Zio tobia

Arithmetical environments semantically anchored

Father Woodland in Italy becomes Zio Tobia (Uncle

Toby)Carlo Marchini (Mathematics

Department of Parma University)

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PremisesPremises

• The starting idea comes from the reading of the The starting idea comes from the reading of the paper:paper:

• Hejný M., Jirotková D., Kratochvilová J. (2006)Hejný M., Jirotková D., Kratochvilová J. (2006)Early conceptual thinking. Early conceptual thinking. Proceedings. 30th PME Proceedings. 30th PME

(Vol. 3, pp. 289-296).(Vol. 3, pp. 289-296).

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PremisesPremises

• The paper The paper abstract states: abstract states: A pupil’s mathematical development is aimed at a procedural rather than a conceptual style of thinking. Both types are characterised and we illustrate the consequences which neglecting conceptual thinking can bring. We describe a fairy tale context, which enables us to investigate conceptual thinking, its diagnosis and development of pupils of Grade one. Action and clinical research was carried out and some mental phenomena describing the thinking processes of pupils in the given context were found.

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PremisesPremises

• Father Woodland (FW) is a fairy tale figure who Father Woodland (FW) is a fairy tale figure who looks after different animals and organises tug-looks after different animals and organises tug-of-war games. of-war games.

• The weakest animal is a mouse (The weakest animal is a mouse (MM). ). Two mice are as strong as one cat (Two mice are as strong as one cat (CC). ). A cat and a mouse are as strong as a goose (A cat and a mouse are as strong as a goose (GG). ). A goose and a mouse are as strong as a dog A goose and a mouse are as strong as a dog ((DD). ). Other animals are introduced in a similar way. Other animals are introduced in a similar way. Each animal is Each animal is represented by a picture, an icon represented by a picture, an icon and a letter.and a letter.

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PremisesPremises

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PremisesPremises

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PremisesPremises

wordword picturepicture iconicon letterletter definitiondefinition

mousemouse MM

catcat CC =goosegoose GG =dogdog DD =

Pictures by D. Raunerova

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FW mathematical environmentFW mathematical environment • From: Hejný, M., Jirotková, D., Kratochvílová, J. (2006). From: Hejný, M., Jirotková, D., Kratochvílová, J. (2006). Early Early

conceptual thinking, conceptual thinking, Proc. 30th PME Proc. 30th PME (Vol. 3, pp. 289-296). (Vol. 3, pp. 289-296). PraguePrague

• This context is suitable for developing the following This context is suitable for developing the following mathematical concepts and competenciesmathematical concepts and competencies::– early number senseearly number sense– understanding of the difference between a quantity (expressed understanding of the difference between a quantity (expressed

in units) and a number (expressed in pieces)in units) and a number (expressed in pieces)– pre-concept of equations pre-concept of equations – pre-concept of divisibility, the lowest common multiple and pre-concept of divisibility, the lowest common multiple and

greatest common divisorgreatest common divisor– conceptual thinking in pupils not only at the elementary levelconceptual thinking in pupils not only at the elementary level– solving methods of linear equationssolving methods of linear equations– solving of Diophantine equationssolving of Diophantine equations

• It is also a diagnostic tool enabling to characterise both It is also a diagnostic tool enabling to characterise both cognitive and meta-cognitive styles of pupils.cognitive and meta-cognitive styles of pupils.

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FW mathematical environmentFW mathematical environment • An important feature of this environment is that it An important feature of this environment is that it

presents itself simply, and pupils accept without presents itself simply, and pupils accept without problems, the relational thinking connected with equalityproblems, the relational thinking connected with equality..

• Literature documents that an understanding of equality as a relation is crucial to the devel-opment of algebraic thinking (Alexandrou-Leonidou & Philippou, 2007; Attorps & Tossavainen, 2007; Puig, Ainley, Arcavi & Bagni, 2007).

• Here we focus on formal number sentences, building on the work of Molina, Castro & Mason (2007) and, in particular, on relational thinking – a term that Molina et al. (2007) borrow from Car-penter, Franke & Levi (2003

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FW mathematical environmentFW mathematical environment • The student employs relational thinking if s/he• “makes use of relations between the elements in

the sentence and relations which consti-tute the structure of arithmetic. Students who solved number sentences by using relational thinking (RT) employ their number sense and what Slavit (1999) called “operation sense” to consider arithmetic expressions from a structural perspective rather than simply a pro-cedural one. When using relational thinking, sentences are considered as wholes instead of as processes to carry out step by step.” (Molina et al., 2007, p. 925)

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Some questions ISome questions I

Consider the FW environment; in your Consider the FW environment; in your opinion:opinion:

• Q1) Which other mathematical topics Q1) Which other mathematical topics could it inherently embed?could it inherently embed?

