The Use of
Logic in Teaching Proof: Draft
Susanna S. Epp
Department of Mathematical Sciences
Introduction: Even rather simple proofs
and disproofs are built atop a normally unexpressed substructure of great
logical and linguistic complexity. For example, in [7] I described a number of the many reasoning processes needed to
establish the truth or falsity of the following statements: (1) The square of
any rational number is rational; (2) For all real numbers a and b, if
a > b then a2 >
b2; and (3) For all real
numbers x, if x is irrational then –x
is irrational. The article cites evidence that a significant number of students
taking college mathematics courses do not bring with them an intuitive feeling
for the logic required to evaluate such statements and argues that some
explicit instruction in logical reasoning is needed in courses that require
students to engage in proof writing. However, because proofs and disproofs of
even elementary statements require a substantial base of understanding, a
“clarifying” analysis for a proof may be so complex that if students could
understand it, they would not need it in the first place. Figuring out how to
present proof construction simply enough to be intelligible yet detailed enough
to be effective is one of an instructor's greatest challenges.
The following sections contain ideas for helping
students learn to construct simple proofs and disproofs. Most are approaches I
have used myself and for which student reaction has been positive. Others are
ideas for which colleagues have reported success.
For students who come to a course with reasonably
good intuition for logical principles, merely seeing them stated and working a
few examples can be a pleasure – like the delight of the Moliere character who
learned one day that he’d been speaking prose all his life. For many students,
however, simple exposure to principles of logic is not sufficient to counteract
deeply ingrained incorrect patterns of thought, and follow-up instruction is
needed to illustrate the uses of the principles in mathematical contexts. Thus
Section I contains not only suggestions for how to take advantage of having
provided students with a brief introduction to basic principles of logical
reasoning before requiring them to make serious attempts at mathematical proof
but also advice for how to help students develop a firmer and deeper grasp of
reasoning principles as proof and disproof of various mathematical topics are
discussed. Section II offers additional strategies to guide students through
their initial proof efforts and lead them to see the desirability of expressing
proofs with care, and Section III discusses additional ways to help students
come to learn the need for proof.
I. Building on Initial
Coverage of Logical Principles
Using Puzzle Problems: To make the transition
from elementary logic to proof, it can be helpful to assign puzzle problems,
such as Raymond Smullyan's knights and knaves.[20] These puzzles posit an island where each inhabitant is either
a knight, who always tells the truth, or a knave, who always lies, but it is
impossible to distinguish knights from knaves by their appearance. Each puzzle
describes a situation in which certain inhabitants make certain statements, and
the goal is to figure out who is lying and who is telling the truth. When
solutions are discussed in class, quite a number of students make it clear that
they do not have a natural feeling for the kind of indirect reasoning needed to
solve most of the puzzles. Nonetheless, almost all students seem to enjoy the
puzzles, and working on them helps develop a basis of intuition for proof by
contradiction. Discussing the solutions serves to illustrate how inference
rules are used in practice and helps students develop a sense for the flow of
deductive reasoning, which they will use later in mathematical proofs of all
types.
Using Natural Deduction
Proofs:
John Barwise and John Etchemendy developed computer software called Tarski’s
World, named after the logician Alfred Tarski, to represent situations in a
world that consists of a grid containing a number of geometric shapes in a
variety of positions. The accompanying courseware [2,3] shows students how,
among other things, to produce natural deduction proofs of statements about the
shapes in the world. Work of Lee and
Stenning [11] supports the view that
use of these materials improves students’ ability to reason deductively.
Another teacher who uses instruction in natural deduction to prepare students
for reasoning in more general
environments is Richard L. Morrow, a middle school mathematics
coordinator with advanced training in logic.[16] When Morrow first
taught geometry to a group of gifted eighth graders, his students finished the
book a month before the end of the year. Thinking to fill in the extra time,
Morrow began the course the next year with a few weeks study of formal logic,
focusing on student construction of natural deduction proofs. While his
students said they found the work difficult at first, they eventually all
succeeded, and to his amazement they then learned the geometry so much faster
that they still finished the book a month early.
