Any maths teacher uses symbols

such as ’ ’ to indicate implication, ‘ ’ to indicate a concept which is repeated without

limitation and the symbol ‘?’ to indicate the data to look for. In creating a cartoon I believe it is

very important to start with a historical context, which is known or easy to recognise. In this way

the reader can use the space-time rules for inserting an experience into a personal background, in

which they know how to move.

In this way, as McCloud says, cartoons are a means of communication and expression in which the

public is a spontaneous and conscious collaborator and clousure is the object of change, movement

and time: on the edge of the margin, human imagination takes two separate images and transforms

them into one idea. The recognition of a context takes place through the recalling of personally-

arranged maps which lead us to complete the action in our mind and it is important that the images

used are the most evocative possible. If the experience narrated is a common life experience, then

more people can interpret it in the same way and with the same (or at the most, similar!) emotional

state, otherwise it is necessary to bear in mind the possible and different interpretations which each

individual can have regarding their own cultural experiences. Obviously, all these elements would

not be necessary if the basis of the interpretation was not the reader’s creativity or imagination

which continuously functions to reconstruct the entire story.

When we look at a cartoon or read a story, we are interpreting and reconstructing a framework

which conceals precise and rigorous rules. In attributing logical meaning to a written sentence, we

need to use the grammar correctly and then construct the sentence correctly, not only regarding the

syntax but also the semantics. I believe that when interpreting a narrated story with images,

temporal and spatial contexts represent the syntax whilst our backgrounds represent semiotics.

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Clousure, the correct use of onomatopoeia and the intermediary tools used by the teacher have the

very difficult role of recalling the most from our background, with the aim of providing semantic

meaning to the sequential images in cartoons.

Following my experience and attempting to construct a possible grammar for a mathematical

cartoon, the following can be considered:

Obviously syntax and semantics and, therefore, the spatial-temporal context (including our cultural

baggage) lead us to an interpretive framework of the experience in a relationship of order between

elements of the cartoon which allow us to organise them into well-defined temporal contexts.

Reading static designs in dynamic terms (ie continuously) allows us to visualise the story in our

minds. In creating cartoons, it is necessary to take care with the description of this historical

context since it can serve as a guiding thread in passing from one image of an other. But this is not

the only strategy; it is also possible to use the main character to leap from one context to another.

The choice of the main character is a very import moment and the features of their character can be

essential in developing the story, recalling events and introducing new ones. Recognising the

context, the reader knows how to move within it and can, therefore, recognise the sequence of

images, recognising the semantic meaning inherent in their sequences.

…but what are the characterising elements of the grammar of this invisible art?

Scott McCloud’s book, Invisibile Art, was particularly useful to me when formulating a possible

grammar for mathematical cartoons. The geniality and creativity with which this author explains

the art of cartoons through the exclusive use of a cartoon highlights how images, if well organised,

- SYNTAX - SPATIAL- TEMPORAL CONTEXT - SEMANTICS - PERSONAL CULTURAL BAGGAGE - ADJECTIVES - GRAPHIC TECHNIQUES - ADVERBS - ICONIC CODE - PRONOUNS … - GRAMMATICAL - PHENOMENON OF CLOUSURE CONSTRUCTION - ONOMATOPOEIA - MEDIATORY MATHS TOOLS

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can tell us and even explain something to us. Their use constitutes a scale of iconic abstraction,

together with the figures and words which make up the vocabulary of a language called ‘cartoon’.

But how can this triangular relationship assume semantic and syntactic meaning? What is the

mechanism in our mind which shapes the concept and facilitates the recognition of what we are

looking at? What is it enables the images, discretely read, to continuously join up and form a story

in our mind? I have one answer to this question ……. our human experience and cultural

education!

Starting with what McCloud calls ‘the plane of the figure’, used for describing an illustrated

vocabulary plus cartoon, I have tried to list the possible symbols (meant as evocative elements),

founded on the grammar of a cartoon, which discusses mathematics. Below is the plane of the

figure, described by McCloud:

Plans of the figure 1 2 3 FACE

reality language meaning

Where the dotted lines represent:

1. the margin of the retina

2. the borders of language

3. the conceptual margin

The specific image in his book is:

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The elements characterising syntactic and semantic understanding are: concepts, ideas

1. ICONS seen as structured and organised language for Inferring mathematical concepts

degree of abstraction regarding the topic

continuous postulate of the illustrated image

2. CLOUSURE meant as recall of internal background

evocative tool of specific neuronal maps for recognising the context and for passing from one context to another

3. The serial and parallel presence of words and figures which contribute to creating a boom and sending back of information in the cartoon

logical guiding thread, passing from one cartoon to another in recalling spatial and temporal contexts.

4. HISTORICAL CONTEXT AND as evocative tool in an easy to recognise MAIN CHARACTER recognisable environment

the most possibly familiar environment for introducing teaching experience

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effective time of the experience

5. TIME subdivided into reading time of the experience

representational time of the experience

assimilating time of the experience

An observation which can be made is the following: it is necessary to bear in mind a suitable

equilibrium between all the elements used as an excessive amount of words and figures expressing

the same information could create confusion. If the images used are too evocative, in certain cases

this could introduce elements of misunderstanding into the text. These images could erroneously

recall a particular somatic marker and intimate that pre-constituted conceptual maps can be used. In

this way, it would not be possible to modify one’s own interpretative framework, which would

instead be anchored in a semantic field which was not appropriate to understanding and the use of

new information (misconception!).

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6.2 Textual analysis of a mathematical cartoon : Donald Duck in the land of Mathemagics This cartoon speaks about the queen of sciences in a amusing and absorbing way. The main character is the adorable but unfortunate Donald Duck, who easily attracts attention to itself with his clumsy walk. His irony and charm derive from his problems and his expressions, both which are very human. He has a sweet and tender nature but he is alsovery unfortunate! It is not by chance that he has been chosen as main character. The story begins with a concrete problem about daily life: Donald Duck is full of debts and he must give money to the mercenary Uncle Donald. (a real problematic- situation context).

The icons and words used aim exclusively at evoking a familiar context, one that is easily recognisable with our cultural education. Donald Duck is desperate and in the end he is worn out by the mathematical calculations he does in trying to pay off his debt to his Uncle; he curses against mathematics. “.. how I wish that mathematics had never been invented…….” But at that point the mathematics genie comes out of nowhere (secondary character) who calls Donald Duck and takes him to a magic village……..

Donald Duck finishes up in a mathemagic world, surrounded by purely mathematical images and experiences in well-defined historical contexts:

Initially, the only guiding thread is the chronological story of the events (time of the experience connected to the reading time and time for describing the story). The genie immediately shows to the little blasphemer what the world was like before the conscious use of mathematics:

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But where did mathematics come from? Was it not maybe discovered and created by man for improving living conditions and for fun? The constant presence of a famous and historical context holds the attention and interest of the reader, who is amazed at discovering the mathematical regularity of everyday life. The cartoons are alternated with a chronological sense of events (the story as guiding thread) and various historical contexts:

The described environments make superficial references, using images and specific icons; they are very effective for encouraging understanding and the recognition of the context. Mathematical topics are introduced, according to utilitarian features or simply for fun:

In an moving sequence of historical images and mathematical relationships, the author uses the lively figure of Donald Duck who challenges the genie, asking him about a possible connection between mathematics and music (the personality of the main character as guiding thread).

In this way the author presents a sequence of cartoons which allow him to reuse the story as a guiding thread for speaking about Pythagoras and his secret sect.

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Very striking and funny cartoons are followed one after the other and completed in the presentation by a symbol chosen, not by chance, by Pythagoras for identifying his sect of followers: the pentagram. In order to explain the properties of this figure studied by Pythagoras, the author suggests a brief sequence of cartoons which aim at maximising the visualisation and presenting of the properties of the golden section. Making the best use of iconic codes and graphical techniques, the author describes the image of the pentagram……….

……. and then he takes it to pieces, demonstrating its graphically-described mathematical proportions and how they have influenced art and architecture in later centuries; this is described below a small collection of cartoons:

Mere words would never have been able to do all this! But the influence of mathematics is not only in the creation of man: the author shows that we can find the proportions of the pentagram in nature in the symmetry of growing organisms (placing side by side similar images). There follows the following series of cartoons:

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Having discovered the wondrous combination of the perfect angles of snowflakes with the microscope, Donald Duck is challenged by the mathematics genie to visualise a perfect circle, given that it is impossible to find in nature (the personality of the secondary character as a guiding thread). After much effort, Donald Duck is able to visualise perfect circle and at that point the genie asks him to put a triangle inside and “…….make the triangle rotate like a helix……. and the circle like a flag”. Obviously, Donald Duck should already know how a helix rotates and the circle like a flag, otherwise such a splendid image would not have come into his mind:

But the genie continues asking Donald Duck the purpose of that ball…… and he asks him to cut it up………

and again... At this point the author demonstrates by comparison (using the genie’s words) what is obtained if only the triangle was rotated like a flag and what would be obtained by cutting it up. Thus rotating solids and their uses are introduced:

But mathematics is not only hard work. It is also play! It is the basis of many sports…….. where a score is kept and where you play in a geometrically defined space:

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…But much of the fun in mathematics does not require fancy equipment but some arithmetical knowledge and a pencil and paper

You can have fun with numbers and this can be a great hobby, especially if you know some of the tricks! The author finishes by highlighting that the most interesting aspect is the mathematical one! In a mathematical world, you cannot fail to see the magic of numbers (another secondary character who is particularly iconic and introduced on an ad hoc basis) and it will be precisely this character who suggests to Donald Duck the trick by which he can beat Uncle Donald!

The Magician suggests various mathematical tricks to Donald Duck and in the end he chooses those which would give him the most satisfaction in beating Uncle Donald: the chessboard trick! Donald Duck wakes up from his dream and sets off immediately for Uncle Donald’s house:

Donald Duck asks his uncle to accept his house in paying off his debt, asking his uncle to accept the chessboard for the difference: he puts a one cent coin on the first square, two on the second square, four on the third etc until he has piled up coins on each of the 64 squares; the latter is what Donald Duck will take in paying the difference.

Intrigued by the apparent simplicity of the calculation, Uncle Donald accepts his nephew’s proposal and signs the contract. Uncle Paperon soon realises the trick and the impossibility of gathering up the sum of money; he finds himself being forced to accept that the debt has been paid off by his mathemagic nephew!

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The cartoon concludes with the problem of geometrical progression and this notion does not create

problems for the reader. At the very least, the sweet smell of partial success could induce the reader

to spontaneously review the images if the topic has not been well understood.

As a narrative sequence, the cartoon is bound to the time in the ‘mathematical’ story and various

historical leaps often occur in a real context, which is easily recognisable by the reader with a

logical continuity which originates in the placing together of two images. However, the main

character must be is loveable and intelligent, thereby including psychologically disturbed

connotations, which in the end belong to all of us. These characteristics are fundamental in rousing

the reader’s curiosity and interest in organising a sequence of topics. Also, the choice of secondary

characters must not be underestimates. The choice of the genie, as a guide through time, and at the

end the choice of the Magician have not been made by chance. Throughout this cartoon the author

has frequently used Donald Duck’s splendid presumption and curiosity in coherently passing from

one topic to the next in the absence of historical consequences! However, a particular figure was

required for the conclusion of the story, one who would conceal all the magic and astonishment

inherent in the logical rigour of a mathematical calculation, which is apparently simple!

