Factors of integration of dynamic geometry software in the teaching of mathematics

Colette Laborde

Laboratoire Leibniz-IMAG, University of Grenoble, FRANCE

The reflections which will be presented here, result partly from experimental teaching sequences of geometry for the first year of French senior high school (15-16 year old students), and from research carried out in several countries about the use of Cabri-geometry, in particular from case studies of students solving mathematical tasks in a Cabri-geometry environment.

Experimental teaching sequences

They have been designed and carried out in classroom for three years. They are based on the use of Cabri-geometry (software version and application of the TI 92). The students were given a TI 92 for the whole academic year. It implies that they could use it at home and not only for geometry. They also worked for some activities on computers in a specific computer room during part of the maths hours. The fact that more or less the same dynamic geometry program was available in both environments (computer and calculator) allowed this double use of technology. The availability of the geometry application on a hand held device certainly contributed to its integration under several aspects : the students could decide on their own to use it , the teacher could give home work or class assignment to be done on the TI 92 (the files were easily transferred and collected on the calculator of the teacher).

It must be stressed that these students were not involved in a mathematics oriented class. At this level of schooling the classes gather all students independently of their achievement in any subject matter. The choice of a major is made on the following year when students are 16-17 year old.

I - Why integrating new technology in the teaching of maths ?

The philosophy of such an integration is not the use of technology for itself but for supporting, improving and changing the learning of geometry through various possibilities of computers which have very often been presented: exploration of a great number of cases, possible variations on the parameters of the problem, visual or numerical feedback from computer.

The computer may work as a catalyst for generating questions ("what if … ?"), for creating counterexamples, in a word it may foster a more scientific approach from students. The computer allows the design of new kinds of tasks revealing well the meaning of theoretical objects.

An additional aimof integrating new technology is to develop the students’ ability of choosing the relevant tool to use it in a relevant way for solving a problem.

The case of dynamic geometry software

Dynamic geometry software based on direct manipulation offer a microworld in which theoretical objects and relations (sometimes very complex from a conceptual point of view) can be visualized and physically manipulated. Such kind of environments offers the possibility for students of constructing knowledge in action and not only by having recourse to language:

For example, a geometrical transformation like point symmetry can be used as a tool for constructing a parallelogram. When students are given the task of constructing a parallelogram, they all do it by using parallel lines and the parallelogram collapses when moving a vertex until three vertices are collinear. The construction by point symmetry appears as more powerful because preserving the parallelogram even when it is flat. The environment Cabri-geometry reveals thus the power of point symmetry under two aspects : the operational aspect as a construction tool and the generality of this tool allowing a construction resisting to a limit case.

The property of symmetry of the parallelogram in a paper and pencil environment is mainly used for proving, dynamic geometry software allows the students to experience this property in action before using it at a more formal level (Laborde 1995).

Dynamic geometry software may be used for creating intriguing visual phenomena which are not expected by students. The only way of explaining those phenomena is the recourse to theory. Prediction activities are from this point of view also interesting: the student may become aware of the inadequacy of his/her expectations when confronting his/her prediction with the observed result on the computer. Ex : what will happen to the image of a polygon through a translation if you move the vector of translation ? or how to modify the vector of translation so that a circle and its image become tangent ?

This brief description of possible uses of technology in mathematics teaching reveals the importance of the design of tasks in which the computer is used. This leads us to attempt at defining more precisely the idea of integration.

II - What does mean integration of new technology in mathematics teaching?

Integration may be realized in several ways to a more or less extent. We would propose a gradation in the extent of integration in a increasing order of integration :

- scarce use of technology for activities not really related to the teaching content: it may happen that teachers use computers for creating sessions on open problem solving, three or four times the year but these sessions are not part of the course and quite no reference is made to these sessions during the ordinary lessons;

- use of computers for sessions of activities on previously introduced notions: the computer offers opportunities of "hands on" sessions in which students are more autonomous ; but the notions which are objects of teaching have been introduced in an usual way in a paper and pencil environment;

- the teacher introduces mathematical content through technology and not only using technology on previously introduced mathematical notions : it implies that reference situations based on technology are created which may affect strongly the meaning of these notions.

