PROOF,
PROOFS, PROVING AND PROBING:
RESEARCH RELATED TO PROOF
David A Reid
Acadia University
Research
on proof and proving in mathematics education makes use of several different
meanings for the words “proof” and “proving”. In some cases this can
lead to seeming contradictions in research findings.
This paper identifies four current usages under the headings “the
concept of proof”, “proofs”, “proving” and “probing”. These refer respectively to a belief that proof leads to
absolute certainty, the written style of published proofs, deductive
reasoning, and quasi-empirical investigations in mathematics. Areas in need of
additional research are identified in terms of these four meanings.
One
research study (Maher and Martino 1996 p. 195) reported a fifth grader's
elegant proof by cases…. Proof by contradiction is also possible with young
children. (NCTM 2000 p.59)
The
majority of the [high attaining 14 and 15 year old] students were unable to
construct valid proofs in [the domain of number and algebra]. (Healy &
Hoyles 2000 p. 425)
If
… all the propositions [mathematics] enunciates can be deduced one from
another by the rules of formal logic, why is not mathematics reduced to an
immense tautology? The syllogism can teach us nothing essentially new, and, if
everything is to spring from the principle of identity, everything should be
capable of being reduced to it. Shall
we then admit that the enunciations of all those theorems which fill so many
volumes are nothing but devious ways of saying A is A! (Poincaré 1905 p. 5)
Even
within the context of such formal deductive processes as a priori
axiomatization and defining, proof can frequently lead to new results. To the
working mathematician proof is not merely a means of a posteriori
verification, but often also a means of exploration, analysis, discovery and
invention. (deVilliers 1990 p. 21)
The past ten years has seen an increase in attention paid to proof and proving by researchers in the psychology of mathematics education and in documents such as the National Council of Teachers of Mathematics’ Principles and Standards (NCTM 2000). Surveying this literature can be a dizzying experience, juxtaposing claims that secondary students do not prove (e.g., Fischbein & Kedem 1982, Senk 1985, Healy & Hoyles 2000) with claims that primary school children do (e.g., Zack 1997, Maher and Martino 1996, NCTM 2000), and definitions of proof and proving that include “convincing argument” (Hanna, Balacheff, and Pimm 1991 p. xxxiii), “a means of coming to understand” (Schoenfeld 1982 p. 168), “investigating using deductive reasoning” (Reid 1995 p. 7), “a stylized minuet which the author dances with his
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readers
to achieve certain social ends”(Davis 1972 p. 259) and “arguments
consisting of logically rigorous deductions of conclusions from hypotheses”(NCTM
2000 p.56). In this paper I will
clarify and classify the many meaning of proof and proving in mathematics
education research, both as a way of organizing the results of past research
and as a guide to areas that are particularly in need of further research.
In so doing I am building on the work of Godino and Recio (1997) who
described some of the meanings proof has in the domains of research in
mathematical foundations, mathematics, the sciences, and in classrooms.
I
will classify the technical meanings of proof and proving in mathematics
education research under these four headings: “the concept of proof” “proofs”
“proving” and “probing”. Each
of these headings captures a meaning of either “proof” or “proving” or
both that has appeared in the research literature.
Briefly, here I use “the concept of proof” to refer to a set of
beliefs about the nature of proof and proving and their role in mathematics,
“proofs” to refer to writings that conform to the expectations of
professional mathematicians, “proving” to refer to deductive reasoning,
and “probing” to refer to investigating.
The concept of proof
Most
professional mathematicians would say that the existence of a proof for a
mathematical statement means that statement in true an absolute sense.
In the words of Fischbein and Kedem (1982) “a formal proof of a
mathematical statement confers on it the attribute of a priori universal
validity.” As proof is
central to mathematics, this makes mathematics different from all other areas
of human activity.
Some
research on secondary school students’ proving and proofs is in fact focused
on the question “Do students understand the concept of proof?”
The answer reported in most studies is “No” (Bell 1976, Braconne
& Dionne 1987, Fischbein 1982, de Villiers 1992, Finlow Bates 1994, Senk
1985, Healy & Hoyles 2000). Perhaps this is not surprising given that this
concept is precisely what makes mathematics unlike every other experience
students have had of the nature of knowledge.
Most other fields of human activity don’t have absolute truths, and
those that do (e.g., some religious practices) base those truths on faith or
authority, not on proof.
