Instrumentation of ICT-tools: the case of algebra in a
computer algebra environment
by Paul Drijvers* and Onno van Herwaarden+
*
Freudenthal Institute, Utrecht University, Utrecht, The Netherlands
+
Subdepartment of Mathematics, Wageningen University, Wageningen, The Netherlands
This paper describes a classroom experiment using hand held computer algebra for the
learning of algebra. During a five week period students of the ninth grade (14-15 years old)
used a symbolic calculator for solving systems of equations that contained parameters. In doing
so, the aim was to develop the conception of a parameter as a means for generalization. The
instrumentation schemes for the mathematical tasks involved, however, were not easy to the
students. The appropriate use of the technology seemed to be closely related to the
understanding of the mathematical concepts behind the procedures. The idea of the
instrumentation of information and communications technology tools turned out to be fruitful in
the interpretation of the students’ behaviour.
1. Introduction
Computer algebra systems find their ways to the mathematics classroom more and more, in
hand held as well as in desk-top format. When the use of information and communications
technology tools (ICT-tools) such as Derive in education was seriously considered for the first
time, early in the nineties, optimism dominated the debate. The technological device would
open new horizons that had been inaccessible so far, and would provide opportunities for the
exploration of mathematical situations. This way, the tools would facilitate the students’
investigations and discoveries.
Since then, however, things turned out to be more complex. Even if these ideals are still worth
while and in some circumstances were (partially) effectuated, the optimism nowadays has more
nuances. It becomes clear that an efficient and successful use of an ICT-tool is not self-evident
and that the appropriate skills have to be acquired by a process of instrumentation that can be
long and difficult. It has been argued by many researchers (e.g. see Guin and Trouche, 1999)
that the differences between the mathematical objects and their representations in the
idiosyncratic micro-world of the technological tool demand efforts of the student and the
teacher. In this journal the link between technical work in a computer algebra environment and
conceptual reflection was stressed by Lagrange (1999c).
In the study that is presented here, the instrumentation is considered in the case of the learning
of algebra using computer algebra: students study the notion of parameters in algebraic relations
and solutions. First, the aim of the research is explained. Then we present the theoretical
framework with an emphasis on instrumentation, followed by the development of the didactical
scenario. After some details on the experiment and the research method, an instrument
utilisation scheme is described, that concerns the solution of systems of equations that may
contain parameters. A discussion and a conclusion finish the paper.
2. Aim and background of the research
The study presented here is part of a research project called ‘The learning of algebra in a
computer algebra environment’.
The general research question and the two sub-questions of this project are:
How can the use of computer algebra promote the insight in algebraic operations and
concepts?
1. How can the use of a computer algebra system contribute to a higher level
understanding of parameters as they appear in algebraic expressions and functions?
2. How does instrumentation of computer algebra take place and what is the role of the
relation between machine technique and mathematical conception?
1
This paper deals with the second sub-question in the context of the understanding of parameters.
For a better understanding of the background of these questions we need to explain the approach
of algebra in The Netherlands. The introduction in algebra during the first years of secondary
education (students of 12 – 15 years old) takes place in a careful way. Much attention is paid to
the exploration of realistic situations, to the process of mathematization and to the development
of informal problem solving strategies. Variables, for example, are often words that have a
direct relationship with the context where they come from. The translation of the problem
situation into mathematics is an important point, whereas the formalisation is delayed. Although
this is a good pedagogical policy in general, as a consequence the algebraic skills such as formal
manipulation are developed relatively late and to a limited extent. This may cause some
difficulties at upper secondary level, especially for students at the highest and most theoretical
level. Therefore, a reconsideration of algebra education is taking place, and the role of computer
algebra is a part of that.
The conception of the parameter can be a suitable subject to try to bridge the gap between the
informal, ‘close-to-the-context’ initiation in algebra and the more ‘vertical’ formal algebraic
operations and concepts. Variables and parameters are in the heart of algebra and a rich
conception of them is indispensable for success in advanced mathematics. A parameter can act
as generalized number, unknown, changing quantity, et cetera, but these roles are played on a
higher level than is the case for ‘ordinary’ variables. Therefore, a parameter can be considered
as a ‘meta-variable’ that is conceptually more difficult than a variable. Concerning the relation
with computer algebra, Guin and Trouche (1999, p. 205) and Drijvers (in print) pointed out that
an adequate use of computer algebra tools requires making explicit the different roles of the
letters to a further extent than while working with paper and pencil.
