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= X
= X · 4 
= X · 4 + 10
Many of these students were able to represent the relationships in the word
problems in traditional algebra language. Collaborative and parallel studies
(with similar results) have been carried out by Teresa Rojano in Mexico
(Rojano & Sutherland, 1993; Sutherland & Rojano, in press).
184
THE ROLE OF PROGRAMMING
6. PROGRAMMING AS A MEANS OF EXPRESSING AND
EXPERIMENTING WITH MATHEMATICAL IDEAS
Within all of the studies discussed in this chapter, we have made video- or
audiotape recordings of groups of students as they work in pairs on the pro-
gramming activities. The programming language itself and the ways in
which students interact with the language and use it in their talk to commu-
nicate with their peers play an important role in the student constructions.
Most of the problems presented to the students are challenging in that they
do not know how to solve the problem before working at the computer. So,
for example, in Logo, students might be constructing a general Logo proce-
dure to produce geometrical images in proportion without knowing rules for
ratio and proportion. These rules are constructed by the students as they
work at the computer. They learn from the visual image on the screen that
"take does not always work . . . times is better." In this sense, they are aware
of the global geometric constraints of the problem: "well the two sides there
stay the same . . . it would still be the same distance between here and
here." When constructing the function and inverse function shown in Figure
3, students used the spreadsheet to help them find the rule. The majority of
the 10-year-old group of students and the 14- to 15-year-olds (with low
mathematical attainment) did not immediately program the correct rule for
this problem. Many of them entered a rule of the form "A3 - 0.5" to produce
the Y values, and then, when they copied this rule (in the column labelled Y)
realized that this was not correct. They usually tried out a number of other
rules before finding the correct one (of the form "A3/2"). But after this
experimental work at the computer, both groups improved on these types of
problem in a post-test carried out away from the computer. This may seem a
trivial problem, but it illustrates the important idea that students can
negotiate a general rule whilst working at the computer.
The idea of experimenting in mathematics is new and contentious. As
Epstein points out:
Originally, experimenting would have been doing calculations with a pen and try-
ing out various special cases of a theorem you think might be true. Then when

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ROSAMUND SUTHERLAND
185
you've found enough cases to convince you that it is true you try to prove it. This
is the method Gauss used a lot. His private notebooks are just covered by huge
numbers of calculations. (quoted in Bown, 1991, p. 35)
Epstein goes on to discuss how mathematicians have traditionally hidden
this experimental work:
A typical example is a 140-page paper I wrote and won a prize for. The whole
thing is based on computer work but the paper just goes on and on with theory . .
. the whole direction of the research, how I decided which thing to try and do next
was determined experimentally. (quoted in Bown, 1991, p. 35)
Programming is an ideal environment for developing an experimental math-
ematics. Different languages and problems allow the student to experiment
with different types of object. In a spreadsheet, the focus of experimentation
can be with the algebraic code, or with the graphical representation, depend-
ing on the type of problem. The language used will depend on the problem
and will include such environments as Cabri Géomètre (Laborde & Strässer,
1990) and computer algebra systems like Maple. In the past, we have not
paid enough attention to how students justify the results of their experimen-
tation (actually, in the traditional mathematics classroom, it has often been
the teacher or the answers in the book that provide the justification).
Students are much more likely to invest time in a proof if they are con-
vinced (by means of experimentation) that their conjectures are correct.
Programming involves the use of a formal language, and this language can
be the basis for justification and proof, but students will not do this sponta-
neously. Here again, the teacher will have a critical role.

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7. A CONCLUDING REMARK
In the future, students are likely to have their own portable computer, which
will be powerful enough to support a range of programming environments.
The majority of students will not spontaneously use their computers for
mathematical experimentation unless this is supported by the culture of the
school mathematics classroom. With this support, there will be more stu-
dents like Sam who learned to program at home and at the age of 10 said:
there's quite a lot of maths involved in it. I did a program that calculates your age
. . . it's still a bit faulty at the moment . . . but what it does you enter in your age in
years and the date . . . well just the date and the month that you were born and it
calculates the year you were born and how many years and days old you are.
Of course there are standard and efficient algorithms to calculate age from
date of birth, but, for Sam, it was important to construct the program for
himself. Interactive programming offers the potential for trying out and
refining problem solutions, and all the evidence from classroom work sug-
gests that students are remarkably successful at this activity. I suggest that
most of the potential of programming within mathematics education will be
lost if teachers over-direct students' problem solutions by an overemphasis
on pre-written macros, standard algorithms and work away from the com-
puter. In my work in schools, I have focused on relatively unsophisticated
uses of computer programming, because I believed that these needed atten-
tion. This work has shown that students can construct programs and experi-
ment mathematically, but rather more work still needs to be done to flexibly
integrate these activities into the mathematics curriculum.
186
THE ROLE OF PROGRAMMING
REFERENCES
Bown, W. (1991). New-wave mathematics, New Scientist, 131(1780)
Fletcher D. (1992). Foreword. In W. Mann (Ed.), Computers in the mathematics