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6.3 Components of affective learning from a didactic-neurophysiological point of view

The neuro-scientific point of view was very effective in trying to understand how a person, in

addition to being a product of genetic processes, is the result of their own perceptive experience.

The use of games and cartoons can be considered as impulses which are found in a system of

gratification and reinforcement. From a neuro-scientific point of view, the use of games and

cartoons concerns the production of endorphins and, therefore, a state of well-being. From a

didactic point of view, one can hope to create a dependence on these particular tools if their use is

located in a system founded on ‘necessity’: the necessity to learn by having fun. Behaviour also

depends on needs and wishes. Internal needs provide motivation and they orientate themselves in

particular directions with the aim of reaching or avoiding certain objectives.

When we begin to interpret a cartoon or sequence of cartoons, images are perceived or recalled in

our minds with obvious difficulty, according to M Ferreri’s two possible frameworks (see below).

The difficulty of trying to model such a complex process can be seen in the fact that the modelling

could lead us to think that the transmission of information occurs in a sequential way and with a

certain temporal scansions. In reality this is not the case. All the information is transmitted to

various neuronal regions with the same temporal scansion (or almost). It is then propagated and

branches out with an interpretation of their own and parallel decoding, which is contemporaneous

from region to region. All this implies the impossibility of being able to say what happens before

and after but only what happens in each single part of the brain inside, according to what has been

discovered and experimentally verified.

Pre-fontal associative areas

Primary sensory areas

anterior gingolato Pre-time areas

hypocampus

Thalamus

Input which activates an index

Associated sensory areas

attention

Space

time

Activation of topographically organised map

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When reading cartoons, perceptions of sight, hearing and touch and somato-sensorial perceptions

pass through the thalamus. All the information, except that relating to smell and taste, arrive in the

thalamus which focuses our attention (our pre-attention has already happened in the upper

collicolo). This information creates an environment and therefore a context in which the narrated

story or game will take place. In this environment SNDs and pre-descriptions are recalled, which

are in an hierarchical organisation in parallel to cerebral networks to as to be subject to the

constraints of their particular context. This fact can have more or less strong repercussions on the

amydala which records bodily variations throughout the experience. Information from the thalamus

arrives at the visual cortex, which elaborates the information. After the experience has passed from

the associative areas, it arrives at the hippocampus and becomes conscious. It is the hippocampus

which contextualised and allows the experience to be recalled by the memory. It is thought that the

cingulum is the agent which established relationships of order in the various events; it connects the

hippocampus, which provides the spatial context, to the pre-frontal dorso-lateral areas; the latter

contextualises the temporal experience, thus encouraging connections of order to be established

regarding changes in context from one cartoon to another.

Attempting to mark out an interpretative clue in the reading and decoding processes of an image or

a sequence of images, I believe that one can dare to say that:

thalamus

Primary sensory areas

Pre-frontal associative areas

Rappresentazioni toograficamente organizzate (mappe)

Hippocampus entorinal area

torinale

Frontal gingolato anteriore

pre-frontal areas

time

space

arrangement of neuronal frameworks (indices)

Attention

Sensorial inputs

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The visual experience through the visual system is decoded according to its shape, colour

and the spatial place which it occupies. This event, if it takes place in a system of gratification and

reinforcement, will produce an increase in endorphin and will develop into an emotion (a variation

in bodily state) which will motivate the reading of the story and cartoon.

This variation, in homeostasis on a bodily level, influences the amydala and, therefore, the

emotional characteristic; it acquires a stronger imprint, creating a state of necessity. In this

environment SNDs are recalled and contextualised by the reader, who internally recreates the

spatial-temporal context, activating the memory through maps.

Recalling the modular theory of the brain (Michael Gazzaniga):

“… the human brain has a modular organisation, that is, it is organised into

relatively independent functioning units, which act in parallel…….…….. This

modular activity often acts independently of our verbal awareness…” “Being aware

that the mind has a modular organisation means that certain behaviours must be

accepted by their capriciousness and that a particular conduct may not originate in

our conscious thought processes”.

With this meaning cultural, environmental and personal processes are fundamental in mathematical

experimentation and, therefore, the process of indexation which each one of us performs during

their own cultural experiences. Each experience implies more contexts and, in the absence of a

context, there can be no memory and learning. In order to be able to recognise an experience and

understand it, it is important that an experience becomes part of our SND and that the experience is

analysed with our maps. If this does not occur, it should be possible to use pre-existing SND to

create a suitable map for decoding the experience. Another important experience is that when we

see or read of an action, the pre-motor areas command the motor areas, thereby leading to an action.

When reading an image, these areas prepare us for the action which will or will not take place,

inducing ‘virtual’ bodily participation. The amygdala records such variations and a somatic marker,

(probably) tied to it, induces an action which will show our state of mind (for example, a

spontaneous laugh). With this meaning the evocative power of cartoons merits consideration: in the

discrete interpretation of iconic images, we find ourselves absorbed only emotionally but also

bodily since we can observe the bodily states of characters (for example, tiredness).

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6.3.1 The teaching aim: the importance of contextualisation in teaching

I believe that it is very important to consider learning as a superimposition of temporally- and

spatially- contextualised experiences with the aim of being able to programme and structure

effective teaching. Teaching experiences recall descriptions which have been topographically

organised in a personal cultural and sensorial context. The grammar of cartoons and clousure

favour the recalling of the process of contextualisation, which takes place in our mind. Images,

mimed actions and clousure recreate a framework which inserts itself perfectly into our pre-

descriptions, thereby making the experience more direct and absorbing. Whilst instantaneously

recognising the context, the reader knows how to move in the context. Generally, the following is

valid:

Ø The presence of IICCOONNSS:

-creativity (parallel thought)

- flash-backs (short-term and long-term memory)

- categories,

classifications

Ø sensorial reading in ppaarraalllleell to the experience told

by regions in a spatial-temporal discrete context.

SSCCHHEEMMAA ((‘‘ssttrruuccttuurree’’))

Ø the sseerriiaall nature of cartoons Relations of equality

( narration, telling a story) and order

Ø cclloouussuurree

observing the parts ..…

and perceiving the whole....

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Images immediately provide a context, an interpretative framework which is the contrary of a

written text, which the learner must create for themselves by reading. It is important that

information in a game or cartoon is described with a simple and correct temporal and spatial

scansion so as to encourage the recognition of images in pre-existing SNDs in the learner. If

these images are not recognised by the pre-existing SNDs, this information must facilitate the

creation of other SNDs on a superior level with more characterisation. The correct use of models

in teaching in well-organised (temporally and spatially) contexts can re-enter into our natural way

of learning. By means of a parallel elaboration of data, they encourage the activation of arranging

maps or the searching and elaboration of other maps, which are functional to the understanding of

a motivational state. For this reason I have attempted in teaching (which envisages the use of

Maths Comics) and activities (which envisage the ‘Guess the number’ game) to analyse and

structure action phases to encourage the most of not only an effective devolution of the problem

but also an effective understanding of the teaching aim.

A qualitative analysis of protocols and learners’ behaviour during the activities has revealed the

willingness for learners to actively participate and be motivated (studying and having fun) in

negotiating their meaningful comparisons in various aspects of mathematical activity. The

presence of a possible somatic marker in the experience participates in the choice of this or that

neural framework. From a teaching point of view, the presence of possible somatic marker can be

useful in evaluating the type of mistakes made by the learner and the difficulties encountered in

understanding a topic, which makes generalisations or which recalls a previously-taught topic.

The teacher should always take into consideration the fact that, when teaching, something is being

modified inside the listener and the teacher should, therefore, be careful to connect one map of

experience activated by the learner with the mathematical concept being taught. If the

information is inserted onto pre-existing maps, which recall similar experiences, understanding is

enhanced and, therefore, greater integrated pre-mnemonic acquisition.

6.4 A teaching activity for introducing the use of cartoons in geometry: Math Maps An example of a teaching activity for introducing the use of cartoons in geometry is that of math maps. These visual summaries which provide an overview and details of geometrical concepts, using the technique of mind mapping. A mind map is a connected pattern or related ideas and it was

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originally developed by the well-known learning expert Tony Buzan in the 1970s; it has become an extremely popular method for visual learning. • starting with a central focus, a mind map is built up by branching out from the center, attaching

key words to the lines. • similar concepts are grouped together on main branches and key words, symbols and colours are

used to represent ideas. Arrows can be used to connect certain elements on different parts of the map.

• You will notice that a large amount of information can be represented in a small space. The ideas are arranged in such a way as to show relationships which can be made between concepts, thereby encouraging global thinking.

Having a visual map to summarize a topic is much like seeing a picture of a puzzle before you start to assemble the pieces. It allows you to see the big picture and to group concepts, as you would before starting a puzzle. This process allows a large amount of significant information to be represented in a small area. Each section of the maths map kit presents a different geometrical concept in a logical fashion, beginning with the most basic Preview Map and proceeding in interchangeable steps as students progress through more complex maps, including the Overlay Map, Skeleton Map, Half-Developed Map and the Detailed Map. What are Maths Maps? Structure: One of the key aims of Maths Maps is to provide a structure or framework that allows students to see relationships and make meaning (like a maths comic!). This structure is similar to an index system in a library. It allows concepts to be filed away in memory compartments, making retrieval of the concepts much easier. The activities are designed so that students can explore this structure and later develop their own mind maps.

Geometry Overview Key Words: Key words help to simplify the presentation of information on mind maps. Starting from the centre, the key words link together concepts as they are read. Key words are repeated on many main branches, allowing patterns and connections to be discovered on the maps.

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Key word concepts include: types, relationships, gradients, lines, symmetry, area, volume, angles. Symbols Maths map symbols help to encode information, imprinting it on your visual memory and making recall easier. A standard set of symbols is used throughout maths maps, encouraging connections between maps to be explored and discovered. Maths map symbols include: internal angles, external angles, lines, graphic equations, points, distance, area and volume. Introduction to the activities Activities involving Math maps can easily be integrated into your lesson planning. The activities described on the following pages provide an excellent opportunity for students to gain an overview of the concepts within a topic: to get a glimpse of the big picture. They will also assist learners in making the all-important connections for meaning.

Suggestions Activity 1 Overview The simplest and possibly most effective activity involves a classroom discussion based on the preview maps. This activity provides students with a breakdown of geometrical concepts. Through classroom discussion, students can visually sort already familiar ideas into appropriate categories. This gives an opportunity to briefly discuss how some of these concepts are interrelated, and provides the teacher with insight into the students’ prior knowledge of geometry. PREPARE:

• either a transparency of the geometry overview or create the equivalent map. • enough copies of the geometry overview for each student. DIRECTIONS: 1. Either display the transparency or post the display map at the front of the

classroom. Give each student a copy of the geometry overview map. 2. Draw students attention to the eleven topics covered on this map, ensuring that

they know the meaning of each key word and associate this with the graphic image.