3. Theoretical framework
The theoretical framework of this study consists of three elements: results of research on the
learning of algebra, the philosophy of Realistic Mathematics Education and the theory on the
instrumentation of ICT-tools.
3.1 The learning of algebra
Much has been written on the learning of algebra in general and, more specifically, on the
notions of variables and parameters (e.g. see Küchemann 1981, Usiskin 1988, Wagner 1981,
Warren 1999). Tall and Thomas (1991) provide a short overview of difficulties with algebra in
general and stress the relevance of algebra:
… there is a stage in the curriculum when the introduction of algebra may make simple
things hard, but not teaching algebra will soon render it impossible to make hard things
simple. (Tall and Thomas, 1991, p. 128)
Concerning variables, one can find classifications of the different roles of variables such as
unknown, indefinite, generalized number, dynamical variable and parameter in many
publications (e.g. Usiskin, 1988). Küchemann (1981) distinguishes between the variable that is
changing dynamically and the variable that is representing a set.
This distinction can also be applied to the parameter, that can be considered as a ‘sliding
parameter’ or as a ‘family parameter’1. Some studies specifically address the parameters.
Bloedy-Vinner (1994) stresses the hierarchy of substitution and the implicit quantifier structure
that is often involved while using parameters. These ‘hidden quantifiers’ also are described by
Furinghetti and Paola (1994), who also state that parameters are conceptually more difficult than
variables. They refer to the long history of the development of the parameter concept (see
Harper, 1987).
With respect to the learning of algebra, much attention has been given to the dual character of
1
Terminology of Van de Giessen, personal communication, 2000
2
mathematical concepts, that have a procedural and a structural aspect. Sfard (1991) uses the
word reification for the ‘objectivation’ of processes:
Again and again, processes performed on already accepted abstract objects have been
converted into compact wholes, or reified (from the Latin word res - thing), to become a
new kind of self-contained static constructs. (Sfard, 1991, p. 14)
This has been elaborated for the case of algebra by Sfard and Linchevski (1994). Close to the
idea of reification is the encapsulation as it is described by Dubinsky (1991). Dubinsky states
that encapsulation of processes into objects is an important step in reflective abstraction. He
suggests that performing processes using a computer may stimulate its encapsulation. Repo
(1994) elaborated this for the case of differentiation using a computer algebra system and
reports positive results.
As a third means of representing the bilateral nature of mathematical entities we mention the
procept as it has been developed by Gray and Tall (1994). ‘Procept’ is a contamination of
process and concept. The authors stress the flexibility that learners of mathematics need in order
to be able to deal with the ambiguity of mathematical notations: in 3+5, the + may be an
invitation to perform the process of addition, whereas in a+b the + is only a symbol that defines
the object ‘the sum of a and b’.
It is our conviction that the theories of reification, encapsulation and procept are very relevant to
the learning of algebra and to the instrumentation of computer algebra tools. As Monaghan
(1994) pointed out, the mathematical entities in such an environment may tend to have a
structural character, whereas the processes are more distant to the objects than while working in
a paper-and-pencil mode. Also, a flexible way of dealing with letters is required. Usiskin (1988)
pointed out that a letter in a computer algebra environment is not a generalized number or a
variable with a certain role; it is nothing more than a symbol. For the technological tool ‘all
letters are equal’. For the user of the tool, however, this is often not the case. ‘Some letters are
more equal than others’, so to say. The ability to manipulate algebraic expressions in a computer
algebra environment requires that the user be conscious of the different roles of the letters that
are involved.
Several studies have been devoted to the use of the computer for the improvement of the
learning of algebra in general, and that of the concept of the variable in particular. Tall and
Thomas (1991) use a computer environment as a generic organizer of examples and nonexamples. They claim that the understanding of algebraic concepts had significantly improved.
The graphic calculator was used by Graham and Thomas (2000) to stress the concept of the
variable as placeholder for numerical values. Boers-Van Oosterum (1990) also claims to
improve the conception of the variable by using different software packages. Brown (1998) uses
a computer algebra environment for generalization of patterns, for solving equations step by
step and for solving number problems. None of these studies uses ICT-tools for the learning of
the concept of the parameter, as we are doing in the project described here.
