using geometrical models in a process of reflective thinking in learning and teaching mathematics

A. GAGATSIS AND T. PATRONIS

U S I N G G E O M E T R I C A L M O D E L S IN A P R O C E S S OF

R E F L E C T I V E T H I N K I N G IN L E A R N I N G A N D T E A C H I N G

M A T H E M A T I C S

ABSTRACT. In this paper we investigate how geometrical models can be used in learning and teaching mathematics, in connection with the development of a process of reflective thinking, which we study first in general. Some more specific questions - arising from the use of geometrical models in the classroom - have led us to an experimental study, the results of which are presented and discussed in the paper.

I N T R O D U C T I O N

The models used so far in mathematics education can be roughly divided into two categories; both of them have the same final aim (improvement of learning and teaching mathematics), but there is a sharp difference, both in nature and use, between the two of them. The models of the first category are generally characterized as models of the learner's mind or of the learning process. These models are almost exclusively used by researchers and educators; they cannot be used in the classroom, or by the learners themselves, since usually the learners are not aware of their own mental operations and procedures, at least in the way that these are described in the models. On the contrary, the models of the second category reflect the intuitive processes involved in the subject-matter which is to be taught, and thus they could be suitable for use by the learners at a certain stage of the development of their thinking. Geometrical representations and concrete models of this kind have indeed been introduced in classes by inspired teachers and mathematics educators; yet they have not been extensively studied from a theoretical point of view.

Skovsmose put forward three alternative views on mathematics and mathematics education (Skovsmose, 1985): structuralism, pragmatism and process orientation. Adopting, here, the process-oriented approach, we try to investigate how geometrical models can be used by students and teachers in a process of reflective thinking; this could help understanding and consciousness in learning and teaching, in the sense of Kilpatrick (1985). But we need first to study carefully the development of reflective thinking in the mathematical activity of students, of teachers and of mathematicians in their own research. Intuition plays an important role in this development. This role is crucial, in particular, for young subjects: for them intuitive thinking necessarily precedes reflective thinking and can help its

Educational Studies in Mathematics 21: 29-54, 1990. �9 1990 Kluwer Academic Publishers. Printed in the Netherlands.

30 A. GAGATSIS AND T. PATRONIS

evolution. How can this be done? As we shall see, geometrical models may be of much help in this direction, especially when they have been used first spontaneously by the child, as (implicit) models of action.

These, and other related questions, are discussed in Sections 1, 2 and 3 of our paper. In subsequent sections we report an experimental study of a specific family of geometrical models, namely the models of polygonal shapes. These models have been extensively used in the teaching of geometry, but there are some psychological questions arising from their use. In an attempt to answer these questions and to investigate these models, as models of action for the young children, we organised an experiment, the results of which are presented and explained from the point of view outlined in Sections 1 and 2.

1. A PROCESS OF REFLECTIVE THINKING IN MATHEMATICAL ACTIVITY

Skemp (1971) distinguishes between two levels of functioning of intelligence: the intuitive and the reflective. At the intuitive level, he says, we are aware, through our senses, of data from the external environment, which are "automatically" classified and related to other data, but we are not aware of the mental processes involved in this activity. At the reflective level, the intervening mental activities become the object of introspective awareness. Skemp makes clear that this distinction is a matter of difference in the task (or goal) of the subject:

Multiply 16 by 25. (i) What is the answer? (ii) Now explain how you did it? To answer the second question involves turning your attention from the task itself to your mental processes involved in doing i t . . . Being able to do something is one thing; knowing how one does it is quite another.

Piaget (1971) also pointed out that the operational structures of intelligence are not conscious to children, although they direct their activity. The learning of mathematics requires the student to reflect consciously on his (or her) own mental structures and procedures. But the question arises of how, actually, this can be one. The answer is not easy, and the history of mathematics education during the last three decades shows that there is no direct way to attain this goal. The "structuralist" trend of the innovators in mathematics curricula was followed by complete failure and disappoint- ment. It seems that only by observing, noticing things, asking questions, one has good opportunities for reflection; in fact this is the beginning of a process of reflective thinking in mathematics. How could we help the development of such a process in students' minds? Clearly we have first to

U S I N G G E O M E T R I C A L M O D E L S 31

explore this process and understand it as a whole. Therefore we are led to study the process of reflective thinking in mathematicians' activity; this means that we will study the various ways and stages in which mathematicians become aware of their own mental procedures.

Speaking about a "process of reflective thinking" we can refer to any domain of experience and any kind of procedure, either mental or mechan- ical or natural, like problem solving, running, computing, approaching a limit by means of some algorithm, reading a text, teaching, making research, writing a poem or composing a symphony; but reflective thinking cannot be indentified with anyone of these procedures. What we mean is a process through which the subject observes and conceives such a procedure as above and tries to understand it and/or explain it to others. Thus the stages of a process of reflective thinking should not simply indicate the progress of the solution of a problem, but rather the degree of awareness of the subject about the whole theme. For example, the process of invention in mathematics and other fields remains unconscious in it greatest part; yet it has been the object of introspective awareness by eminent mathematicians, as Henri Poincar6 and Jacques Hadamard (1949).