3. Choose a topic to briefly discuss with students, e.g. solids, and ask students for all the ideas they think are related to solids.

4. Continue through each topic and, whenever possible, link concepts which

students supply to various parts of the map. Finally, let them re-draw their newly constructed map.

Activity 2 Recreating the maps The purpose of this activity is to provide students with pre-exposure to the structure and concepts in maths maps through a creativity activity. The end result of this activity is a huge colourful classroom wall display which can be used as an ongoing reference for students and teachers.

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PREPARE: • copies of the detailed maps which will make up the essence of the huge display

map. • supply a sheet of flip chart paper for each group. DIRECTIONS: 1. introduce this activity by posting the geometry overview at the front of the class.

Briefly outline each of the eleven topics. 2. Divide the class into eleven teams, assigning each team to one of the eleven

maths maps topics (shapes, solid lines, triangles, transformations, curves, quadrilaterals, circles, points, angles, trigonometry). Give each team a Detailed Map which makes up their section. For example:

or

3. Students use the flip chart paper to create their own version of the map, incorporating their own key words, symbols and colours.

4. Draw attention to the colouring guidelines. The key is to assign one basic colour to each main branch. Advise students to use light colours for colouring over key words and formulae boxes to allow these to be read at a distance.

5. When the teams have finished, they can present their learning to the others. Post each team’s map around the classroom.

Activity 3 Topic Preview

This activity allows students to attach details of a maths map (a skeleton map) to a template.

Students write details onto their skeleton map as the teacher discusses and attaches those details onto a transparency of the skeleton map.

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PREPARE: • A transparency of the skeleton map for the topic you wish to preview. • enough copies of the skeleton map for each student to have one. • one copy of the half-developed map for your reference.

DIRECTIONS: 1. Display the skeleton map on the screen and hand a copy out to each student. 2. Discuss briefly each main branch, ensuring that students are familiar with the key

words. 3. Add details to the branches with fine transparency markers using the half-

developed map as your reference. Discuss the details with students as you write. Allow students to attach the details to their own maps.

4. Emphasize the connections between concepts on different parts of the map. Connections can be physically linked by joining the concepts with an arrow.

5. Add a pair-share or team activity after the lecture to encourage discussion of the topic. Then let students redraw the maps from memory.

Activity 4 Topic Recall

A topic review using a skeleton map involves students recalling details from memory and attaching these details onto the skeleton map structure. At the end of the activity an overlay of these details is displayed on the overhead projector screen.

PREPARE: • A transparency of the skeleton map for the topic you wish to review • A transparency of the detailed map • Enough copies of the skeleton map for each student to have one • One copy of the detailed map for your reference DIRECTIONS: 1. Display the skeleton map on the screen and hand a copy out to each student.

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2. Discuss each branch with students by asking leading questions. Use the half-developed map as a reference. Ask students to attach the details to each branch as you discuss the structure.

3. When students are ready, align the overlay transparency and allow students to make any additions or changes.

Activity 5 Creation Maps

In this activity teachers and students explore a variety of key topics by building their own maps. Through this process students develop flexibility in their thinking by learning to piece together mathematical concepts in different ways. The following is a list of possible topics to explore in more detail:

area, Pythagoras, symmetry, circle geometry, locus, volume, perpendicular, gradient, graphs

PREPARE: • Blank sheets of paper (enough for each student). DIRECTIONS:

1. Students build their own maps in teams of 2 or 3. Encourage them to use any available resources, including other students, CDs, books or notes.

2. Give them 30-60 minutes for this activity. When finishe, they can share with other classmates.

Brainstorming (some suggestions as a springboard for more ideas)

Activity 6 Interview:

Teachers can bring in practitioners for students to interview, looking for the maths’ applications, which are related to their jobs.

Activity 7 Build Models:

A project assignment might be for the students to build 3-D models illustrating real- life applications for the topics.

Activity 8 Predictions:

Ask students to predict future applications of the concepts illustrated in the maps. Use current events as an example.

Activity 9 On-Going Mapping:

The preview map can be given to students at the beginning of the unit of study to fill out and take notes on as they progress throughout the topic. Students can add their own illustrations as reminders. The detailed map can be shared with students as a “you are here” map.

Activity 10 Re-Sort:

Ask students to find additional categories to re-sort the topics. The topics might be: easy to complex, numeric to spatial or practical to theoretical.

Activity 11 Student Review:

Use the preview maps as a student review technique. This can be done individually or in groups. Have students fill in on the preview maps what they have collectively learned.

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Then present the maps to the class or hang them on the wall during review.

Activity 12 Peer Teaching:

The detailed maps are great resources. Advanced students can share them with less advanced students as a quick reference or to help them create their own detailed mind map.

Activity 13 Bulletin Board:

The detailed map can be used as a large poster or bulletin board that students add to as they make new connections. Preview or skeleton maps can also be used on a bulletin board. Assign each team in the classroom a “branch” of the bulletin board to finish. This can be used for research or review purposes.

Activity 14 Test Review:

Students can reconstruct their mind maps as a method of review. They can create mapping possibilities to turn in as a form of student self-assessment.

Activity 15 Deck of Cards:

Copies of the skeleton maps can be presented like a deck of cards to each team along with pieces of string. Each cards includes a vocabulary word, equation or a picture from a specific topic. It is the team’s job to construct their own mind map using the card and the string. They can attach this to butcher paper and share it with the whole class.

Activity 16 Connected the Maps:

See how the students connect all of the separate topical maps together. The maps have overlapping ideas. For instance, volume is on the maps titled triangles, quadrilaterals, circles and solids. Could the students build a 3D display, perhaps a mobile?

Activity 17 Previous Learning:

On the first day of school the teacher might hand out a poster sized sheet of paper and simply let students construct their own math mind maps based on their previous learning.

Activity 18 Preview:

Prior to each unit or chapter, students or teachers can introduce a concept by displaying a map of that chapter.

Activity 19 Class to Class:

Working in group or pairs, students from other classes can make mind map overviews of the respective course topics to be shared between classes.

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Appendix 1

The discrete and the continuous:

aspect and methods for a philosophy of mathematics

The author Willem Kuyk (1982) believes that the history of the fundamental principles of

mathematics can be considered as the sum of converging efforts to bring together the varied and

multiple aspects of mathematical activity, as expressed below:

1. the role of ‘logic’ or ‘naïve logic’, the natural and spontaneous capability of man to define

concepts about things, concepts about concepts about things etc;

2. the role of language as a tool and carrier of communication;

3. the role of (formal) logic as a historically determined set of rules of thought (which have

been more or less formalised), created with a specific aim of ordering and organising,

coherently and deductively, concepts belonging to specific realms of thought;

4. the role of constructive thought, taken to mean the construction of new entities by given

entities so that the characteristics of these new entities can be derived from the former;

5. the role of intuition, understood as a the prompt gathering of the proof of things, that is, as a

way of arriving at knowledge, which is independent of deduction;

6. evaluating the problem of existence in mathematics of ‘primary knowledge’ as opposed to

‘derived’ or ‘secondary knowledge’ (for example, is the concept of numbers primary or does

it refer to a set, as followers of Kant maintain);

7. following deductive routes, an explanation of the fact that the application of mathematical

methods often brings forth true knowledge (but approximate) of the world, that is, the fact

that the fundamental mathematical properties of numbers and the knowledge of topology

and analysis can be applied in many non-mathematical spheres.

There are two different ways of dealing with these issues: explicitly and implicitly.

To make explicit the functions of the above-mentioned factors is to philosophise or theorise about

mathematics. Most of the existing theories about fundamental principles take into consideration

everything that is called ‘mathematics’. When such a discipline or some of its salient characteristics

are considered typical for ‘doing’ mathematics in its entirety, a philosophical reductionism takes

place, which has left mathematics as the only discrete point of view, neglecting the philosophy of

geometry.

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The implicit way of dealing with fundamental aspects of mathematics derives from the practice of

mathematics. When operating in fields such as the theory of numbers, topology, geometry and

algebra, mathematics is spontaneously concerned with the above-numbered issues. Consequently,

mathematical reasoning, which mathema tics applies, is nearer to that which we have called logic in

the meaning of 1. above, than the formal logic mentioned in 3. above.

Two types of mathematical existence

Two different types of combined existence are legitimately present in mathematics: geometric and

discrete. If this assertion is made, the following questions must be answered:

1. Do two different theories of sets correspond to the two different types of mathematical

existence? It is clear that such a question is tied to the so-called theory of the

‘arithmeticisation’ of geometry, which states that to each classical, geometrical statement

there is a corresponding arithmetic one. It can be seen immediately that this theory is a

corollary of the Kantian idea of continuously reconstructing concepts, starting with

discrete concepts, or rather rebuilding geometrical concepts, starting with the concept of

sets.

2. If two different types of sets operate in classical geometry and analysis, do these theories

of sets depend on the ‘nature’ of the entities, which are found in one whole? How can

the concept of sets be distinguished?

For Hilbert the pure concept of continuity connected pure mathematics on the one side and applied

mathematics and physics on the other. According to Brouwer and Weyl, to speak of a pure concept

of continuity means introducing actual infinity into mathematics. They conceive continuity as

something which can potentially be constructed by discrete notions. This concept of continuity

corresponds to the Kantian one in that Kant also believed that the continuum could be constructed

by starting with discrete notions but it differs in that Kant accepted the hypothesis of the capability

of the human mind to build beyond a ‘potential infinity’, to an actual infinity.

I very much agree with the hypothesis that there are (at least) two different fundamental ways of

obtaining concrete discrete and continuous bases in epistemological research, like the research of

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Jean Piaget (Piaget, 1969). This research confirms how so-called spatial intuition can be developed

in the evolutionary process, relatively independently of the concept of natural numbers. Indeed,

Piaget called a line, a curve etc ‘a spatial symbol’ which takes the place of real, physical-spatial,

bodily transformations. Piaget believed that purely spatial concepts in our mind may have been the

result of impulses and physical transformations. Piaget’s discoveries legitimised the hypothesis that

mathematics is based on two fundamental insights – that of the continuous and the discrete – each

of which probably corresponds to a particular type of existence: existence meaning numbers and

discrete entities, and existence meaning spatiality and continuous entities.

Language, the theory of sets and mathematical complementarity

In tackling the discussion regarding the relationship connecting the two different and fundamental

concepts in mathematics, we must spare a thought for the role of language. In formulating a system

of axioms in a particular discipline, it is necessary to create a language which is relatively

independent of the domain of entities (models), to which the language will be applied. This

comprises a relationship between three functions: construction, thought or intention (yet to be

formulated) and axiomatic language; from time to time each of these precedes the others.

The ‘anticipatory function of language’ is that ‘leap forward’ which occurs when the operating of a

formula (or on a set of formulae) brings about a result which could not be thought to be true prior to

the operation. A characteristic of complementary thought is to consider the discrete and continuous

as complementary aspects of the world. According to the types of problems dealt with in

mathematics, we tend to presuppose existence on the basis of a geometric-continuous world or to

consider existence or computability with a discrete numerical meaning. Nor is the fact that some

mathematical disciplines depend to a great extent on the treatment (intuitive) of geometrical

continuities (an n-dimensional Euclidian topology, classical geometry etc) an example while, at the

other extreme, some disciplines are only concerned with (sets of) discrete entities (the algebraic

theory of numbers, numerical analysis etc).

