The Casyopée project: computer symbolic computation for
students’ better access to algebraic notation and rich
mathematics
Jean-Michel GELIS jm.gelis@laposte.net
Jean-Baptiste LAGRANGE Jb.lagrange@reims.iufm.fr
Equipe de Didactique des Mathématiques, Université Paris VII
Summary 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 project, with the integration of a dynamic geometry module. This
paper illustrates by example the possible ways of approaching and solving a problem,
involving enactive manipulation (within the dynamic geometric window), and
numerical, functional, formal and graphical approaches .
Symbolic calculation to help students access algebraic representations
A common concern by mathematics educators, is that generally, secondary students
only get acquainted with paper/pencil algebraic representation and hence lack
proficiency in this area. Thus, many students 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 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 begin 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. It is based on a
transpositive perspective: mathematicians did not abandon the algebraic notation, but
rather developed specific computer artefacts (symbolic calculation, mathematical
writing systems…) to help in their everyday practice of this notation.
Symbolic calculation (generally referred to as Computer Algebra Systems, CAS) is now
available for everybody’s computer or calculator. The Casyopée project takes into
account the potentialities that CAS can offer to teaching/learning, but also considers the
difficulties of classroom use of standard CAS established by research studies like Lumb
et al. (2000).
The Casyopée project starts from the potentialities of CAS outlined below:
a. Going beyond mere numerical experimentation and accessing the algebraic notation.
Dynamic geometry 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, such as it does for
mathematicians now.
b. 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.
c. 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
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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.
Designing a CAS environment
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 & 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.
The Casyopée environment that we designed from these principle differs from standard
CAS, which operate mainly on symbols, as each object in Casyopée has a clear status
with regard to the curriculum: real number, function, parameter… Functions are defined
on the set of real numbers or on reunion of intervals. While standard CAS’ window is
just a 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 ‘dynamic geometry’ (DG) module and facilities to
help students to link geometrical dependencies and algebraic representations. Thus
enactive exploration and the modelisation are not separated from the algebraic work and
students can go back and forth between algebra and geometry, building and testing
models. The example in the following section will demonstrate these capabilities, using
a problem that can be solved by 10th grade students.
An example of problem solving
Curricula encourage the approach of functions via problem solving, where students can
explore functional dependencies in a non-algebraic field and look for properties of these
dependencies. Exploration can be done “enactively” in the non-algebraic field 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 dependant 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 can learn about functions by approaching a formalisation of a
functional relationship. They can explore the dependency by calculating values of the
dependant 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.
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The problem
oAB is a rectangle triangle and C a free
point on the segment [oC]. The parallel
to (oB) passing through C intersects
[AB] in D. The parallel to (oA) passing
through D intersects [oB] in E.
How does the length EC vary when C is
moving on [oA]? Is there a minimal
value?
Figure 1:
Casyopée
The
figure
built
within
Setting the problem in Casyopee
It is always better to keep in mind a possible generalisation of a problem. That is why
we first draw a figure within Casyopée using two parameters a and b to represent the
triangle’s sides. The points A and B are consequently defined by means of their
coordinates respectively (0,a) and (b,0). Figure 2 displays views of the Casyopée
environment.
Figure 2a: View of the menu bar and main icons of Casyopée. Note that the parameters
a and b are integrated in the icon bar as soon as they are defined.
Figure 2b: Definition of the parameters
a and b.
Figure 2c: Menu allowing the figure
building.
Figure 2:Some features of Casyopée’s interface.
Studying a particular case
Studying particular cases is often used in mathematics to investigate problems.
Consequently, we first restrict ourselves to the isosceles triangle’s case, obtained when
parameters a and b are equal to the same numerical value, which can be easily set with
the help of the track bars in the main menu of Casyopée. Figure 3 shows all the actions
that can be done with the help of the three parts of Casyopée: its main window, the
Dynamic Geometry and Geometric Calculations ones. This screen is a synthesis of all
possible approaches. A teacher who wants to create a session for his/her students will
chose, among all these possible actions, the relevant ones in view of his/her objectives
and the student’s abilities.
The different approaches of the dependency between oC and EC, are based on
complementary views on the notion of a function:
 A numerical co-variation: when activating Geometric Calculations, a user can
explore the numerical values of EC. By dragging the point C on the segment [oA], a
user can conjecture EC’s variation with respect to oC. It is then easy to find that the
distance EC reaches a minimal value when C is approximately on the midpoint of
[oA].