On
the other hand, unpublished data from Vicki Zack’s grade five classroom
(Zack 2000) suggests that some of her students are not far from understanding
the concept of proof. Without
being told any definition of proof one student in her class responded to the
prompt “Write about what you think about proof and proving” with “I
think proving means showing that your answer is correct and it can’t be
wrong.” Others made similar responses, suggesting that they associated the
idea of absolute certainty with proof. Reid (1995) and Healy (1997, Healy
& Hoyles 2000) suggest that school curricula may in fact undermine
students’ understanding of the concept of proof.
The question of how to inculcate this concept of proof into students is one that has not been researched extensively. Fawcett (1938) reports a successful teaching
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experiment
in teaching the concept of proof. The question can be asked, however, whether
it is in fact desirable that students should understand the concept of proof. Some mathematicians, sociologists of mathematics, and
philosophers of mathematics (e.g., Davis 1972, Lakatos 1976, Tymoczko 1986 ,
Crowe 1988) assert that understanding of the concept of proof must be limited
by the knowledge that the proofs produced by fallible mathematicians are
different from the proofs referred to in the concept of proof.
Mathematicians make mistakes. Their
mistakes are sometimes discovered, and sometimes aren’t.
The theorems that they purport to have proven then are true not in an
absolute sense, but only probabilistically, with the odds of their truth
depending of the thoroughness of the checking of the proof.
As checking is also a process limited by the fallibility of the human
beings or machines involved in it, no proof can be, on theoretical grounds,
said to be 100% correct. In
practice of course many elementary proofs have been checked so many times and
in so many ways that we can be more sure of them than of any other human
knowledge. The case for recent,
complicated, proofs is of course different, as Davis (1972) notes “Most
proofs in research papers are unchecked other than by the author” (p. 259).
Proofs
There
are sections of writing in mathematics textbooks and journals, which are
called “proofs”. They are
characterized by a particular form and style.
The proofs of school texts are different from the proofs of
professional mathematics journals, but there is sufficient unity in the styles
to justify the use of the same term for both.
Within the category of “proofs” however, it has been useful to make
distinctions between differing levels of formality of proofs.
Lakatos (1978 p. 61) points out that the proofs of professional
mathematicians can be described as pre-formal, formal, or post-formal. Pre-formal proofs appear in working notes and conversations,
and can involve hidden assumptions, analogies, and informal language and
notations. Blum & Kirsch
(1991) use almost the same term “preformal” to describe proofs produced by
students that use common “intuitions” as hidden assumptions. Formal proofs
are suitable for publication. It is useful to extend Lakatos’ classification
slightly to distinguish between semi-formal proofs, which leave gaps in the
argument and which make use of common assumptions without comment, and
completely formal proofs in which every step and assumption is specified.
Most published proofs are semi-formal, simply because without them
proofs would become forbiddingly long. Filling
in the gaps is part of the skill of reading proofs, a skill that may be
pre-requisite to writing semi-formal proofs (Selden & Selden 2000).
Lakatos’ last category, post-formal proofs, describes proofs in
meta-mathematics about the nature of formal proofs, e.g., Gödel’s Theorem.
Writing proofs is a goal of the recent reform documents: “High school students should be able to present mathematical arguments in written forms that would be acceptable to professional mathematicians” (NCTM 2000 p. 58). Blum and Kirsch (1991) advocate the use of pre-formal proofs in schools as a stage toward semi-formal proofs. In fact, pre-formal proofs seem to come naturally to children
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(Anderson,
Chinn, Chang, Waggoner and Yi, 1997), suggesting that further research into
pre-formal proofs and the transition to semi-formal proofs would be useful.
Teaching
students to read and write proofs “that would be acceptable to professional
mathematicians” may be complicated by the difficulty mathematicians
sometimes have in deciding whether a proof is acceptable. In published proofs
“the line between complete and incomplete proof is always somewhat fuzzy,
and often controversial” (Davis
& Hersh 1981 p. 34). If all
published proofs were completely formal, this issue would not arise, as
determining the completeness of a proof would be a purely computational
problem, and students could be trained to use machines to check proofs.
Checking semi-formal proofs is an aspect of mathematical activity that
may turn out to be difficult to teach, and the way mathematicians check proofs
is worthy of additional research.
