Learning to Prove: A Taxonomy of Objectives
Vincenzo Liberatore
Department of Computer Science,
Rutgers University,
Hill Center, Busch Campus,
New Brunswick, NJ 08903.
E-mail: liberato@paul.rutgers.edu
Abstract
We consider the educational objective of learning how to prove a
statement in a mathematically rigorous way and we break down this
educational objective into several partial objectives following
the standard taxonomy of [Bloom56]. Three advantages stem from
this approach: first, the partial objectives are more manageable
in the process of curriculum development; second, it is easier to
individuate the teaching strategies that are most effective for
each educational objective; and, finally, we discover some objec-
tives that are often neglected, but are in fact surprisingly leg-
itimate - these include the rote learning of proofs, the abili-
ty of interpreting mathematical statements, the analysis of
proofs and proof techniques, and the ability of reading the
current literature.
1. Introduction
A most important educational objective for most students in the
technical and scientific disciplines is to grasp the method of
proving a statement mathematically and rigorously; in this sense,
mathematical proofs are contrasted with experimental results and
with assertions by intuition.
In practice, most students at the undergraduate level are not ex-
posed to operational techniques of mathematical proving. In most
courses, proofs are expounded by the instructor, but the students
are not required to know those proofs, to modify them, or to
create new proofs. The principal educational objective is the
application of the theorems and, as far as mathematical reasoning
is concerned, the sole outcome of this method is the intuition
that it is possible in principle to prove a statement mathemati-
cally. The main assumptions of this paper are that the ability
to prove a statement mathematically can be learned just like any
other ability, and that this broad objective can be broken down
into partial objectives exactly like any other educational objec-
tive. The emphasis of this paper is on the general logical
thinking necessary to prove mathematical statements; the applica-
tion to mathematical logic is a particular instance of this
methods.
The educational objective "to learn how to devise a convincing
mathematical proof" is extremely broad; for this reason, it only
points out a general policy and a criterion for evaluating curri-
cula. This broad objective should be more accurately defined if
we desire to determine the specifical goals to be achieved in a
course and if we wish to define more clearly the outcomes of the
educational process. In this paper, we attempt to break down
this objective in partial objectives following the standard tax-
onomy of [Bloom56]. Throughout the paper we will recall defini-
tions and observations from [Bloom56] so as to make our presenta-
tion accessible to a wider audience. However, we briefly recapi-
tulate the main points of the taxonomy in the remainder of this
section.
The taxonomy "is intended to provide for classification of the
goals of our educational system" [Bloom56], and for convenience
it is shortly sketched in the table below. Our definitions are by
necessity very short, and our examples are far from being exhaus-
tive; we refer to [Bloom56a] for more details.
Objective Description Example
- --------- ----------- -------
Knowledge recall knowing technical terms
Comprehension translation, predicting continuation
explanation, and of trends
predicting consequences
Application usage in new and apply science principles to
concrete situations new situations
Analysis the breakdown into distinguishing facts from
constituent elements hypotheses
Synthesis the putting together of mathematical discoveries
elements so as to form
a whole
Evaluation judgements about the comparison of major theories
value of learning
Although this taxonomy is not a hierarchy of objectives, the ob-
jectives reported later in the hierarchy will often be built on
the achievement of previous educational objectives. Also, some
taxonomy objectives are partly overlapping [Bloom56a]; we will
observe an example of this phenomenon in the appendix.
We remark that we do not make any recommendation about the
development of a curriculum because curricula would have to take
into account, in addition to the educational goals reported in
this paper, also the information available about the students,
and in particular issues regarding their present development,
needs, and interests. However, we believe that in any mathemati-
cal course a subset of the objectives below should be emphasized,
the choice of the specific subset depending on the students' pro-
file.
2. Knowledge
The first educational objective that we examine pertains the ac-
quisition of knowledge, that is, the development of the ability
to recall information about proofs and proof techniques.
Knowledge, as defined in the taxonomy, involves mainly the
psychological processes of remembering; by contrast, the objec-
tive examined in the following section will involve processes
like relating, judging, and reorganizing. Knowledge is often
neglected as an educational objective in the context of teaching
mathematical reasoning. Nevertheless, knowledge can be deemed a
legitimate educational goal as it is a prerequisite to all other
advanced educational objectives in mathematical reasoning. For
example, knowledge of established proof techniques form a reper-
toire of methods for attacking mathematical and non-mathematical
problems.
A very basic educational objective is the knowledge of proof ter-
minology, classification, and methodology; words like theorem,
lemma, corollary, proof, entailment, and contradiction are a sam-
ple of crucial phrases and the methods these terms define are
fundamental.
