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Research funded by
ESRC, Project Number R000236178
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Background
Research in mathematics education has consistently
highlighted students' difficulties in engaging with
formally-presented, analytical arguments and understanding
how these differ from empirical evidence (Balacheff, 1988,
Bell, 1976, Chazan, 1993 and more recently papers to PME 22,
e.g., Arzarello et al, 1998, Furinghetti & Paola, 1998).
The current National Curriculum for mathematics in England
and Wales prescribes an approach to proving, partly as a
response to these student difficulties, in which the
introduction of formal proof is delayed until after students
have progressed through stages of reasoning empirically and
explaining their conjectures largely in the context of
data-driven investigations (see Hoyles, 1997). This approach
to introducing proof to school students has been the subject
of considerable criticism. To provide systematic evidence in
this debate, we started a research project in 1996 to
describe how high-attaining students who have followed this
curriculum conceptualise proving and proof in mathematics
and to explore ways to address any difficulties through new
teaching approaches.
The research comprised two phases: a paper
and pencil survey of the conceptions of proving and proof
held by 2,459 students aged 14/15 years at a high level of
mathematics attainment, (about the top 20%), followed by two
computer-integrated teaching experiments in geometry and in
algebra. This paper is concerned with phase 2 and aims to
present the principles underlying the design of the teaching
experiments and the major findings from their evaluation. It
will incorporate findings from phase 1 (see Healy &
Hoyles, 1998) only in so far as they informed the design and
analysis of the teaching experiments.
Methods
Our phase 1 analysis showed that students, even in this
high attainment band, had a limited view of proof and this
had a significant and negative influence on their competence
in proving. Yet the majority of students recognised that a
valid proof should be general and valued arguments they felt
convinced and explained. Most students felt that
formally-expressed arguments would receive the best marks,
but few used deductive reasoning, with formal arguments
occurring very rarely.
Our teaching experiments in algebra and
geometry aimed to build on the evident strengths of our
students in narrative explanations and help them develop a
multi-faceted view of proof, which included verification,
systematisation and deduction (see de Villiers, 1990), along
with more formal presentation. Given that to construct on a
computer requires (1)
explicit attention to the processes used, we hypothesised
that students would be better able to formalise explanations
derived from computer-based activities.
Teaching experiments were piloted with 6
students in three schools (one mixed, one boys, one girls)
and modifications made on the basis of feedback from
students and teachers. These included more precise
procedures for the collection of systematic written data
from the students and the imposition of a common structure
on both experiments; i.e. students were to construct
mathematical objects on the computer, identify and describe
the properties and relations that underpinned their
constructions, use the computer resources to generate and
test conjectures about further properties, and make informal
explanations as to why they must hold. This was to be
followed by a teacher-led introduction to writing formal
proofs using paper and pencil examples, where students would
be helped to organise the arguments generated during the
computer activity into logical deductive chains in the
appropriate formal language. Finally, the students were
given a challenging computer construction to be explained
and proved, followed by an opportunity to use the properties
proved to explain why a subsequent construction was
impossible (named here for reference the possible/impossible
construction).
Each teaching experiment comprised 3
lessons and 3 homeworks and was conducted by the two
authors. A total of 15 students, 5 from each school,
undertook both experiments within a 5-month period
(2).
The students were chosen by their teachers according to our
criterion of high-attainment, but also so that the group
experience would be beneficial both individually and
collectively. All students completed the phase 1 survey
before the teaching experiments and were interviewed
immediately after them, when they were given the opportunity
to review some of their survey responses as well as to
reflect on the teaching experiments.
The mathematics teachers of the student
group in each school completed the school survey
(3)
as well as the multiple-choice proof survey questions. They
were also interviewed to provide further contextualising
data on the school, class and mathematics curriculum
followed, and the individual students involved in the
experiments.