3.2 Realistic Mathematics Education
A second theoretical element is the philosophy of Realistic Mathematics Education (often
abbreviated as RME) that has been developed and is still ‘under construction’ at the Freudenthal
Institute. We restrict the discussion of RME here to two aspects, the mathematization and the
use of informal strategies. More detailed descriptions can be found in Freudenthal (1973),
Treffers (1987), De Lange (1987), Gravemeijer (1994) or at http://www.fi.uu.nl/en/rme.
Mathematization plays an important role in the theory of RME. The learning of mathematics
often includes the organisation of phenomena by means of progressive mathematization. Two
kinds of mathematization are distinguished: horizontal mathematization concerns the process of
modelling a real world situation into mathematical representations and vice versa, whereas
vertical mathematization concerns the process of attaining a higher level of abstraction within
mathematics itself. Mathematization takes place by means of a reinvention process that is
guided by the teacher and by instruction materials. We can hope, now, that a computer algebra
system is an environment that supports the vertical mathematization.
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In RME much importance is attributed to the informal strategies that students develop. An
informal strategy is apparently the most natural way for the student to attack a problem. The art
of teaching is to make the most of these intuitive approaches and to take them as starting points
for the mathematization and for a careful formalisation. The question is, however, whether
computer algebra supports the gradual formalisation of informal strategies (see Drijvers, in
print).
3.3 Instrumentation
The idea of instrumentation emanates from cognitive ergonomics (Rabardel, 1995) and concerns
learning to use (technological) tools. French mathematics educationalists (Artigue 1997,
Lagrange 1999abc, Guin and Trouche 1999, Trouche 2000) have applied this concept to the
learning of mathematics using ICT and that is, of course, what we will be doing too. First a
global explanation.
A tool is not automatically a useful instrument. A hammer, for example, initially is a
meaningless thing, unless you have used it before, or have seen somebody else hammering.
Only when the idea appears that it can be used to hit a nail, if there are nails available and
maybe someone who can help you to learn carpentry, the hammer becomes a valuable and
useful instrument. The user will develop skills to use it in a handy way and may even learn to
see in what circumstances a hammer would be useful. Such a process in which a ‘bare’ tool, an
artefact, develops into a useful instrument is called instrumental genesis. Everybody who has
once hit his/her fingers knows that this process can be long and painful.
Rabardel speaks about an instrument when there exists a relationship between the tool, the user
and the task. The user has to develop skills to use the tool for specific tasks, and also has to
know for what kind of tasks the instrument is appropriate and how this could be set about. This
latter part, the ‘how’, is condensed in the form of instrument utilisation schemes or, shorter,
instrumentation schemes. The development of these schemes is an important aspect of
instrumentation. Think for example of the moving of a paragraph in a word processing
environment: the ‘cut-and-paste’-instrumentation scheme is applied efficiently with a sequence
of keystrokes and/or mouse movements, whereas this would have been much effort when you
were a novice user.
The word ‘scheme’ can be considered as synonym to ‘schema’ that is often used in cognitive
psychology. We refer to the following description in Cottrill et al (1996):
A schema is a coherent collection of actions, processes, objects and other schema’s that
are linked in some way and brought to bear on a problem situation. (Cottrill et al, 1996, p.
172)
We should realise that an instrumentation scheme consists of more than a technical sequence of
actions that one performs on a machine in order to obtain a certain goal. The other part of the
scheme is mental and consists of the mathematical objects involved, the problem solving
strategy and the motivation for the machine actions. For example, suppose a student needs to
solve an equation using a graphing calculator. S/he can draw the graphs of the functions
corresponding to the two sides of the equation and then apply the intersect procedure. To be
able to do so, s/he first has to see the relation between the solution of an equation and the
intersection point of two graphs. Such mental mathematical conceptions are part of the
instrumentation scheme, and can even grow further during the development of the scheme.
Technical skills and algorithms on the one hand and conceptual insights on the other are
inextricably bound up with each other in the instrumentation scheme.
If we take into account this conceptual part, it is not surprising that the instrumentation process
does not take place automatically and without problems. For a teacher who uses ICT in the
mathematics teaching, it can be worth while to pay explicit attention to the development of
instrumentation schemes. Also, the concept of instrumentation can be valuable while trying to
understand the difficulties met by students, as will be shown later in this paper.