Proving
However
researchers use the words “proof” and “proving”, there is usually some
connection with deductive reasoning. There
has been extensive research on children’s reasoning.
Quite young children have been observed to be reasoning deductively
(Anderson et al. 1997, Zack 1997, Maher & Martino 1996, English 1996, Reid
1998) but their reasoning seems to depend strongly on context.
Additional research on the contexts in which children find deductive
reasoning useful is needed.
An
important question related to deductive reasoning is how the reasoner’s
awareness of her own reasoning affects the reasoning process. I have called
this awareness “formulation” and some research has begun to explore its
effect (e.g., Mok 1997, Reid 1995 1997).
Probing
Lakatos
(1976) describes a process of “proof-analysis” which is part of the cycle
of proofs and refutations that he claims is “a simple pattern of
mathematical discovery” (p. 127). For
Lakatos, the purpose of a proof is to provide a formulated target for
criticism and examination in light of counter-examples that refute it.
“Proving” for Lakatos is probing, testing the truth of a statement.
This usage is also current outside of mathematics, for example in the
phrase “the exception proves the rule” which sums up Lakatos’ attitude
towards counter-examples in mathematics nicely.
Exceptions force us to probe into the meaning of our rule and a proof
of it gives us an opportunity to see what point of weakness in our
understanding has allowed the exception to arise.
A
number of researchers have done work based on Lakatos’ understanding of
proving as probing, including Lampert (1990) and Balacheff (1991).
Lakatos claims that his version of the nature of proving is
incompatible with the concept of proof described above, which implies that
researchers working from either perspective need to be careful they are not
misunderstood as working from the other.
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Proof on the playground
As
Godino and Recio (1997) point out, outside mathematics the words “proof”
and “proving” are used in other ways.
The overlap between the everyday usage of “proving” as probing and
that usage in mathematics has already been noted.
The other common everyday usage of “prove” is in the sense of
verify. When Othello says
Villain, be sure thou prove my love a whore,
Be sure of it; give me the ocular proof:
Or by the worth of man's eternal soul,
Thou hadst been better have been born a dog
Than answer my waked wrath! (Act III, scene 1)
“proof”
and “prove” refer to establishing
the truth of a statement. Proving
in this sense is identical to verifying, and a proof is a verification.
Children on the playground may use “proof” to refer to physical
force, verbal abuse, social pressure, empirical evidence, or any other means
that convinces someone else that a statement is true.
This usage is sometime used in mathematics education research, e.g.,
Sowder & Harel’s (1998) “proof schemes,” which include
justifications based on authorities, symbol manipulating, appearances, and
examples.
Conclusion
Here
I have described five usages of the words “proof” and “proving,” four
specific to mathematics and one everyday usage.
I have distinguished between “the concept of proof”, the belief
that mathematical proofs ascribe absolute truth to the statements they prove;
“proofs”, the text objects that appear in mathematics journals an
textbooks; “proving”, deductive reasoning; “probing”, testing and
refining mathematical statements using a process that include proving and
proofs in the previous two senses; and the everyday usage of prove to mean
verify by whatever means.
In
providing these descriptions I hope to offer a way of clarifying mathematics
education research into proof and proving, which can be at times confusing due
to the varied use of terminology. I
do not claim that I have made every possible distinction, only that these
distinctions are useful in looking at contemporary research.
I
have also endeavoured to raise some questions worthy of additional research
within each of the four mathematical usages of “proof” and “proving”.
These questions include:
·
Should
students understand the concept of proof or should mathematics be presented as
a human activity without access to absolute truths?
·
Do
children in the early grades already understand the concept of proof, and if
so how can teaching practices in the later grades be changed to reinforce
rather than undermining this understanding?
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·Should
students be taught to fill in the gaps when reading semi-formal proofs, and if
so, how? How do mathematicians do this, and does their ability to do
so have implications for teaching?
·
Can
students’ preformal proofs be used as a basis for teaching them to produce
semi-formal proofs?
·
How
is children’s deductive reasoning dependent on context?
·
What
is the relationship between the formulation of deductive reasoning and the
reasoning itself?
· Is Lakatos’ cycle of proofs and refutations, in which proofs and deductive reasoning are used to probe into mathematical concepts, a useful model for mathematics teaching? What implications does it have for teaching the concept of proof?
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