- in addition of content introduced through technology, the teacher "institutionnalizes" the notions to be memorized by referring to some elements of the computer environment

It is clear that the latter ways of integrating computers require a long term use of technology. There must be no need for the teacher to repeat how to use some or some command.

A greater degree of integration is fulfilled if the student on his/her own may have recourse to technology to solve a problem given outside the computer/calculator context.

III — The process of integration

III.1- A long process

Our experience of designing scenarios based on Cabri and integrating them in the regular course of the teaching shows that it takes a long time before reaching the adequate tasks taking advantage of the computer environment and finding the good time management in the classroom. It turned out that the composition of the team of teachers involved in the project revealed well the difficulties and successive phases of integration since those teachers presented various degrees of experience in either teaching mathematics or using technology.

We propose to describe the process of integration in terms of equilibrium processes according to a Piagetian metaphor. Integrating technology introduces perturbations in the teaching system. By teaching system we understand the system made of several

elements mutually interacting, around three poles, the teacher, the students and knowledge. We assume that the teaching system is subject to several constraints (time, societal choices regarding curriculum, inner structure of the mathematical domain of knowledge, conceptions and ideas of students…) and that it evolves from an equilibrium state to another one by making choices. When a new element is introduced, the system is perturbed and is lead to make new choices for insuring a new equilibrium. Following Piaget who proposes three kinds of reactions of an individual to cognitive perturbations, we propose the following reactions

- reaction alpha : ignoring the perturbation

- reaction beta : integrating the perturbation into the system by means of partial changes

- reaction gamma : the perturbation is overcome and looses its perturbing character.

We consider that there is a real integration of technology when the teachingsystem is reacting according to the gamma reactions. But our claim is that the process leading to this kind of reaction is a long term process depending on several factors. Some of them play an important role. Let us mention two of them: the features of the computer environment, the tacit learning hypotheses and beliefs of the teacher.

III.2 - The role of the computer environment

Numerous research papers stress how the context shapes the students’ solving strategies when they are faced with a problem. Sharing this point of view, we also claim that the problem itself is shaped by the context. Features of a computer environment may play a crucial role on the type of task the students have to solve and affect thus the possible strategies and the cognitive outcome of the task.

Let us give an example taken from our own experience of writing tasks in a computer environment.

One of the first work sheets designed by one of the teachers involved in the project dealt with the sum of the angles of a convex quadrilateral. It is interesting to note that this is not part of the French syllabus. When starting using Cabri, this teacher chose to address a point slightly external of the curriculum (behavior partly belonging to reaction alpha). Because of the type of computers available in his school he had to use Cabri-geometry 1. He gave the following task to the students:

"Construct any convex quadrilateral ABCD. Mark and measure the interior angles of this quadrilateral by means of the menu items "Mark an angle" and "Measure" of the menu "Miscellaneaous". Note the measures in the table below. Calculate the sum of the four angles and note the result in the table. Move A, B, C or D and note the new measures, then calculate the sum. Repeat it several times. What do you notice ? Make a conjecture and prove it"

In this activity, the computer is only used for discrete measures. The student must add the measures on his sheet as soon as he moves one of the vertices of the quadrilateral because there was no calculating facility on Cabri 1. The statement of the task did not offer an opportunity of analyzing the behavior of the quadrilateral when one of its vertices was continuously dragged. The computer was used as a box from which quadrilaterals could be randomly taken each one after the other. It could be easily be replaced by a table of values of angle measures of quadrilaterals. The change brought by the use of a computer is reduced to almost nothing.