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4. Development of the didactical scenario
In the first phase of the research we made a conceptual analysis of the phenomena variable and
parameter. We conceive a parameter as an ‘extra’ letter in an algebraic expression, formula or
function that is a generalization over a class of formulas, over a family of functions or
expressions. The parameter is considered as a meta-variable: change of its value does not affect
a single point, but the complete graph, and the parameter generalizes not only over a domain but
over a class of mathematically isomorphic problem situations.
This resulted in the identification of three main steps in the learning trajectory:
1. The change of value of a parameter:
A first important step in the didactical scenario is the change of the value of a certain
number due to changing circumstances in the problem situation. The word ‘parameter’
is not needed yet. The graph is replaced by another graph.
2. The introduction of a parameter to generalize a relation or a solution:
After having solved several problems that are mathematically isomorphic, we hope to
evoke the need for generalization so that the problem will be definitely solved. The
parameter can serve as a means for such generalizations. The graphical model is a
‘family’ of curves.
3. The study of the dynamic effect that the change of a parameter has on a ‘family’ of
curves: Students are asked to graph functions for several values of the parameter. Then,
they will consider the effect of changes of the parameter. The dynamical aspect of the
change of the parameter is studied. The graphical image is the animated graph.
In the second and the third step we see the importance of the reification theories. In the case of
the generalization, problems that seem to be different turn out to be members of one family and
one can describe the solution of all these problems with one formula. In order to consider a.x+b
as a generalization over 3x+5, –x+99.9, …, the latter formulas need to be seen as objects. Also,
as the general solution will be calculated, the result has to be considered as an outcome instead
of a process to calculate the numerical value.
In the third step, the change of the value of the parameter does not affect one point (as is the
case if the value of a variable changes) but a complete graph. The graph thus has to be perceived
as a whole, as an object that is affected by the changing value of the parameter.
The example presented in section 6 concerns the second step, the parameter as a means for
generalization of the solution of a system of two equations.
5. Experiment and research methodology
The experiment took place during the spring of the year 2000 in two classes of about 25
students each. The students were 14 – 15 years old and were preparing for the highest, preuniversity level of upper secondary education. Streaming according to future interest had not yet
taken place: in the two classes we find future students of the exact stream and the language or
society stream together. After this year, they will split up into four profiles: culture and society,
economy and society, nature and health, and nature and technical science.
As computer algebra tool the students had a symbolic calculator, the TI-89, that they used at
school and at home. As the experimental period was relatively short (5 weeks with 4 lessons of
45 minutes each week) the process of instrumentation had only begun. Because these students
had no previous experience with technology such as graphing calculators, using this type of
handheld technology was really new to them. For reasons described above, their knowledge of
formal algebra was quite limited. For example, the general solution of a quadratic equation had
not been taught so far, so the help of the symbolic calculator was needed. We wondered whether
performing procedures in the computer algebra environment would contribute to the
development of insight in algebraic substitution and in the distinction of the different roles of
letters. Using the machine, the students don’t have to worry about the calculations and this
might enhance a more global conception of the procedures.
The experiment started with a pre-test. The students then worked through a booklet called
‘Introduction to the TI-89’ that had been developed for this occasion and that aimed at getting
accustomed to the machine and preparing them for the relevant machine techniques. The ‘main
5
dish’ consisted of the booklet ‘Variable Algebra’ in which the notion of the parameter was
developed along the lines indicated above. The experiment was finished with a post-test.
The research method was mainly qualitative. The most important data were the observations
that were made by the observer who attended the lessons. Audio recordings were made as well
as field notes on the observation forms that had been developed. Furthermore, written work of
the students was gathered, as well as the results on the investigation tasks that were in the
booklet. After the experiment, the teachers and some students participated in the evaluation of
the teaching sequence.
The data were analysed by coding the classroom incidents according to previously defined
categories. This way the categories that were the most dominant came up clearly. The
instrumentation aspect that we will address in the next section was one of them.
6. Parameters and generalization: The ISS utilisation scheme
6.1 Idea and exercise
An important element of the conception of a parameter is the parameter as a means for
generalization. The following exercise that was given to the students may clarify our intentions
on this point.
