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).


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.


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).


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.


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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