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Somepedagogicalaspectsofproof
ArticleinInterchange·February1990
DOI:10.1007/BF01809605
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Interchange, Vol. 21, No. 1 (Spring, 1990), 6-13
Some Pedagogical Aspects of Proof
Gila Hanna
The Ontario Institute for Studies in Education
In discussing the role of proof in mathematics education it seems helpful to distinguish
among different perceptions of proof. It is particularly useful, I believe, to consider the
following three aspects, around whichi will organize my comments.
1. Formal proof: proof as a theoretical concept in formal logic (or metalogic), which
may be thought of as the ideal which actual mathematical practice only approximates.
2. Acceptable proof: proof as a normative concept that defines what is acceptable
to qualified mathematicians.
3. The teaching of proof: proof as an activity in mathematics education which serves
to elucidate ideas worth conveying to the student.
Formal Proof
A formal proof of a given sentence is a finite sequence of sentences such that the first
sentence is an axiom, each of the following sentences is either an axiom or has been derived from preceding sentences by applying rules of inference, and the last sentence is the
one to be proved. Such a proof is, in principle, mechanizable, and eliminates the
psychological aspects of proof. The formal approach to proof was developed, in fact, to
eliminate the need for recourse to intuitive evidence and human judgment, both seen as
potential sources of serious error.
Formalism has a long history. The Greeks made a distinction between the content of
an argument and a formal structure based upon the rules of thought and independent of
the content. Aristotle and Plato may have held different views on the genesis of the formal
rules of thought, as they did on the nature of universal properties and laws in general,
but they both recognized the usefulness and power of such rules.
The promise offered by a separation of content and form was pursued by other
philosophers, notably by Leibniz (1646-1716) who believed that it was possible to elaborate
a characteristica universalis, a symbolic script with the power to express all human thought
without ambiguity. Knowledge could then be transmitted through an algebra which would
allow one to distinguish clearly between fact and fiction, truth and falsity. Should any
arguments arise, they could be easily settled: "Gentlemen, let us calculate!" Such a language,
in his view, would make truths "stable" "visible;' "irresistible," and "mechanical."
The formalization of mathematical knowledge in particular was further elaborated by
Frege (1848-1925) who devised a formal conceptual language (Begriffsschrift) embodying
explicit criteria for the validity of chains of inference. Frege also attempted to demonstrate
that all arithmetical notions could be reduced to purely logical concepts -- an approach
which Hilbert (1862-t943) took several steps further. In his Grundlagen der Geometrie
(1899), Hilbert showed that to understand the properties of concepts such as point, line,
and plane one need not rely upon knowledge of objects existing outside of mathematics.
These concepts could usefully be assumed within a formal axiomatic system that defined
what signs and combinations of signs were valid expressions and what constituted a valid
derivation. New truths about these geometrical concepts could then be found by mere derivation from the axioms and definitions of the system.
6
Interchange Vol. 21/1 e'I'he Ontario Institute for Studies in Education
SOME PEDAGOGICAL ASPECTS OF PROOF
7
Within such a formalism, there clearly can be no appeal to a metaphysical definition
of truth: the truth of a statement hinges entirely upon the axioms and the internal consistency of the system itself. Thus, to demonstrate the full power of the formal approach
to mathematics, it is necessary to be able to establish definitively, for any given formal
system, that it is or is not consistent. But this requirement proved to be unattainable when
G6del showed, in 1931, that the formalization of a theory does not ensure that its consistency can be definitively established. Furthermore G6del demonstrated that in a consistent formal system there will be theorems (true statements) which are not provable within
the system: such a system will always be incomplete. The implication is that a formalization of a branch of mathematics, for example, can never encompass all of" its theorems.
There are other limitations which a strict formalism might impose on mathematics.
Tymoczko (1986) underlined the potential sterility of a purely formal approach by describing
how it might change the work of the mathematician: The " . . . ideal mathematician chooses
a certain axiom system to investigate.., turns on valid formal deductions in the appropriate
formal s y s t e m . . , is often characterized as a game player, playing with formal systems"
(p. 45). Tymoczko pointed out that if all mathematical activity were to fit this description,
then mathematics would become a very limited science, oblivious in fact to mathematical
content. Alternatively, mathematics would become a very broad science indeed, redefined
as the investigation of formal theories in general - - in which case, as Quine put it,
mathematics might as well investigate formal theories in Greek mythology.