Using Disproof by
Counterexample:
In any course that asks students to write proofs, one can start by giving
students statements to identify as true or false, asking them to justify a
determination of true “as best as you can” and to support an answer of false by
providing a counterexample. One reason for beginning in this way is that most
students find it easier to understand and construct disproofs by counterexample
than to understand and construct even simple direct proofs. A second reason is
that the more experience students have in seeing that a single counterexample
disproves a universal statement, the more likely they are to understand that a
general argument is needed to show that no counterexample exists. Finally,
offering students mathematical statements whose truth or falsity they have to
determine themselves helps make the point that proof and counterexample are
first and foremost problem-solving tools.
Direct Proof: Identifying
the Starting Point and Conclusion to Be Shown: The most important initial point to
communicate to beginning students about proving a universal statement is that
they will have to move from something they suppose to something they must show.
It then becomes natural to
The most common type of mathematical statement is
universal and condition, having the form
For all elements x in a certain set, if <hypothesis>
then <conclusion>.
A
direct proof of such a statement has the following outline:
The
amazing thing about this proof technique is that merely by reasoning about a
single element x, one deduces that
the conclusion follows from the hypothesis for every element of the set – which is typically of infinite size. The
validity of the reasoning is determined by the fact that x is arbitrarily chosen, or “generic,” which means that it has all
the characteristics and only those characteristics common to every other
element of the set. Hence everything one deduces about it is equally true of
every other element of the set, and thus a descriptive name for this type of
reasoning is generalizing from the generic particular.
A dramatic way to emphasize the power of this proof
method is to show how one can use it to structure proofs involving terms one
does not even understand. For instance, given the statement “For all toths T, if T has a rath then every wade of T
is brillig,” the starting point and conclusion to be shown for a proof would be
“Suppose T is any toth that has a
rath. We must show that every wade of T
is brillig.” This transformation may seem obvious to a mathematician, but it
does not come naturally to many students.
Yet as students venture further and further into realms of mathematical
abstraction, instinctive ability to use the transformation becomes increasingly
essential to their success.
Recognizing the “Suppose”
and “To Show” in Proof by Contraposition and Proof by Contradiction: Once one has introduced
proof by contraposition and proof by contradiction as well as direct proof, one
can help students understand the differences among them by pointing out that
while for each method there is something supposed and something to be shown,
these “somethings” are dramatically different in each case. In a direct proof
one supposes one has a particular but arbitrarily chosen object that satisfies
the hypothesis, and one shows that this object satisfies the conclusion. In a
proof by contraposition one supposes one has a particular but arbitrarily
chosen object for which the conclusion is false, and one shows that for this
object the hypothesis is also false. In a proof by contradiction one supposes
that the entire statement to be proved is false, and one shows that this
supposition leads to a contradiction.
Use of Definitions: Mathematically speaking,
the most important part of a statement’s proof is how one gets from the
hypothesis to the conclusion. For most
of the proofs undergraduate students are asked to construct, the majority of this
task is achieved through a logico-linguistic analysis of definitions. The
reason is that the inner structure of a straightforward, or routine,
mathematical proof is largely determined by the meanings of the terms.
Note that, although they are frequently stated less
formally, definitions are actually bidirectional. For instance, for n to be an even integer means that “n is even if, and only if, n equals twice some integer.” Thus if we
know that n is even, we can deduce
that n equals twice some integer
(from the “only if” part of the definition), and if we know that n equals twice some integer, we can
deduce that n is even (from the “if”
part of the definition).