Data analysis
To cope with the complexity of all the data collected
(i.e. students' and teachers' survey responses, worksheets
completed by students during class and homework sessions,
students' computer work, two sets of observation notes,
transcripts of final interviews, school questionnaires and
transcripts of teacher interviews), we adopted a cyclical
and iterative process of analysis, during which we
constructed student case histories, first by considering
student and school variables, and then weaving in
consideration of group, task and software. We also
documented how students responded to the computer and how
they viewed it as a tool to learn mathematics. Finally, we
sought to take account in assessing progress of how the
students interacted and worked together during the teaching
sessions.
Results
Each individual student case history provides a rich
research study in itself, but here we focus on general
trends and school differences rather than individual
responses and progress.
Deductions from
properties
As described earlier, in both geometry and algebra, the
students were asked to undertake a possible/impossible
construction, as summarised in Table 1 below.
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Construction
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Necessary Property
(explained & proved)
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New Construction
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Geometry: Construct with Cabri a
quadrilateral in which the angle bisectors of two
adjacent angles cross at right angles. Write down
its properties and prove them.
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One pair of opposite sides must be parallel
(i.e. trapezium).
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Predict if you can construct a triangle in which
two adjacent angle bisectors cross at right angles.
Predict yes or no, try to do it, and then explain
why your prediction was right or wrong.
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Algebra: Construct 4 consecutive numbers
in Expressor, write down any properties of their
sum and prove them.
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Sum of 4 consecutive numbers is even but not
divisible by 4
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Predict whether you can find 4 consecutive
numbers that add up to 44. If yes, write them down,
if no, explain why it cannot be done.
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Table 1: Using properties to make predictions:
possible/impossible constructions
All the students managed to complete successfully the
first construction--albeit after time and help. In geometry,
11 students predicted they could construct the triangle.
After trying to do this, all realised it was impossible and
13 could explain why, using the parallel property identified
in the quadrilateral construction. In algebra, 6 students
predicted it was possible to find 4 consecutive numbers
adding up to 44. After trying to find these numbers, all
then realised it was impossible, but only 6 referred to the
necessary property they had proved about non-divisibility by
4, and 4 of these still did not regard this as adequate
refutation and went through a calculation process, i.e.
showed 4n + 6 was not divisible by 4.
Overall, we found the students' responses
to be surprisingly consistent, i.e. despite being utterly
convinced of the necessity of the property in the initial
construction, few students used it to deduce the
impossibility of the new construction. Having noted this in
respect to this particular task, we searched our observation
data and found numerous instances of similar problems in
making inferences from deduced properties, which will be
shown in the presentation. Overall we conclude that the
students were poor at making deductions from properties they
had discovered, explained, and even proved.
Evaluation of visual
arguments
In phase 1, we had analysed whether students assessed a
visual argument in algebra and in geometry to be general or
specific and found rather more thought them specific than
general: 50%, 31% compared to 21%, 18%
(4)
. In analysing the responses of our 15 case study students,
we found similar proportions. Clearly many students were
unfamiliar with the power of visual representations and
their potential to serve as generic examples, a conclusion
supported by our process data: for example, the presence of
a picture in a request for a geometry proof confused some
students about whether it was general or not. Visual
arguments were also frequently attributed lower status than
other forms, and described as 'simple' or 'daft', even by
those who acknowledged their explanatory power.
Summarising the case
studies
Apart from the general trends above, we found differences
in progress across the schools, so report other findings
categorised by school. First we present the school profiles,
brief characterisations of each school, mathematics
department, and teacher.
School A
School A was mixed-sex and comprehensive with
mixed-ability teaching practised in the mathematics
department. Students in Year 10 studied 3 hours of
mathematics/week, and specialist interest in mathematics
was encouraged by additional after-school activities, in
extra sessions, and visits to outside school mathematics
lectures. Proof was not taught as a topic until Year 11.
The computer was rarely used in mathematics lessons. The
teacher clearly valued different expressions of proofs,
recognised students could construct good 'intuitive
proofs' but had difficulties with formalising them. It
was also apparent that the students were encouraged to
take risks and challenge themselves mathematically.