With the intention of classifying the disciplines listed below into two categories, it is worth

mentioning their epistemological origin and final objective in the discrete domain and theories of

numbers, and those which come from (and regard) the continuous and geometrical domain; the

table below has been taken from Kuyk’s book (1982, p.146):

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(a) Discrete (Algebra) (b) Continuous (Topology)

Theory of Sets - Logic

1. groups, rings, graphs etc groups, rings, topology

2. algebraic geometry analytical geometry

3. algebraic topology analytical topology, varieties

4. Lie’s algebraic theory Lie’s theory

5. algebraic bundle theory analytical bundle theory

6. algebraic number theory analytical number theory

7. algebraic varieties analytical (and differential) varieties

8. combinatory theory of probability combinatory theory of probability

9. theory of addition (series) integral calculus

10. finite differential equations differential equations

11. linear algebra linear analysis (Banach’s algebra)

12. finite geometry classical geometry

13. processing science

The Theory of Categories

According to Kuyk, the theory of sets does not reduce the continuous to the discrete, but it can be

considered a theory that formulates rules for treating geometric and arithmetic entities. From this

point of view, therefore, the theory of sets regulates the use of mathematical language in that it is

concerned with the formation of newly-formed wholes, starting with the wholes of given entities.

That the theory of sets formulates rules for using language is clear from the fact that the theory of

sets is the only discipline in mathematics which, throughout the course of its history, there have

been problems like: “To what point can the existence of mathematical entities be supposed?”

Fundamentally, the answers to these question go in two different directions:

Ø a Kantian-realist-formalist position which maintains that you can push where you want to;

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Ø the other position, including the idealist-intuitionist and logician-realist, which establishes

limits on language, with the theory of sets depending on cosmological considerations.

According to Kuyk, having a point of view which balances the above seven aspects of mathematics,

in such a way that none is favoured, makes us believe that the theory of sets must make explicit the

implicit “naïve theory of sets”, which operates on a level of mathematical experience

(epistemological) yet to be formalised. This means that the a posteriori Piagetian point of view can

be accepted, according to which, during the process of mathematical learning, each individual’s

mind passes through a totality of natural levels of abstraction in such a way that each ‘subsequent’

level is dependent on the ‘previous’ one.

An outline of a Theory of Complementary Sets

The complementary orientation is reconstructed from the wish to comply with the theory of sets,

which is naïve and spontaneous on epistemological and classical mathematics levels, and which

complies with the former unity of pure and applied science. Historically, only when mathematics

became more abstract and the disposition for constructing sets of sets of sets etc…….of sets of, let

us say, natural numbers, did it become evident that those who chose an a priori conception of

mathematics had to make a choice between one attitude (philosophical), which took them either in a

Kantian-realist or constructivist-idealist direction. As Poicarè said: it seems that the a priori concept

would compel philosopher mathematicians, at least regarding the problems of existence in

mathematics, to direct their real research either towards the world of the continuous and geometry

(that of Kant etc) or towards the world of discrete entities and the theory of numbers

(constructivism etc). The mathematics of complementary inspiration, which considers mathematics

in a posteriori way, constructed from ‘raw material’ (natural and continuous numbers, as analytical

qualities in the world) and which generally conceives the theory of sets as a formulation of rules of

language, is not forced to choose between two positions.

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Appendix2

How the brain constructs the visual image…

One of the main questions we could pose is: “what does to see mean?”. David Marr’s answer is the

following: the act of seeing is the process through which starting from images we discover what

exists in the world around us and where it is positioned. Mortimer Mishkin and his working team

were the first one who affirmed that these two assignments are processed by different anatomic

parts/routes. The (of) what identification implies the presence of two different subsystems, one of

which contains the information relative to the shapes of the objects whilst the other one contains the

information relative to the colours. Both information terminate in the inferotemporale cortex,

which is the essential part concerning the recognition of the shapes.

The position (where) of the objects in the space is instead the assignment of a third system, the

termination of which is in the back parietal cortex, in a complex of areas which is essential referring

to the organisation of the space.

The fact that the visual analysis implies the presence of different ways acting in parallel instead of a

certain number of stations positioned one after the other has become clear only in recent times. The

integration of the information is obtained in an interactive and not progressive way.

First of all, let’s see how the brain stores the sensory experiences through the visual perceptive way.

The results of some important experiments let us suppose that the visual information is gradually

elaborated along the nerve route. The simple observation of the external world, the recognition of a

face or of a landscape imply a very complicated faculty of analysis.

In fact, the nervous information related to shapes, movements and colours are not analysed by a

single nervous way hierarchically organised; on the contrary, they are analysed by at least 3 (and

even more perhaps) analysis ways which have parallel positions and different interaction levels.

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The visual perception is a creative process

Whenever we make a movement or whenever the light environmental conditions change, measures,

shape and brightness of the images which are projected on the retina change too. In spite of this our

perception of dimensions and colours of the objects doesn’t change.

Starting from the empirical conceptions of the 17th and 18th centuries and particularly John Locke’s

and George Berkeley’s conceptions, who saw the perception as an additive synthesis of the single

basic sensations, we then arrive to the modern conception affirming that the perception implies

some active and creative processes which pass the simple acquisition of the sensory information.

At the beginning of the present century, the backers of this conception are Max Wertheimer, Kurt

Koffka and Wolfgang Kohler; they gave origin to the so-called Gestalt psychology . The Gestalt

psychologists used to explain the perception laws affirming that the visual system organises the

image analysis following some congenital rules relative to the structure, the shape, the distance and

the movement of the objects situated in our field of vision. In fact, if some points are positioned in a

certain direction and are one near the other, the nervous system sees the whole structure as disposed

in that particular direction according to a continuous and dynamic organising process.

For example, let’s consider Victor Vasarely’s image in which this perceptive decision is present.

Both black and white images may be alternatively seen as figure or as bottom :

Another example is Mauritius Escher’s repeated image representing fishes and birds (1971), where

the two different figures have the same profile:

Concerning this strategy of the “who wins catches all” Mauritius Escher writes: “our look uses to

stare at single objects. But in that same moment when we’re looking at the object, everything

around it is reduced to a mere bottom… the human eye and mind are not able to do two different

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things at the same time, and have thus to spring continuously and rapidly from one thing to another

one”. The dichotomy between figure and background highlights one of the basis of the visual

perception: the attention may focus only a part of the images, all the rest has to shade in the bottom.

Looking at an image, what catches our attention is shape and profile. Consequently, we are able to

recognise clearly an object or a visual scenery even when this one is presented us as a mere

approximate drawing without shades or colours:

Optical illusions, which may also be said as "bad readings" of the different information by our

brain, do clearly show that the nervous system applies to the sensuous information a series of

classifications concerning the structure of the visual world. In Muller-Lyer's classic optical illusion

we can see how two segments having the same length seem not to be equal:

The fact that we still continue to see the line ending with the two inwards appendixes shorter than

that ones having the outwards appendixes depends from a certain kind of logic which is proper of

our brain and (probably!) from the fact that the experience has taught us to use the template of the

objects as token of their dimensions!

Beside the innate logic and the previous experience of our nervous system, spatial relations

interposing between an object and its bottom help us to interpret the meaning of the images, too.

This means that we evaluate the dimensions of an object by comparing it with the other objects

surrounding it.

The role of the light source…

When we don't have a correct information about the provenance of the light source, the curvature of

the object, for example, results equivocal. In fact, looking at the picture below, we can see spheres

or hollowness depending on the provenance of the light that we assume:

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Each time our brain has to interpret the kind of curvature of a whole of objects, it seems to assume

that the illumination of the entire scene comes from a unique source. This probably derives from the

fact that the Human Being has had its evolution in a natural environment in which the only source

of light existing was the sun and consequently our nervous system assumes that each source of light

should always come from the top and not from below.

The anatomic characteristics of the visual ways

Let's shortly analyse the anatomic ways and the psychological tests which demonstrate how the

analysis of shapes, movements and colours comes to an end along three different ways in parallel

positions which run from the retina to the lateral geniculate body and then from here to the cortex.

The axons of the retina's gangliar cells constitute the optic nerve which projects in a tidy manner to

the thalamus's lateral geniculate body.

The lateral geniculate body projects in turn to the primary ipsilaterale visual cortex or V1 which is

situated in Brodmann's area 17 (also called striated cortex). As a consequence of these tidy

projections the striated cortex contains a complete map of the retina. This map is not the only one;

in fact there are some other ones of them in the visual areas group belonging to the upper level

which are situated in the extra striated areas (in these areas there are about 32 different

representations of the retina):

retina

nervo ottico chiasma

collicolo superiore

corpo genicolato laterale

radiazione ottica

Corteccia visiva primaria

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These extra striated areas are different first of all because of the kind of selectivity shown by their

neurons towards the different characteristics of the stimuli.

It's then possible to outline them in this manner: the information starting from the thalamus arrive to

the visual cortex which is formed by a primary visual area V1 which initiate the elaboration (the

V1 and V2 areas have not much specialised functions) and by associative visual areas

V5 fundamental in relation to the analysis of the movement and of the sense of profundity

V4 related with the recognition of the colours and with the bearings of the edges of the objects

V3 fundamental for what concerns the analysis of shape and movement

Therefore, the visual system has several different representations of the retina's receptive structures

and each of these representations analyses a different aspect of the visual information.

The division of the information concerning the different aspects of the images starts in the retina

which contains two different populations of gangliar cells which are respectively constituted by big

cells (also called magnocellulari or of M type) and by little cells (also called parvicellulari or of P

type). The two group of strata in turn give origin to three main ways which extend from the lateral

Percezione del colore

Forma e colore

Movimento

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geniculate body to V1, that is the primary visual cortex, and then to V2 and then finally to the other

extra striated cortices.

The first of these ways originates from neurons situated amid the parvicellulari strata of the lateral

geniculate body and forms the blob parvicellulare way, specialised in the perception of the colours.

The second of the parvicellulari ways, or interblob-parvicellulare way, is instead specialised in the

perception of the shapes. This way, just as the way deputed to the analysis of the colours, ends in

the inferotemporale cortex too. The neurons of this system are sensitive to shapes and to the

bearings of the images, which constitute important elements for the perception of the shapes; they

also have a high resolutive power and this characteristic seems really to be very important for

observing the details of still objects. After all it seems to be a system deputed to interpret what can

be seen. The lesions of the inferotemporale cortex cause the appearance of deficits in the

recognition of complex objects and of faces.

The third way (or magnocellulare way - thick stripes) is specialised in the analysis of movement

and of spatial relations among the objects; it also gives a contribute to the stereoscopic vision. This

way is then deputed to give the answers about where an object is. The neurons of this system are

relatively insensitive to colours and have a very little faculty of perception of still objects.

