Int. J. Cont. Engineering Education and Life-Long Learning, Vol. 18, Nos. 5/6, 2008 575

The Casyopée project: a Computer Algebra Systems environment for students’ better access to algebra Jean-Baptiste Lagrange* and Jean-Michel Gelis Equipe de Didactique des Mathématiques, Université Paris VII, Case Courrier 7018-2, Place Jussieu, 75251 Paris Cedex 5, France E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: Casyopée is an evolving project focusing on the development of both software and classroom situations to teach algebra. This article first describes the general motivation for the project, then its objectives. It is entering a new phase of development as part of the European ReMath project1, with the integration of a Dynamic Geometry module. This article illustrates by example the kind of problem that Casyopée can help to solve and how, and the working environment it creates for students. Keywords: algebraic learning; computer algebra systems; CAS; Casyopée; computer symbolic computation; dynamic geometry; DG; modelling. Reference to this paper should be made as follows: Lagrange, J-B. and Gelis, J-M. (2008) ‘The Casyopée project: a Computer Algebra Systems environment for students’ better access to algebra’ better access to algebra’, Int. J. Continuing Engineering Education and Life-Long Learning, Vol. 18, Nos. 5/6, pp.575–584. Biographical notes: Jean-Baptiste Lagrange is a Professor of Mathematics Education. He started the Casyopée project in 2001 after conducting research studies about the educational use of Computer Algebra Systems. Another field of interest is researching on how teacher use technology. Jean-Michel Gelis is a Doctorate in Computer Science. He investigated particularly computer representations to support students’ algebraic thinking. He participated in many projects of implementation of these and joined the Casyopée project in 2005.

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Introduction

A common concern by mathematics educators is that after secondary studies, students generally have a very poor conception and mastering of algebra. Thus, they have no means in which to access the richness of mathematics that could help them to understand the role of mathematics in today’s society. Technology should offer better access to algebra and rich mathematics. Some researchers are looking for new technological Copyright © 2008 Inderscience Enterprises Ltd.

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representations easier or more motivating in themselves. For instance, using a spreadsheet could help students to represent situations and to solve problems without having to directly confront algebraic notation. It could then allow the student to reach algebraic thinking. The Casyopée project (Lagrange, 2005) takes another orientation. Its motivation is to use technology to provide students with a means to access existing algebraic representations, built by mathematicians in the preceding centuries. Symbolic calculation (generally referred to as Computer Algebra Systems, (CAS)) is now available for everybody’s computer or calculator. The Casyopée project starts from the potentialities of CAS outlined below: 1

Going beyond mere numerical experimentation and accessing the algebraic notation. Dynamic Geometry (DG) or spreadsheets certainly offer students means for modelling situations and experimenting, but CAS offers expressive means much closer to ordinary mathematical notation and much more powerful. Computer algebra could help students to go beyond mere numerical experimentation.

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Focusing on the purpose of transformations rather than on manipulation. In paper/pencil, algebraic manipulations and transformational skills are necessary in order to get a given form, possibly hiding the interest of the outcome, as compared to the initial form. Basic capabilities of CAS (expand, factor…) help students to choose a relevant transformation for a given task.

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Connecting the algebraic activities. Students’ use of CAS in an experimental algebraic activity would also help to better articulate the different algebraic activities. Yerushalmy (1997) studied a classroom exploration of the asymptotic behaviour of functions with the help of a graphing tool. Students had to link the perceptive evidence of an asymptotic line on the graph and the partial fractional expansion of the function. But the graphing tool was of no help when deriving the expansion and the students had to resort to paper/pencil polynomial division. Few students could do it alone and, as it was a long process, and they lost view of the goal. If students had used CAS, they could have freely explored algebraic transformations looking for one that corroborated their graphical observation.