On the other hand, Hadamard, as also Bouligand (1962) have pointed out that an intuitive approach to a mathematical problem - even at a high degree of abstraction or generality - is analogous with the act of recognizing a person by its general characteristics (physiognomy or Gestalt). In the words of Bouligand:

� 9 An adaptation therefore takes place. As the geometer becomes progressively more familiar with the abstract objects that he studies, he comes to have a concrete idea of them almost as real as his memories of real objects.

This means that, in many cases, mathematicians create "mental images", which are present in their work and help them, and which may become the object of conscious reflection.

In a recent study (Kaldrimidou, 1987) among mathematicians, the use of "mental imagery" as above was systematically explored. It was found that indeed, mathematicians, in their research activity, develop some representations of an individual character (different from the standard mathematical representations), which in psychology are called "mental images", The role of these images is to lead the subjects in their action as internal models formed by the interaction of the subject with the object of research. There are individual differences in using these images: some mathematicians use them explicitly and consciously, but some do not. In general it is very hard to communicate such images; some mathematicians try to communicate

32 A. G A G A T S I S A N D T. P A T R O N I S

them to their university students, in order that students can take advantage of them in understanding the mathematical concepts and processes. An- other important finding was that these mental images are elaborated, during the activity of the subject, from a fragmentary and unclear form to a global and clear one. They take part in a process of reorganisation of knowledge and reconstruction of one's conceptions of a problem.

Here are three quotations from subjects' responses in Kaldrimidou's research:

(i) . . . Because in the beginning, if one does not know well some notion, one "sees" it in a fragmentary way.

(ii) I am speaking of images that serve as a counterexample, as situations in which one knows that something is going wrong.

(iii) . . . there is an unexpected renovation of i m a g e s . . . I think that this renovation is related to renovation in the scientific sense of the term.

What is most interesting here, from our point of view, is that these three responses describe different levels of awareness of a subject on a mathematical problem and on the concepts involved. In response (i) we have a "beginning" stage, where the subject's knowledge is very limited and "fragmentary"; being in such a situation, one usually is not aware of what in fact one is searching for, although one may be guided by primary conceptions and intuitions. In response (ii) the situation has changed: counterexamples are found, which prove that some of the subject's intuitions are not compatible with real facts; at this level one becomes aware of the existence of some error, and one is forced to re-examine the situation (at least a part of it). In this way one is led to a (partial) reorganization of one's knowledge - which now becomes more meaningful and less "fragmentary" - and to a (partial) reconstruction of one's conceptions similar to "renovation in the scientific sense" (response (iii)). At this level one has (partly) understood the nature of the problem and one knows (or feels intuitively) that one is on the right way; may be it will need a long chain of calculations, or of small deductive steps, but this is just a matter of time; the subject can "see" globally the solution through elaborated mental images; the plan is already complete!

The work of a teacher of mathematics is offering opportunities for a similar development of awareness. There is, in the literature, a characteris- tic example: Polya I describes how he started thinking about the process of solving problems. Polya's discovery that the plan or the process of the solution may be represented by a graph (joining the data to the unknown) belongs to a still higher level of awareness. It is a product mainly of

USING GEOMETRICAL MODELS 33

reflective thinking, i.e. thinking about one's own mental procedures and structures. At this level one not only knows that there is a plan which is going to be successful, but one also knows why this plan is going to be (or was) successful. However, it is obvious that this level has been preceded by a stage of questioning and of hard work on the whole subject again.

Summarizing, and taking into consideration the existing analogies in the work of students, teachers, mathematicians and educators, we can describe the main stages of a process of reflective thinking in mathematical activity as follows:

Stage O: Initial thoughts, primary intuitions and conceptions on a subject- matter or a problem; starting the work with "local" success or failure; unclear and "fragmentary" mental images; making observations "at random". Stage 1: Reflecting on the subject and trying to understand, i.e. to organize the new experience into previously existing intuitive structures; classifying observations, analysing wholes into parts, reflecting on them, recalling other similar examples, finding counterexamples, questioning former beliefs and conceptions. Stage 2: Discovery and (partial) understanding; finding and/or justifying a rule; finding an explanation for some error; recombining parts of a decom- posed whole into new wholes; interpretation and (partial) reorganization of the new facts according to previous structures; "completion" of mental images and deriving a plan of the solution or of the proof; intuitive feeling of certainty for the success of the plan. Stage 3: Introspection; trying to see "what is all about" i.e. reflecting on the process of the solution, the logic of the proof and one's own mental structures and processes; chequing or testing one's own results (or conclu- sions) into other problems or fields; examining analogies and setting up new questions; analysing the whole situation again, but at a higher level; questioning again; questioning "questioning"; epistemological dispute. Stage 4: Full awareness; understanding the underlying logic; illumination of the whole subject; becoming aware of one's own mental structures and processes; widening the old structures, transforming or fully rejecting them (and constructing new ones); radical reorganization of ideas, possibly on new foundations; making sound generalizations and extensions; constructing and formulating new theories.

It is necessary to emphasize the recursive character of these stages: each one strongly depends on - and in fact uses - all stages before it.


2. G E O M E T R I C A L MODELS AND THEIR USE

Several authors have discussed the meaning and role of models in mathematics education. We do not attempt to give here a survey of these different (and sometimes conflicting) views. We shall confine ourselves to the views of Fischbein and Brousseau, which we will discuss in brief, with respect to our considerations of Section 1.