School B
School B was a girls-only comprehensive where again
mixed-ability teaching was practised in the mathematics
department from entry, but there were few 'extra'
mathematics sessions. Students in Year 10 were taught
2.75 hours of mathematics/week with proof addressed
through investigations and coursework. The computer was
not used regularly in mathematics lessons, although there
were two machines in each classroom. The teacher thought
of herself as an educator "not a mathematician" and
fostered a collaborative and nurturing approach where
students would not be faced with unnecessary
challenge.
School C
School C was boys-only, with the boys set by attainment
in mathematics from Year 8. 95% of the top set sat the
higher GCSE paper. The students studied 3 hours of
mathematics/week and proof was taught through
investigations. The computer was not used in mathematics
lessons, although there were well-resourced laboratories
in the school. The teacher was a highly-qualified
mathematician with no formal teaching qualifications, who
described himself as computer-illiterate. He fostered a
strong emphasis on exams in his department, and
discouraged coursework as he felt the higher achievers
spent too much time on this.
We then summarise the background, process and outcome
profiles of the 15 students grouped by school. For the
background profile (Table 2) we used five dimensions to
capture the most salient aspects: KS3 test score
(5)
; views of proof and constructed proof score as assessed in
the proof survey; and prior knowledge of algebra and
geometry as assessed in the survey and from teacher
interviews and classwork
Though the process profile (Table 3) was
harder to draw up, careful analysis of the case histories
led us to draw out 3 dimensions, which though overlapping,
could usefully be distinguished: reaction to the computer
work as evidenced in observations and student interviews;
nature of interactions with the computer and with peers; and
modes of expression used in explanations and in proof (i.e.
narrative, symbolic, visual).
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School
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Summary Description
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KS3 Test
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A
B
C
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4 level 8; 1 level 7
5 level 7
3 level 8; 2 level 7
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Views of proof
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A
B
C
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Only referred to proof in algebra
Only described proof as verification
Unclear about proof and what it was for
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Constructed
proof scores
(range 0-3)
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A
B
C
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6 0's; 2 1's; 5 2's; 6 3's. Highest score
for a formal proof 3
11 0's; 6 1's; 1 2; 2 3's. Highest
score for a formal proof 0
3 0's; 9 1's; 2 2's; 6 3's. Highest
score for a formal proof 1
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Algebra
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A
B
C
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Mainly for expressing generality, and finding
specific unknowns
Very little algebra
Some experience with algebra manipulation and
finding specific unknowns, very little in
expressing generality
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Geometry
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A
B
C
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Very little geometry
Very little geometry
Knew a few geometry facts
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Table 2 : Background profiles of school groups
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School
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Summary Description
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Reaction to
computer
work
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A
B
C
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Enjoyed the challenge of building and
investigating computer constructions
Found computer constructions difficult and took a
long time to build them
Focused on procedures "how to get it constructed"
rather than structure of the constructions
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Nature of
interactions
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A
B
C
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Experimental and collaborative
Insecure when unsure, helped each other
Not very experimental, highly competitive
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Modes of
expression
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A
B
C
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Used many modes of expression flexibly
Slow to adapt to new ways of working, dependent on
others and teachers to validate methods and
outcomes; used variety of forms of expression but
made only fragile connections between them
Liked to finish quickly using only one mode of
expression; generally prioritised formal and
rejected visual
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Table 3: Process profiles of school groups
Finally, for the outcome profiles (Table 4), we
distinguished five dimensions: competence in proof in
algebra and geometry as evidenced in homeworks and final
interviews, and reactions to the role of the computer and
the teaching experiments as evidenced in students' written
evaluations following each teaching session.