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Casyopée: aims and structure

In spite of the above potentialities, standard CAS give little support to the students’ work when solving a problem, and classroom use is then difficult to master for teachers (see Lumb, Monaghan and Mulligan, 2000). Thus, a central option in the Casyopée project has been to develop a software environment embedding a symbolic kernel, rather than to use a standard CAS. The goal is to build an open computer environment that students could master and easily link with paper/pencil mathematics. It also aims to give a clear statute to algebraic objects, helping students to keep clear of erratic behaviour often observed in the use of standard CAS and to concentrate on the relevant objects in problem solving (Lagrange, 2000). The Casyopée environment that we designed from these principles differs from standard CAS, which operates mainly on symbols, as each object in Casyopée has a clear status with regard to the curriculum: real number, function, parameter, etc. Functions are

Casyopée project: a CAS environment for students’ better access to algebra 577 defined on the set of real numbers or on reunion of intervals. While standard CAS’ window is a mere memory of commands and feedback, Casyopée’s interface displays the objects relevant for a problem – real numbers, functions with their definition and standard forms (factored, developed…), equations, parameters and their properties. These objects are dynamically updated as in a spreadsheet when a change is made. With the extension developed within ReMath, Casyopée is no longer limited to algebra. It has been extended to include a DG module and facilities to help students to link geometrical dependencies and algebraic representations. Thus, enactive exploration and modelisation are not separated from the algebraic work and students can go back and forth between algebra and geometry, building and testing models. Next, we explain the structure of Casyopée (Figure 1), and then we demonstrate the capabilities by using a problem that can be solved by 10th grade students. Finally, we will expose a series of lessons that we are currently experimenting. Figure 1

Casyopée’s structure

Casyopée’ main window is a symbolic module built in accordance with the above principles. Literals can be defined as parameters and animated. There is a list of real numbers that allow defining a set of definition for functions. For instance, the set can be made of the three values – ’, 0 and ’, which allows to create the set of definition for the function xĺ1/x as the union of the intervals ]’, 0, ’ [ and ] 0, ’ [ or the set of definition for the function x ĺ¥x as the interval [0, ’ [. Functions have properties (signs, variations) that a student can prove using basic algebraic theorems. Actions – creations, calculations and justifications – in this window are menu driven, and powered by a symbolic user-transparent kernel. Graphs, tables are also available for numeric

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exploration. Casyopée records the actions and results in a notepad based on Latex that the user can edit in order to give account of his work, especially the proof. The DG window first offers standard DG systems feature: creation and animation of geometrical objects. It offers in addition a wealth of symbolic facilities. For instance, parameters or functions of the symbolic window can enter into the definition of geometrical objects. Then, the window offers means for modelling dependencies between measures, by allowing the user to create ‘geometrical calculations’, to select a distance as an independent variable and finally to create a geometrical function, the dependent variable being a geometrical calculation. The rationales for that and the way it operates will be demonstrated by an example in Section 3.

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An example of problem solving

Curricula encourage the approach of functions via problem solving, where students can explore functional dependencies in a non-algebraic domain and look for properties of these dependencies. Exploration can be done ‘enactively’ in the non-algebraic domain as well as by using numerical (tabular) and graphic representation. Conjecturing and proving properties like extrema provide a motivation to find an algebraic formulation of the dependency. Here is an extract from the French official curriculum for upper secondary level (tenth grade, ‘Seconde’): “It is possible to study for instance geometrical situations, the independent variable being a length and the dependent variable an area. The problem is then often to look for a maximum, a minimum or simply a value.”

As Lagrange (2005) explained, working on a situation of the type recommended by this extract, students could learn about functions by approaching a formalisation of a functional relationship. They can explore the dependency by calculating values of the dependent variable at given values of the independent variable and then explicit the dependency by expressing the relationship in specific registers. Expressions of the relationship can then be used for the task of finding and proving extrema. Here, we take an example from a presentation in ICTMT8. Gravina (2007) demonstrated ‘dynamical visual proofs’. Here, the idea is to look at how algebraic manipulation could help students to tackle such a problem without loosing view of the geometrical situation, helping them, at the end, to find a geometrical proof.