Fischbein (1972) considers the "generative" function of models used in teaching as the more important one for intellectual development. He borrows the term "generative" from linguistics, where it was used by Chomsky to describe the grammatical competence of human language. In analogy with this grammatical competence, which permits a child to construct an unlimited number of phrases from a limited number of elements and of rules of combination, Fischbein says a model is useful for productive thinking if it is "generative" in a same sense: using a limited number of elements and of rules of combinations, we must be able to represent, within the model, an unlimited number of situations. Moreover, a model must have a heuristic value, for pedagogical as well as for scientific uses; i.e. it must lead us easily, and independently of the original system it represents, to new information about this system. Fischbein gives as examples of such "generative" models the tree-like diagrams as used in combinatorics and the Euler-Venn diagrams.

The approach of Brousseau (1983, 1985) is different. Construction (and use) of models is considered in the context of three distinct processes:

i) the "dialectics of action", concerning the decisions of the subject for action and a system of elaboration of these decisions, which for a long period remains implicit or unconscious;

ii) the "dialectics of formulation", concerning the explicit description or communication of the subject's conceptions and of the empirical results of one or another decision; and

iii) the "dialectics of validation", by which the subject is led to establish, to prove his (her) assertions before another subject, who tends to accept or to refuse these assertions. In each of these processes the subject constructs and uses, implicitly or exp!icitly, a corresponding kind of models; thus we have models of action (which are, at first, implicit in child's cognition), models of formulation and models of validation; the latter two kinds are explicitly used.

The above distinction of Brousseau is important, because it cannot be neglected without some unhappy consequences. Example: when some teaching activity fails, the teacher (or educator) may try to find some


explanation. In doing so, she may be led to construct some representation, corresponding to an advanced stage of the R-T process (Stage 3 or 4), as a model of formulation or a model of validation; but the same model was implicitly used by her at a lower stage as a model of action. The teacher thinks this invention will be useful to pupils, but it is not so, since this model is completely new for them. It has not been previously a model of action. Thus there is a serious gap here in the development of the R-T process, which is a recursive process; in order to advance, one needs all previous experience, which in our case does not exist for the pupils. The Euler-Venn diagrams or the graph representations, for example, may indeed be "generative" models with heuristic value, as Fischbein says, but only for one who is in a position to make an intelligent use of them.

From the preceding discussion we conclude that, in order for a model to be suitable for didactical purposes, it has to be at first implicit in the subject's cognition or, in other words, a model of action.

Moving now to the concept of a geometrical model, it is well known that thought in science is greatly helped by the use of geometrical pictures and other representations (graphs, diagrams etc.), which have, more or less, a metaphorical sense. They represent some object, or system of objects, or some situation or process, which we want to study, and whose properties are interpreted (or represented) in such a way that the resulting situation is easier to investigate. 2'3 In some of these representations the intrinsic geometric properties of points, lines etc. are all relevant, but in others (as in the example (e) below) this does not hold.

We shall say that a collection ~ of points, lines or other figures in n-dimensional Euclidean space, representing a system I of objects or a situation or process, is a (theoretical) geometrical model of l , if the intrinsic geometric properties of the elements of ~ are all relevant in this representation, i.e. they correspond to properties of the system I . If this condition is satisfied only for the topological properties of lines or figures in Sr then we shall speak of a geometrical model in the wide (or topological) sense.

This intuitive definition will suffice for the purpose of this article. We also note that this definition agrees, in some extent, with the abstract notion of a model as defined in metamathematics, as well as with the requirement that a model should have a heuristic value; because the study of the intrinsic properties of a geometrical model in the above sense leads, independently of the original system I , to new information about 27 (at least in principle).

Some examples: (a) The ancient Greeks - starting from the Pythagoreans - have used geometrical representations in establishing arithmetical proper-


ties, as we learn from Euclid ("Elements", Book 7, 8, 9), from Nicomachus of Gerasa and from Theon of Smirni. 4 This method was further developed by the Arabs; for instance, a geometrical proof of the identity

13 + 2 3 + . . . + l/3 = [�89 + 1)] 2

extracted from the books of algebra "AI-Fachri" is mentioned by Toeplitz (1963, pp. 52-53).

(b) The following geometrical model can be used for solving the para- metric system of equations (in the unknowns x, y)

(1) x + y = s

(2) x . y = p

in the domain of real numbers: Suppose for instance that p > 0. Then for each value of s and p > 0 we have, on the cartesian plane, a straight line representing (1) and a hyperbola (with two branches) representing (2). By examining the possibilities that the straight line and the hyperbola have an intersection at two points, or no intersection at all, or that the straight line is tangent to the hyperbola, one easily obtains the necessary and sufficient condition that the above system has a solution in the reals, i.e. the condition s z > 4p. Thus the model consists here of two families of curves: one of straight lines and one of hyperbolas.