 A formal functional definition : the same Geometric Calculations window allows a
user to define, a function (here named Z), the independent variable being oC and
the dependent variable being EC. Casyopée automatically computes the definition
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

interval [0,a] and the algebraic expression helping the user in this important and
difficult phase of modelisation. Note that the extremities of the definition interval
are mentioned in the relevant part of the window (up and left, between -  and  )
and that parameters a and b are kept in the algebraic expression.
A graphical representation: Casyopée’s interface displays the usual graphical and
tabular representations of a function. Note that the graph can also be defined within
the geometrical window. Consequently, the vertical line passing through C indicates
the corresponding precise point of the curve representing the value of EC. A user
can check graphically the conjecture previously explored numerically and
conjecture that the midpoint of [oA] is the position of C for a minimal value of EC.
A means for an algebraic proof: The expression Z(x) is a square root. Students are
more familiar with polynomial forms, so the idea of working on the square is
natural. To prove that the midpoint (corresponding to x=2) is the position for the
minimum of the function, it is possible to compute the difference between this
squared distance (denoted by Z(x)²) and the supposed minimal squared value (here
Z(2)²). With Casyopée, a student can obtain the factorisation of the difference Z(x)²
-Z(2)²., which is a square. Consequently the difference is always positive and 2 is
proved to be the value for the minimal length. Note that a justification module (not
demonstrated here) helps the student to build and write a proof.
Figure 3: An example of a complete study of a particular case (isosceles triangle) of the
problem, involving different points of view.
Returning to the general case
Let us consider now the case of a general triangle. Figure 4 presents the main resolution
steps that could be proposed to the student:
 a numerical and graphical exploration that establishes that the midpoint of [oA]
does not make the distance EC minimal (figure 4a)
 a geometric observation of the position of C corresponding to the minimal value
of EC (see figure 4b): in the rectangle oEDC, the two diagonals EC and oD are
equals. So, minimizing EC turns into minimizing oD, which happens when D is
the orthogonal projection of the point o on (AB).
 the formal point of view (see figure 4c): Casyopée helps to express oG with
respect to the parameters a and b. A user can evaluate oG with the numerical
values of the parameters and prove, in this numeral context, that oG actually
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corresponds to the minimal position. A generalisation can also be done in the
context of uninstancied values of the parameters a and b.
Figure 4b: This
construction allows to set
up the position G of C
corresponding to the
Figure 4a: The middle of
minimal value of EC.
Figure 4c: possible formal
[oA] does not obviously
Distances EC and oD are study and proof: The
correspond to the minimal
equals, so that the
distance oC corresponding
value of EC as shown by the
minimum is specified by to the minimal value is
successive numerical values
F, the point F being the evaluated. The relevant
and by the Z function’s
orthogonal projection of proof can be done, also in
graphical representation.
o on (AB) and G being
the context of uninstancied
the orthogonal projection values of a and b.
of F on (oA).
Figure 4: Exploring and proving in the general case.
Conclusion
This paper focused on design principles underpinning the Casyopée project. It aimed to
show how this computer algebra environment could contribute to a new practice of
algebra involving problem solving, modelisation and multi-representation. Classroom
experimentation is also an important dimension. In a recent experimentation, we
proposed to 10th grade students a similar problem about the variation of a parallelogram
aera, the parallelogram being defined by points moving on a rectangle’s sides. We
observed that working freely on this problem was possible for average achievers thanks
to Casyopée. The difficulties experienced by students were not the 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 students to solve this problem. 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
Lagrange, J.B. (2005) Curriculum, classroom practices and tool design in the learning
of functions through technology-aided experimental approaches. International Journal
of Computers in Mathematics Learning10: 143–189.
Lumb, S., Monaghan, J., & Mulligan, S. (2000). Issues arising when teachers make
extensive use of computer algebra. The International Journal of Computer Algebra in
Mathematics Education, 7(4), 223-240.
Yerushalmy, M. (1997). Reaching the unreachable: technology and the semantics of
asymptotes, International Journal of Computers for Mathematical Learning, Vol. 2.
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Intended theme of this paper
Designing and using Dynamic Mathematics environments
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