Linking computer algebra systems and paper-and-pencil
techniques to support the teaching of mathematics
Onno van Herwaarden and Jo Gielen
Biometris, Department of Mathematical and Statistical Methods
Wageningen University
Dreijenlaan 4, 6703 HA Wageningen, The Netherlands
Introduction
An efficient and successful use of computer algebra systems in mathematics education is not
self-evident. In recent years research has been done on the relation between computer
techniques and conceptual understanding. For instance, attention has been paid to the process
of instrumentation, in which conceptual insights and technical skills come together, see e.g.
Artigue (1997), Lagrange (1999), Guin and Trouche (1999), Drijvers and Van Herwaarden
(2000). In this paper we deal with a related link between technical work in a computer
algebra environment and conceptual understanding. Often students show a lack of conceptual
insight while using the computer algebra system, because, as it seems, they don't incorporate
in a right way the computer techniques into their mathematical way of thinking. In this study
we describe the setup of an elementary calculus and linear algebra course for first year
university students in social sciences. In this course we have integrated the use of a computer
algebra environment into a more traditional course, but with special attention for the
connection between both. In particular, we have made an attempt to integrate paper-andpencil and computer algebra techniques.
Educational setting
Before they entered university, most of the students had taken a curriculum in upper
secondary education that prepares for a study in social sciences at university. That
curriculum contains mathematics courses that place the mathematics in a realistic context,
but in which the algebraic skills such as formal manipulation are not highly developed. The
students were not familiar with the use of a graphing or symbolic calculator. (We note that
this will change in the near future: the graphing calculator is now introduced at upper
secondary level in The Netherlands.)
In our university course the applicability of the mathematics received more emphasis
than its theoretical finesses. Applications relevant for the social sciences were included. Also
the course aimed at conceptual insight rather than at far reaching technical skills. During a
period of six weeks the students attended four 2-hour lessons each week. Three of these
weekly lessons were taught in a more traditional way, in which alternately the teacher was
explaining the mathematics and the students were studying the subject, for instance by
making assignments, but where no computer facilities were available; during these lessons
the students just had a hand held calculator at their disposal without graphing or symbolic
facilities. In the other weekly lesson a computer algebra environment on a PC (Derive 5.0)
was used in combination with paper-and-pencil techniques. The total number of students that
participated in the course was 80.
Aim of the use of the computer algebra and didactical scenario
We supposed the computer algebra to be a good way to facilitate the process of gaining
conceptual insight, see e.g. Heid (1988), Artigue (1997), Drijvers (2000). The amount of time
available for the use of computer algebra was quite limited, though. Therefore it was not
feasible in this course to aim at acquiring a thorough knowledge of the computer algebra
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environment or to develop large skills in using it. Instead, this led us to use the computer
algebra as a support for the learning of the mathematics involved rather than to teach the
computer algebra as a goal on its own. Also, it made us select an easily accessible computer
algebra system (we chose Derive 5.0).
As mentioned above, students often lack conceptual insight while using a computer
algebra environment. A reason for this seems to be that they don’t incorporate the computer
techniques into their mental approach of mathematics. Because the students have learned
mathematics using paper-and-pencil methods and their mathematical way of thinking has
developed in close relation with these methods, one can suppose that a good internalisation
of computer techniques can be reached by an appropriate link with paper-and-pencil
methods. To establish such a link between computer algebra and paper-and-pencil techniques
we have taken the following approach. The 2-hour lessons, where the students don’t have a
calculator with symbolic facilities at their disposal, cover all subjects of the course in a more
traditional (paper-and-pencil) way. In the weekly 2-hour lessons with computer algebra
facilities we have tried to integrate the use of computer algebra and paper-and-pencil work.
In principle we have adopted the following structure for these lessons. The students first have
to make an assignment with paper-and-pencil. Usually, such an assignment is of a type
already dealt with in one of the more traditional lessons, but not demanding too much effort.
Next this assignment has to be solved using the computer algebra system. At this stage
difficulties with computer manipulations are likely to distract the students’ attention from the
mathematics involved. In order to reduce this problem we have written out the necessary
computer algebra commands in rather much detail, sometimes accompanied with the
expected computer algebra output. Then after some similar exercises to obtain more practice
and to master the process, more difficult assignments have to be made by the students using
computer algebra. These may be extensions of former assignments that are too elaborate or
demand too much technical skill to treat without computer algebra. Some of them are
applications relevant for the social sciences. But, when necessary or useful, they are again
preceded or interrupted by corresponding suitable questions for paper-and-pencil to establish
the link with the computer algebra work.
An illustration
We will illustrate here how we have worked out this structure in an example from linear
algebra: the solving of systems of linear equations. In one of the ‘traditional’ lessons the
students have already learned how to row reduce systems of linear equations to echelon form
(‘triangular’ form) by hand, and how to determine the solution from the echelon form. Then
in the next lesson with computer algebra facilities we start with the following system of
equations
 x  2 y  3z  2