(c) Let us transform the shape of an isosceles triangle, by retaining its base and its axis of symmetry fixed; we get an infinity of isosceles triangles (Fig. 1), whose angles vary continuously with the altitude. Let us suppose, moreover, that we have been able to conjecture, after some kind of

I

A K

Fig. 1.


experimentation, measurement, or by careful observation, that the sum of the angles of each of these triangles remains constant (which means that, what is "gained" by augmentation of the angles at the base is "lost" by diminution of the angle at the upper vertex, as the altitude increases). Then we can guess the value of this constant sum by following a kind of "intuitive reasoning": As the altitude tends to infinity, each one of the angles at the base tends to a fight angle, while the upper vertex angle tends to zero: so the whole sum tends to two fight angles, which must be the constant value in question. (A similar argument holds when the altitude approaches zero.) Thus a non self-evident mathematical theorem can be discovered (and one may be convinced, as well, about its truth) by means of an intuitive argument as above. This kind of intuitive thinking, which involves a continuous variation process, probably was first used systematically in mathematics teaching by Castelnuovo, who called it "dynamic intuition", in contrast with the "static" situation of stable, non-varying figures and representations (see Section 4). If we ignore the magnitude of the sides of the changing isosceles triangle of Fig. 1 and we focus only on its shape which is continuously varying, then we obtain the following representation (Fig. 2) of a continuous variation of shape; this is a path in a topological space, that was investigated mathematically by Robertson (1984). We examine this model in Section 6.

(d) The Euler-Venn diagrams can be considered as geometrical models in the wide sense, since they use the fundamental property of a simple closed curve to divide the plane into two regions, but nobody cares about e.g. the geometrical shape of these closed curves.

(e) An example of a representation which is not a geometrical model: By using points and arrows to represent mappings or transformations of sets, we get only a figurative representation, not a geometrical model (even in the wide sense); because if arrows are interpreted e.g. as directed segments (or arcs) in the plane, then the possibility that two of these arcs intersect

line flat segment isosceles

equilateral sharp line isosceles segment

Fig. 2.


does not imply anything for the mappings or transformations in question. Of course a graph in two or more dimensions, representing some relation or process, is a geometrical model in the wide (topological) sense.

3. THE USE OF G E O M E T R I C A L M O D E L S IN C O N N E C T I O N WITH

THE R-T PROCESS

In the mathematical or educational literature, geometrical models usually appear either as models of formulation or as models of validation (following the terminology of Brousseau).

As a model of formulation, a geometrical model is characterized by the language used with it. For the mathematician, this language is intimately connected with intuition, since it suggests new facts, which are "'trans- ferred" from the model to the system or situation it represents; thus geometrical language plays here a guiding role, it is a heuristic tool, s as e.g. it is the case with examples (a)-(c) of the preceding Section.

As a model of validation, a geometrical model may also play a decisive role. In fact the example (a) of Section 2 has been used not only as a model of formulation, but perhaps mainly as a model of validation for arithmetical facts that were inductively discovered.

Now, what can be said about geometrical models as models of action? Mathematical papers and textbooks give us very little information on this subject, because mathematicians and textbook writers usually make an effort to delete from knowledge all subjective or affective characters, all "useless" reflections, intuitions or "errors" committed (unless they want to warn the reader to be careful!). Yet some of the mental pictures of mathematicians seem to be implicit geometrical models which guide their action, as it results from our discussion in Section 1.

An example from classroom experience, which can be analyzed in terms of the R-T process, is furnished by the results of a remarkable teaching experiment on the topic of "the handshake problem": find the number of handshakes between n people, if everyone of them shakes hands with everyone else once only. Billington and Evans worked on this problem with children whose ages varied f rom 6 to 15 (Billington and Evans, 1987). The authors observed, in the process of solving this problem, several levels of knowing, which are very interesting and are related to the R-T process. The authors were convinced that some general strategies (or abilities) are natural to children and should be developed, not taught. The following abilities were observed in levels:

(i) process information and "own" the problem (ii) make predictions and test them


(iii) symbolise (iv) tabulate (v) illustrate their mental pictures and/or physical actions by diagrams

(vi) search for and investigate patterns (vii) see connections

(viii) generalise (find a rule) (ix) establish a proof (know why the rule works). The kind (and style) of work in (i)-(vi) is typical in Stage 0 and Stage

1 of the R-T process. Stage 2 (corresponding to (vii)-(viii)), involved various ideas, algorithms, and interpretations of the situation in geometrical models. Some of the tables worked out by children (see Fig. 3) are very significant, since they indicate the use of an historically well known pattern, namely the triangular number pattern.

As it is well known, the Pythagoreans were giving geometrical forms to numbers, calling them "triangular", "square", "pentagonal" etc. with respect to their additive or multiplicative properties. Thus a "triangular number" is a number of the form

t n = l + 2 + 3 + - . . + n .

In the case n = 7, the triangular number t 7 = 28 is illustrated by the children's drawings in Fig. 3.

6 I,,,~ *nd ~* *,-

B

Ruth, second y~r junior

Fig. 3.


The spontaneous use of the triangular number pattern indicates that this was first a model of action for the children. This model was not introduced by the teacher, but was rather a mental image in them, and it became explicit by their drawings (strategy (v), Stage 1 of the R-T process).