Casyopée project: a CAS environment for students’ better access to algebra 579 The problem: In the angle with vertex o take P a fixed point in its interior. Construct the points A and B in the sides of the angle, such that P belongs to segment AB and the triangle OAB has the minimum area. For the sake of simplicity, we will assume that AoB is a right angle, A being on the x-axis and B on the y-axis.

In addition, we observe that the function is not defined at x = a. Geometrically, it corresponds to the case of the line (AP) not intersecting the y-axis, and therefore, a forbidden value. The user can observe the point of the curve and how it moves when he

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(she) moves the point A in the DG window. He (she) can check that the optimal value of x does not depend on b, by animating this parameter. By animating a, he (she) is able to conjecture that this value is 2a.

A formal proof is easy by defining the function x o f ( x)  f (2a) . After defining this function g, Casyopée can give a closer form. The factorisation is the good option because it helps to prove that this function is positive and then that x = 2a is the optimal value. For that, it is necessary to work on sub expressions. Note that Casyopée gives comments on the proof in the NotePad. A student will be able to write his/her proof in this NotePad using these comments. Coming back to the geometry window, a user can create the optimal triangle oCD and the line passing through C parallel to the y-axis. It intersects (AB) in E. The area of EAC is the difference between the areas of the mobile triangle oAB and the optimal triangle oDC. This is a geometric proof. It can be confronted to the algebraic proof by defining the function oA o d(E, C)d(A,C)/2 and comparing its expression to the expression of g, which is precisely the difference.

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A long-term scenario: approaching functions with Casyopée

This paragraph illustrates how Casyopéee can be used in the classroom for a series of lessons. A problem of optimisation, like the example above is proposed to the students only at the end of this series, after carefully preparing students with regard to algebraic and geometrical knowledge as well as to modelling and to using Casyopée. Here, the scenario is presented. The classroom experimentation is in progress by the time this article is written, as a part of the European project ReMath. This scenario is designed for the 11th grade. A first driving principle is to establish a balance between working on previously learnt knowledge and introducing new concepts. Then, the scenario aims to help students to develop skills for modelling geometrical situations and to reflect about the idea of variables in situations by way of dialectic between algebra and geometry. Casyopée is intended to help student explore situation and to partly discharge them of technical algebraic manipulations. Six 2-hour lessons are planned in three chapters:

Casyopée project: a CAS environment for students’ better access to algebra 581 Chapter 1

Algebra; Lessons 1–3.

Chapter 2

Function and geometry: variable and equation; Lessons 4 and 5.

Chapter 3

Function and geometry; optimisation; Lesson 6.

4.1 Lesson 1: introducing associated functions This lesson is in a regular classroom. Casyopée runs on a computer attached to a video projector. Throughout this lesson, f is a function defined on the set of real numbers. The functions g, h and t are defined on the same set by: gx o f ( x )  2; hx o f ( x  2); tx o 2 f ( x) . They are ‘associated’ to f. The general question is the relationship between the graph of f and the graphs of the associated functions. In addition to study this topic as recommended by the curriculum, students get a first acquaintance with Casyopée. In a first step, f is a function (sinus) whose algebraic properties are not known by students in order to foster geometrical interpretation, rather than algebraic transformation. In a second step, students consider the square function. They can use again the geometrical interpretation and compare with an algebraic transformation. At the end of the lesson, the teacher introduces the reverse problem (target functions). A graph is given. The goal is to find the expression of an associated function with the given graph. Five phases are planned. Phase 1. The teacher presents the associated functions. Phase 2. The students do by hand the following task. x

graph (in blue) the function fxĺsin x

x

use this graph to draw in green (on three separated drawings) the graph of g, h and t

x

draw the correct graph (in another colour) after the collective discussion.