While G6del's findings placed severe limits upon the power of the formal approach, and
others such as Lakatos and Tymoczko have challenged it on philosophical grounds, these
concerns have not diminished the usefulness of formal methods in the treatment of segments
of mathematics or in individual proofs. And deductive rigour, one of the essential elements
of formalism, has acquired the status of an important tool in mathematical work and an
ultimate criterion which mathematicians can fall back on when required. Yet a concern
for precision in definition and rigour in deduction was never unique to the strictly formalist approach. This concern was shared by the togicist, formalist, and intuitionist schools
whose important work on the foundations of mathematics gave enormous impetus to the
growth of mathematics after the turn of the century and has had a lasting effect upon
mathematical practice.
The new stress on precision and rigour was also reflected in the syllabus of the universities, where the axiomatic approach became a common denominator of most mathematics
courses. This trend was furthered in particular by the considerable influence enjoyed by
the group of talented French mathematicians who wrote under the name of Bourbaki:
The axiomatic method teaches o n e . . , to find the common ideas buried under the external
apparatus of detail appropriate to each of the theories considered, to single out these ideas,
and to exhibit them. (Bourbaki, 1971, p. 26)
It is not surprising that these developments also influenced the educators who prepared
the various "new math" curricula for the secondary schools. What is perhaps most interesting is that these educators not only assumed that rigorous proof was reflective of
mathematical practice, but also seem to have held the underlying belief that formal derivations are, in themselves, an aid to understanding and thus constitute a useful didactic device
(Hanna, 1983).
Acceptable Proof
In the last couple of decades both mathematicians and mathematics educators have begun
to reassess the role of axiomatic structures and formal proof. It has come to be more wide-
8
GILA HANNA
ly recognized, in particular, that practising mathematicians have long agreed that proofs
may have different degrees of formal validity and still gain the same degree of acceptance.
This has led to a reconsideration of what the ideal proof should be and of what one ought
to teach in the schools.
Mathematicians freely admit that a proof may lack conviction when it is shown to be
valid by virtue of its form alone, without regard to its content. And when a new mathematical
idea is in fact accepted, it is often the significance of what is proved, rather than the correctness of the proof, which carries the most weight. The acceptance of a theorem by practising mathematicians is a social process which is more a function of understanding and
significance than of rigorous proof.
It was the fallibility of social processes and of the human judgment embodied in them,
ironically enough, which provided the impetus for the development of systems of formal
validation. Yet mathematicians undeniably insist on using their judgment when deciding
whether or not to accept a new proof-- and continue to accept such proofs for their semantic
rather than their syntactic qualities. The criteria which mathematicians appear to apply,
consciously or unconsciously, are that the proof must proceed from specific and accepted
premises, must present an argument that is not flawed, and must lead to a result which,
even if unexpected, seems upon reflection to make sense in the context of other mathematical
knowledge. The implicit assumption may well be that a proof which passes these tests of
acceptability would also pass the test of formal validation, but in most cases an actual formal validation is not regarded as necessary.
It can certainly b e argued, as does Fetzer (1988), that "what makes (what we call) a
proof a proof is its validity rather than its acceptance (by us) as v a l i d . . , social processing,
therefore, is neither necessary nor sufficient for a proof to be valid" (p. 1050). Yet most
mathematicians choose to proceed with their work without concern for the formality of
their proofs. It is true that a proof is considered a prerequisite for the publication of a
theorem, but the proof need be neither rigorous nor complete. In practice, mathematicians
seem to adhere to Bateson's view that "the advantages in scientific thought come from a
combination of loose and strict thinking, and this combination is the most precious tool
of science"
A factor in the general acceptance of a new result is, of course, the value which mathematicians see in it. The same is true of proofs. Mathematicians seem to value most a proof
which has interesting implications for the branch of mathematics in which it is embedded
and yields insights into related fields. A proof is valued for bringing out essential
mathematical relationships rather than for merely demonstrating the correctness of a result.