Later, the children's geometrical representations turned out to be models of validation for their empirical patterns and results:

Using real objects or their mathematical diagrams, they were able to convince themselves and others of the 'correctness' of their answers. This gave them a great sense of satisfaction and creativity. They could do it! (Billington and Evans, 1987)

4. QUESTIONS ARISING FROM THE USE OF MODELS OF POLYGONAL SHAPES

We shall study a particular case of geometrical models used in teaching. We shall need some mathematical terminology. By a "polygon" we will understand a convex polygon, i.e. the convex hull of a finite set of points in the plane. More generally, by a "polytope" P in n-dimensional Euclidean space E n we mean the convex hull of any finite set of points in En; the dimension of P is the dimension of the affine subspace generated by P. We shall consider here only the dimensions 0, 1, 2 and 3. Thus O-polytopes are simply points, 1-polytopes are line-segments and 2-polytopes are (convex) polygons. As we shall see, there is a strong reason for considering polytopes of different dimensions together, namely the possibility of "convergence" of a sequence of polytopes to a polytope of lower dimension; this fact plays an important role from a psychological point of view. Another reason is that every n-polytope has (n - 1)-dimensional faces; thus a polyhedron (3-polytope) has polygonal faces, a polygon has 1-dimensional faces (or sides) etc.

It often happens that we are not interested in a particular polytope as a specific set of points in space, but rather in all polytopes similar to it. Similarity is an equivalence relation in the set of polytopes, which enables us to define mathematically the intuitive notion of "shape": The "shape" of a polytope P is its similarity class, i.e. the set of all polytopes similar to P.

In teaching mathematics, the transformations that leave a figure invariant (as a whole) or transform it to a congruent or similar figure have been used as a very significant tool. The conception of geometry as a theory of invariants of groups of transformations (which, in the case of Euclidean geometry, would be groups of linear isometries or at least similarities) has played a crucial role in this direction. This theory, as well as the classical properties of similar triangles, has a long history in science; it is, therefore, included in all curricula, either in a modem setting or not.

U S I N G G E O M E T R I C A L MODELS 41

Transformations in geometry consist, formally, of a domain and the images of its points, which are obtained according to some quantitative relation or "law". In particular, similarity transformations are characterized by a nonzero real number, the "ratio" of the similarity. 6

This situation is modeled in a satisfactory way in the geoboard. 7 This model is indeed a "generative" one, in the sense of Fischbein (Section 2). As Gattegno (1987) says, "nothing of substance is being stated by a static lattice of nails on a board", except, of course, "the constraints imposed by the lattice on the possible structures that the elastic bands and the nails will reveal". However, there is here a presupposition, that Gattegno clearly points out:

By putting into students' hands a bare board and some coloured elastic bands, we a s s e r t that students already own certain ways of working: - g e n e r a t i n g whatever is brought out of nothing by the stretched bands (or line segments); - s e e i n g truths about the figures l o o k e d a t from certain points of view or with certain

stressings.

But, does this presupposition hold indeed? According to the Van Hieles - who used the geoboard in their research

- at the initial level of development of geometrical knowledge (van Hiele level 1) children distinguish figures by their shape as a whole; they can recognize, for example, a triangle, a rectangle, a square and other figures; but they conceive of the square as completely different from the rectangle or the rhombus, and they do not recognize the rhombus as a parallelogram (see Wirszup, 1976). Thus, at this initial level, which corresponds to State 0 of the R-T process, children do not yet own the second of the abilities mentioned in the above quotation (i.e. "seeing truths about the figures looked at from certain points of v iew. . . ").

Besides the geoboard, other concrete models, such as sets of coloured figures, series of similar boxes etc. have been used from a very early stage of development; they are intended to help the evolution of logico- mathematical structures in a child's thought, but it is doubtful whether they succeed in this task. Geometrically, these concrete models have a striking common feature: they are all characterized by symmetry and similarity (same shape). All items in a series of boxes, for example, must have the same shape, so that it is possible to put anyone of them into any other of greater size. As another example, all triangles in a "Dienes block" must be similar to each other, in order that their shape is clearly distinguished among other parameters (colour and size); thus (since usually the other shapes are the square and the circle) the equilateral triangle is naturally preferred among other triangles as having a maximal symmetry. On the


other hand, according to Piaget and Inhelder (1981), similarity and propor- tionality are constructed in a child's thought gradually, during a long period of intellectual development, where transformations conserving parallels play a crucial role. There are no concrete models in common use, helping the development in this direction.

The approach of Castelnuovo is essentially different from the formal approach to transformations and invariants as described above. According to her view concerning "dynamic intuition", children do not easily observe figures and their shapes when they are steady; but they do observe figures and shapes when they move or vary in a continuous manner. Castelnuovo, working with her young pupils (Castelnuovo, 1972), introduced the ideas of area and perimeter combined together. She used several models of continuously varying polygonal figures, which offered an intuitive, "qualitative" view of the simultaneous variation of area and perimeter, thus helping avoid confusion between these two fundamental measures. As she remarks in her book, "some qualitative arguments in considerations of limit cases can lead the intuition also to discoveries of a quantitative character".

One of these "qualitative" arguments about limit cases is connected with the well known isoperimetric problems:

(P1) Of all rectangles of a given perimeter, find that one which has a maximum area;

(P2) Of all rectangles of a given area, find that one which has a minimum perimeter.

Emma Castelnuovo used, for each of the situations in the problems (P1) and (P2), a corresponding model, consisting of a sequence of paper-made rectangles, which were mapped on a plane coordinate system (Fig. 4a and 4b respectively).