Then a student explains his (her) solution on the white board and the teacher demonstrates Casyopée’s algebraic module and uses this module to discuss the student’s solution, by projecting Casyopée’s graphs upon the student’s drawings. Phase 3. The students do task two by hand. x

use the graph of the function fxĺx² to draw the graph of g, h and t

x

draw the correct graph after the collective discussion

x

what geometrical transformation transforms the graph of f into the graph of g, h, t?

A student operates Casyopée for the discussion. Phase 4. The teacher enters associated functions with parameters, and displays the graphs by animating the parameters. One or more students operate Casyopée for a similar animation. Summary and discussion about the geometrical interpretation follow. Phase 5. The teacher explains the reverse problem. He solves an example and stresses that, when more than one parameter is involved; it is better to animate them successively, rather than together.

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4.2 Lesson 2: target functions This lesson is in a computer room with a computer for two students. Three tasks are about target functions. The students know the target graphs and the reference function (xĺx²). To help them, in the first two tasks, the general parameterised form of the target function is given. They know that in the target functions, parameters will take integer (not necessarily positive) values. Thus they get knowledge about the use of parameters in Casyopée. Task 1. Target function defined by hx o ( x  5)2  2 ; form given hx o f ( x  k )  a . Task 2. Target function defined by tx o ( x  3) 2 ; form given hx o af ( x  k ) . Task 3. Target function defined by ux o ( x  1)2  3 ; form given hx o af ( x  k )  m . Students have to note the correct values of the target functions and explain how the answer could have been (partly) anticipated by way of the geometrical interpretation. Then, they have to solve a ‘Guess my function’ problem: a group of students graphs a function and then delete (hide) the algebraic expression and another group has to discover the function.

4.3 Lesson 3: different expressions of quadratic functions This lesson is also in a computer room. The goal is to learn about equivalent expressions (after factoring, expanding, completing the square) of a quadratic function. Students will use the target function method that they approached in the preceding lesson. Phase 1. One target function and three possible forms are given. Target function: hxĺ2x²  4x  6. Forms given: fx o a( x  d )²  egx o a[( x  k )²  m] and pxĺa(x  u)(x  v). There are two methods, to manipulate the parameters in order that the graphs of the hidden function and the targeted function are the same for each function, or to do the same process for one function and find the others by algebraic manipulations (with or without Casyopée). Phase 2. Find, whereas it is possible, the two others forms of the following functions hx o 5 x ²  4 x  1and hx o 3x ²  2 x  6 . Find the abscissa of the extremum of each function. Here, no target function is involved, in order to foster algebraic methods. Phase 3. Collective discussion and validation, institutionalisation of the three forms, interpretation is possible for the parameters.

4.4 Lesson 4: introduction (dividing a triangle into figures of fixed area) This lesson is in a regular classroom with a video projector. The goal is to consolidate geometrical knowledge on simple situations. It is also to introduce students to Casyopée’s DG window and to the relationship with the symbolic window.

Casyopée project: a CAS environment for students’ better access to algebra 583 Phase 1. The students do by hand the following task In orthogonal normed axis, draw A (0,6); B (2; 5) and the triangle OAB. Problem: we want to split the triangle into two parts of equal areas, not necessarily triangular. How to do it with one scissor cut? How to do it with two scissor cuts? For one cut, one takes the line between a vertex and the middle of the opposite side. One can also cut the line parallel to a side at a ration of ¥2. With two cuts, I being the middle point of [AB] one can cut in OI and IA. This phase ends by a collective discussion with Casyopée: the teacher shows how to enter coordinated points, the middle point, the other lines and the geometrical calculations. Phase 2. The students do by hand the following task Draw again the same triangle, and then take a point M on the segment [OA]. The line parallel to (OB) passing by M intersects [AB] in P; the line parallel to (OA) passing by P intersects [OB] in Q. Draw the figure. Use it to find the graph of functions g; h representing the areas respectively of OAB, and OMPQ. Problem 1. Find M in order that OMPQ’s area of will be the half of OAB’s. Problem 2. Solve the same problem in order that OMPQ’s area of will be the 4/9 of OAB’s, then the third. Problem 3. What happens if the coordinates of A and B are changed? Problem one is easy because M has to be the middle point. Problem two is less easy. Students will become aware of Casyopée’s help for solving equations. The teacher shows how to create a free point on a segment, then a student will operate Casyopée’s geometrical module then create a function to show a solution. The teacher explains the rest of the procedure, particularly about the variable (choice, feedbacks). For problem 3, the teacher shows how to create points with parametric coordinates.