To answer the question “How do I show that the
conclusion follows from the hypothesis?” the prover needs an operational understanding
of the “if” direction of the definitions of the mathematical terms in the
conclusion. For example, to derive the conclusion that a certain quantity is
rational, one needs to show that it can be expressed as a ratio of integers
with a nonzero denominator. To derive the conclusion that one set is a subset
of another, one needs to show that any element in the one set is an element in
the other. To derive the conclusion that a function f is one-to-one, one needs to show that given any elements x1 and x2 in the domain for which f(x1) = f(x2),
one can conclude that x1 =
x2. Similarly, to work
forward from the hypothesis toward the conclusion, the prover needs an
operational understanding of the “only if” direction of the mathematical terms
in the hypothesis. Helping students translate the formal wording of a
definition into such operational terms is one of the most important tasks
facing a teacher in a course introducing students to proof.
One way to help students learn to
use definitions is to try to induce them to see a definition as providing a
test that has to be passed to decide whether something is the case. As soon as
a new definition is introduced, one can introduce a range of examples, phrasing
each as a question. For instance, immediately after defining rational, one can
write “Is 0.873 rational?” and simultaneously ask the question out loud. To a
student’s answer of “yes,” one can write “Yes, because” and look expectantly at
the student. The student may be surprised that additional words seem to be
called for but is generally able to supply the reason without difficulty (or
other students may help out). One can move on to slightly more complicated
examples (Is –(5/3) rational? Is 0 rational? Is 0.25252525… rational?), each
time acting as if it is taken for granted that the student answering the
question will give a reason. Soon students learn to give the reference to the
definition without prompting, and gradually they come to understand the value
of using the definition to answer such questions. By the way, if students dispute that 0 is
even (a common occurrence), one can use the occasion to emphasize that it is
the definition, and only the definition, that determines the answer.
It is also useful to discuss alternative but
logically equivalent ways to phrase definitions because it is often the case
that the truth or falsity of a mathematical statement is more apparent if one
uses one phrasing of a definition rather than another.
Another reason to discuss alternative wordings for
definitions is to compensate for the fact that quite a few students are still in
the process of developing a more sophisticated concept of variable. For
example, one way to state the definition of even is “n is even if, and only if, there is an integer k such that n = 2k,” and in the usual development of many
proofs it is important to be able to use this formulation. However, students
with a naïve understanding of variables and quantification often make mistakes
when they use it. For instance, to prove that the sum of any two even integers
is even, they represent both as 2k,
thereby only considering the case where the integers are the same. To help them
come to a more mature understanding of the definition, it is helpful (1) to
restate it less formally, as was done previously, without using an additional
variable such as k, and (2) to write
it several times using a variable but each time with a different symbol to
represent it, pointing out that it is the existence of the integer k, not the symbol used for it, that is
important.
In [21]
Tall and Vinner introduced the notion of “concept image,” which shed
considerable light on students' understanding of mathematical definitions. A
concept image for a definition is “the total cognitive structure that is
associated with the concept, which includes all the mental pictures and
associated properties and processes.” An overly narrow concept image leads to
mistaken assumptions and may result in incorrect mathematical arguments. For
students to develop concept images adequate to help them effectively evaluate
abstract mathematical statements, they need experience with a broad range of
examples for each newly defined term. They also need to become acquainted with
the diagrams and other visual representations that mathematicians use in
reasoning about the term. These might be arrow diagrams for relations and
functions, the image of a nonspecific real number and its floor sitting on a
number line for the study of the floor function, or a kind of blurry generic
fraction with an indeterminate numerator and denominator for discussions about
rational numbers.
II. Guiding Students’
Fledgling Efforts
No
matter how much one tries to prepare students for the process of writing proofs
on their own, a certain number find it very difficult. It seems that some
students cannot believe that an instructor is serious about demanding coherent
expression, while others simply have difficulty putting all the pieces together
in a way that makes sense. To learn as complex a skill as proof construction,
most students need quite a bit of individual, back-and-forth interaction with
an instructor. To the extent that one cannot act as a private tutor to every
student, one can try to devise effective substitutes. For example, one can
·
have students complete a few fill-in-the-blank proofs as part of the
process of starting to write their own;
·
supply model solutions for some of the homework problems to show
students that their individual work is really supposed to resemble the kind of
proofs that have been modeled in class;
·
suggest that students read their proofs out loud to test whether they
are written in coherent sentences;
·
discuss the kinds of errors often made in writing proofs.