These models represent a continuous variation of shape of a rectangle. In Fig. 4a, for example, we pass continuously from the limit case of a 1-polytope, which is a vertical line segment, to another limit case, which is an horizontal line segment; therefore, the square appears naturally as a particular rectangle in a continuous process. This could be a good reason for changing one's naive conception of the square as "completely different" from a rectangle. But we need first to know whether such models appear, implicitly or explicitly, in children's own perception, thought and action. In particular, what are the conceptions of children on rectangles? Do they conceive, for example, a rectangle of coordinates x = a, y = b in Fig. 4 as having the same shape as the rectangle of coordinates x = b, y = a? And what about the "degenerate" rectangles (or 1-polytopes) as limit cases?


Y

24,

21

18

15

12

9

6

3

yl 8

5

2

9

6 5 4 3 2 1

0 3 6 9 12 15 18 21 24 x 0 1 2 3 4 5 6 9 12 15 18 x

Fig. 4a. Fig. 4b.

5. THE EXPERIMENT; PRESENTATION OF THE RESULTS

In an attempt to give some answers to the questions which we pointed out in the foregoing section, we organized an experiment in the following

phases: - In thefirst phase, a large number of children of 4 - 8 years of age was

asked to draw various geometrical figures (triangles, squares, rectangles). In this way 242 children were selected, who did not have a big difficulty in

drawing such figures. These children were:

39 of age 4 - 5 91 of age 5 - 6 30 of age 6 - 7 82 of age 7 - 8

In total 242 children of age 4-8 .

- In the second phase we performed the main part of our experiment, which was as follows: We asked from all subjects selected in the first phase "to draw a stairway of triangles, each one bigger than the preceding one" and to repeat the same procedure with squares and then with rectangles. We used the word "stairway" as better, for children, than "sequence" or "series". What we wanted to investigate more specifically by this experiment was the extent to which the children would conserve shape, or to what


extent they would implicitly use some model of variation of shape; moreover we wanted to see how all these things would develop with the children's age.

- In the third (and last) phase we interviewed some of the older subjects (aged 7-8). These subjects were asked to draw a "nest of squares", i.e. a square, then a smaller square inside the first etc. and to repeat the same (limit) process with triangles and rectangles. They were questioned about what happens "at the end" (limit) of these processes; they also were asked to reproduce, for each "nest", the figures appearing in it separately, one after the other, thus producing a new "stairway" (this time from bigger figures to smaller ones, until the limit). What we wanted to know about was their understanding of these processes; is there any kind of reflection on these processes?

The results from the second phase were codified according to the following rules:

i) Codification for triangles: "Success (1)" means a more-or-less complete series of triangles, as e.g. in Fig. 5a. "Failure (0)" means a defective series as in Fig. 5b.

ii) Codification for squares and rectangles: We use a scale of three marks (0, 1, 2) as follows:

"2" means conservation of shape, increasing both dimensions of the figure (complete series of figures).

"1" means increasing mainly one special dimension (complete series of figures: possibly producing rectangles in the series of "squares" or a square in the series of rectangles).

"0" means a defective series (very irregular figures or non increasing at all in a regular way).

Of course this classification may be rough, since it classes together different series, e.g. one containing many figures with another one containing much less (see Fig. 6a and 6b); but it permits us a first and total estimation of what is happening.

Fig. 5a. Fig. 5b.


E27

t- - -~

~ '7

Fig. 6a. Fig. 6b.

Here are the results from the second phase (Table I); the same results are pictured in histograms (Fig. 7).

The general picture shows that the results improve with age, but there is a noticeable permanence after 6 years. There are tow strategies developing in children (in general):

a) Conservation of shape, increasing both dimensions of a plane figure at the same time, thus producing a series of similar figures of continuously increasing dimensions (Fig. 5a).

TABLE I

Triangles Squares Rectangles

A. Children of 4-5 years: 0 28/39 (72%) 25/39 (64%) 25/39 (64%) 1 11/39 (28%) 9/39 (23%) 7/39 (18%) 2 5/39 (13%) 7/39 (18%)

B. Children of 5-6 years: 0 36/91 (40%) 34/91 (37%) 30/91 (33%) 1 55/91 (60%) 26/91 (29%) 12/91 (13%) 2 31/91 (34%) 49/91 (54%)

C. Children of 6- 7 years: 0 3/30 (10%) 6/30 (20%) 10/30 (33%) I 27/30 (90%) 15/30 (50%) 3/30 (10%) 2 9/30 (30%) 17/30 (57%)

D. Children of 7-8 years: 0 15/82 (18%) 15/82 (18%) 26/81 (32%) 1 67/82 (82%) 42/82 (51%) 8/82 (10%) 2 25/82 (30%) 48/82 (59%)

4 6 A . G A G A T S I S A N D T. P A T R O N I S

100 -

80-

6 0 -

4 0 -

20-

0

T R I A N G L E S :

1 0

4 - 5 years 1 0 1 o

5 - 6 yea rs 6 - 7 y e a r s 1 0

7 - 8 y e a r s

1~176 SQUARES:

6 o - ~

4 0

2 0 -

0 2 1 0

4 - 5 years 2 1 0 2 1 0 2 1 0

5 - 6 yea rs 6 - 7 y e a r s 7 - 8 y e a r s

100 "

80-

6 0

40

20

0

R E C T A N G L E S :

2 1 0 2 1 0 2 1 0 2 1 0

4 - 5 y e a r s 5 - 6 y e a r s 6 - 7 y e a r s 7 - 8 years

Fig. 7.

b) Increasing mainly one dimension of the figures: In the case of triangles - which are usually made isosceles - this dimension is the altitude to the base (Fig. 8), while for rectangles it is usually the longer side (Fig. 6a). Some children try to increase each dimension separately, in different stages (see Fig. 9); in this case, as also in the case of Fig. 6b, there is a square occurring among the rectangles in a very natural way, as an "instance" in a continuous variation of the shape of a rectangle. In the case of squares, this strategy (which is indicated by the mark 1 for squares and rectangles,

U S I N G G E O M E T R I C A L MODELS 47

Fig. 8.