The Teaching of Proof
In recent years many mathematics educators have actively re-examined the role of mathematical proof in the curriculum, and as a result there has been a trend away from what
has often been seen as an over-reliance on rigorous proofs. In a desire to take into account
the role of proof as a means of communication, and in recognition of the social processes
that play such a crucial part in the acceptance by mathematicians of a new result, educators
have come to place greater emphasis on the concept of proof as "convincing argument."
This trend away from formal proof in the curriculum, and the resulting search for alternative ways of demonstrating the validity of mathematical results in the classroom, has
motivated a number of studies dealing with the problem of teaching proof. Leron (1983),
concerned that most of the formal proofs found in textbooks do a poor job of communicating
mathematical ideas, suggested that such mathematical presentations would be much more
SOME PEDAGOGICAL ASPECTS OF PROOF
9
comprehensible if the proof were structured into short autonomous modules, each emphasizing one particular idea. Deploring the teaching of geometry as narrow and overly concerned with deductive proof, Volmink (1988) argued that mathematics education would be better served if the curriculum were to place greater emphasis on the social criteria for the
acceptance of a mathematical truth, at the expense of the purely formal ones.
Movshovitz-Hadar (1988) elaborates upon six different ways of presenting theorems and
six ways of presenting proofs, in an effort to enhance mathematical understanding through
what she calls the "stimulating responsive method?' Alibert (1988), on the other hand,
prefers to rely on the method of scientific debate, which provides students an opportunity
to discuss the arguments made in the course of a proof. In an extensive study of the processes involved in teaching a mathematical proof, Balacheff (1988) also points to the importance of creating classroom situations in which the student becomes aware of the complexity of the problem and of the necessity to produce valid arguments.
These authors, and others not cited here, have made a substantial contribution to our
understanding of the didactics of proof and have offered specific and interesting new ways
of teaching proofs. In their expositions, however, a proof is viewed primarily as a valid
argument, as opposed to an argument that must be both valid and explanatory. I believe
it would be useful to introduce to the discussion an explicit distinction between proofs that
prove and proofs that explain.
In what follows I will first elaborate upon the concept of proof as explanation, and then
consider its implications for the handling of proof in the curriculum, suggesting that whenever
possible we should present to students proofs that explain rather than ones that only prove.
A proof that explains and a proof that proves are both legitimate proofs. By this I mean
that both types of proof meet the requirements for a mathematical proof, and thus serve
in equal measure to establish the validity of a mathematical assertion. Both consist of
statements that are either axioms themselves or follow from previous statements (and thus
eventually from axioms) as a result of the correct application of rules of inference. They
are not necessarily different in their degree of rigour, and both types would be recognized
as valid by the mathematical community.
There is nevertheless a very important difference between these two kinds of proof. A
proof that proves shows only that a theorem is true; it provides evidential reasons alone.
It is concerned only with substantiation, with what are known as Rationes cognoscendi,
that is, why-we-hold-it-to-be-so reasons. A proof that explains, on the other hand, also
shows why a theorem is true; it provides a set of reasons that derive from the phenomenon
itself: Rationes essendi, or why-it-is-so reasons. A proof that proves may rely on
mathematical induction or even on syntactic considerations alone. But a proof that explains
must provide a rationale based upon the mathematical ideas involved, the mathematical
properties that cause the asserted theorem to be true.
The sense in which I use the term explanation in this context is perhaps best clarified
in contradistinction to that of Balacheff. In his analysis of the cognitive and social aspects
of proof, Balacheff (1988) proposed the following distinctions:
• We call an explanation the discourse of an individual who aims to establish for somebody
else the validity of a statement. The validity of an explanation is initially related to the speaker
who articulates it.
• We call proof an explanation which is accepted by a community at a given time.
• We call mathematicalproofa proof accepted by mathematicians. As a discourse, mathematical
proofs have now-a-days a specific structure and follow well defined rules that have been formalized by logicians. (p. 2)
For Balacheff, then, a proof would seem to be an explanation by virtue of being a proof.