• Q2) Which transversal cognitive Q2) Which transversal cognitive competences could it help to develop? competences could it help to develop?

• Q3) Which school grades would it be Q3) Which school grades would it be suitable for?suitable for?

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Some questions IISome questions II

• Q4) Is the ‘Father Woodland and friends’ Q4) Is the ‘Father Woodland and friends’ tale present in your childhood folklore?tale present in your childhood folklore?

• Q5) In case of negative answer, think Q5) In case of negative answer, think about a substitute environment in your about a substitute environment in your folklore which allows the same folklore which allows the same mathematical development.mathematical development.

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Some personal answersSome personal answers

• From the structural point of view, this From the structural point of view, this environment is a sort of formal algebraic environment is a sort of formal algebraic system with equality: system with equality:

AlphabetAlphabet →→ icons or letters. icons or letters.

Term formationTerm formation →→ juxtaposition, juxtaposition,

Atomic formulaeAtomic formulae →→ equality of two terms equality of two terms

AxiomsAxioms →→ definitions definitions

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• Furthermore the presence of pictures and the Furthermore the presence of pictures and the tale itself give the opportunity to encapsulate tale itself give the opportunity to encapsulate also the semantics of this formal system. also the semantics of this formal system.

• The formal system axioms are not completely The formal system axioms are not completely exhibited. The semantic suggests a hidden exhibited. The semantic suggests a hidden presence of addition (given by juxtaposition) presence of addition (given by juxtaposition) which also allows (thanks to true formulae in the which also allows (thanks to true formulae in the intended semantics) to obtain a richer algebraic intended semantics) to obtain a richer algebraic structurestructure

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• This setting, with explicit (sintax) and implicit This setting, with explicit (sintax) and implicit (semantics) rules is a sort of ‘(semantics) rules is a sort of ‘Eudoxian Eudoxian semigroupsemigroup’ for ‘magnitudes’: ’ for ‘magnitudes’: – Addition: associativeAddition: associative and and commutativecommutative – IntegerInteger multiplesmultiples of magnitudes, of magnitudes,

– Ordering Ordering relationrelation– ‘‘CompatibilityCompatibility’ of ordering relation with addition.’ of ordering relation with addition.

• These aspects would be suitable proposed These aspects would be suitable proposed as example of formal system for secondary as example of formal system for secondary school, but they are intuitively practised by school, but they are intuitively practised by children.children.

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The Italian versionThe Italian version•Starting from HejnStarting from Hejný ý et alet al. (2006), . (2006), during the school during the school year 2007/2008 a team of teachers of Viadana year 2007/2008 a team of teachers of Viadana Primary School, Rossella Guastalla, Maura Previdi, Primary School, Rossella Guastalla, Maura Previdi, Roberta Santelli, under my supervision, prepared a Roberta Santelli, under my supervision, prepared a sort of guide for presenting and exploiting the Italian sort of guide for presenting and exploiting the Italian version of FW environment. version of FW environment.

•In the school year 2008/2009 Guastalla followed In the school year 2008/2009 Guastalla followed and improved that guide applying it in and improved that guide applying it in two grade 1 two grade 1 classes.classes.

•In the school year 2010/2011 the Anna Frank In the school year 2010/2011 the Anna Frank School of Parma experimented the same activity School of Parma experimented the same activity with teachers Losi and Pompignoli.with teachers Losi and Pompignoli.

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The Italian versionThe Italian version

• The fairy tale of FW is completely unknown to The fairy tale of FW is completely unknown to Italian pupils, therefore we had to find a Italian pupils, therefore we had to find a suitable setting for our children.suitable setting for our children.

• Father Woodland becomes Zio Tobia, the Father Woodland becomes Zio Tobia, the character of a children song settled in a farm character of a children song settled in a farm (a translation of the ‘Old Mac Donald Farm’ (a translation of the ‘Old Mac Donald Farm’ song)song)

• FW’s tug-of-war becomesFW’s tug-of-war becomes the animals feedthe animals feed: : for example two mice eat as much as one for example two mice eat as much as one cat.cat.

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The old farm of Zio TobiaThe old farm of Zio Tobia

• The activity has been splitted in two temporal The activity has been splitted in two temporal phases. phases.