Fig. 9.

while strategy (a) is indicated by the mark 2) appears in a frequency which is considerably higher than in the case of rectangles. Children usually produce rectangles in the form ~ when their series of "squares" is prolonged (Fig. 10). Some of the children were sure that a rectangle has always the form D, while the form D belongs to a square!

From the subjects' responses in the third phase we present two interesting cases: EMY and JOHN (both aged 8), who had "successfully" passed the second phase of the experiment and they both had followed strategy (a). Their drawings, produced in the third phase, are presented in Fig. 1 la (for EMY) and 1 lb (for JOHN).

We asked EMY to describe in her own words the variation of the shape of the rectangles. She said that these rectangles gradually "become so thin, that the last one of them is simply a line", while the triangles "end as an inverted T". We asked her what these figures looked like; she replied that the whole process reminded her of some computer graphics she had watched a few days before. One of these procedures was finally producing something like Fig. 12.

Fig. 10.

48 A. GAGATSIS A N D T. PATRONIS

5

L 2x z2 z~ a J

[3

s

A

.1-

Fig. lla.

S---~

c ~ 0 o

Fig. I lb.


Fig. 12.

Although JOHN was asked to repeat the reproducing of rectangles more carefully, he essentially continued to apply the same strategy (last column of Fig. l lb). When he was asked what these figures looked like, he answered at once that the squares look "like a block of flats" and the rectangles "like a skyscraper"!

6. I N T E R P R E T A T I O N OF THE RESULTS; A T H E O R E T I C A L M O D E L

Strategies developed in solving problems may be effective or not; this point is not of much interest for us here. What is most important is to explain these strategies and try to see where they could lead.

The task given to children in the second phase of the above experiment is not an easy one, if it is interpreted as a construction of a series of similar figures in the mathematical sense. As Piaget and Inhelder pointed out, perception of "similar forms" is not at all the same thing as the ability to construct, by appropriate operations, a figure similar to a given other. This problem involves questions of the psychology of intelligence which are different from the problem of perception (Piaget and Inhelder, 1981, p. 372). But, even in the context of this difficult task, we see that it is possible to identify the models which are spontaneously and implicitly used by the subjects in their action and which, potentially, could become the object of reflection.

Thus we interpret strategies (a) and (b) developed by children as two alternative models of action, which both correspond to Stage 0 of the R-T process (Section 1). Children have used one or the other model where it seemed easier or more convenient for them to do so. In fact they had much freedom to choose, since the task was given to them as an "open-ended" one (the word "bigger" does not necessarily mean "same figure on a bigger scale").


We shall see now that these models of action correspond to special cases of a general theoretical model, which we propose as their mathematical interpretation. This general model is the space of all similarity classes of polytopes, which is defined and studied in a mathematical monograph of Robertson. (1984). This is a topological space Y~, whose "points" are the similarity classes (or "shapes") of n-polytopes for all n >__ 0, and which has the following property: if we leave aside 0-polytopes, the resulting space

+ of similarity classes of n-polytopes with n -> 1 is path-connected; this means that given any two shapes in ~ + (which can be e.g. any two polygonal shapes, or, also, the common shape of all 1-polytopes) there is always a path in ~ § (i.e. a continuous mapping of the interval [0, 1] into

+) which joins the two shapes. In other words, any shape of polytopes can be continuously deformed into any other, with the exception of 0-polytopes. Fig. 2 represents such a continuous deformation (or path) of shapes of triangles. For another example see Fig. 13.

It is evident that there exists a close relationship between these geometrical representations and the children's drawings of the foregoing section. The paths of Fig. 2 and Fig. 13 correspond to strategy (b) of the children, while each drawing following strategy (a) is represented in the space Y' by a constant path, i.e. a path confined into one only similarity class. It is interesting to note that, according to Robertson's theory, the similarity class of 0-polytopes is a separate singleton path component in the space ~ ; thus, "in principle", strategy (a) can never reduce a square into an (ideal) point, while strategy (b) can "finally" reduce it to a line-segment, as a limit case of rectangular shapes.

In fact there is another special case of this theoretical model, which has already appeared in psychological research. This case results from the affine transformations of a rhombus, as perceived by children of age 5-8 (see Piaget and Inhelder 1981, pp. 351-370). This fact leads to a new observation: In the case of Fig. 2 and Fig. 13, as well as in all cases of children who followed successfully the strategy (b) in our experiment, we have also (in

s q u a r e s

l, II II r e c t a n g l e s

Fig. 13.

1 - p o l y t o p e s


general) a conservation of parallels; we can say that in all these cases, if {P,10_<t< l} is a p a t h of polytopes, then, for every t 2 and t 2 with 0 < t I =< t 2 < 1 there is an affine transformation between Pt~ and Pt2.