The visual information are conveyed in different nervous ways

It has been possible to study the perceptive contribute of the parvicellulare system by reducing the

influence of the magnocellulare system through the use of equiluminous* stimuli, that is stimuli

which diversify only for the colour and not for their luminosity degree. Two equiluminous colours

have contrasts of colours but not of luminosity.

The researches which have been effectuated on the reactions to equiluminous stimuli in the Human

Being enable us to conclude that in conditions of equiluminosity the perception of movement is

strongly compromised. Even perspective, dimension relative to the object, perception of the field

profundity, relation figure-bottom and optical illusions tend to disappear in conditions of

equiluminosity.

To make the figures show up from the bottom it seems that our nervous system should organize the

visual scenery into groups of coherent elements, each of which has got particular values of distance,

luminosity and structure. Moreover, if an image makes a movement each group of elements will

also have a specific direction and a specific moving speed. In this way, the characteristics of the

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movement could represent a further evaluation element too, which allows to distinguish an object

from the others.

Even though today there is a wide agreeing concerning the fact that the visual system uses analysis

in parallel, the precise modalities through which this analysis is distributed among the three ways

which we have described are still object of discussion, and it's not still known how many basic ways

of the system does really exist (2 or 3). Another discussion point concerns the interactions which

come into being among the different ways. Many researchers highlight that however much one of

the ways might be particularly specialized in the analysis of a determined property of the visual

image, the other ways in all probability contribute to the analysis of those same properties too,

because of the same connections which establish between one way and another one.

The clinical observations talk about the elaboration in parallel of the visual information

With regard to the opinion that the different aspects of the visual perception are positioned in

different brain areas, at the beginning of this century, Sigmund Freud affirmed that the inability of

certain patients in recognising determined objects is due to some alterations of the cortex which he

called agnosias. Patients having agnosias of the sense of profundity were not able to estimate the

profundity or the thickness of the objects. Everything seems totally flat.

The study of the patients who consequently to a cerebral stroke are not able anymore to recognize

the physiognomies (prosopagnosia) has proved the existence of a specific way deputed to the

recognising of the faces. These patients have got lesions of the inferotemporale cortex and are able

to identify a face in its wholeness or its different parts and are even able to recognize the particular

emotional expressions of a face, but they are not able to identify a person by observing his face.

What misses in these patients is not the notion of the identity of a person, but only the connection

between the features of a face and the specific identity of that same person. For example, whenever

they have to recognize a friend, this patients have to base on his voice or on other non-visual

elements.

What nervous operations do determinate the appearance of a coherent and unique perception?

The role of the main map or salient map

How is it possible to draw a coherent and unique perception from the information relative to

colours, shapes, movement and sense of profundity since these different elements are analysed by

different nervous ways?

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Complex visual images come into being gradually, through the activity of different ways positioned

in parallel and able to analyse every single characteristic of the images such as movement, solidity,

shape and colour. Consequently, to make it possible to perceive at the same time the entire

combination of the visual characteristics which are present in the field of vision in every instant, it

is necessary that independent groups of cells, each having a different function, might temporarily

act in association. This hypothetical mechanism, not very clear till now, is called mechanism of the

connection. Ann Treisman’s and her collaborators’s psychological researches as wells as those one

by Bella Julesz have separately shown that the come into being of these connections requires a

process of attention. They tried to explain the process which allows to concentrate the attention on a

single object present in the visual field as well as the characteristics thanks to which that object

shows up onto its bottom.

First of all, they noted that the basic faculties of an object, such as luminosity, colour and direction

of the lines succeed in showing up the contour lines of an object. Treisman and Julesz observed that

when the borders of an object are constituted by elements which are very different from those ones

of the bottom, its contour lines do show up nearly automatically within 50 ms. Starting from these

observations they have hypothesised that in the visual perception two different processes come into

play. At the beginning there's a pre-attention process which inspects the external objects and shows

the coarse peculiarities of an object.

The elementary differences between one object and another one are detected exclusively in terms of

image or of borders. A next attention process directs the attention onto the peculiar characteristics

of an object, selecting and pointing out combinations of peculiarities which in a first time were

segregated in maps of different characteristics. This process acts a strategy of the type who-wins-

catches-all which allows to exalt and analyse certain characteristics of an object, whilst other

peculiarities and other objects are totally ignored. Treisman deems that the single properties of the

objects are codified in different maps of characteristics positioned in different parts of the cerebral

cortex. Treisman has hypothesised the existence of a main map or salient map which codifies

exclusively the general characteristics of the image allowing us to distinguish the object which we

want to pay attention to from its bottom.

Once the main characteristics of an object have been represented in the main map, all the other

details may be controlled, analysing them in the other different single maps of characteristics where

they are stored. For this reason the total recognising of the object is possible only as the different

details, situated in different maps of characteristics, are put or connected together.

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In didactic terms we can think that in the process of recognising of an object a cataloguing activity

happens which starts from the general properties and goes then to the more detailed and specific

properties.

The evaluation elements which allows us to understand how the consciousness of the external

world comes into being

The problem concerning the selective attention was defined for the first time in 1890 by William

James in his Principles of Psychology:

Million things…occur at my senses which will never be a part of my conscious experiences. Why? Because I

have no interest in them. My experiences are what I'm interested in… Everybody knows what the attention is.

It consists in the fact that one of the numerous objects upon which our mind might stop clearly catches our

mind.

Its essence may be resumed with the words focusing, conscious concentration of our thought. It entails that we

neglect some things so that we can attend exclusively to others.

Even though our visual system contains numerous ways in parallel which are able to elaborate at the

same time information having a different nature, the quantity of these information which reaches

superior centres of analysis in the cortex is limited by the mechanism of the selective attention. The

selective attention filters some images, while augments our perception of other images. In this

strategy of the type who-wins-catches-all some stimuli gain importance inside our conscience,

whilst other suddenly disappear after a light perception.

It's interesting thinking that exploring the processes of the visual attention we will be able to define

the nervous processes of this particular kind of awareness. Probably the biological study of the

different parts of the consciousness might give us an idea at least of some of its more complex

factors, such as will, projecting and self-consciousness.

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Appendix3

Perceiving shapes and movement

The visual field is the part of the external world which is seen by both eyes without moving the

head. It's possible to determine a right half and a left half of the field of vision. The left half-field or

left half of the field of vision, will project its own images onto the nasal half-retina of the left eye

and onto the temporal half-retina of the right eye. The right half-field, or right half of the field of

vision, will project its own images onto the nasal half-retina of the right eye and onto the temporal

half-retina of the left eye. The light coming from the central part of the visual field strikes both eyes

and forms the binocular area. In each half of the field of vision exists a monocular area too (see

picture A):

The light coming from the temporal region of the half-field of vision will strike only the nasal half-

retina of the homonym eye, because the rays of light are stopped from the nose which makes them

impossible to reach the eye of the opposite side (picture B):

The optical disc is the area of the retina which doesn't contain photoreceptors and is therefore

totally insensitive to the light. As the optical disc is situated in the middle of both fovea , the light

175

coming from every part of the binocular region can never fall upon both discs and for this reason we

don't usually become aware of the existence of the blind spot. The blind spot can however be seen

by closing our right eye and fixing with our left eye the little cross positioned in the upper part of

the following picture:

The retina projects to the lateral geniculate body

The optical nerves of the two eyes meet at the level of the optic chiasm (chiasma opticum). Here the

fibres originating from the nasal half of each of both retina cross passing to the opposite cerebral

hemisphere, while the fibres coming from the two temporal half-retina on the contrary doesn't cross.

The fibres thus separated forms the right and left optical traits (lines).

The left optical line will contain a complete representation of the right visual hemisphere. The right

optical line will then contain a complete representation of the left visual half-field.

The right and left optical lines project to three under-cortical areas. Of these three areas, only the

lateral geniculate body is deputed to elaborate the visual information necessary for the perception. It

constructs a retinotopica representation of the counter-lateral half of the field of vision.

Consequently, the lateral geniculate body owns a nervous map of the retina.

The ways of the "where" and of the "what"…

As we have already seen the magnocellulari and parvicellulari ways of the lateral geniculate body

give origin to three different ways positioned in parallel which extend to the primary visual cortex

and from here on they reach the upper level visual cortex in relation to the perception of shapes,

colours and movement. The magnocellulare way seems to be related to the localisation and to the

movement of the visual images, in other words it lets us know where an object is. The

parvicellulari-blob and parvicellulari-interblob ways seems instead to be related to the analysis of

shapes and to the vision of the colours; they therefore let us know what an object is.

It's thought that the first steps of the perceptive analysis of the visual stimuli depend from

complicated mechanisms of convergence of the connections which establish among the cells of the

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primary visual cortex. At each level, each cell has an ability of elaboration superior than the ability

of the cells belonging to the lower levels.

At lower levels of the system the gangliar cells of the retina and of the geniculate body principally

analyse the contrasts of luminosity inside little receptive fields having circular shape.

In the retina and in the geniculate body what is important concerning the stimulus is its position

inside the receptive field.

The primary visual cortex is organised in columns and strata

As we already said, the cells of the primary visual cortex react in a selective way to the stimuli

having retina localisation and specific bearing axes .

The neurons of the primary cortex which receive afferences from the parvicellulare-interblob way

and from the magnocellulare way are organized in columns. All neurons belonging to a same

column react to stimuli with the same bearing axis. Each bearing column contains also complex

cells, that is that they receive direct connections from the simple cells of the same column. The

interconnections among the different kind of cells of a same column allow the analysis of more and

more complex characteristics of the visual information.

Therefore, the first very necessary step for the analysis of the shape of the objects entails the

splitting of the images situated in the visual scenery into linear segments of different bearing.

Thanks to several researches it has been possible to determine that the bearing columns are spatially

organised in radial direction and not linearly. The different bearing axes develop radially, just like

wheel spokes, starting from a centre of bearing and each bearing axis is represented only once in

each ray. The columns of bearing are regularly interrupted from cells belonging to the region of the

blob (which are areas of the cortex where the information concerning the colours is analysed).

Next to the columns deputed to the analysis of the axes of bearing and to the information of the blob

which are related to the vision of the colours, the striated cortex has a third component constituted

by the columns of ocular domination, which are at the head of the binocular vision and make the

perception of the field profundity possible. A complete sequence of right and left domination

columns, of columns of bearing and of the group of blob contained in them occupies a part of the

cortex called hypercolumn . Each hypercolumn contains the necessary nerve structures to analyse

the information coming from a certain point of the retina and therefore it corresponds to an

elementary analysis model.

177

It’s thus possible affirming in the end that the organisation of the primary visual cortex is based on

two connection systems which reciprocally intersect: functional columns locally formed by the

connections between the different strata of the cortex and functional columns having the same

reaction properties and constituted by connections which establish at distance among the different

columns.

Beyond the primary visual cortex: the representation of faces and of other complex shapes

originates in the inferotemporale cortex.

What are the processes which make it possible that our visual perceptions of things or persons

become unique and complete?

