©Geoffrey
Roulet
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The words "proof" and "prove" do not appear in recently
released Ontario (Canada) mathematics curriculum documents
(Ontario Ministry of Education, 1999; Ontario Ministry of
Education and Training, 1997, 1999) until the university
preparation courses of Grades 11 and 12 (ages 16-18 years).
This is a continuation of a curricular perspective (Ontario
Ministry of Education, 1980, 1985) that views the concept of
proof as not appropriate for students less than 15 years of
age, and of an instructional tradition that often introduces
mathematical deduction as a highly formal process with
little connection to other forms of reasoning. In a number
of ways this negative situation for the teaching and
learning of proof is the outcome of a three decades-long
embrace of Piagetian theories by those providing curriculum
leadership.
During the 1960s, Piaget's description of the
processes of intellectual development became the cognitive
theory of choice for those charged with developing
recommendations for new directions in Ontario's education
system generally (Hall & Dennis, 1968) and within the
subject of mathematics (K-13 Geometry Committee, 1967;
Mathematics Committee K-6, 1965). Calls for: curricula that
acknowledged children's stages of cognitive development, the
use of concrete materials, experiential learning through
student activity, and personal discovery of concepts, were
reflected in new and significantly revised policy documents
(Ontario Ministry of Education, 1975, 1980). Although
subsequent Provincial Reviews of mathematics teaching and
learning, showing continued widespread use of
teacher-centred direct instruction, suggest that most of the
Piagetian ideas failed to take hold, two aspects of Piaget's
theory, structuralism and the age ranges for the stages of
intellectual development, in a narrow and restricted
fashion, found favour with teachers and continue today to
inform their instructional planning.
Piaget, claiming that the "algebra of logic"
provides "a precise method of specifying the structures
which emerge in the analysis of the operational mechanisms
of thought" (1953, p. xviii) makes a connection between the
predicate calculus (logic) and the processes of "natural"
reasoning. In fact, Piaget goes further and states that "the
mother structures of the Bourbaki", the set theoretic
foundations of mathematics, "correspond to coordinations
that are necessary to all intellectual activity" (1970, p.
27). Studies of thinking in disciplines considerably distant
from mathematics, research in the history and ethnology of
mathematics, and accounts of mathematical activity by
present day research mathematicians, all contradict this
identification of formal mathematical deduction as the sole
model for mature rational thought. Despite this evidence,
mathematics teachers find Piaget's equivalence between
thinking and mathematical structure reasonable and pleasing.
It identifies a mode of thought in which they feel at home
as the pinnacle of intellectual development; a state towards
which, it is assumed, all humans are headed, although some
may not achieve this final target.
In Ontario, a simplistic and overly prescriptive
interpretation of Piaget's theories has had effects beyond
encouraging the view that mathematical formalism emerges
from natural mental activity. Piaget and Inhelder set out a
theory of intellectual development (1969) that describes a
child's increasing mental abilities in terms of a sequence
of levels and stages, each with an observed average age
range. Between the ages of eleven and fifteen, children are
seen to progress from a concrete operational level, where
"operations relate directly to objects and to groups of
objects", to "the beginning of hypothetico-deductive or
formal thought" (p. 132). Following this picture of
cognitive development, Ontario's 1985 secondary school
mathematics guideline asserted "that the ability to think
hypothetically, although encouraged by experience, appears
to be a product of maturation" (Ontario Ministry of
Education, 1985, p. 41), and held explorations of proof
until Grade 10 (age 15 years). Once this age is reached, the
guideline invites teachers and textbook authors to take a
formal approach with the advice that, "attention should be
directed to the role of the building blocks of a proof
(undefined terms, definitions, assumptions), with geometry
used as a model for a logical system" (p. 40). Assuming that
their students have collectively reached Piaget's stage of
formal operations, Grade 10 teachers often lead geometry
lessons that employ painful detail to prove the obvious.