2 x  5 y  8z  7 .
3x  8 y  10 z  9

First the students have to solve this system with paper-and-pencil. Subsequently we let them
solve the system with computer algebra using Derive’s Solve > System command, and
compare the answer with the result they had obtained by hand. To let them not get stuck in
the computer manipulations we have described in detail the actions and commands they have
to carry out. Besides we have included a text box with the output [x = –3  y = 1  z = 1] that
will appear on the screen.
Next we ask the students to consider the two systems
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 x  2 y  3z  2
 x  2 y  3z  2


 2 x  5 y  8 z  7 and
 2 x  5 y  8z  7 ,
3x  8 y  13z  10
3x  8 y  13z  12


where the left system has no solution and the right system infinitely many. First the students
have to solve these systems with paper-and-pencil. The approach is the same as for the first
system. By row reduction the students obtain the following echelon forms
 x  2 y  3z  2
 x  2 y  3z  2


y  2 z  3 and
y  2z  3 .




0  2
0  0


From these echelon forms they deduce the solution (or the absence of a solution), in the
second case expressing x and y in the (independent) variable z. Before proceeding to the
computer algebra approach we focus the students’ attention on the fundamental property
concerning the possible number of solutions of systems of linear equations, i.e. no, one or
infinitely many solutions. This fundamental property was not treated in the more traditional
lesson. But from the paper-and-pencil work that the students have just carried out row
reducing the systems to echelon form, they can see the validity of this property (that we don’t
prove formally).
Then we let the students solve these two systems using computer algebra. Treating
the left system using Derive’s Solve > System command yields the output [], from which the
students easily establish that the system has no solution, in correspondence with their paperand-pencil outcome. But when the right system is treated with the Solve > System command,
it turns out that Derive doesn’t yield the set of solutions in that case. In fact, it returns an
echelon form of the system of equations, though a slightly different one than the students had
obtained by hand.
Result:




To establish the link between the computer algebra and paper-and-pencil approach, we now
ask the students to copy Derive’s echelon form into their exercise book, to express x and y in
the free variable z using this echelon form, and to compare the answer with the result already
obtained with paper-and-pencil. Finally, the students are told that they can easily let Derive
carry out also this last step. Changing the solve variables [x, y, z] in the command line #1 into
[x, y] makes Derive solve x and y, i.e. express x and y in the variable z. The students then
arrive at the following output, in agreement with the result they had obtained by hand.
Result:




Subsequently we let the students continue with some similar exercises to master the
process and to see the fundamental property on the possible number of solutions confirmed
in those examples. Next we turn to some applications. First a closed Leontief input-output
model for an economy with three sectors is treated, in which again paper-and-pencil
techniques and computer algebra are combined. This is followed by a Leontief model, but
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now for an economy of five sectors, and by two other applications from macroeconomics,
where the students have to deal with three equations with five unknowns, respectively four
equations with seven unknowns. Where necessary or useful these exercises are also
combined with questions for paper-and-pencil.
Reflection
After this rather detailed illustration of the structure we adopted for our integration of paperand-pencil and computer algebra techniques, we now want to reflect on it. We expect this
integration to be helpful for the conceptual understanding of the mathematics involved. One
obvious reason is that in our structure the students work out basic assignments in two
different ways: using paper-and-pencil and using computer algebra. Not only that repetition,
but especially the treatment of those assignments using two different approaches may be
expected to support the mental conception. With respect to both approaches we have tried to
facilitate this process. On one hand, the paper-and-pencil assignments during the computer
lessons are mostly not elaborate, because in this situation we do not want to focus on
technical problems, but on the basic concepts. On the other hand, the computer algebra
actions and commands are carefully guided at this stage, because we want the students not to
get stuck in the computer manipulations, but to focus on the connection between paper-andpencil and computer algebra methods.
We think, though, that we can say more of it. In some cases computer algebra
commands can be seen as a ‘summary’ or an ‘abstraction’ of a paper-and-pencil process. For
example, Derive’s Solve > System command summarises, as it were, the whole paper-andpencil process of solving a system of equations into one compact whole. We think, there is a
link here with the theory of reification of Sfard (1991). She uses the word reification
(litterally ‘thing-making’) for the ‘objectivation’ of a process. Processes are converted into
compact wholes, to become a new kind of self-contained static constructs. Close to the idea
of reification is the idea of ‘encapsulation’ as described by Dubinsky (1991). He 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. We
think the way we carry out the transition from paper-and-pencil to computer algebra
facilitates the process of objectivation and that way also the process of internalisation by the
students of the computer algebra techniques into their mathematical way of thinking.
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