• First school semester: establishement of the First school semester: establishement of the semanticssemantics– visit to a neighbouring farmvisit to a neighbouring farm– musical education with ‘The old farm’ song (rhythms, musical education with ‘The old farm’ song (rhythms,

clapping hands)clapping hands)– Linguistic education animals names and calls Linguistic education animals names and calls – artistic education.artistic education.

• Explicit mathematical aspects were not involved Explicit mathematical aspects were not involved in this first stage.in this first stage.

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The old farm of Zio TobiaThe old farm of Zio Tobia

• Second school semester: mathematical Second school semester: mathematical exploitation of the environment.exploitation of the environment.

• Use of the teachers’ guide for Use of the teachers’ guide for – Arithmetic learningArithmetic learning– Ordering relationOrdering relation– Use of symbolismUse of symbolism

– Early concept of equationsEarly concept of equations– Attention to the language developmentAttention to the language development

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wordword picturepicture iconicon letterletter

topo (plural: topi) TT

gatto (plural: gatti) GG

oca (plural: oche) OO

cane (plural: cani) CC


• The examples of protocols I will show come from The examples of protocols I will show come from our school activity. A translation of the previous our school activity. A translation of the previous table could be useful to understand them better:table could be useful to understand them better:

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The Italian versionThe Italian version•The Italian version makes easier The Italian version makes easier the contemporaneous presence of the contemporaneous presence of the meanings of number as quantity the meanings of number as quantity and number as magnitudeand number as magnitude

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Examples from Italian guideExamples from Italian guide•The first three sheets are devoted The first three sheets are devoted to a familiarization with the four to a familiarization with the four animals, their drawings and icons. animals, their drawings and icons. •Use of masks, dramatizing, posters Use of masks, dramatizing, posters is required.is required.•The fourth sheet presents the The fourth sheet presents the definitions of animals in the Italian definitions of animals in the Italian version.version.

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With animals picturesWith animals pictures

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With iconsWith icons

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With numbersWith numbers

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With Cuisenaire-Gattegno’s rodsWith Cuisenaire-Gattegno’s rods

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Preferred representationsPreferred representations

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Some questions IVSome questions IV

• Q9) Focus on the two type of writings:Q9) Focus on the two type of writings:– C = 2GC = 2G

– I thought a lot about how much they can eat. A I thought a lot about how much they can eat. A mouse (1), a cat (2) a goose (3 because you mouse (1), a cat (2) a goose (3 because you must line them all) and a dog (4).”must line them all) and a dog (4).”

Are there differences in the use of Are there differences in the use of numbers?numbers?

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A theoretical difficultyA theoretical difficulty

• The paper of HejnThe paper of Hejný ý et al.et al. (2006) presents the (2006) presents the drawingdrawing

and the writings and the writings

{CCG} ~ {DM}, {CCG} ~ {DM},

{CXX} = {DD}.{CXX} = {DD}.

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Some questions VSome questions V

• Q10) Are these types of drawing and Q10) Are these types of drawing and writing commonly used in your schools for writing commonly used in your schools for the representation of other mathematical the representation of other mathematical concepts? Which ones?concepts? Which ones?

• Q11) How would you avoid the possible Q11) How would you avoid the possible conflicts due to these representations?conflicts due to these representations?

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The corralThe corral

• 1A1A Matilde and Matilde and 1B1B Nicolò: “ Nicolò: “Instead of a corral Instead of a corral we could just draw a circle”we could just draw a circle”

• 1A1A Filippo and Filippo and 1B1B Davide: Davide: “Instead of the corral “Instead of the corral gate we could just draw some lines”gate we could just draw some lines”

• 1A1A Ahmed proposed for gates a symbol very Ahmed proposed for gates a symbol very similar to the ‘sharp’ button, #, on the mobile similar to the ‘sharp’ button, #, on the mobile keyboard.keyboard.

• Tiredness suggested the use of letters or Tiredness suggested the use of letters or numbers instead of pictures or icons.numbers instead of pictures or icons.

• Finally, the accepted representation was like Finally, the accepted representation was like #C C G#.#C C G#.

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Zio Tobia’s activity resultsZio Tobia’s activity resultsAssessment X is a (rational) number in the interval [0,10]

which results from 10 individual tests. The sample of 39 pupils presents the following distribution

Zio Tobia results

13%21%

15%

51%

0%

10%

20%

30%

40%

50%

60%

X <4 4< X <6 6< X < 8 8 < X

Assessment

rela

tive

fre

qu

ency

Zio tobia

Education

Transcript of Zio tobia