Frequently the focus is on the format or physical layout of
the proof rather than the underlying thinking.
Studies (Lawson & Renner, 1974; Shayer &
Wylam, 1978) have challenged the predicted pace of
intellectual development taken from Piaget's work and shown
that within traditional school programs the majority of
students at age 15 are still working at a concrete
operational level and thus can be expected to have
difficulties with formal proofs. In fact, this situation
persists throughout the secondary school years. On the other
hand, carefully designed programs that provide introductory
experiences in formal reasoning for students in the seventh
and eighth grades can accelerate pupils' progress through
Piaget's stages (Adey & Shayer, 1994).
Although Ontario's new curriculum documents
restrict the study of proof to senior level university
preparation courses, they, in all years, call for increased
attention to mathematical thinking. From the earliest grades
teachers are to "ensure that students talk about their
reasoning" (Ontario Ministry of Education and Training,
1997, p. 6), and beginning in Grade 7, students are to
"engage in inductive and deductive reasoning as they make
conjectures and seek to explain why they are valid" (p. 76).
Taking these directions as an invitation, a project has been
initiated to develop and test mathematical environments in
which Grades 7 and 8 students collectively participate in
activities that provide initial experiences with proof.
A mathematical environment begins with a context
that poses a problem or from which questions naturally
arise. A rich context motivates pupils to ask, "Why" and
"What if" questions so that subsequent mathematical
explorations belong to the class as a whole rather than
exclusively to the teacher. The problem itself and the
materials (blocks, counters, graph paper) and tools (rulers,
compass, calculators, computers) available to the class
suggest potentially valuable methods for exploring the
questions that arise. Students are grouped and seated in
arrangements that encourage conversation and debate about
the problem and experiments. This discussion allows students
to test out tentative ideas and construct arguments in a
collaborative setting before presenting their conjectures to
the whole class. Thus, while on the surface it may appear
that the class is participating in an open-ended activity,
in fact, the teacher has structured the environment to
restrict the range of options and focus the students'
efforts.
Since the teacher and textbooks do not serve as
authorities, verifying ideas that arise from the class, the
context, materials and tools must provide the means for
students to test conjectures. In particular, there needs to
be some way for the pupils to construct examples of the
phenomena under discussion to provide evidence for their
hypotheses or counter-examples that can lead to revised
conjectures. Class activity moves back and forth between
group work, where ideas are developed and tested, and whole
class discussion, where tentative theorems or problem
solutions are presented and defended.
In these social constructivist environments,
communication is markedly changed from that in a traditional
mathematics classroom. The pattern is no longer that of a
teacher posing questions, students answering, and then
teacher in turn assessing the pupils' responses. Now the
talk becomes a conversation involving the problem,
materials, tools, all the students and the teacher. Both
questions and the validity of answers come from the
environment. All participants are responsible for posing
problems, suggesting solutions, testing conjectures, and
constructing arguments. Taking Vygotsky's view that,
"functions are first formed in the collective as relations
among children and then become mental functions for the
individual" (1981, p. 165), it is hoped that such activities
will help students develop the formal operations employed in
the construction of mathematical proofs.
In experiments conducted to date, class progress
has generally followed that outlined by Lakatos (1976) in
his analysis of the processes of mathematical discovery.
Experiments that produce multiple examples of the question
situation reveal patterns from which pupils make
conjectures; "I think that ... always happens". Groups then
spend time constructing and testing further examples or
attempting to develop arguments for their conjectures.
Students' competitive natures often motivate the desire to
locate counter-examples to the conjectures posed by their
peers. In doing so, they are involved in specialization
(Mason, Burton & Stacey, 1985), the construction of
particular cases that highlight the details of the problem
situation that will need to be addressed in any future
argument or proof. The location of counter-examples or
failure to produce complete arguments for a conjecture
result in new, revised hypotheses. Through repeated cycles
the class moves toward a generalization and supporting
reasoned arguments or the initial stages of a proof.