If Cabri II or its implementation on the TI 92 is used, it is possible to compute and display the sum of the angle measures. The student may be asked for observing what remains invariant in the diagram in the drag mode. He/she may thus observe that not only the sum remains constant but also that three vertices remain invariant when one of them moves. This is the starting point of establishing a relation between the invariance of the sum and a geometrical analysis of the quadrilateral. Recognizing an invariant triangle in the quadrilateral may prompt the use of the property of the sum of the angles of a triangle. One can notice how the drag mode in the second context may be the source of linking a numerical invariant and a geometrical one and how this calls for previous knowledge of the student. For these reasons we consider that the computer is more integrated in the second case.

Following Kaput (1994) we think that "technology without curriculum is only worth the silicon it is written on" but we would add that curriculum with inadequate use of technology is only worth the text processor it is written with.

But again finding the adequate environment and elaborating an adequate use for a specific learning aim is not trivial and needs a deep knowledge of the environment.

III.3 — Learning hypotheses — Epistemological beliefs

When designing teaching based on the use of technology, the teacher or the researcher may not be able to predict as he does in traditional environments the possible outcome of the situations proposed to students. According to the learning hypotheses or teachers implicit beliefs underlying the design of teaching, it may lead to various deviations.

Let us present two extreme cases:

i) Case of constructivist hypotheses, such as:

- Students learn when they are faced with tasks for which math notions are efficient tools of solutions

- Feedback coming from the situation may favor an evolution of solving strategies more than a judgement coming from the teacher

Feedback of dynamic geometry software may be from this point of view very rich in that it allows an interaction between the visual and the theoretical aspects of geometry.

The teacher may rely too much on feedback of computer or give a too large place to activities in which students work on computer, and propose tasks of a greater complexity with respect to corresponding paper and pencil tasks.

The teacher underestimates the complexity of the task, the time needed for the student to solve the task because he has no reference point in his past experience. He overestimates the possibility of interpretations by the student of feedback given by the software.

ii) Epistemological view of geometry as intrinsically linked to paper and pencil

The teacher does not rely on computer based activities as allowing learning of geometry and in addition to computer activities proposes similar paper and pencil tasks without being aware that sometimes a paper and pencil task is less demanding in terms of knowledge and allows perceptive strategies instead of strategies based on theoretical properties. We encountered this situation in our teachers group. One of them, who was an experienced teacher but novice in using technology in her teaching designed a scenario for introducing the students to a new geometrical transformation: "dilation". This was a great change for her but she reduced the effects of the change by two behaviors :

- giving only problems which could be given in a paper and pencil environment

- systematically asking twice the same construction task, the construction in Cabri then in paper and pencil environment. Her behavior can be assigned to a beta reaction: she reduced the effects of the perturbation by performing only partial changes in the teaching.

This behavior is often linked with the conception of a paper and pencil environment as not a context. Knowing how to perform something in paper and pencil environment would be the warrant of a decontextualized knowledge. Noss and Hoyles (1996, p. 48) propose an alternative view of abstraction as not necessarily linked to decontextualisation and "as a process of connection rather than ascension". They add that the "situated, the activity based, the experiential can contain within it the seeds for something more general" (p.49).

Another reason for such a behavior may be also explained by the institutional context. Even if all kinds of calculators are allowed in our French national evaluation, all tasks are given in a paper and pencil environment. The teacher sticks at this context to be sure that students are able to perform the tasks in this latter environment.

IV — Evolution of choices in the design of scenarios

IV.1 Evolution of the tasks

During the last three years, successive versions of the same scenarios have been written after they have been achieved in classroom. The evolution of their features can be summarized as follows.

The first versions called often for immediate visual observations and generalization by inductive reasoning. The need for proof was less important than in a paper and pencil environment. The first versions gave a great role to measuring. For example the first version of the scenario "Dilation" asked the student to construct the image of a segment by a dilation with given ratio and to compare the length of the image and of the initial segment; this was to be done with several ratios. Again as in the case of the quadrilateral, the activity called only for discrete measuring and did not use the animation facility of the numerical display of the ratio.

