Published Mathematics in School, May, 1999, 19-21.
Students’ views of proof
Celia Hoyles and Lulu Healy
Mathematical Sciences
Institute of Education, University of London
The role of proof in school mathematics, perhaps more that any other topic, is the source of
endless discussion amongst mathematicians, teachers and researchers, as is clear simply from
a brief review of past issues of this journal (see for example Barnard, 1989; Watson, 1995;
Steward, 1996; and Rodd, 1997). The current Mathematics National Curriculum for England
and Wales (DfE, 1994) prescribes an approach to proving that is considerably different from
that adopted by other countries; for example, Italy, Taiwan, to name but two, where
Euclidean geometry is still the dominant context for teaching deductive reasoning. Our
process-oriented approach aims to make proof more accessible, with students progressing
through stages of reasoning largely in the context of investigations 'outside' the more contentrelated attainment targets. Students' justifications are usually expected to draw on
experimental results, with theoretical arguments not required until the highest levels of the
curriculum are reached. The question is what exactly do our students learn in terms of what
it means to prove a mathematical statement, and how do they go about this task?
In an ongoing research project1, we set out to address these questions by conducting a survey
of high-attaining Year 10 students’ views of and competencies in mathematical proof. A
proof questionnaire was administered to 2,459 high-attaining Year 10 students (aged 14-15
years) from 94 classes, in 90 schools. All the students were from top mathematics sets, or
chosen as high-attaining by the mathematics departments, and none had scored less than 5 in
their Key Stage 3 tests (the average was 6.56). The schools were spread across England and
Wales and included mixed and single sex schools in different locations (urban, rural and
suburban), operating a range of forms of selection procedures on entry. As well as completing
the proof questionnaire themselves, the teachers of the classes involved also provided
information about the school, the class and, particularly, the approaches they had adopted to
help their students learn to prove. To analyse the data, we used descriptive statistics to
describe patterns in students' constructions and evaluations of proofs, followed by multilevel
modelling using data from the school questionnaire to identify factors associated with
performance and how these varied between schools. The questionnaire covered proof in
algebra and geometry, but in this paper, we focus only on algebra and report our findings by
discussing responses to a selection of questions from the survey. (For a detailed description
of the project, see Healy and Hoyles, 1998).
1
Justifying and Proving in School Mathematics, funded by ESRC, Project Number R000236178
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Published Mathematics in School, May, 1999, 19-21.
After first describing what they thought proof was, students were asked to evaluate a range of
mathematical arguments presented in different forms. They also had to choose the argument
they believed would most closely resemble their own approach and the one they thought
would receive the best mark. Figure 1 shows the first of these multiple-choice questions in
the algebra section, and Table 1 presents the distribution of student choices according to these
two criteria.
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Published Mathematics in School, May, 1999, 19-21.
A1.
Arthur, Bonnie, Ceri, Duncan, Eric and Yvonne were trying to prove whether the
following statement is true or false:
When you add any 2 even numbers, your answer is always even.
Arthur’s answer
a is any whole number
b is any whole number
Bonnie’s answer
2a and 2b are any two even numbers
2+2=4
4+2=6
2a + 2b = 2 (a + b)
2+4=6
4+4=8
2+6=8
4 + 6 = 10
So Arthur says it’s true.
So Bonnie says it’s true.
Ceri’s answer
Even numbers are numbers that can be
divided by 2. When you add numbers with a
common factor, 2 in this case, the answer
will have the same common factor.
So Ceri says it’s true.
Duncan’s answer
Even numbers end in 0 2 4 6 or 8.
When you add any two of these the
answer will still end in 0 2 4 6 or 8.
So Duncan says it’s true.
Eric’s answer
Let x = any whole number,
y = any whole number
x+y=z
z–x=y
z–y=x
z + z – (x + y) = x + y = 2z
Yvonne’s answer
So Eric says it’s true
So Yvonne says it’s true
From the above answers, choose one which would be closest to what you would
do if you were asked to answer this question.
From the above answers, choose the one to which your teacher would give the
best mark.
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Published Mathematics in School, May, 1999, 19-21.
Figure 1: The first multiple-choice question in algebra; question A1.
% of students choosing the argument
Criterion for Choice
Own approach
Best Mark
Duncan
Arthur
Eric
Ceri
Yvonne
Bonnie
29
12
2
17
16
24
7
22
41
18
9
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Table 1: Percentage distribution of students’ choices on question A1.
Table 1 shows that students were clearly influenced in their choices by whether they were
selecting an argument as closest to their own approach or for the best mark2. Nearly a quarter
of the students (24%) selected Bonnie's empirical argument for themselves although the most
popular choice was Duncan's exhaustive argument, chosen by 29% of the students. Formal
arguments expressed in algebra were relatively unpopular, with only 12% opting for Arthur
and 2% for Eric. The arguments that could be described as focussing on the structure of even
numbers, presented in narrative (Ceri) or visual (Yvonne) forms, were selected by a similar
number of students (17% and 16% respectively).