25
31
The two right-angle edges of a rectangular triangle together have a length of 31
units. The hypotenuse is 25 units long.
a. How long is each of the right-angle edges?
b. Solve the same problem in case the total length of the two edges is 35 instead of
31.
c. Solve the problem in general, that is without the given values of 31 and 25.
The students had no experience with systems of equations so far and they did not know a
systematic way to deal with them. However, the beginning of the teaching sequence indicated
that they did have some ideas of informal strategies to attack such problems.
Our idea was that, after taking the concrete dimensions as a starting point and repeating the
method for a different case, the student would discover that the solution method is always the
same and that the student would be able to express the solution in a general form using
parameters. This way, we hoped, the student would experience the power of algebra and that of
parameters in particular.
6.2 The ISS utilisation scheme
In the instrumentation phase that preceded this exercise, we proposed to the students an
instrumentation scheme that we called Isolate-Substitute-Solve, or, abbreviated, the ISS scheme.
In the actual situation of the question a, the equations are x + y = 31 and x2 + y2 = 252, where x
and y stand for the lengths of the two perpendicular edges. Figure 1 shows the ISS utilisation
scheme in this case on the TI-89.
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Figure 1: The ISS scheme on the TI-89
First, one chooses one of the variables, in this case y, to be isolated in one of the equations (I).
The result is then substituted into the other equation using the ‘with’-operator |. This is step S.
In the resulting equation, there is only one variable left (here x) and a second solve-command
(S) provides the answer, in this case x = 7 or x = 24. The calculation of the corresponding values
of y is no longer difficult.
6.3 Difficulties
In reality students had many problems with the ISS scheme that we had not foreseen. Even with
concrete dimensions, before parameters had appeared, the instrumentation was not evident at
all. What were the most frequent obstacles?
First the isolation step. Students had conceptual difficulties here in using the command ‘solve’.
To them ‘solve’ means finding a solution, whereas 31 – x is an expression, not a solution. A
solution should be a concrete number, and by preference a decimal number. The students’
conception of solving had to be extended to ‘expressing one of the variables in terms of one or
more others’.
Figure 2: Difficulties with the Isolation
Furthermore, the students did not feel the need to add a letter at the end of the solve-command
(see left screen of Figure 2). To them, solve(x2 – 3 = 0) would have been enough, and they are
right. They do not realise that the case x2 + y2 = 252 is different because there is a choice to be
made explicit. By the way, the isolation of x in x2 + y2 = 252 provides a response with ‘and’ and
‘or’ because the TI-89 takes into account the domain of the variables (see right screen of Figure
2). That output is hard to understand for the students. However, after some time most of the
students were able to perform the isolation step.
Figure 3: Difficulties with the Substitution
The substitution was the hardest point. The first line of the left screen of Figure 3 shows how
some students tried to substitute a non-isolated form. Note that in set theory the notation
{x2 + y2 = 252 | x + y = 31} would be OK. Another error that occurred frequently was the
substitution of an isolated form into the equation from which it had been derived (Figure 3 left
screen). Of course, the message ‘true’ caused some confusion and often the students did not
understand the logic behind it.
The third step, the solution of the resulting equation, did not cause many troubles in cases where
parameters were not involved. In one of the two classes, however, the teacher demonstrated how
one could do the substitution and the solution (the second and the third step of the scheme) in
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one line:
solve(x2 + y2 = 252 | y = 31 – x, x)
The problem with this nested form of two commands is the choice of the letter at the end: with
respect to which letter do we need to solve? The right screen of Figure 3 shows what happens if
the wrong letter is chosen: the substitution is performed but the letter y does not appear anymore
in the equation, so the equation is returned in an equivalent form. Some of the students
considered this as a means of substitution and they persisted in this error. After the
‘substitution’ they continued with a second solve-command, this time with respect to the right
letter. In this way, the ‘short-cut’ is as long as the method presented in Figure 1, and even more
complex. This problem solving behaviour gives evidence of a limited insight in the idea of
substitution and in the meaning of ‘solve’, which is of course a foreign word to the Dutch
students.
6.4 Variations on a scheme
Apart from the errors that students made in the performance of the ISS utilisation scheme, they
also invented some variants that we had not foreseen.