The later versions introduced two new kinds of tasks :

- tasks in which the environment allows efficient strategies which are not possible to perform in a paper and pencil environment

- tasks raised by the computer context, i.e. tasks which can be carried out only in the computer environment.

Example of the first kind of tasks in the scenario "Vectors" : "Construct a triangle ABC from the given points B, C and G centroid of triangle ABC." A can be constructed as satisfying the vectorial relation :

vector GA + vector GB + vector GC = vector 0. (1)

Fig. 1 - Construction of the image of A’ through a point symmetry around G.

Fig.2 — Result of the construction

A is constructed as the symmetrical point with respect to G of the endpoint of the vector sum of the two vectors GB and GC (Fig. 1 & 2).

Cabri contributes to linking the algebraic aspects of vectors to the geometrical aspects Relation (1) is restricted in paper and pencil environment to algebraic calculations while in Cabri it also receives a geometrical meaning since it is a tool of construction. It offers a new connection in the conceptual field of vectors (Vergnaud) or in the*web* of vectors (Noss & Hoyles, op.cit.).

The second kind of tasks involves mainly two categories of tasks :

- the "black box" situations

- the prediction tasks

In the black box situations, the students are given a diagram on the screen of the computer and they are asked qustions about it. This kind of situations was used in our scenarios for introducing new transformations. A point P and its image P’ through the unknown transformation were given to the students. They could move P and observe the subsequent effect on P’. Students were asked to find the properties of the unknown transformation by means of this black box. In such a task students must themselves ask questions about the transformation :

Does it preserve collinearity ? Does it preserve distance ? Does it have invariant points ?

Cabri can be used to design experiments and get empirical answers. For example, one may redefine P as belonging to any given straight line and obtain the image of this line as the Locus of P’ depending on the variable point P. Two specific tools of Cabri are used "Redefinition" and "Locus". It presupposes that the students not only master their use but also decide to use them. This decision actually involves mathematical knowledge : the fact that the image of a figure is the set of images of points of a figure; this is often completely implicit in our curricula but it presents a conceptual cut (even obstacle probably from both cognitive and didatical origin) with the view of a figure as an entity and not as a set of points.

Fig.3 — Redefining the given point P…

Fig.4 - …as a point on … Fig.5 — a line

Fig.6 —Locus of P’ when P is moving on the line

Fig.7 — Locus of P’ when P is moving on a circle

Such a task offers a very different point of view on the notion of geometrical transformation. Instead of studying the effects of a known transformation, students are asked to characterize the transformation by means of its properties. Of course this may be an attractive task only if some exotic transformations and not only the usual ones are given to students. Theorems of invariance receive a new meaning in this kind of task : they are tools for identifying the category which the unknown transformation belongs to. An effect of this kind of task is that students may understand why to study all these theorems about invariant elements of transformations. The invariance properties become remarkable phenomena instead of being the routine.

Prediction tasks are also specific to computer environment. Where should be the center of dilation for obtaining the image of a given circle tangent to this circle ? When the prediction of the students turns out to be wrong, this is a good opportunity of asking "why is it so ?" and calling for proof.

The design of scenarios revealed us that there are ways of integrating the computer in the problems given to students. The computer is an essential part of the meaning of the problem. At the beginning of our work, technology was mainly used as illustrating mathematical facts. It is only in a second phase that teachers used it as source of problems. This is not without raising problems for teachers:

- the possibility of going beyond the usual curricula

- the tension between the mathematical and the instrumental

- problems of time management

IV.2 - Tension between the mathematical and the instrumental

A computer based teaching involves mathematical knowledge but also knowledge of how to use the software, to perform operations on the software. There is a tension between both kinds of knowledge (Artigue 1997). The teacher is faced with questions as : to what extent must the statement of the task refer to what is to be done in terms of technical operations in order to avoid time consuming on technical aspects ? or should the task statement not refer to the technical aspects in order to allow students’ autonomy ?