The analysis of the visual images doesn’t stop in the primary visual cortex at the level of the

complex and simple cells. It in fact represents only the entry cortical step. The clinical observations

of the Human Being as well as the experimental studies on Monkeys make us think that the analysis

of the visual information allowing us the recognising of the shapes is deputed to the parvicellulare-

interblob system (V1). From here on, the information passes to V2 and finally to V4, which is a

cortical area containing neurons sensitive to both shapes and colours. From V4 this way projects in

the end to the lower temporal cortex, and this is the cortical region which is most strictly connected

to the analysis of what a visual image is.

The movement of the images in the field of vision is analysed by a particular neural system

Given that we are continuously in movement, we need to receive precise information about how the

objects move relatively to us. Even when we don't move or the objects which interest us don't move,

Ipercolonne Gruppo colonne di orientamento (forma) Gruppo colonne di dominanza Formazioni dette blob (colore)

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the images projected onto the retina move all the same, because neither our eyes nor our head are

ever perfectly still. The visual system disposes of two ways to recognize the movement: one is

related to the movement of the images and the other one is related to the movement of the eyes and

of the head.

If our eyes don't move, the image of an object in movement crosses the retina. The information

relative to the movement is transmitted to the brain by the discharge in succession of different

retinal receptors. On the contrary, when the eyes follow an object the image of the object in

movement remains still in a point of the retina and the nervous system evaluates the entity of the

movement of the object on the basis of the entity of movement made by eyes and head.

Many invertebrate (such as the frog) are not able to see the objects unless they move. The

movement of an object in the visual field could be displayed comparing the different positions that

its own image assumes onto the retina in successive moments. Theoretically speaking, then, the

visual system should be able to distinguish the movement of an object comparing, instant after

instant, its successive localizations onto the retina.

The apparent movement or optical illusion, may be receipted as "true" as the real movement. Film

images are a clear example of this optical illusion. The impression of the movement is effectively

due to the rapid succession on the screen of images which are really immobile. The real movement

determines the successive discharge of the photoreceptors positioned along the trajectory traced by

the image onto the retina. The apparent movement might also be more convincing than the real one,

and it constitutes the perceptive basis upon which film projections are based. If we alternatively and

at opportune intervals turn on and off two lights having A and B position, our perception is that of

one light moving between two points. Considering the absence of a successive discharge of

photoreceptors, this example is a prove of the existence of a particular system deputed to the

recognition of the movement.

Some psychophysical observations have suggested the idea that the information regarding the

movement are analysed in two successive steps. The first concerns the movement towards a single

direction. At this step of the analysis each single active neuron records only the movement running

in orthogonal direction as regards the bearing axis of its own receptive field.

At the second step the global movement of the entire object is analysed. The neurons active at this

second analysis step integrate the different components of the movement displayed by the neurons

which are active in the first analysis step. These neurons are therefore called selective neurons for

the components of the global direction of the movement.

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The global movement of an object is evaluated by neurons which are sensitive to the total shape of

the object, on the basis of information coming from neurons which are selective for the single

components of the direction of the movement. In this part of the cortex there is a retinatopical map

containing information relative to the speed and to the direction of the images coming from the

counter-lateral half-field of vision. Just as the other cortical areas this one is organised in columns

too, and the neurons belonging to each column are activated by the movement towards a certain

direction, whilst they don't react to movements towards other directions. The kind of discharge of

the neurons of these selective columns for the movement takes also probably part in forming the

perceptive judgement of the movement itself.

It's possible nowadays studying at a cellular level the problems related to the cellular attention

The studies of the visual attention at a cellular level have demonstrated that the attention entails

either an augmentation of the discharge frequency of cells responding to the object for which they

show interest or a reduction of the discharge frequency of neurons responding to objects which are

ignored.

On the basis of some studies, Robert Wurtz and his collaborators have proposed that, as a person

pays attention to a visual stimulus, the cells of the back parietal cortex responding to that object

begin to discharge at a high frequency.

The sensory experience and the making of the visual circuits

The visual perception is the synthesis of a coherent image of the specific information which the

central nervous system receives concerning shape, movement and colour of the objects, and is

mediated by the nervous cells which, connecting reciprocally in an accurate manner, form

numerous nervous ways positioned in parallel.

The synaptic connections may be changed all the adult lifetime long by the nervous activity caused

by the experience. This fact suggests us that there is probably a continuity between the nervous

processes and the processes mediating the learning activity.

The influence exerted by the environment on the central nervous system and consequently on the

behaviour, varies depending on one's age. Generally speaking the abnormal environment

experiences have deeper effects during the first steps of the postnatal growth than during the adult

lifetime. Following to some studies it has been observed that the sense precocious privation alters

both perceptive development and development of nervous circuits.

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The association, the co-operation and the competition…

In the primary visual cortex the association, the co-operation and the competition constitute the

mechanisms through which two populations of afferent fibres distribute themselves without any

superimposition to a nervous common space.

The development of the columns of ocular domination depends from the competition among the

afferent fibres from both eyes and from the co-operation among the fibres coming from the same

eye. The co-operation is necessary to the synchronisation of the neuronal activity.

Probably each cortical area and each of its strata own a proper determined time for the segregation

of the connections and therefore a proper critical specific period. The experiences which interfere

with the development of a primary sense region of the central nervous system, such as the loss of

the sight, have repercussions on the behaviour when they happen during the first steps of the

postnatal growth; on the contrary, other experiences such as the social isolation exert their action

later. These differences in the maturation times of a cerebral region would explain two well-known

characteristics of the knowledge and of the behaviour development:

(1) certain faculties, such as those one inherent to the language, the music or mathematics, generally

have to be well developed before the puberty in order to show totally; and (2) certain traumatic

lesions which occur in determined period of the postnatal life may well alter some aspects of the

perceptive and character development.

During the first steps of the development of the social behaviour there is a critical period

The researches about the development of the columns of ocular domination have supplied the

referring context to understand how different and more complex experiences of the senses in the

first steps of the growth might be able to modify the circuits and the structure of the central nervous

system. Just as the normal development of the sense faculties which depends from the possibility to

do specific experiences during particular critical periods of the cerebral growth, the normal

development of the behaviour depends from the possibility to do particular social activities during

the course of certain specific steps of the growth too.

The researches on the development of the sensory systems give us a significant example of how the

genetic factors and the experience interact during the maturation of the brain and of how a privation

of the environment might seriously alter the growth processes. Children grown in isolation are

socially misfit, and generally with no solution. These abandoned children are often dumb and

unable to learn to speak.

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Appendix 4 An interview with Claudio Stassi (designer-advertising cartoonist)

Palermo, 20/02/03

Me: Hi, I’m here to ask you some questions about “the comic strip” but before I would like you to

tell us something about yourself and the reason why you decided to draw comics.

- Hi, my name is Claudio Stassi, I’m 26 and I’ve worked with different editors for some years.

These are Lo Sciacallo Elettronico that is a Milanese Azer Edizioni and I also work together with

the Grafimated Cartoon of Palermo for the creation of cartoons.

I decided to create cartoons because I’m so fond of them that if they were a woman I would merry

her but since they are only drawings and they are unreal. The only thing I can do is or to draw them

imagining to go into that unreal world or to wait that they could go out like Roger Rabbit making

me happy.

Me: Well…. This is impossible..

- It’s still impossible but with the virtual reality..

Me: Ok,.. I think it’s better to talk about my drawings.

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- They are good for what concern the point of view but they are some points of references that you

had to consider.

First of all what is a comic strip? A comic strip is a sequential story so planed side by side, a

vignette close to another one, that tells you a story, a “transcribed” story in a sequence, in a free

form, because there are many authors that use rhythms and when I talk about rhythms I also refer to

cinematographic ones: variable rhythms , someone uses a show rhythms , someone else a fast one, it

depends on how you want to tell the story and how the story is.

So…when you draw a drawing in a similar way actually it isn’t a regular comic strip but rather

some illustrations or vignette. A vignette next to another one is a comic strip because it tells you

something in a sequence.

Me: So… what does it have to tell mainly? …is these a cause and effect that links them?

- As a picture has to communicate those feelings that the author wants to express; the comic strip

has to express what you want to say:

It’s by for the simplest way to communicate.

Me: And what happens when you can’t use the words?

- If you can’t use any words you can use a special language called icon effective either for the

presentation of something or to tell something. For example if you have to use some words as “gun”

instead of writing gun, you can drew a gun! If you have to give the impression of the gun shooting,

you can use an onomatopoeia that in the semiotic language is the noise of the object you are using,

but not only…it is also the noise of the object you throw out the window, of the object you break, of

the punch you give on a wall, or also, the croaking of a frog, or the sound of a goose… why is

Donald Duck called Duck in English? Because the sound of the young goose is duck.. duck…!

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Me: Are the symbols used to represent these sounds universal?

- Not at all universal. These are some different sings but the comic strip uses a language that joins

everybody, in fact if I read a Japanese comic strip I know that instead of writing Sbam, they write

Stum ( to mean that you are throwing someone on the ground or on the wall ), so Stum is used in

their language when Sbam in ours. The symbols are icons that refer to a universal language.

Me: To represent the ideal you want to express, however you have to use a sequential logic.

- Of course.

Me: It has to cause in the reader an emotion since I choose this or that comic strip according to the

emotions…

- …but we can’t speak about emotions… fundamentally the comic strip, according to the icons

used, tells what you want to say, emotions regard another kind of story, for example Bratz tells you

want happens to Colto Maltese and in the same time make you live what was happening to Colto

Maltese.

Me: What are the rules to follow? If so, is these a logic or a grammar of the comic strip?

- In the comic strip these are some specific rules to follow. First of all it’s necessary to use the

pigeonholing language for the vignette. For a linear and well-balanced communication it would be

better to use the Bonelli size: that are the six vignettes organized in three series, two for each one

and at the most you could link the dower one to the middle one or you could also alternate the upper

ones to the lower ones. All three ones are lower to be linked.

A very important thing, then, is the character! To arouse a reader’s interest in what he’s reading it’s

necessary to create a good character that follows him during the reading.

If you use a series of different characters the reader may forget the previous ones. In all the comics

all over the world like Martin Mistero, Mafalda, Lupen, Linus, there is a first character that holds up

the others.

Me: What are the characteristics of the first character?

- For example, he must be well characterized! You have to be immediately fond of the character,

for example Lupen, here we call him in this way but in Japan he’s called Monkey Punch that means

“monkey”! In fact he looks like a monkey!

Me: Could the mouse be a character loved by a child?

184

- Yes, think about Diddle! It’s a mouse. But consider a bunny: if you want to use an animal that

speaks you have to make it humane! If you use a mouse they could look at it as something

disgusting! A mouse wants to explain me subjects! While if you make it humane the mind work

out: “ah! He’s a humane that looks like a mouse that wants to explain me, in a funny way, what he’s

telling”. Mickey Mouse is a mouse that has had a great success because he is a humanized mouse

that tells us things, this is bizarre not comic. It became comic in a second time for its language, it’s

more bizarre for the excess of the paradoxical because he tells us what he himself couldn’t say.

So it’s really important o build the character and to tell through him what you want to tell.

Me: Then how do you have to say it?

- Well, first of all you have to create a subject…

Me: Are these any strategies to express concepts or ideas without using words?