Additional
strategies are discussed in greater detail below.
Student Critiques of Proofs: A number of textbooks for
“bridge” and discrete mathematics courses contain exercises asking students to
determine whether a proposed proof for a given statement is valid or not. Campbell and Baker [4] developed activities that take these exercises one step
further. Each activity “consists of a
given statement and several different proposed proofs of that statement,” some
of which are valid and some of which are not. Students are divided into groups,
and each group is given “one of the statement’s proposed proofs, with
directions…to determine if the proof is an acceptable argument,” and, if so, to
answer the following questions:
1) “What type of logical
argument did the author use (direct, contradiction, contrapositive)?
2) How well written is the
proof?
3) Was it easy to follow? Why
or why not?
4) Can you think of some specific
details which would make it clearer? If so, what are they?”
If
students determine that the proposed proof is not an acceptable argument, they
are asked to “identify all the major problems” they find with it. Each time a
group finishes evaluating one proposed proof, it is given another, until each
group has critiqued the entire collection. In the next class period, the
students and the instructor discuss the various groups’ critiques, “both on the
level of identifying major issues, as well as minor problems such as style and
clarity.” Campbell reports that “having a variety of proposed proofs, all of
the same statement, seems not only to help the students in recognizing certain
logical errors, but also in developing a language of their own, recognizing
that a statement can be correctly proven in a variety of ways, and learning the
importance of reviewing one’s work with a careful and objective eye.” She also
comments that students have benefited by becoming aware of the importance of
format and of making proofs reader friendly.
Whole-Class Proofs: One technique for increasing student involvement in the
proof-development process is for a teacher to do the writing on the board but
have the students supply the individual steps. Richard L. Morrow [16] reported that when he uses this
approach, he allows each student to give only one step so that as many students
participate as possible. He wrote that “everyone gets to absorb the step,
including its genesis or motivation, reason and role in the proof” and stated
that the process makes it so that he “can
1)
demonstrate how
to go to the final steps and work backwards, when getting stuck approaching the
proof from the beginning,
2)
knowingly allow a proof to head off in the
wrong direction and ask for suggestions on what to do when we get stuck –
something which is sure to happen to many students when working alone,
3)
demonstrate the
value of marking up a diagram before writing down the steps,
4)
show the value of
getting a holistic view of the situation before putting down the series of
steps – the right brain is especially useful in geometry proofs,
5)
watch faces and
judge how well the class or individuals are doing,
6)
demonstrate that
proofs do not need to be perfect or elegant to work,
7)
let students know
that everyone (or nearly so) is in the process of learning to do these things.”
Identifying the Crux of a
Proof: Many
of the proofs one asks students to develop depend on a single central idea.
Starting the proof-development process by trying to identify it accords with Leron's
[12] suggestion to work down from a
“top-level view of the proof.” For other proofs, however, one may only come to
realize the essential features after plowing mechanically through its details.
Coming to see the crux of a proof in this way occurs, therefore, during the
part of the problem-solving process Polya refers to as “looking back.”[17] A practiced mathematician can
easily reconstruct a lengthy proof just by recollecting its essence, but
students often have difficulty when told the main idea because they are still
struggling to master the underlying logic of proof construction. Becoming aware
that it is possible to reconstruct proofs from a few central ideas can help
motivate them to develop facility with the more routine aspects of mathematical
argumentation.