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School
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Summary Description
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Proof
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A
B
C
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Ended with multi-faceted views of proof,
geometry as well as algebra
Extended their views of proof to include
explanation as well as verification
Limited "internal" sense of proof; "external
conviction" view still predominated
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Algebra
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A
B
C
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Split between those who learnt algebra routines
and those who connected algebraic to other forms of
explanation
Began to express algebraic relationships in context
of microworld
Learnt algorithmic approaches to constructing
algebra expressions
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Geometry
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A
B
C
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Learnt geometry facts, could separate explicitly
givens from properties to be deduced, still some
difficulties in constructing complete chains of
argument
Learnt geometry facts, began to distinguish givens
from properties to be deduced, still experienced
problems with local deductions
Still found it difficult to organise geometry facts
into logical steps within a formal proof
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Evaluation of
role of computer
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A
B
C
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Algebra: saw connection between programming
and
Geometry: helped to identify and see properties but
not prove them
Reaction same in algebra and geometry: i.e. helped
to be accurate and locate properties; few links
between constructing and proving
Strong emotional reactions--hated or loved
software; allowed work to be done more quickly and
made it easy to construct things; few links between
constructing and proving
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Evaluation of teaching
experiments
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A
B
C
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Writing formal proofs described as hard with
specific problems identified--but enjoyable
Writing formal proofs described as generally hard
with no specific details
Learning to use computer described as hard and
proof mentioned infrequently
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Table 4: Outcome profiles of school groups
School differences
The summary data shows that the teaching experiments were
most successful in school A, had some success in school B
and were least successful in school C. In school A, students
made progress in algebra, geometry and most crucially
developed a multi-faceted and connected sense of proof. This
is illustrated in the case-history of one student,Tim from
school A:
Tim enjoyed the computer work because 'it
was different'. In his evaluations, he described the most
enjoyable parts in algebra as 'programming', 'watching the
programs work' and 'proving'. When asked he clearly saw a
strong connection between proving and the computer work on
algebra.
T I liked the programming stuff - that
helped [to write proofs] because it sort of
showed how it was constructed so… It helped prove
because it showed you how they were made....how that
construction was made step by step.
In geometry, this link between informal and formal proof
was more tenuous. The computer work helped Tim 'see'
relationships, but not to prove them formally, although it
may have performed an important role in satisfying his need
to be convinced.
T Well you could actually see if they were
congruent - you could take however much you were allowed
to take and actually make a triangle. If it was congruent
then you could tell it was.
C Tell it how?
T Just by seeing....
C And did that help you write your formal
proofs?
T Not really - not the formal stuff. ....Well it
made it more enjoyable.
We conjecture that factors in the overall success in
school A were that students were used to the experimental
approach required by our activities, they had an adequate
knowledge base to engage with the activities, and they were
willing to share knowledge and help each other. In school B,
the students started from a weaker knowledge base than
anticipated in the activities, and although they were
beginning to appropriate a broader conception for proof,
they needed more time to consolidate this and gain
confidence in experimenting with the software to learn
algebra and geometry. This group of girls was much less
secure &emdash; they were willing to share but lacked the
competence and confidence to do so. Finally, in school C,
the students viewed the activities more as learning to use
the computer software than to explore the mathematics. They
seemed more used to learning procedures and techniques than
striving for conceptual understanding. As a 'typical' group
of high-achieving boys, they were competitive and preferred
to work on their own.
Individual variation
Clearly, individuals within a school group varied on all
the dimensions of the three profiles, but in each school
there was at least one individual (and rather more in School
A) who adopted a flexible approach to proving which
interweaved verification with seeking understanding and
explanation. From our case histories, it was clear that
interaction with the computer helped students to make and
maintain these connections &emdash; they were able to
reflect on the steps they had made in constructing their
explanations and re-use these steps in deductive arguments.
In these cases, internal conviction was achieved using a
combination of empirical and anaytical methods, logical
arguments were constructed and expressed in a variety of
ways (including formal expression) while keeping narrative
explanations in mind. We also found that in each group there
was at least one student (and rather more in School C) who
ended with a view of proof that prioritised 'external' over
'internal' conviction. These students moved quickly from
informal argumentation to the production of a formal proof
and in the process lost touch with the sense of the problem
to be proved. We also found (particularly amongst students
who had planned to drop mathematics after it was no longer
compulsory), a tendency to be satisfied with their narrative
arguments and remain unconvinced of the need for any formal
proof. In both cases, we interpreted the responses to be a
consequence of the mismatch between our goals, teaching
style and experimental activities and the students' views of
mathematics and of proof.