- There is a common usual procedure for all, there is a plot (the idea, a stroke of genius!). After the

plot you had to trash out, to emphasize and then there is the linear story that explain you, using

technical and more specific terms, what it’s happening.

Me: And what could I do to optimise the time? Being in a classroom I have often a limited time.

- It depends on the difficulty of what you have to teach, it depends on the boys’ faculty to learn,

more time you use simpler the difficulties will be.

185

Me: Can the colours give emotions to the readers? Can these ones be used to bring up a subject to

the reader?

- Well, if you draw your drawing in black and white the youth get too bored. You have to be able, to

use colours, you have to colour the character and to use vivid colours that make the reader enjoy

himself. If you use different colours to colour two roads it means that the two roads are different.

Me: Do you think that the use of one or two characters (the hysteric mouse and the bizarre mouse)

that interact inside the same difficulties situations could create “emotions” (echo..) or could help to

understand the difficulty? Why?

- The situation could be different. You could create the dull one and the more intelligent one that

explain things but then could be a little matter: if the little boy while reading identify himself with

the dull character, it’s terrible! While, if you make the ideal character speak to the child it becomes

more interesting, in this way the character that you are looking at, starts speaking to you. At this

point you shouldn’t use the dull character, but a character that walks with the child as a little dog. In

that situation he use the little dog to a multiplication, he shift its snout and makes it become and

addition, he takes away a piece of it nose so it becomes a subtraction, it starts crawling so it

becomes a division…

In this case these is a fellow traveller that follows him in his way. So it becomes nicer.

Me: Is a subject dealt with by a comic strip more interesting than one dealt with by a book?..

- Surely! It’s more direct! You remember it better, because there is someone that is speaking to you,

while what about the power of books?

If you read a book you had to imagine what you are reading, the comic strip is a simpler means and

it’s also considered less artistic by critics, because it has a too direct language, there is nothing that

let you to bring forward yourself with it. It is as it were a cold means. It’s not true, the reader takes

part in the comic strip. When you see an image, for example, if you see a person with an axe in his

hand, and another person that is going to be killed by that one… in the next vignette you see… the

night, some buildings, the full moon and in the sky you read “AAAAAAAAAHHH…” you deduce

that man has been killed.

But between a vignette and the other there is a censorship that divides them, and when you have

heard it, because you have hear the person shouting, you have decided how he has been killed… by

an axe, and where he has been hit, in his heart? In his head? Has his leg been reduced to a pulp? He

gave him a single chap of his axe or a hundred? It isn’t cold, it is a means where you become the

186

author, not only the drawer and the scriptwriter are explaining you what they want to say but you

become together with them the author of what they are telling you.

Me: How can I make the implication be understood? And the cause and effect consequentiality?

- It’s more or less when you have done in the drawing with the plot!

Me: But I use a little arrow in the plot!

- Well, now I think you should improve this drawing…

You should use the vignette instrument rather than the comic strip. They are absolutely different,

also the time is different… by a vignette you have to express in the best possible way the problem.

For the Pythagorean Theorem you can use different vignettes.

187

Me: How could I reduce to a minimum the written sentences?

- You should link the character that is doing what you have to describe, to the thing he is doing.

Then you add a balloon where you insert the description of the thing the boy has to look for, and a

second balloon with the problem is. In the way you’ll had a perfect triangulation.

Me: Well, I thank you very much for giving me a moment of your time, and I leave you to your

work. Thank you.

- You are welcome, it was a pleasure for me. Hi and good luck teacher...

188

Appendix5 Teaching Units:

‘ The Cartoons of Clamat ’

Class I of a lower secondary school

Teaching Unit: THE PROBLEM OF POTS

Specific Objectives:

Ø Check of the knowledge of the notions of multiply and under- multiply;

Ø Check of the knowledge of the notions of repartition;

Ø Check of understanding of the notions of repartition;

Ø Check of ability of make a division in two figure and with a decimal quotient;

Pre-requirement:

Ø Knowledge of the notions of multiply and under- multiply;

Ø Knowledge of the notions of repartition;

Ø To make division in two figure;

189

Teaching Unit: THE PROBLEM OF THE ROADS

Specific Objectives:

Ø Check of the knowledge and of understanding of the direct formula for the

calculations of the perimeters and the areas of the following flat surfaces:

• Triangle;

• Square;

• Rectangle;

• Rumble;

Ø Check of the knowledge and of understanding of the indirect formula;

Ø Understanding of the notions of equivalence;

Ø Check of the capacity to confront of the properties of the geometric figures: the

ability of research of equal angles and sides;

Pre-requirement:

Ø The notion of congruency;

Ø Knowledge of the measurement of length:

Ø Direct and indirect formula for the calculations of the perimeters and the areas

190

Class II of a lower secondary school

Teaching Unit: THE PROBLEM OF POTS

1-Power 2-Divisibility

Specific Objectives:

Ø Integer and decimal

Ø The four operations and properties

Ø Decimal metric System

Subjects:

Regards to the Power:

Ø Definition of power

Ø The power with exponent 0 or with exponent 1

Ø Some particular cases of the elevation to power

Ø Size order

Ø Properties of the power

Ø Arithmetic expressions with powers

Ø Mention about square and cubic root

Regards to the Divisibility:

Ø Multiply and partition

Ø Multiply of a number

Ø Partition of a number

Ø Criteria of partition

Ø Prove of the 9 of the four operations

Specific Objectives:

Ø To understand the subjects

Ø To use these concepts in the problematic situations

Ø To know and to understand the criteria of divisibility and/or of the properties of the

power

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Class II of a lower secondary school

Teaching Unit: THE PROBLEM OF THE ROADS

Surface of the Triangle and of the Quadrilateral

Subjects:

Ø Figure with the same extension

Ø Measurement of the surface of the rectangle

Ø Measurement of the surface of the square

Ø An important relation between the measurement of the side and the diagonal of a square

Ø Measurement of the surface of the parallelogram

Ø Measurement of the surface of the rumble

Ø Measurement of the surface of the triangle

Ø A way to calculate the measurement of the surface of the triangle: the Erone’s formula.

Ø Measurement of the surface of the trapezium

Ø Measurement of the surface of a quadrilateral with perpendicular diagonals

Ø Problems about the equal perimeter and the equal extension

Pre-requirement:

Ø The fundamental geometrical elements.

Ø The Geometry as description of the physic world.

Ø The angles.

Ø The straight line.

Ø The polygon, the triangle, the quadrilateral.

Ø Important points of the triangles. Criteria of congruency of the triangle.

Ø The transformations to the same metric and the direct and indirect congruency.

Ø From the congruency to the same extension. The same extension of the figures: the same

extension for addition and the same extension for difference.

Ø Measurement of the surfaces of the polygons.

192

Specific Objectives:

Ø To understand the concepts of equivalence of plane figures.

Ø To apply this concept of equivalence for the calculus of the surfaces.

Ø To know and to understand the formulas for the calculus of the surfaces of the polygons

Ø To know and to understand relationship between the same perimeter and the equal-

extension

In particular to strangers learners:

Ø To check of the knowledge of the subjects

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Class III of a lower secondary school

Teaching Unit:

THE PROBLEM OF THE LADDERS

The Pythagoras Theorem

Subjects:

Ø An important case of the same extension:

ü Different way to check the Pythagoras theorem

ü Arithmetic analysis of the Pythagoras theorem

ü Inverse of the Theorem of Pythagoras

ü Terns of Pythagoras

Ø Application of the theorem of Pythagoras to polygons:

ü To rectangle

ü To square

ü To trapeze

ü To rumble

ü To parallelogram

ü To triangle

ü To rectangle triangle with the angles of 30°, 45° e 60°.

Ø Application of the theorem of Pythagoras to the elements of the circumference

Ø Quantity commensurable and incommensurable

194

Pre-requirement:

Ø The fundame ntal geometrical elements.

Ø The geometry as a tools for the description of the physic world.

Ø The angle.

Ø The straight lines in the plane.

Ø The polygons, the triangles, the quadrilaterals.

Ø The circumference and polygons inscribed and circumscribed at the circumference.

Ø The equal-extensions of the figures.

Ø Measurement of the surface of the triangles and of the quadrilaterals.

Specific Objectives:

Ø To know and to understand the theorem of Pythagoras

Ø Capacity of recognise the primitive terns of Pythagoras and know derive them.

Ø Capacity of apply the theorem of Pythagoras at the geometric figures studied

In particular to strangers learners:

Ø To check of the knowledge of the subjects

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Class III of a lower secondary school

Teaching Unit: THE PROBLEM OF THE SURFACE OF THE HOUSE

The solids

Subjects:

Ø Measurement of the total and lateral surface of the right prism.

Ø Measurement of the total and lateral surface of the rectangle parallelepiped.

Ø Measurement of the total and lateral surface of the cube.

Pre-requirement:

Ø The geometric elements in the space.

Ø Generality about polyhedron.

Ø The prism.

Ø About prism: the parallelepiped.

Ø About rectangle parallelepiped: the cube.

Ø About polyhedrons: the pyramid.

Specific Objectives:

Ø To know the fundamental concepts of the solid geometry.

Ø To know distinguish polyhedrons and solids from curved surface.

Ø To acquire general knowledge about the prism.

Ø To know and to understand the concept of equal-extensions between solids.

Ø To acquire and to understand concepts and procedures for the calculus of the relative

measurement of a polyhedron.

Ø Capacity of apply the formula for the calculus of the surface of the polyhedrons.

Ø Capacity of to apply the indirect formula.

In particular to strangers learners: To check of the knowledge of the subjects.