Using Informal Explanations: Hodgson and Morandi [10]
report success following an idea of Mason, Burton, and Stacey [14] to have students first develop an
informal explanation to convince a fellow student of the truth of a statement
before trying to write a proof formally. Initially, the student verbalizes the
explanation, using a tape recorder to refine it until a fellow student finds it
convincing. Then the student writes up the explanation carefully. Only after
completing these steps does the student rewrite the explanation, filling in any
necessary details and using standard mathematical language. In their article,
Hodgson and Morandi follow a student through the process as she develops a
proof that for all integers n, n(n+1)(n+2) is divisible by 6.
Student Presentations: Having students present
proofs from homework assignments for the rest of the class at the board is
especially effective if started in the very first class period after proofs
have been assigned. It is important, however, to make sure to preserve the
self-esteem of the presenters. One can thank them for being good sports when
they volunteer and point out that to the extent that they make mistakes,
discussion about them helps everyone in the class avoid similar errors in the
future. If a student’s proofs are good, the other students see that the demands
made by their instructor can actually be met by one of their own kind. If a
student’s proofs contain mistakes or sections that are not well expressed, an
instructor can ask for suggestions for improvement from the rest of the class.
A ploy is to ask students to imagine they are a research team for a large
company and that if they can collectively come up with a perfect answer they
will get to share a sizeable bonus. After the class has finished its critique
and some changes have been recorded, the instructor can take a turn, using the
opportunity both to comment on significant errors that have gone undetected and
also to show students the kinds of things the instructor will be looking for
when grading students’ work.
When I use this technique, I discuss small details
as well as larger issues, but I try to put my criticisms in perspective,
explaining frankly that certain corrections are more important than others, but
that I also care about what might seem to be relatively minor points. For
instance, if a student’s proof states that a certain number, say n, is even because it equals 2k, I would ask what was missing. Most
likely, based on the emphasis I had previously placed on definitions, one of
the other students would tell me to add “for some integer k.” I would agree, pointing out that, for example, 1=2×(1/2) and
yet 1 is not even and adding that it is not enough for n to be 2 times something – that something has to be an integer.
My primary reason for engaging in these kinds of
critiques is to provide immediate feedback on students’ proof writing, but an
important secondary reason is to counteract student anxiety about how their
proofs will be evaluated. Since there is more than one right way to construct
any given proof and since different instructors may well have different
standards of correctness, I feel obliged to try to give my students a sense of
the range of proof styles I consider acceptable and to indicate which parts of
a proof I consider most important. So when I critique student proofs, present
my own, and write proofs at the board that have been developed collaboratively
with members of the class, I discuss alternative ways of expressing the steps
that I would consider acceptable. I also talk about conventions of mathematical
writing, such as giving only part of the reason for a certain step, enough to
indicate that the writer of the proof has considered and resolved the issue but
not so much as to overload the proof with unnecessary verbiage. In addition, I
point out that the amount of detail included in a proof varies considerably
depending on the intended audience. In my courses I generally suggest that
students address their proofs to a fellow student who has missed the last few
classes.
At
Rewriting Proofs: Requiring students to
rewrite proofs until they are correct is a useful way to help students improve
their proof-writing skills. In a large class it may be impossible for an
instructor to find time to provide suggestions for improvement for the majority
of assigned problems, but it may be possible to make sure that students rewrite
at least one of each type of proof that is assigned. Nancy L. Hagelgans,
1) Have students submit
double-spaced, word-processed drafts electronically, except for first drafts
taken from tests.
2) Write comments in pencil.
3) Comment on the appropriateness
of the proof method or the lack of evident method.
4) If the method is
appropriate, comment on the argument.
5) If the argument is valid,
comment on the English composition.
6) Mention the good points: “A great
first sentence!” “Clear organization!” “Good choice of method of proof!”
“Excellent proof so far!”
7) Have conferences outside
class with a few of the weakest students after several drafts.