Figure 4: Variations on the scheme
As shown in the left screen of Figure 4, some students isolated both equations with respect to
the same variable (often y) before the substitution. Probably this stems from the way they
worked before they had the machine: with paper-and-pencil they were used to writing formulas
in the form y = … . In the case presented here, this strategy may lead to problems, because the
result of the isolation of the equation of second degree looks terrible: two solutions (one of
which is not valid in the context) and two domain restrictions. To continue with the scheme, the
student needs to choose the right expression out of the four ‘candidates’, but some students
managed this.
The second variation is a real short-cut. In one of the classes the students discovered that the TI89 can solve a system of two equations simultaneously by using ‘and’ (see Figure 4, right
screen). Although this is very practical in simple cases, the power of ‘and’ is limited, so some of
these students were left without alternative in more complex systems, and that was what
happened at the final test…
A third variation consisted of the repetition of the whole ISS procedure to find the value of the
second variable:
first
solve(x2+y2 = 252 | y = 31 – x, x)
and then
solve(x2+y2 = 252 | x = 31 – y, y).
Not very efficient but it works, although one does not know which value of x corresponds to
what value of y.
To finish this section we do not look at variations of the ISS scheme but at variations between
the two classes. The students could do the ISS scheme in different ways: the step-by-step
method (Figure 1), the nested form described in section 6.3 and the combination method using
‘and’ (Figure 4 right screen). The nested form was presented by the teacher of class A, whereas
the combination method was discovered by students of class B. One of the assignments at the
final test was about a situation that was quite similar to the triangle example. The distribution of
the different methods reflects what happened in the classroom quite well (see Table 1). As was
stated by Kendall and Stacey (1999) the teacher’s privileging indeed affects students’
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behaviour.
Method
Step-by-step
Nested
Combination
Other
Frequency
class A
0
19
0
7
Frequency
class B
6
8
4
7
Table 1: Different methods in different classes
6.5 And how about the parameters?
Question c of the exercise concerns a generalization using parameters. After the difficulties in
the concrete cases, one might expect that this would be even a harder point. For some students,
the parameters did not seem to make much difference; to them, the ‘sting’ was more in the ISS
instrumentation scheme. For others, however, the presence of parameters was an extra
complication. The next three examples illustrate this.
John and Rob had no problems introducing parameters in order to generalize the equations.
They wrote:
a2 + o2 = p2
o+a=s
o=s–a
a=s–o
Then they entered into the machine:
solve(o2 + a2 = p2 | o = s – a, o).
The wrong letter at the end! The response of the machine was:
0 = –s2 + 2.a.s – 2.a2 + p2
The boys solved this a second time with respect to a:
solve(0 = –s2 + 2.a.s – 2.a2 + p2, a)
This time John explained the choice for the letter a at the end: “You want to know the a”.
This way they found the solution for a expressed in the parameters s and p. Their solving
strategy did not seem to be confused by the presence of the parameters.
This was different for Sandra. The exercise was this time to calculate the dimensions of a
rectangle with given perimeter and area. Using the viewscreen she tried to demonstrate to the
class how the corresponding system of equations could be solved:
b+h=s
b * h = p.
First Sandra entered:
solve(b + h = s | b = s – h, b)
The machine replied: true.
Rob commented: “She did not use the p, she did not use the second equation.”
Sandra changed the command into:
solve(b + h = 20 | b = s – h, h)
Before the generalization the value of the parameter s had been 20, and apparently she felt the
need to return to the concrete case. The result, s = 20, is logical but Sandra does not notice that.
Then she realises that Rob was right, but she entered b + h = p instead of b * h = p:
solve(b + h = p | b = s – h, h)
The machine replied s = p, which is also reasonable. At the end Sandra entered
solve(b + h = s | b = p/h, h)
and that gave the right answer.
Sandra’s behaviour gives the impression that the parameters were an extra, complicating factor
in the problem solving process. In earlier situations she had shown that she was able to apply the
ISS scheme correctly in cases without parameters.
The third example concerns Erna who tried to solve a variation of the problem of the rectangular
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triangle. The question now was how big the joint volume of two cubes could be, if the total
length of the edges is 20 units.