Our data also suggest that in part these
individual variations can be accounted for by reference to:
the adequacy of an individual's knowledge base as evidenced
in the background profile &emdash; particularly crucial in a
group where sharing/helping was not prevalent; a readiness
to explore different ways of presenting mathematical ideas,
an attitude to learning mathematics which included
problem-solving and experimentation as appropriate learning
strategies; and an approach to technology that did not
preclude seeing computer interactions as relevant to
appropriating mathematical ideas. Of course, fulfilling
these conditions cannot guarantee success &emdash; the
composition and dynamics of the group, and the nature of its
interactions also serve as intervening influences &emdash;
but not fulfilling them is likely to inhibit progress.
Conclusions
Our results show that the computer-integrated teaching
experiments were largely successful in helping students
widen their view of proof and in particular link informal
argumentation to formal proof--a transition known to be
problematic. Not all students however made the anticipated
progress, pointing to the well-known complexity of the
cognitive and metacognitive processes that need to be
appropriated in learning to prove.
Our research also highlighted rather
particular problems our students had with respect to local
deductions and visual reasoning. Since these problems
emerged in all 3 schools and in both algebra and geometry,
we are led to conclude that these two processes are given
rather little emphasis in the current curriculum for younger
students--a conclusion supported by an analysis of the
National Curriculum and by data from our teacher interviews.
Our students are simply unused to making deductions and
predictions &emdash; an unfamiliarity which our research
shows inhibits their capacity to engage with the demanding
complexities of proof at higher levels.
Finally our study identified considerable
variability in responses between students and between
students grouped by school--despite identical activities and
teaching. We interpret this differential progress at the
school level to be related to the match or mismatch between
our expectations of the students' prior knowledge and their
actual knowledge, and our goals and constructivist
orientation and those of the mathematics department as
mediated through the student group. There were also other
variables that influenced success, such as attitude to
computers and their role in learning mathematics among
students and staff, and the gender mix and internal dynamics
of the student group. Taken alongside evidence from previous
research that more 'traditional' methods of teaching proof
have variable success, our findings suggest that teaching
can make a difference to students' competence in proving but
the same activities and the same teaching approaches
inevitably will not be equally effective in all schools or
with all students.
Notes
In algebra, we
built a microworld in Microworlds Logo; in geometry a
dynamic geometry system, Cabri. [Back]
We planned to have 3 pairs of
students per school, but in each case one student did not
attend every session. [Back]
The school survey was used to find
out about a school, its curriculum and the mathematics
teacher of the class selected to complete the proof survey.
[Back]
First percentage is algebra, second
geometry. [Back]
Level 8 is highest grade attainable
in Key Stage 3 (KS3) tests (about the top 1/2%). These tests
are administered nationally at the end of Year 9 (ie age
13/14 year old students). [Back]
References
Arzarello F. et al (1998) A model for analysing
the transition to formal proofs in geometry. Proc.PME 22
University of Stellenbosch, S. Africa. v.2, 24-31.
Balacheff N. (1988) Aspects of proof in pupils'
practice of school mathematics. in D. Pimm (Ed.),
Mathematics, teachers and children, 216-235. Hodder
& Stoughton.
Bell A.W. (1976) A study of pupils'
proof-explanations in mathematical situations.
Educational Studies in Mathematics v.7, (July)
23-40.
Chazan D. (1993) High school geometry students'
justification for their views of empirical evidence and
mathematical proof. Educational Studies in
Mathematics v.24, 359-387.
Furinghetti F., Paola D. (1998) Context
influence on mathematical reasoning. Proc. PME 22
University of Stellenbosch, S. Africa. v.2, 313-320.
Healy L., Hoyles C. (1998) Justifying and
proving in school mathematics: Technical report on the
nationwide survey. Institute of Education, Univ.
London.
Hoyles C. (1997) The curricular shaping of students'
approaches to proof For the Learning of Mathematics
v.17 n.1, pp 7-16.
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