Some
variations she suggests are to have the whole class discuss selected first
drafts that are projected on a screen, have student pairs discuss and write
comments on each others’ first drafts in class, and have students write
comments on copies of selected first drafts. Another variation is to have
students work on proofs in groups in class and go from group to group reviewing
their work and offering hints on how to correct it.
Addressing Process Issues: To help students cope with
the often frustrating enterprise of mathematical discovery, one can encourage
class discussion about the psychological aspects of the process. For instance,
if a few students have found a counterexample for a mathematical statement that
stymied a majority of the class, one can ask the successful students to share
the thoughts that went through their minds when the counterexample occurred to
them. One can also point out that mathematical discovery may involve emotional
ups and downs, that even the best mathematicians find mistakes in their
arguments which force them to abandon one approach and seek another. For
example, work of Schoenfeld [18]
supports the view that while successful problem solvers are persistent, they
readily change to new approaches when previous ones do not appear to be
working, though they might eventually return to a previous approach if a new
attempt seems unsuccessful.[1]
To assist students in structuring their time when
they are trying to determine truth or falsity of a mathematical statement, one
can suggest that they begin by imagining they actually have an object or
objects satisfying the conditions described in the hypothesis. They can then
ask themselves whether the conclusion must necessarily follow. If, after some
effort, they do not see why this must be so, they can explore the possibility
that the statement might be false by trying to think of elements that satisfy
the hypothesis but not the conclusion. If this effort also fails, they can
posit a situation where the hypothesis is true and the conclusion is false and
try to derive a contradiction. If this method also seems to lead nowhere, the
very process of having tried it and the other approaches may have resulted in
insights that could lead to greater success when one of the previous approaches
is tried again.
III. Motivating the Need for
Proof
A
common use of proof is to affirm the general truth of properties that one has
seen to be true in some cases, thereby coming to understand the essential
reasons why the property always holds. While all introduction-to-proof courses
try to convey this point, in courses where exploration and experimentation play
a major role, it is the primary way the need for proof is introduced. For
example, in the
When students are skeptical about the need for
proof, a particularly effective way to motivate it is to have them evaluate
statements about whose truth or falsity reasonable people might reasonably
disagree. Fortunately, there are more such statements than one might think
because what is obvious to a mathematician is not always obvious to a student.
It is also possible to find relatively elementary statements upon which most
people would need to reflect in order to reach a definitive answer. Such
statements are especially effective when used for student presentations in
class. For instance, consider the statement “For all integers a, b,
and c, if a divides bc then a divides b or a divides c. If one assigns a homework problem
asking students either to prove or provide a counterexample for this statement
and then uses it for class discussion, it is common for one part of a class to claim
it is false and another to say it is true. Once when two students from each
group were chosen to go to the board to present their solutions, the result was
one false proof, one partial “proof,” one incorrect counterexample, and one
correct counterexample. Such an outcome is a powerful argument for the
importance of careful reasoning, especially if one points out that the ability
to come up with correct answers to such mathematical questions provides the
theoretical foundation to be able to engineer airplanes that do not crash,
develop encryption systems to keep transmission of credit card information
secure, and so forth.
This approach was developed as a formal teaching
method, called “scientific debate,” by a group of mathematics educators in
The approach is taken even further in courses that
use the “
Conclusion
A few years ago I had an experience with one
particular class that made a special impression on me. The class was unusually
small, only twelve students, and was the second quarter of a sequence. The
previous quarter had dealt with logic, an introduction to direct and indirect
proof, mathematical induction, and elementary combinatorics, all interwoven
with various computer science applications. The second quarter was to cover set
theory, function properties, recursion, some analysis of algorithms, relations
on sets, and an introduction to graph theory, also with an admixture of
applications. The class met only once a week but in three-hour sessions.