Field tests so far have shown that it is
possible to develop mathematical environments that support,
with limited teacher intervention, inductive processes in
which pupils experiment and generate conjectures. It has
proven more difficult to encourage students to formalize
their work and construct summary deductive arguments.
Children appear to agree with Mandelbrot's (1992) view that
the house building of mathematical exploration is much more
satisfying than the house-cleaning of mathematical
proof.
The presence of counter-examples encourages
pupils to question the hypotheses and arguments advanced by
peers. When counter-examples are difficult or in fact
impossible to construct, challenges to faulty or incomplete
reasoning do not naturally arise. Treating mathematics as an
empirical science, the class in this situation usually takes
the conjecture to be proven. Grades 7 and 8 pupils do not
appear to possess the skepticism that motivates a careful
analysis of the arguments advanced by others. Thus the
teacher is forced to provide the calls for further
development and clarification of students' reasoning.
Although this questioning can provide a valuable model of
the care required in the development of proofs, it can also
appear to students to be part of a rather meaningless game.
A remaining challenge in the project is to find a balance
between rigour and meaningfulness.
References
Adey P., Shayer M. (1994) Really raising
standards. London: Routledge.
Hall E. M., Dennis L. A. (Co-chairs). (1968)
Living and learning: The report of the provincial
committee on aims and objectives of education in the schools
of Ontario. Toronto: Ontario Department of
Education.
K-13 Geometry Committee (1967) Geometry:
Kindergarten to grade thirteen: Report of the (K-13)
geometry committee. Toronto: Ontario Institute for
Studies in Education.
Lakatos I. (1976) Proofs and Refutations: The
Logic of Mathematical Discovery. Cambridge: Cambridge
University Press.
Lawson A. E., Renner J. W. (1974) A
quantitative analysis of responses to Piagetian tasks and
its implications for curriculum. Science Education
58(4), 454-559.
Mandelbrot B. (1992) Fractals and the Rebirth of
Experimental Mathematics. In H-O. Peitgen, H. J rgens, &
D. Saupe (Eds.) Fractals for the Classroom: Part One:
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Springer-Verlag.
Mason J., Burton L., Stacey K. (1985)
Thinking Mathematically (revised ed.).
Wokingham, England: Addison-Wesley.
Mathematics Committee K-6 (1965) Mathematics:
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programme (K to 6). Toronto: Ontario Curriculum
Institute.
Ontario Ministry of Education (1975) Education in
the primary and junior divisions. Toronto: Author.
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curriculum guideline for the intermediate division.
Toronto: Author.
Ontario Ministry of Education (1985) Curriculum
guideline: Mathematics: Intermediate and senior
divisions. Toronto: Queen's Printer for Ontario.
Ontario Ministry of Education (1999) Mathematics
2000: Grades 11 - 12: Curriculum policy document
(draft). Toronto: Author. available: http://www.enoreo.on.ca/ecoo/curric/math_main.htm
Ontario Ministry of Education and Training (1997)
The Ontario curriculum: Grades 1-8: Mathematics:
1997. Toronto: Queen's Printer for Ontario.
Ontario Ministry of Education and Training (1999)
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1999. Toronto: Queen's Printer for Ontario.
Piaget J. (1953) Logic and psychology.
Manchester: Manchester University Press.
Piaget J. (1970) Structuralism (C. Maschler,
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1968)
Piaget J., Inhelder B. (1969) The
psychology of the child (H. Weaver, Trans.). New York:
Basic Books. (Original work published 1966)
Shayer M., Wylam H. (1978) The distribution of
Piagetian stages of thinking in British middle and secondary
school children. II - 14- to 16-year-olds and sex
differentials. British Journal of Educational
Psychology 48, 62-70.
Vygotsky L. S. (1981). The genesis of higher mental
functions. In J. V. Wertsch (Ed.) The concept of activity
in Soviet psychology (pp. 144-188). Armonk, NY:
M.E. Sharpe.
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