Does the student fail because of mathematics or of the technical aspects ? This tension may be solved only if the use of technology is a long term use. In this latter case, the use of the environment may become an object of teaching in relation to mathematical teaching. Of course the strength of the interaction between mathematics and using the software depends very much on the features of the software.

Our teachers introduced the tool "Trace" as a tool for easily giving evidence of an invariant point. For example when a variable line is passing through a fixed point, the trace of this line shows clearly this point. This is for example a good way of displaying that the sum MC of two vectors MA and MB is passing through the midpoint segment AB for any point M.

Fig.8 — Trace of the sum of the vectors AM and MB when M is moving on the screen

At the end of the year students became able to use on their own the tool Trace for displaying an invariance phenomenon like in the dilation black box task mentioned above.

This lead the teachers to include technology in their synthesis phase. Knowledge pointed out by the teacher could refer to some related aspects of Cabri .

Example in the scenario "Vectors": The notion of direction of a vector is not easy to understand by students. There is a paradox that this notion is understood as soon as the notion of vector is acquired but to acquire the notion of vector, the students need to understand the notion of direction. The use of the two kinds of pointers in Cabri showed to the students that when the vector is moved by the usual pointer , its coordinates remain unchanged while they are changed when the vector is rotated by the pointer "Rotate" although its norm is unchanged. So the teacher introduced the Cabri property : coordinates of a vector are modified by the Rotate pointer which lead to the usual property "coordinates of a vector depend on the direction of the vector". The pointer played a mediation role in vygotskian terms.

In the same way, the degree of freedom of points was introduced by the teachers as linked to their geometrical status.

IV.3 Time management

Our first versions of scenarios were time greedy. It is a quite usual phenomenon that any kind of teaching innovation provokes inflation of time. Schneider (1998) reported on a teaching based on the use of the TI 92 about logarithms and exponentials which took 40 hours of teaching instead of the usual nine hours. This led us to seek for simple critical computer based tasks which may be very efficient in terms of learning and to have less recourse to long and open tasks.

Ex:

Construction of a straight line passing through a given point whose directing vector is given.

We also attempted to find an optimal balance between what is demonstrated by the teacher on the LCD display and what is done by students, between what is ready made and given on the calculators to students and what has to be done by the students with the software. We preferred after one year to give the macro-construction of the multiplication of a vector by a number to be explored and interpreted rather than to be constructed by the students themselves.

After sometime teachers decided to give homework requiring the use of technology. It not only solved some time problems but also gave a more institutional status to the use of technology for the students.

When examining the changes occurred over the last three years, it seems that the use of technology became a more usual and ordinary element of the classroom, less spectacular, calling for less complex tasks but involved in more events of the school activities of the student, possibly beyond the only class. So assessment in the classroom in the last year included tasks in computer based environments.

V- Mottos for a curriculum integrating new technology

• Technology to design learning situations in which mathematics is constructed by the student as a way of describing and modeling behaviors (s.c. black box situations)

• Constructing mathematics for predicting, producing and explaining mathematical phenomena reified through technology

• Technology to link multiple representations (algebraic, geometrical, numerical, stochastic)

• A long term use of technology for creating a culture** **

References

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didactiques liés à l’utilisation d’environnements informatiques
pour l’apprentissage *Educational Studies in Mathematics* Vol33, n°2,
133-169

Laborde C. (1995) Designing tasks for Learning Geometry in a computer based
environment, in: *Technology in Mathematics Teaching - a bridge between teaching
and learning,* Burton L. & Jaworski B. (eds) (pp.35-68), London: Chartwell-Bratt

Noss R. & Hoyles C. (1996) *Windows on mathematcal meanings* Dordrecht:
Kluwer

Schneider E. (1998) Lecture given at the conference "Symbolic and geometric calculators in the teaching of mathematics" Montpellier 14-17 may 1998

Vergnaud G. (1991) La théorie des champs conceptuels. *Recherches
en didactique des mathématiques* , 10/2.3, pp. 133-169