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instructions to generate sequences as illustrated by the following example:
Follow the instructions to find all the square numbers between 0 and 100
179
FOR NUMBER = 1 TO 10
PRINT NUMBER * NUMBER
NEXT NUMBER
END
10
20
30
40
In the strand related to problem-solving, it states that students are expected
to identify and obtain information necessary to solve problems. This is elab-
orated as: When trying to draw repeating patterns of different sizes using
Logo, realize the need for a procedure to incorporate a variable, and request
and interpret instructions for doing it.
The whole nature of this UK National Curriculum is such that it frag-
ments mathematics, and, as can be seen from the above example, ideas from
computer programming have become so fragmented as to be almost point-
less. But computer programming in schools predates the National
Curriculum, and I am optimistic enough to believe that some of the absurdi-
ties in this new curriculum will change with time. Over the last 10 years,
computer provision in schools has changed dramatically. Ten years ago, we
had to provide the computers in order to carry out our research in the class-
room. Nowadays, we can easily find schools with adequate computer pro-
vision. The school in which I recently completed a project has three com-
puter rooms full of networked computers and a computer in each mathemat-
ics classroom. Many secondary schools in the UK now have good computer
facilities, but the mathematics teachers still need considerable support to

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For a number of years I have been working on the ways in which pro-
gramming influences students' developing use and understanding of alge-
braic ideas. This work was initially influenced by the considerable research
on students' learning of algebra (e.g., Küchemann, 1981), which reported
that students find it difficult to understand that a letter in algebra can repre-
sent a range of numbers and to accept “unclosed" expressions in algebra
(e.g., x + 4). Most of this work on childrens’ understanding of algebra was
influenced by a Piagetian perspective. The implicit assumption often made
was that if students cannot perform satisfactorily on certain algebraic tasks,
then they have not reached the stage of formal operations. Results from
work in computer programming environments conflict with many of the es-
tablished results on the learning of traditional algebra (Sutherland, 1992;
Tall, 1989)
180
THE ROLE OF PROGRAMMING
make use of these facilities for teaching and learning mathematics
(Sutherland, Hoyles, & Noss, 1990).
3. COMPUTER PROGRAMMING AND LEARNING ALGEBRA
4. LOGO PROGRAMMING
Our first study carried out with the programming language Logo
(Sutherland, 1989) as part of the Logo Maths Project (Hoyles & Sutherland,
1989) showed that, with Logo programming experience, students develop a
different view of literal symbols from those developed within school alge-
bra. Tall also found similar results working with the BASIC programming
language (Tall, 1989). In the programming environment, students know that
any name can be used for a variable, that a variable name (either a word or a
literal symbol) represents a range of numbers, and readily accept the idea of
working with unclosed, variable-dependent expressions. Moreover, many
students can use these programming experiences and more traditional alge-
bra situations (Sutherland, in press). But the most important result from this
work, which influenced the direction of our ongoing research, was that the
algebra understandings that students develop depend very much on the na-
ture of their Logo programming experiences, and this is influenced by the
way the teacher structures the classroom situation. In retrospect, this seems
like common sense, but, at the time, the prevalent theoretical view, influ-
enced by the theories of Piaget, was that algebraic understandings depend
more on the developmental stage of the child. Initially in the Logo Maths
Project, we had been cautious about introducing the idea of variable to stu-
dents because of an awareness of the negative attitudes many students have
about algebra. So, in the first instance, we waited for students to choose
goals that needed the idea of variable, and only changed this strategy when
it became clear that most of them would not do this spontaneously. The de-

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velopment in our teaching approach and how it changed within two subse-
quent projects has been described in Sutherland (1993).
When a whole class of students are working on computer programming
activities, they can be actively engaged in their own process of problem-
solving. The teacher's role ought to be one of providing problems to be
solved, or letting students choose their own problem, giving support with
syntax, discussing a problem solution, but essentially devolving much of the
responsibility to the students themselves. It seems that the crucial factor
here, from the point of view of mathematics education, is that the students
construct a problem solution themselves. This contrasts with the idea of giv-
ing students a preprogrammed algorithm, which is more prevalent in the
teaching of BASIC than in the teaching of Logo. Presenting students with
standard solutions is also part of school mathematics practice, and Mason
(1993) has criticized the fact that, in much of school algebra, students are
presented with someone else's solution to a problem and are not given the
opportunity to construct their own solutions. Interactive programming lan-
guages provide an ideal setting for students to construct their own programs,
so it is interesting to question why teachers so often provide programming
solutions for their students, either in the form of pre-written macros or
standard algorithms. It may result from a lack of confidence, on the part of
the teacher, that students will be able to construct their own programs –
often a projection of the teacher's own lack of confidence and expertise onto
the students. Another reason relates to the "mainframe mentality" and the
idea that a program solution must be planned away from the computer.
ROSAMUND SUTHERLAND
181
5. A SPREADSHEET ENVIRONMENT – EXCEL
More recently, I have been working with the spreadsheet Excel with groups
of 10-year-olds, 11- to 13-year-olds and 14- to 15-year-olds. Here I will dis-
cuss the work with the older group of students who were chosen because
they had all experienced considerable difficulty with school mathematics –
many of them were disaffected with mathematics and disaffected with
school, and all of them had very little previous experience of algebra. All
students were interviewed at the beginning and end of the study in order to
trace their developing use of algebraic ideas. The majority of the 14- to 15-
year-olds could not answer any of the pre-interview questions that focused
on the algebraic ideas of: expressing generality; symbolizing a general rela-
tionship; interpreting symbolic expressions; expressing and manipulating
the unknown; function and inverse function. All of the students had great
difficulty in expressing very simple general rules in natural language (e.g.,
“add 3”), and none of them were able to answer questions on inverse func-
tions. The majority were unfamiliar with literal symbols exhibiting the clas-
sic “misconceptions” reported in a number of algebra studies (e.g.,
Küchemann, 1981). For example, Jo thought that the higher the position in
the alphabet the larger the number represented. This clearly related to expe-

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182
THE ROLE OF PROGRAMMING
riences from primary school: “A starts off as one or something . . . when we