Now let us come to the third phase of the experiment. The task given there was different and, perhaps, more definite. But it appeared that at least two totally different attitudes are possible in this respect. One is the attitude of EMY: In each case of figures she follows carefully the limit process and tries to understand it; in the case of triangles, she produces an "inverted T"; in the case of rectangles, she passes clearly to strategy (b). Furthermore, EMY reflects on the whole process again, when she is recalling the similar procedure of computer graphics. It seems that this experience played an important role in the development of her way of thinking and looking at things. Recognizing of similar processes is a remarkable progress; it can be maintained that EMY has advanced to a step further in the R-T process, namely to Stage 1.

On the other side we have the attitude of JOHN: He continues to use strategy (a) for the new task, in all cases of figures. His model is so strong for him that he follows it in all cases, even contrary to what he sees: when he is drawing the "nest" of rectangles, it is clear that he follows a non-constant path of rectangular shapes, whereas in reproducing the same rectangles as a "stairway" he follows a constant path. JOHN does not even recall any similar process with that of the experiment; he simply recalls similar objects, having the shape of a square ("block of flats") or of a "long" rectangle ("skyscraper"). Thus in JOHN's case we do not yet see reflective thinking, but only immediate reactions and operations at a primary-intuitive level.

In this way, our theoretical model made it possible to distinguish between a "reflective" case of mental operations and a "primary-intuitive" one. Reflective thinking yielded a change in the strategies followed by the subject, when there was a change in the external process; on the contrary, operating without reflection did not yield any change at all.

7. CONCLUSION AND FURTHER RELATED QUESTIONS

In Sections 1-2 of this article we described the stages of a process of reflective thinking (the "R-T process"), which leads from primary intuitions and initial thoughts to full awareness of a mathematical subject. In connection with this process, we discussed geometrical models and we examined the conditions under which these models can be helpful in learning and teaching. We concluded that, in order for a model to be


suitable for (explicit) use in the classroom, it must first have been used implicitly by the children as a model of action. Our discussion did not include one important aspect, namely the socio-cognitive aspect of the learning process, where the models of formulation and the models of validation find their proper place. For example, it would be very interesting to know, whether the simultaneous appearance of different levels (mathematics acted out and mathematics observed) in a heterogeneous learning group (Freudenthal, 1978) could help the development of the R-T process, s

In our experimental study, two particular models of polygonal shapes were identified as alternative models of action for children of age 4-8. The first of these models, which was a "constant path" in the space of shapes, is more classical from a mathematical and didactical point of view. It corresponds to exact geometrical forms, which remain invariant under similarity transformations. The second model, a "continuous path" of varied polygonal shapes, is less classical. What we have in this case is essentially a topological deformation of geometrical figures, performed by a continuous family of affine transformations; parallels are conserved, but not the shape (exact form) of a figure. This model of action corresponds to Castelnuovo's teaching technique and justifies her view about "dynamic intuition". Although most children apply the first or the second model rather occasionally, it seems that it is possible to use the second model for changing their naive conceptions of a square and rectangle. But we need to organise special teaching experiments in this direction. We need to test, in some way, the attainment of higher levels in the R-T process. According to our experimental results there is a permanence in the strategies used by children between 6 and 8 years. Would we have the same results with still older children? Does this imply that a certain teaching intervention is needed before 7 years, since after this age a stabilization of conceptions takes place? We need much more research in order to answer such questions.

NOTES

i See Polya (1962, Chapter 7: Geometric representation of the progress of the solution). 2 Thorn (1982) examines the benefits from constructing geometrical models in order to describe intuitive processes, instead of using simply the natural language. It happens very often, Thorn says, that a geometrization offers a global view which is not possible in verbal description, because of the fragmentation which is inherent to the latter. Another advantage of geometrical models is that they enable the thinker to keep his object at some distance from himself, by mapping it in space. 3 Gritfiths (1971) says that "insight" into a mathematical theory seems to be related to the realization that the theory has a model in physics, geometry or some other familiar or accessible part of mathematics.


4 Informations from Stamatis (1969). 5 A very interesting case arises when a lower-dimensional system "models" a higher-dimensional one: See Polya (1957), about the method of "inference by analogy"; also the excellent presentation (from a pedagogical point of view) of the "Fourth Dimension" by C. H. Hinton, Speculations on the Fourth Dimension. 6 As an example of reaction to this view we quote, here, the following passage from Hewitt (1986), which is also an interesting example of introspective awareness and reflection on one's mental images:

"Whilst working on some problems, I found it useful to picture various situations in my mind . . . . I suddenly realised what I was doing. This was the link between my transformations and my brother's ellipses and circles. This was geometry. What both had previously lacked for me was movement. Sure, a rotation is a movement, but before, it had just started here and ended up there, I had never watched it between."

7 See e.g. O'Daffer and Clemens (1976, Chapter 5, Group of problems 5.2). s Another direction is followed by Robert and Tenaud (1987) at the University level; students discuss, in small groups, the methods they use in solving geometrical problems.

R E F E R E N C E S

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Department o f Mathematics

University o f Thessaloniki

Thessaloniki, Greece

A. G A G A T S I S

Department o f Mathematics

University o f Patras

Patras, Greece

T. P A T R O N I S

using geometrical models in a process of reflective thinking in learning and teaching mathematics

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