The picture of student responses was almost reversed when students were choosing the
argument they believed would receive the best mark. Here, formal arguments were favoured
by the majority (63%), with Eric's incorrect argument with its references to x and y, even
more popular than Arthur's. A very small percentage of students (only 3%) believed that
Bonnie would receive the best mark. The exhaustive argument (7%) and the visual one (9%)
were also rather unpopular, while in contrast, the percentage of students selecting Ceri for
best mark (18%) was very similar to the percentage who chose this argument for their own
approach.
So how did we interpret these quantitative data? First was to use other information from the
questionnaire since following each multiple-choice question, we had put questions asking
students to evaluate each of the arguments presented. Did they think it was correct? Did they
believe that it was true in general, or only for a specific case or cases? These evaluations
were used to help us to understand what attracted students to particular arguments. An
example of the format used as it applied to Arthur’s argument is shown in Figure 2 below.
agree
don’t know
disagree
Has a mistake in it
1
2
3
Shows that the statement is always true
1
2
3
Arthur’s answer:
2
The variation in choices was highly significant: = 1741.5, df = 5, P < 0.0001.
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Published Mathematics in School, May, 1999, 19-21.
Only shows that the statement is true for some even numbers
1
2
3
Figure 2: Evaluations of Arthur’s answer
In analysing the evaluations of all of the arguments presented, we found that students were
least successful in evaluating the two formal algebraic arguments, especially in the case of
Eric, where only 12% realised it was invalid. Maybe the difficulties students experienced in
evaluating formal presentations contributed to their unpopularity as a choice for their own
approach. The results from the multilevel modelling suggest this could well be the case as
students tended to choose arguments for themselves which they were able to assess correctly
according to the three statements shown in Figure 2. This general tendency, however, did not
extend to empirical arguments like Bonnie's. When these were evaluated, the majority of
students (64%) were aware that they only verified that the statement was true for some cases.
Yet, if students had actually chosen Bonnie’s as their own approach they were less likely to
make this correct evaluation (42%) than those who had not (70%). This suggests that
encouraging an empirical approach, even if systematic, might have negative consequences in
terms of being able to assess the generality of an argument.
Students were also asked to judge if they found each of the arguments convincing and
whether it helped explain the statement. How this was done is shown in Figure 3 below.
agree
don’t know
disagree
Shows you why the statement is true
1
2
3
Is an easy way to explain to someone in your class who is unsure
1
2
3
Arthur’s answer:
Figure 3: Assessing whether Arthur’s answer explains or convinces
We then compared the scores on these assessments with the choices students had made. Our
models showed that students were more likely to choose an argument that convinced them
and helped them to explain as their own approach than one that did not, and the likelihood
increased further if they felt it did both. Once again, this could be a factor contributing to the
relative unpopularity of the algebraic arguments, since 53% of students felt that Arthur's
argument neither explained why the statement was true nor would make it clearer to someone
who was unsure, and this percentage rose to 63% for Eric's argument – perhaps not
surprisingly as it is totally wrong!
The responses to all the multiple-choice questions revealed the same patterns as outlined
above: only a few students believed that their teachers would value empirical arguments, and
the majority felt that algebraic proofs would get the best mark. These same proofs on the
other hand, were selected infrequently as the way students themselves would approach the
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Published Mathematics in School, May, 1999, 19-21.
problem, as students looked for arguments that they believed to be both general — and they
found it hard to assess the generality of a algebraic proof — and helpful in explaining the
mathematics involved — which formal proofs just did not do.
Our findings give support to those who have conjectured that students think 'algebra is proper
and suitable for proof but words (and pictures) are not' (see for example, Perks & Prestage,
1992). But they add another perspective. Algebra is proper and suitable as a proof for the
teacher while other forms are perfectly adequate for the student — especially if they explain.
Maybe we should introduce algebra earlier, but not only as a means to generalise but also as a
language to explain alongside words and pictures? Explaining in algebra can give meaning
to the symbolism, and provide a robust platform from which students come to appreciate the
power and flexibility offered by the algebra notation system for constructing and
manipulating mathematical objects.
So far we have concentrated on how students' evaluations of arguments are connected with
the choices that they make. We found from our models that a number of other factors,
including some concerned with teaching and curriculum, were also involved. This points to
the perhaps obvious 'truth' that what we decide to teach — or not to teach — and how we
choose to organise our teaching makes a crucial difference to what our students learn. In our
next paper we will explore these issues further.
REFERENCES
Barnard, T. (1989). Is the proof in the practice? Mathematics in School, 18, 4, pp9-13.
Department for Education (1994). Mathematics in the National Curriculum London: HMSO.
Healy , L. and Hoyles, C. (1998) Technical Report on the Nationwide Survey: Justifying and
Proving in School Mathematics Institute of Education, University of London.
Perks, P. & Prestage, S. (1992) Why don’t they Prove? Mathematics in School, 24, 3,
pp43-45.
Rodd, M. (1997). Teaching Proof. Mathematics in School 26, 2, pp8-10.
Steward, D. (1996). Three Tasks, Mathematics in School, 25, 2, pp14-17.
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