The system of equations was x + y = 20 and x3 + y3 = d. The question was which values d could
reach. Erna wrote down:
x + y = 20 and x3 + y3 = d
solve(x+y=20, y)  y = 20x
solve(x3+y3=d, y)  y = (x3d)1/3
y = (x3d)1/3 | y = 20x
solve(20 x = (x3d)1/3, d)
d = 60x2  1200x + 8000
solve(d = 60x2  1200x + 8000, x)
15d (2000 )  300
 15(d  2000 )  300
or x 
x
30
30
I cannot figure out what’s d!
In the second and the third line we see two isolations; this is one of the variations we mentioned
before. Then the substitution is OK, but after that there are two solve-commands, one with
respect to d and the second with respect to x. This gives the impression that Erna no longer
knows what she is doing. Copying the results from the machine onto the paper leads to
mistakes, and she ends with a somewhat desperate sentence. The parameter d, whose role shifts
from that of a parameter to that of the unknown in the problem, may have confused her.
7. Discussion
Let us look back at these experiences with the ISS instrumentation scheme in the light of our
theoretical framework, which contained as major elements the theories on reification, on
Realistic Mathematics Education and on the process of instrumentation. Because the latter states
that machine techniques and mathematical conceptions are closely related, we summarise these
two elements for each of the steps in the ISS scheme.
The first step in the scheme is the isolation (see Table 2). It is clear that this requires a broader
conception of ‘solving an equation’ and of what is a solution of an equation. The idea that one
solves an equation with respect to a variable may be new to the students and it is reflected in the
syntax of the command. Also, the students have to get used to the idea that an expression can be
a solution. This is reflected in the fact that the machine uses the same command for isolation
and for solution. To be able to see an expression as a solution, the expression has to be
considered as an object, and we will speak about that further. Some students, by the way,
preferred to do the isolation in their heads, which is a good strategy in simple cases.
Mathematical conceptions
Isolate = find a solution
An expression can be a solution
Solve can be done with respect
to different letters
Machine procedures
‘solve’ for isolation
The syntax of ‘solve’ requires a
letter
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Table 2: Conceptions and procedures for Isolation
Mathematical conceptions
A formula can be considered as
an object instead of a process,
an entity instead of an action
Machine procedures
‘|’ for substitution
Only isolated forms can be
substituted
Don’t substitute into the same
equation!
Table 3: Conceptions and procedures for Substitution
The second step, the substitution, probably was the hardest point. The relation between
conceptions and procedures in Table 3 is so tight that one can wonder whether the second and
the third cell of the second column would also fit in the first column.
Our remark on the formula as an object refers to the idea of reification. We think that the theory
of reification provides an explanation for the difficulties with the substitution. To be able to
understand a substitution, a student should have a mental image of an expression that can be
‘picked up’ as a whole to be put into the place of a variable. In order to understand what
happens in the case of question a in x2 + y2 = 252 | y = 31 – x the student should consider 31 – x
not as a ‘task’ or as an action to subtract x from 31. Instead, it has to be seen as a static object, as
an entity that can replace the variable y, as is symbolised in Figure 5.
.
Figure 5: Mental image of a substitution
Some students were able to explain that only an isolated form could be substituted. For
example, Rob wrote down: solve(o^2 + a^2 = 625 | o + a = 31, a)
Teacher:
How is that with that vertical bar?
John:
You have to put one apart, you have to put either the o or the a apart.
Rob:
You can only work on one letter?
John:
I think, if you have got the vertical bar, you are allowed to explain only one
letter, so o =
Later in the teaching sequence a student showed a good conception of substitution. The students
used braces to graph a family of curves: each different value of the parameter provided another
graph of the family. Many students, however, found it difficult to enter the expressions with
braces as we proposed, such as y1 = {1, 2, 3, 4} – x. One of the students invented a slightly
different notation (see Figure 6). He substituted the set of values for the parameter, and he thus
related the parameter to the value set, which is exactly what happens mathematically.
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Figure 6: Another form of substitution
The third and last step of the ISS scheme was the solution, which was the easiest part if there
was only one variable left in the equation. In case there were parameters, or if the nested form
was used that combines the substitution and the solution in one line, the difficulty was the
choice of the final letter in the solve-command (see Table 4).
Mathematical conceptions
In case of parameters or nested
form:
which letter plays which role?