The small size of the class and the length of the
sessions gave me a chance to work with students more intimately than usual. I
began each period by having students discuss in groups of three or four the
homework they had prepared for that day and went from group to group talking
with each at length. Overall the class atmosphere was excellent, and several
students showed the kind of eager, enthusiastic intelligence that is a
teacher's joy. What surprised me was that as the course moved from one topic to
the next, almost all the students who had attained a relatively sophisticated
level of achievement in dealing with a previous topic made it clear that they
felt they had to struggle to succeed with the next. Yet as we worked through
their questions and difficulties, they ultimately performed excellently with
the new topic as well. Their
understanding of general methodological principles clearly made it easier for
them to learn the new material but it did not make it trivial for them.
This experience brought home to me more effectively
than any before that abstract mathematical thinking is not something that
either one is able to do or one is not able to do. Because of the experience I
have become especially conscious of the need to respect my students and never
to act surprised by their questions. Even when a student asks a question whose
answer I have already discussed, I try to respond to it as if it were fresh.
After all, nobody can concentrate 100% of the time when new ideas are coming in
fast and furiously. In all likelihood the student was not mentally prepared to
absorb the answer when I previously addressed the question. For the student to
formulate the question means that they have thought about the issue, want to
know the answer, and are probably ready to understand it. That is cause to
celebrate. It may also be that clarifying the issue at this point in the course
(if possible in a slightly different way from that presented earlier) will give
the other students in the class greater insight also.
My main advice to those teaching courses whose goal
is to develop students’ mathematical reasoning powers is to play an activist
role but recognize that achieving success is a long-term process. I have sometimes
been surprised when students who in my view fell far short of achieving the
levels of accomplishment I strive for tell me how valuable they found the
course in helping them do better work in their other courses or (I am always
pleased to hear) in their jobs.
The analogy I like to draw is of a child learning to
walk. It takes months of daily effort for most children to take their first
steps and several more months until they actually become steady on their feet.
When a child is trying to move from one stage to the next in learning to walk
and has failed several times, we don't say “Forget it.” We remain calm, good
humored, and encouraging. And when the child finally succeeds, we spare nothing
in expressing our delight.
References
1.
Alibert, Daniel and Michael Thomas. 1991. Research on Mathematical Proof. In Advanced Mathematical Thinking, David
Tall, ed.
2.
Barwise, Jon and John Etchemendy. 1994. Hyperproof. Palo Alto, CA: CSLI Publications.
3.
Barwise, Jon and John Etchemendy. 1999. The
Language of First-Order Logic.
4.
Baker, Diane and Connie Campbell. 2004. Fostering the Development of
Mathematical Thinking: Observations from a Proofs Course. PRIMUS (14) 4, 345-353.
5. Chalice, D. 1995. How to
teach a class by the modified
6.
Cobb, George, et al. 1997. Laboratories in Mathematical Experimentation : A
Bridge to Higher Mathematics.
7.
Epp, Susanna S. 2003. The Role of Logic in Teaching Proof. Amer. Math.
Monthly (110)10,
886-899.
8.
Hagelgans, Nancy. 2005. Learning to Prove by Rewriting. Presentation at the
2005 Joint Mathematics Meetings,
9. Henderson,
Pete. 2006. Nifty Examples for Discrete Mathematics.
http://blue.butler.edu/~phenders/dmresources/
10.
Hodgson, Ted and Pat Morandi. 1996. Exploration, Explanation, Formalization: A
Three-step Approach to Proof. PRIMUS (6)
1, 49-57.
11. Lee, John and
Keith Stenning.
1996. Cognitive Processes Involved in Learning Logic. Proceedings of the International Symposium
on Teaching Logic and Reasoning in an Illogical World.
12.
Leron, Uri. 1985. Heuristic Presentations: The Role of Structuring. For the Learning of Mathematics 5 (3),
7-13.
13. Levine, Alan and Ben
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[1] The Nova progam “The Proof,” which describes Andrew Wiles’ discovery of a proof for Fermat’s Last Theorem, provides a powerful example for the effectiveness of this strategy.