10
GILA HANNA
Yet surely not all proofs have explanatory power. One can even establish the validity of
many mathematical assertions by purely syntactic means; with such a syntactic proof one
essentially demonstrates that a statement is true without ever showing what mathematical
property makes it true. Thus I prefer to use the term explain only when the proof reveals
and makes use of the mathematical ideas which motivate it, Following Steiner (1978), I
wilt say that a proof explains when it shows what "characteristic property" entails the
theorem it purports to prove. As Steiner put it:
an explanatory proof makes reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evident that the results depend
on the property. It must be evident, that is, that if we substitute in the proof a different object
of the same domain, the theorem collapses; more, we should be able to see as we vary the
object how the theorem changes in response. (p. 143)
•
.
.
The following example illustrates the difference between a proof that proves and a proof
that explains:
Prove that the sum of the first n positive integers, S(n), is equal to n(n+l)/2.
A proof that proves
Proof by mathematical induction:
For n=l the theorem is true•
Assume it is true for an arbitrary k.
Then consider:
S(k+l) = S(k) + (k+l) = n(n+l) + (n+l) = (n+l)(n+2)
2
2
Therefore the statement is true for k+l if it is true for k.
By the induction theorem, the statement is true for all n,
Now, this is certainly an acceptable proof: it demonstrates that a mathematical statement
is true. What it does not do, however, is show why the sum of the first n integers is n(n+l)/2
or what characteristic property of the sum of the first n integers might be responsible for
the value n(n+l)/2. (Proofs by mathematical induction are non-explanatory in general.)
Gauss's proof of the same statement, however, is explanatory because it uses the property of symmetry (of two different representations of the sum) to show why the statement
is true. It makes explicit reference to the symmetry, and it is evident from the proof that
its result depends on this property:
A proof that explains
Gauss's proof is as follows:
S
S
.
.
= 1+ 2
= n + (n-l)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
+ ...
+ ...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2S = (n+l) + (n+l) + . . .
S _ n(n+l).
2
+ n
+ 1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
+ (n+l) = n (n+l)
Another explanatory proof of this same statement is, of course, the geometric representation of the first n integers by an isosceles right triangle of dots; here the characteristic property is the geometrical pattern that compels the truth of the statement. We can represent
the sum of the first n integers as triangular numbers (see Figure 1).
SOME PEDAGOGICAL ASPECTS OF PROOF
.
,:.
1
1+2
11
,fi
1+2+3+4
1+2+3
Figure 1
The dots form isosceles right triangles containing
S(n) = 1 + 2 + 3 + . . . + n d o t s
Two such sums S(n) + S(n) give a square containing n 2 dots and n additional dots because
the diagonal of n dots is counted twice. Therefore:
2S(n) = n 2 + n
S(n)-
n2 + n
2
i_ n(n+l)
2
Another explanatory proof would be the representation of the first n integers by a staircaseshaped area as follows: a rectangle with sides n and n + l is divided by a zigzag line (see
Figure 2).
n+l
The whole area is n(n+l), and
the staircase-shaped area,
1 + 2 + 3 + . . . + n only
half, hence n(n+l)
2
4
4
1
Figure 2
Both Gauss's proof and the geometric representation show that one can adopt an explanatory approach to proof in the classroom without abandoning the criteria of legitimate
mathematical proof and reverting to reliance on intuition alone. What one must do, rather,
is to replace one proof, of the non-explanatory kind, by another equally legitimate proof
which has explanatory power, the power to bring out the mathematical message in the
theorem.
In their paper "Wann ist ein Beweis ein Beweis?" (When is a proof a proof?.), Wittmann
and Mueller (1988) characterize explanatory proofs - - of which they furnish an interesting
example - - by a term which makes reference to content (inhaltich-anschaulich). They state:
"In German we use the term "inhaltich-anschaulicher Beweis" in order to indicate that
this method of demonstration calls upon the meaning of the term employed, as distinct
from abstract methods, which escape from the interpretation of the terms and employ only
the abstract relations between them." Their terminology reminds us again that proofs that
12
GILA HANNA
rely on mathematical logic alone can never provide explanatory reasons. Such proofs look
only at the form of the argument: their whole raison d~tre is in being content-free.
