INNOVATING DIDACTICS OF MATHEMATICS COURSES

M. M. Marjanovic

• The text of this lecture has been adapted for a thirty minute

talk and also for the video-beam projecting – the main theses

are pointed out with some comments to follow them. This, of

course, causes some discontinuities that exist in the text.

• First we argue for the teacher training faculties as being the

right places to start with innovations and how their courses of

subject didactics have to be organized if a profound knowledge

of the teaching matter is a goal. We use the term “integrated

course of didactics” to mean the one which combines the

subject matter with some relevant topics from history of a

discipline, educational heritage and psychology.

• We discuss in detail a couple of topics from psychology,

leaving everything else just sketched.

The right places where to start with innovations in education are the teacher training faculties and

theirdidactics courses.

The reasons supporting this opinion are:

• Some members of the faculty staff are normally considered to be experts in education at the high national level,

• When an innovative project is implemented, these experts can professionally contribute to distinguishing between its good and possibly bad points,

• being the places of dissemination of knowledge, these faculties and their courses are the basic structures for the transfer of new ideas to both, pre-service and in-service teachers.

Capability of a teacher consists of his/her profound knowledge of the teaching matter and of his/her art

of acting on the scene of the classroom.

• Here we concentrate on the former aspect of teacher’s skills, confining our considerations to the case of teaching mathematics. The latter aspect is, however, equally important and it includes issues as organization of the classroom, collaborative work, encouraging productive learning, holding discussion and so on and so forth.

• Let us note that when we speak of teaching matter, we take this term to mean that not only the subject matter is under consideration, but also the whole description of the ways how a piece of knowledge has been synthesized and didactically transformed.

When the goal is a profound knowledge of the teaching matter, then a course of didactics must

combine subject matter with some related facts from history of mathematics, educational heritage and

particularly psychology.

• Selection of topics from psychology is quite intriguing a matter

and is dependent on each particular subject. The arguments for

their relevance have to be given relative to the specificity of the

subject matter. In the case of mathematics at primary and middle

levels, we suggest the following list of topics:

• Function of the eye; Principles of perception; Comprehension

of perception; Cues for seeing depth; Concepts as tripartite

entities – corresponding class of examples, mental image and

name (or labeling symbol); Hierarchy of concepts according to

the degree of abstractness;

• Definitions as sentences which determine a

concept via another one of higher degree

plus differentia specifica; Bruner’s modes of

enactive, iconic and symbolic representation;

Pictograms and ideograms; Ontogenetic

development of speech – egocentric speech of

the child and inner speech (Vigotsky);

Structures, cognitive schemes and systems of

concepts; Spontaneous and scientific

concepts.

• Each of these topics deserves to be

included in a course of didactics

independently of it if a general course of

psychology already treats them or not.

L.S. Vigotsky

J. Bruner

For instance, perception is a topic which a psychology course inevitably includes, but in the course of didactics basic facts should be used to clarify some issues essential for the teaching matter:

• Retina image (the part of the world toward which the eye is turned projects on retina background in the form of a faithful image),

• Focusing attention (through that world roams the glance, focusing the narrow range of sharpest vision now on this and then on that object),

• Interpretation of the seen (perception is not mere seeing, but also consists in fitting stimulus material with the inner representations existing in the observer’s mind).

• When the mechanism of perception has been acquired, the attention should be directed to the mathematical content. Understanding of the function of geometric drawings is particularly important and we take an example to illustrate this point.

• What does the following drawing  represent?

• In this case, three different interpretations of the same sensory

input are possible

(a) Majority of observers are apt to see a cube. Some of them may

recognize the drawing as an iconic sign of the cube that already

exists in their experience, the others interpret the seen by

activating corresponding mental image that exists in their mind.

(b) If the context requires seeing a plane figure, then the drawing

is uniquely interpreted as a hexagon with three radii of its

circumscribed circle.

(c) Let us notice that when the drawing is perceived as a cube,

then the point in the middle is seen as the nearest vertex of the

cube. Employing some effort that point can be seen as being

furthest backward. Then the whole percept changes dramatically

and we see a space angle having three rectangular faces.

• 

• Another topic which is extremely interesting for mathematics

teachers is the seeing in three dimensions. Since the retina

resembles a surface upon which a two-dimensional image is

formed, a quite intriguing question is how do we see depth? There

is a number of cues which enable us to do it and they are of the

two kinds: monocular ones that allow a single eye to see some

depth and binocular ones that arise in looking at the same object

with both eyes at once.

• Of special interest for a mathematics teacher are the following

two monocular cues:

- Overlap, when one object blocks part of the view of another

object, making the latter looks further backward,

- Linear perspective, when the further away an object is, the

smaller will be its image on retina (i.e. the smaller it will look).

• Thought the geometric drawings are ideogram (iconic signs

representing concepts), the overlap serves as a cue allowing the

drawings of some solids look more convincingly as being situated in

the space.

(a) (b) (c)

(a) No overlap exists. Either the face 1 or 2 can be seen as being

closest to the observer.

(b) Overlap makes the face 1 look as being closest to the observer.

(c) Overlap makes the face 2 look as being the closest to the

observer.

1

2

1

2

• Linear perspective as a cue for three-dimensional vision was thoroughly studied by the Renaissance artists and used as a new mighty technique of painting. Theory of perspective became a subject taught in painting schools together with the other liberal arts. Gaining a solid mathematical basis, that theory was converted into projective geometry. This is an interesting and instructive example of connection between perception and conception, worth being included in the courses of didactics of mathematics.

Annunciation - Leonardo da Vinci

Extremely important for the artists and of some significance for the teachers of mathematics when they are drawing pictograms (iconic signs representing real objects) are monocular cues:

• Haziness, when distant objects appear hazy,

• Shadows, when two objects seen under different angles, one of them has a smaller projection on retina and looks somewhat shadowy, but here we leave out the technical details.

Of binocular cues we only include:

• Retinal disparity, meaning that the image of an object on retina is slightly different for the two eyes.

Leonardo da Vinci was the first person to diagram this effect:

What does not see one eye is seen by the other, an effect which makes

an object seem like being hovering in the space.

The core content of a didactics course should be didactical analysis of the main teaching themes. That analysis should be proceeded with didactical synthesis showing how the content of these themes is didactically transformed.

The idea of didactical analysis is due to H. Freudenthal who has clearly

exemplified it in his well known books on education. We point out the

main components of this important didactical method:

• Presentation of the subject matter in statu nascendi (in the state of creation), • Evolution of concepts and their systems trough historic epochs,• Examination of the subject matter from a higher point of view, when the content is presented in a structured form and logically very well organized way.

The main components of didactical synthesis are:

• Determination of levels of abstraction for

each stage of education,

• Means of representing concepts (iconic, symbolic) and the ways their meaning is established,

• Gradual formation of concept systems (structuring instead of the use of ready-made structures).

• Detailed shaping of the concrete units of the subject matter as the best method of verification of the involved ideas and arguments.

H. Freudenthal

• Selection of the main teaching themes depends certainly on the personal view point but, according to our opinion, they should be:

Sets and mappings; Number systems; Variable – the key idea of algebra; Logic; Geometry – inherent, visual, stage of local deductions; Euclidean geometry.

• Elaboration of these themes should, of course, be adapted for teachers at all tree levels – primary, middle and secondary.

At the end, I wish to express my opinion that without a profound understanding of the teaching matter there is no art of transferring knowledge.