4.5 Lesson 5: application; dividing a triangle into figures of fixed area This lesson is also in a computer room. The goal is to learn modelling geometrical situations, and to reflect about the notion of variable. The problem is the following: Problem In orthogonal normed axis, create A (0, a); B (b; 0) and C (a; b). (a and b being parameters in [0,10]). How to choose a point M in order that triangle BMC’s area is the third of rectangle ABCD’s? Students use Casyopée to create the figure and then a geometrical function representing the area of MNPQ. They have to fill in a table with choices for the variable and the two feedbacks before and after the definition of a function. The goal is to make students aware that xM and yM are variables, and that yM is a suitable variable whereas xM is not. After that students have to solve the problem in the symbolic window, and then to come back to the geometric module for the interpretation and perform a double check by moving M. In a further work, students have to draw from the previous work a method to cut the triangle or a trapeze into three equal area figures: a triangle and two trapezes.

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By the end of the lesson a discussion highlights and compares the different expressions of the function depending on the choice of the variable.

4.6 Lesson 6: functions and geometry, optimisation This lesson is also in a computer room. The goal is to approach optimisation by using all functionalities of Casyopée. Students have first to explore the situation in the DG window, then to model by a function using the facilities for choosing the variables and calculating the function, then to use algebraic methods to find its maximum, and finally to double check their solution by coming back to the DG window. Problem: a, b and c, positive parameters. A (-a, 0) and B (0, b) and C(c, 0). Find a rectangle (MNPQ) with M on [OA], Q on [OC], N on [AB] and P on [BC] with a maximal area? Further work: How to choose M in order that MNPQ will be a square?

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Conclusions

This article focused on design principles underpinning the Casyopée project and on the possibility to create a working environment for students by way of a six sessions scenario. It showed how this environment could contribute to a new practice of algebra involving problem solving, modelling and multi-representation. Classroom experimentation is in progress and will be analysed and reported in further publications. Nevertheless, we are able to share some observations about lessons in the long-term scenario. We observed that working freely on problems both in algebra and in geometrical modelling was possible for average achievers thanks to Casyopée. Students did not experience usual difficulties with algebraic modelisation and manipulation. Also, students did not experience difficulties related the complexity of standard CAS. This does not mean that is was easy for them. But rather that the difficulties they experienced were deeply linked to the notion of a function: interpreting a co-variation, making sense of the interval of definition and of the algebraic expression, linked to several representations.

References Gravina, M.A. (2007) ‘Drawing movement and insights for the proof process’, Paper presented in the Cd-Rom proceedings of the 8th International Conference on Technology in Mathematics Teaching. Hradec Kralove. Lagrange, J.-B. (2000) ‘L'intégration d'instruments informatiques dans l'enseignement: une approche par les techniques’, Educational Studies in Mathematics, Vol. 43, pp.1–30. Lagrange, J.B. (2005) ‘Curriculum, classroom practices and tool design in the learning of functions through technology-aided experimental approaches’, Int. J. Computers in Mathematics Learning, Vol. 10, pp.143–189. Lumb, S., Monaghan, J. and Mulligan, S. (2000) ‘Issues arising when teachers make extensive use of computer algebra’, Int. J. Computer Algebra in Mathematics Education, Vol. 7, pp.223–240. Yerushalmy, M. (1997) ‘Reaching the unreachable: technology and the semantics of asymptotes’, Int. J. Computers for Mathematical Learning, Vol. 2, pp.1–25.

Note 1

STREP, 6th framework programme ‘Representing Mathematics with Digital Media’.