The distinction I have made between proofs that prove and proofs that explain is similar
to that which Bolzano makes between "making certain" (Gewissmachung) and "building
a foundation" (Begruendung). He argues that making certain requires no more than a formal demonstration, while building a foundation demands an approach which also provides
insight into the connections among ideas.
Much has been made of the importance of convincing in mathematics education (Hirst,
1989), but a proof that convinces need not be a proof that explains: it is certainly possible
to be convinced that a statement is true without knowing why it is true. Thus it is not its
ability to convince which distinguishes the explanatory proof, convincing though we would
expect it to be.
Nor is an explanatory proof distinguished by its degree of validity: one would expect
such a proof to be an acceptable proof, as that term has been defined above. But the focus
of an explanatory proof is clearly upon understanding, rather than upon the deductive
mechanism.
Abandoning non-explanatory proofs in favour of (equally valid) explanatory ones would
not make the curriculum any less reflective of accepted mathematical practice. In fact, as
I have argued elsewhere (Hanna, 1983), mathematicians, including those who have recourse
to purely syntactic methods, are really more interested in the message behind the proof
than in its syntax, and see the mechanics of proof as a necessary but ultimately less significant aspect of mathematics. Thus there is no infidelity to the practice of mathematics if
in mathematics education we focus as much as possible on good mathematical explanations, highlighting for the students in our proof of a theorem the important mathematical
ideas that lead to its truth.
Summary and Implications
As mathematics educators it is our mission to make students understand mathematics. It
is my contention that in support of this mission we should give a more prominent place
in the mathematics curriculum to proofs that explain.
If we stress explanatory proofs our primary concern is for insight, of course. But there
is another important reason for such a stress. Students of mathematics, unlike mathematicians, have yet to learn the relative importance of different mathematical topics, and may
easily be misled on this point by a pervasive classroom emphasis on the deductive
mechanism. (Deductive mechanisms are an important facet of mathematics in themselves,
however, and require their own classroom treatment.)
A first step in shifting to the use of explanatory proofs is for educators of teachers to
stress that mathematical understanding is the goal, and furthermore that understanding is
much more than confirming that all the links in a chain of deduction are correct, that in
fact the completeness of detail in a formal deduction may obscure rather than enlighten,
and that understanding requires some appeal to previous mathematical experience.
Another important step, and a challenge for curriculum developers, is to identify suitable
explanatory proofs as alternatives to the many non-explanatory ones now in use.
Note
Some of these ideas were presented at the 13th Annual Conference of the International Group for
the Psychology of Mathematics Education.
SOME PEDAGOGICAL ASPECTS OF PROOF
13
References
Alibert, D. (1988). Towards new customs in the classroom. For the learning of mathematics, 8(2), 31-35.
Balacheff. N. (t988). A study of pupils'proving processes at the junior high school level. Unpublished paper presented at the Joint International Conference 66th NCTM and UCSMP Project, Chicago.
B0urbaki, N. (1971). The architecture of mathematics. In E Le Lionnais (Ed.), Great currents of
mathematical thought (Vol. 1). New York: Dover.
Fetzer, J. H. (1988). Program verification: The very idea. Communications of the ACM, 31(9),
1048-1063.
Hanna, G. (1983). Rigorous proof in mathematics education. Toronto: OISE Press.
Hirst, K., & Hirst, A. (Eds.). (1989). Proceedings of the Sixth International Congress on Mathematical
Education. Budapest: Janos Bolyai Mathematical Society.
Leron, U. (1983). Structuring mathematical proofs. American Mathematical Monthly, 90(3), 174-185.
Movshovits-Hadar, N. (1988). Stimulating presentation of theorems followed by responsive proofs.
For the learning of mathematics, 8(2), 12-19.
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135-151.
Tymoczko, T. (1986). Making room for mathematicians in the philosophy of mathematics. The
Mathematical InteIIigencer, 8(3), 44-50.
Volmink, J. (1988). The role of proof in students' understanding of geometry. Unpublished paper
presented at the AERA Annual Meeting, New Orleans.
Wittmann, E. C., & Mueller, G. (1988). Warmist ein Beweis ein Beweis? Unpublished paper presented
at the Annual Meeting of Mathematics Teachers, Germany.