Machine procedures
‘solve’ for solution
Choice of the right letter after
the comma
Table 4: Conceptions and procedures for Solution
The point of the choice of the letter in the nested form may be linked to the conception of
substitution, because if one really understands what the substitution does, it is clear with respect
to which variable the equation should be solved. One of the students developed a nice notation
to make this clear by placing an extra pair of brackets to indicate that first the substitution is
done:
solve((x2 + y2 = 252 | y = 31 – x), x).
Since she did this, she never made a mistake in the choice of the letter anymore.
If there are parameters in the equations, it is important that students are aware of the different
roles of the letters. We saw that some students found it natural to use parameters to generalize a
relation or procedure, whereas others seemed to be confused by several letters in one
expression, each having a different role.
How about Realistic Mathematics Education in the light of the experiences in the classes? If we
take the problems with the ISS scheme into account, we might think that it may not correspond
to an informal problem solving strategy. In the test before the experiment some students
indicated that a strategy of ‘fair-share’ was natural to them. This strategy consists of dividing
the total of 31 into two equal parts of 31/2 for each of the edges of the triangle. As this doesn’t
give the desired result, because (31/2)2 + (31/2)2 is not equal to 625, they tried to make one edge
longer at the cost of the other. Some students expressed this orally by saying “31/2 +
something” or “31/2 + a little bit”. The formalisation of this approach means that one has to
give that ‘something’ a name and that a difficult substitution has to be entered:
x2 + y2 = 252 | x = 31/2 + a and y = 31/2 – a
The resulting equation should be solved with respect to a, and the result can be added to x. This
formalisation is apparently too abstract, and this mathematization is ‘too vertical’. Anyway,
generalization of solution methods is a matter of vertical mathematization that can be difficult.
The computer algebra device, on the other hand, often requires that mathematical conceptions
have been made abstract by means of vertical mathematization before they can be entered into
the technological instrument.
8. Conclusion
Instrumentation of ICT-tools refers to the relation between mathematical concepts and technical
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skills. Both are involved in the acquisition of instrumentation schemes that are in the heart of
the instrumental genesis. In this paper we described how the idea of instrumentation helped us
with the interpretation of students’ behaviour.
The acquisition of the proposed instrumentation scheme was not easy for many of the students.
The main mathematical conceptions involved are the idea of substitution and the different roles
that can be played by the letters in an expression. The latter point is especially important when
solutions are generalized by means of parameters. The technical errors that students made seem
to be related to limitations in the understanding of these mathematical conceptions. It is
important that the development of the conceptions keeps step with the technical progress.
Concerning parameters, the conclusion is that the use of parameters may complicate the
situation, but that the extent to which students suffer from that differs. Maybe even more then in
concrete cases, it is important that students are able to perceive formulas as objects. If a formula
invites to a calculation process in the eyes of the student, s/he will find a general solution
containing parameters hardly satisfying.
Beforehand, we hoped that performing procedures with the machine and not by hand would
enhance the understanding of the global mathematical conceptions behind the procedures. This
became reality only to a limited extent. Probably an approach in which the students would also
do substitution and isolation with paper and pencil would lead to a better result. Meanwhile, the
teachers reported that they could benefit of the students’ machine experience with isolation and
substitution while treating the solution of quadratic equations after the experiment.
As a teaching consequence of these conclusions, we would conjecture that the development of
both the technical and the conceptual side of the instrumentation schemes deserve explicit
attention. This can be done by means of classroom discussions and demonstrations, so that the
instrumentation process gets a more social character than when it is supposed to come about for
each individual student.
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Biographical Notes
Paul Drijvers was a pre-service teacher trainer for fifteen years and now is researcher at the
Freudenthal Institute, a research group on mathematics education. His main research interest is
the use of technology, and, more specifically, computer algebra, at upper secondary level. He is
actually preparing a thesis on the learning of algebra using computer algebra. Besides his
research activities Paul is involved in in-service teacher training.
Onno van Herwaarden has been working at the Subdepartment of Mathematics of Wageningen
University since 1982. A large part of his tasks concerns the teaching of mathematics, where he
has a special educational interest in the propaedeutic phase. During a sabbatical at the
Freudenthal Institute he participated in the research that led to this paper. Apart from his
educational interest he is involved in mathematics research in the field of biomathematics.
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