196

Appendix6 A-priori analysis:

Class: III° Primary school THE PROBLEM OF THE ROADS

1.1 I don’t know. 1.2 It wants to catch the cheese 1.3 It wants to eat the cheese 1.4 How to catch early the cheese 1.5 I don’t understand it 1.6 It doesn’t know which road taking 1.7 It wants to find the road 1.8 The road 1.9 The cheese 1.10 He can’t eating 1.11 It is thinking 2.1 I don’t know 2.2 There is the cheese 2.3 There are the mouse and the cheese. 2.4 There are the mouse and two roads. 2.5 There is the mouse that wants to catch the cheese and there are two roads. 2.6 There are two different roads. 2.7 There is a long road and a short road 2.8 There is a road to do to catch the cheese 2.9 I don’t understand it 2.10 It’s beautiful 2.11 There is a mouse, two roads and the cheese 2.12 It intends to eat the cheese 2.13 The interrogative point of Clamat 2.14 It intends to have the cheese and it wonders which road to choose 2.15 It eats the cheese 2.16 I don’t know. 2.17 I don’t understand it 3.1 He has to measure them 3.2 Choice at random 3.3 He has to take the shorter road 3.4 He has to take the longer road 3.5 He has to take the yellow road 3.6 He has to take the green road 3.7 He has to eat the cheese 3.8 He has to find the right road 3.9 I don’t know. 3.10 I don’t understand it 4.1 Because in this way it arrives to the cheese 4.2 Because in this way it arrives early 4.3 Because it likes it 4.4 Because it is the same 4.5 Because it is hungry 4.6 Because I measure it and bit is the shorter 4.7 Because in the first drawing is evident that the yellow road is the shorter 4.8 To eat the cheese

197

THE PROBLEM OF THE POTS 1.1 I don’t know. 1.2 I don’t understand it 1.3 It has to paint 1.4 It doesn’t know how pots it needs. 1.5 It doesn’t know how bits of wood it has to paint 1.6 It thinks 1.7 The paint 1.8 How it has to paint 1.9 I don’t know. 1.10 I don’t understand it 2.1 There are some pots and some bits of wood to paint 2.2 With 1 pot it paints 2 bits of wood, with 2 pots it paints 4, with 3 it paints 6. 2.3 It doesn’t know what to paint 2.4 It doesn’t know how many pots using 2.5 It wonders how many pots using. 2.6 Clamat needs half a pot 2.7 Clamat smokes 2.8 Clamat thinks: «how many pots I needs?» 2.9 The paint and the pots 2.10 There are some flowers 2.11 It thinks 2.12 It wants to paint the bits of wood 2.13 I don’t know. 2.14 I don’t understand it 3.1 It has to divide the number by 2 3.2 It has to multiply 3.3 It has to add 3.4 It has to count the pots 3.5 It has to count the bits of wood 3.6 It has to buy many pots 3.7 It has o paint the all fence 3.8 It has to take a pot of paint 3.9 It has to use 5 pots 3.10 It has to use 7 pots 3.11 It has to take 10 pots 3.12 It needs of the pots 3.13 The problem is how many pots it needs 3.14 I don’t know. 3.15 I don’t understand it 4.1 Because it is in this way. 4.2 It does it to see hw many pots to use 4.3 To paint the fence. 4.4 Because with 4 pots we obtain 8 and with 5 we obtain10. 4.5 It makes this because it has solved the problem. 4.6 Because it needs them to paint the bits of wood and that is left it will use this another time 4.7 Because 10 pots are used. 4.8 Because it has 10 the bits of wood. 4.9 Because it needs the pots. 4.10 Because it think. 4.11 Because in this way the result is right.

198

Class: V° Primary school

THE PROBLEM OF THE ROADS

1.1 It doesn’t know how road to choose. 1.2 It has to eat the cheese. 1.3 It doesn’t know how road to choose to arrive to the cheese. 1.4 It doesn’t know how road to choose to come first to the cheese. 1.5 It doesn’t know how is the measurement of the two roads. 1.6 It doesn’t know which is the shorter road to come first to the cheese. 2.1 The yellow road and the green road. 2.2 The yellow road and the cheese. 2.3 The measurement of the roads and the cheese. 2.4 The perimeters, the areas, the side of the geometric figures and the cheese. 2.5 The perimeters, the areas, the side of the geometric figures. 2.6 The cheese, the two roads and the specific data. 2.7 The two roads, the cheese and the thoughts of Clamat. 2.8 The perimeters, the areas, the cheese. 3.1 To understand which is the road shorter. 3.2 To understand which is the road shorter by calcula the measurement of the geometric

figures. 3.3 To calculate which is the road shortest. 3.4 To measure the two roads to take the shortest. 3.5 To calculate the measurement of the polygons. 3.6 To calculate all the numbers. 3.7 Taking the yellow road to arrive to the cheese. 3.8 Taking one of the two roads. 3.9 To calculate the data for to find the road more short. 3.10 To calculate the length of the segments. 4.1 Because the yellow road is the shortest. 4.2 Because it wont to arrive early to the cheese taking the road shortest. 4.3 Because calculating the road more short it will arrive early to eat the cheese. 4.4 Because measuring the two roads it can know which is the more short. 4.5 Because measuring the sides of every polygons it find the road shortest to arrive early to the

cheese. 4.6 Because measuring it can take the road more short. 4.7 It has to take the first road because it is the shortest and it arrives early to the cheese. 4.8 Because in this way Clamat arrives early to the cheese. 4.9 Because the yellow road goes right to the cheese while the green one does many curves.

199

Classes: I°, II° e III° of lower secondary school.

THE PROBLEM OF THE POTS 1.1 To paint the fence. 1.2 To find the numbers of pots necessary. 1.3 To solve the problem. 2.1 N° 1 pot for N° 2 bits of wood. 2.2 N° 2 pots for N° 4 bits of wood. 2.3 N° 3 pots for N° 6 bits of wood. 2.4 Those draw. 2.5 Those of the problem. 2.6 65 bits of wood. 3.1 To divide the numbers of the bits of wood by 2. 3.2 To divide the numbers of the bits of wood by 3 3.3 To divide 65 by 2. 3.4 It is the division. 3.5 I use the calculator. 4.1 Because if for N° 2 pots is necessary a pot then I have to divide 65 by 2. 4.2 Because for two bits of wood it is necessary only a pot. 4.3 Because it is more easy. 4.4 Because it is the only solution.

THE PROBLEM OF THE ROADS 1.1 I don’t know. 1.2 To eat the cheese. 1.3 To hurry up to catch it. 1.4 To understand the road more short. 2.1 The data are that written. 2.2 The measure of the single segments to find. 2.3 The sides of the geometric figures of the road. 2.4 Those written. 3.1 To measure the single segments. 3.2 To calculate the measurement of the single segment. 3.3 The comparison of the sides two for two. 3.4 To observe the road more linear. 3.5 To measure the roads. 3.6 To solve the problems. 4.1 Because I measure the single segments and then comparison and plus them. 4.2 Because the first road is more linear then second. 4.3 Because the first road is more long then second. 4.4 Because it is more easy. 4.5 Because it is the road shortest.

200

THE PROBLEM OF THE LADDERS 1.1 To catch the cheese. 1.2 To find the ladder less long to arrive to the cheese. 1.3 To calculate the hypotension 1.4 To calculate the length of the ladder. 1.5 To find the right ladder. 1.6 To get back the cheese. 1.7 To arrive to eat the cheese. 1.8 To arrive on the table and to take the cheese. 2.1 The measurement of the catheti. 2.2 The height and the distance from the table. 2.3 The distance between Clamat and the column. 2.4 Those I have just wrote. 3.1 To calculate the measurement of the hypotension. 3.2 To calculate the distance that there is between Clamat and the cheese. 3.3 To apply the Pythagoras theorem. 3.4 The length of the ladder. 4.1 It is a rectangle triangle. 4.2 Because the ladder measures tot. 4.3 Because it must to find the height of the ladder for taking the cheese. 4.4 Because it is more easy. 4.5 Because it has to apply the Pythagoras theorem.

THE PROBLEM OF THE SURFACE OF THE HOUSE 1.1 To measure the Surface of the house. 1.2 To recognise the geometric figures that arrange the development of the house. 2.1 The data are those wrote. 2.2 The measurement of the sides of the single geometric figures that arrange the house. 2.3 The sides of the geometric figures. 3.1 To calculate the area of the single figures. 3.2 To calculate the area of the single figures and add them. 3.3 To compare the single figures and add them. 4.1 Because the total surface is calculated adding the single surfaces. 4.2 Because I calculate the single surfaces, I compare them and I add them. 4.3 Because Clamat has to find the total surface.

201

Appendix 7 Learner-secretary’s protocol for the ‘Guess the number’ game

202

203

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- Malara N.A. & Navarra G., 2000, “Percorsi esplorativi per promuovere il pensiero algebrico

attraverso problemi”, L’Educazione Matematica, anno XXVI, serie VI, n°1, vol.2, pag. 7-21.

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Productions.

- Mc Gregor M., 1991, “Making sense of Algebra Cognitive. Process Influencing Comprehension,

Deakin University press”, Geelong, Victoria, Australia.

- Minsky Marvin, 1989, La società della mente, Adelphi.

- Nemirovsky et al., 2003, “Perpetuo-Motor Activity and Imagination in Mathematics Learning”,

Research Forum, proceeding of PME 27, Hawai-i, 1, 101-135.

- Núñez E.R., 1995, What brain for God’s-eye? Biological naturalism, ontological objectivism, and

Searle. Journal of Consciousness Studies, 2 (2): 149-166.

- Núñez E.R., 1999, Could the future taste purple? Reclaiming mind, body and cognition. In R.

Núñez & W.J.Freeman (eds.), Reclaming Cognition: The Primacy of Action, Interaction, and

Emotion. Thorverton, U.K.: Imprint Academi

- Peirce Ch. S., 1931, Principles of Philosophy, Hartshone and Weiss, Cambridge, Mass.

- Peirce Ch. S., 1991, Peirce on Signs. Writings on Semiotics, University of North Carolina Press.

- Pellitteri Marco, 1998 , “Sense of Comics, La grafica dei cinque sensi nel fumetto”, ed. La

Grafica Castelvecchi.

- Piaget Jean, 1969, Logique et connaissance scientifique, in “Encyclopédie de la Pleide”,

Gallimard, Parigi.

- Pisano L., “Liber Abaci”, edizione curata da Baldassarre Boncompagni, Roma, 1857.

- Pourbaix Frederic, 2001, “Mathématique et bande dessinee (Mathematics and comic strips),

proc. in Mathématique et Pédagogie, p. 19-33.

- Radford L., 2002, “Algebra as tekhne, Artefacts, Symbols and Equations in the classroom”,

Mediterranean Journal for Research in Mathematical Education vol.1(1), pag. 31-56.

- Radford L., 2002, “The seen, the spoken and the written. A semiotic approach to the problem of

objectification of mathematical knowledge”. For the Learning of Mathematics, 22(2), 14-23.

- Radford L., 2000, “Signs and Meanings in Students’ Emergent Algebraic Thinking: a Semiotic

Analysis”, Educational Studies in Mathematics, vol.42, n.3, pag. 237-268.

209

- Rizzolatti G. et al., 1990, Neurones Related to Reaching-Grasping Arm Movements in the Rostral

Part of Area 6 (Area 6a), in “Exp. Brain Res.”, 82, pp.337-350.

- Rizzolatti G. e Arbib, 1998, Language Within Our Grasp, in “TINS”, 21, pp.189-193.

- Saussure F. De, 1916, Cours de linguistique générale, Laterzia, Bari 1970.

- Schultz W. e Dickinson A.., 2000, Neuronal Coding of Prediction Errors, in “Annu. Rev.

Neurosci.”, 23, pp.473-500.

- Shannon C.E. e Warren W., 1949, The Mathematical Theory of Communication, Illinois

University Press, Chicago.

- Spagnolo F. & Ferreri M., 1994, “L’apprendimento tra emozione ed ostacolo – l’errore nella

comunicazione delle matematiche, intersezione tra problemi dell’apprendimento, insegnamento e la

neurofisiologia”, proc. Quaderni di ricerca in didattica, GRIM, n°4, Palermo.

- Spagnolo F., 1999 , “Insegnare le matematiche nella scuola secondaria”, c.ed La Nuova Italia.

- Sperber D. e Wilson D., 1986, Relevance: Communication and Cognition, Blackwell, Oxford,

Harvard University Press, Cambridge, Mass.; trad. it. La pertinenza, Anabasi, Milano 1993.

- Stanislas Dehaene, 1999, “Il pallino della matematica”, c.ed. Scienze, Oscar saggi Mondatori.

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