In the category of knowledge, the objective that most educators
would probably agree on is the acquisition of the major ideas and
general schemes and patterns exploited in proofs. For example, a
broad objective is the knowledge of the structure of a proof by
induction. The generality of these objectives often causes great
difficulty in learning because students are not familiar with the
specific proofs that the general schemes intend to describe
[Bloom56]. We therefore conjecture that more basic educational
objectives should precede the learning of universal proof
schemes.
The preliminary objective we assume is the knowledge of specific
proofs (objective 1.12 in [Bloom56]), which is to say that students
should be able to recall some proofs in all their details. This
objective demands that a list of proofs be given to the students,
and that the students then be asked to repeat some of those
proofs in a detailed and rigorous way. In passing, we remark
that currently this objective is earnestly pursued in some Euro-
pean universities where freshmen in Engineering departments have
to memorize between 100 and 120 mathematical proofs for each one
of their basic courses in Calculus and Algebra. We believe that
a lighter version of such a requirement might be extremely useful
in basic mathematical courses if accompanied by the simultaneous
testing for the comprehension objectives that we will describe
below.
3. Intellectual Abilities and Skills
In no field is knowledge alone commonly regarded as the final
educational objective. In this section, we examine the educa-
tional objectives that aim at the development of intellectual
abilities and skills.
3.1. Comprehension
At this level, the student should be able to explain passages in
the proofs, possibly paraphrasing them and giving examples, and,
conversely, she should be able to extract a proof from a set of
examples and rewrite arguments into a more compact form. She
should be able to defend her exposition of a proof against criti-
cal comments and to give examples of each proof passage.
The student should be able to translate symbolic formulas into
verbal form and vice versa. A very common problem for students is
to determine what is exactly given by the hypothesis of a state-
ment, and what exactly is required to be proven. In particular,
an educational objective is to enable the student to translate
the statement of a proposition into the mathematical entities
whose existence is given or required. The student can sometimes
accomplish this simply by replacing the definitions for the terms
that appear in the proposition statement, but often she will have
to examine the relationships among the given mathematical enti-
ties. More complex educational objectives are impaired if the
student is not able to determine what exactly is given in the hy-
pothesis and what exactly has to be proven [Bloom56]. Moreover,
the student should be able to grasp the sequence of implications
and quantifications of the statement she has to prove.
The student should be able to distinguish among warranted, unwar-
ranted, or contradictory conclusions drawn from the premises.
Test scores differentiating between "crude errors", "going
beyond the premises", and "over-caution" could be especially
useful in monitoring a student progress [Bloom56].
A student might sometimes come across a proof step that can be
applied to situations of more general nature than the proof of a
certain particular result. The student should be able to gen-
eralize such passages to applications not strictly needed in the
proof, but immediately obtainable, and she should be able to do
so even if these applications are drawn from a domain different
>from the one addressed by the proof.
She should also be able to carry out a proof when the proof tech-
nique to be used is suggested to her. Hints and directions need
not be limited to suggestions for a good approach for solving the
problem, but could also point out that some strategies are unpro-
ductive [Gagne85].
The author has gained some experience in teaching a class that
emphasizes among others the objectives listed in this section,
and we believe that these objectives are not unrealistic for col-
lege freshmen and sophomores. Moreover, we think that the stu-
dents gained a better understanding of proof methodologies and we
conclude that this method was generally successful. We will now
describe the most critical problems that students found in the
process of pursuing these objectives. A first difficulty is en-
countered when students try to express an argument in a general
verbal form; in particular, students tend to have problems when
asked to justify a proof step with some general argument rather
than by examples. Problems also arise when students have to jus-
tify a proof passage by citing the correct previously known
result. Another common difficulty is the comprehension of ques-
tions that are phrased differently from those in previous assign-
ments. All these problem sometimes confuse the students on what
is required of them and special care has to devoted in formulat-
ing questions that test the achievement of the comprehension ob-
jectives.
3.2. Application
In the taxonomy, the application objectives is the goal of apply-
ing what the student has learned, but with the following remark:
the major difference between the objective of comprehension and
application is that now the student should be able to carry out a
proof without being prompted to use any specific method, so that
he will have to find a way to attack the problem on his own. The
achievement of some of the educational objectives pointed out in
section 2 and 3.1 is a necessary prerequisite of the application
objectives [Gagne85].
Testing for the application objective requires some special at-
tention. Tests consist of requiring the students to prove a
mathematical statement; usually, these statements can be selected
>from material unknown to the student or can be similar to a prob-
lem known to the student, but with a new slant he is unlikely to
have thought of. A test should also present the student with
some preliminary questions to ascertain whether he has correctly
understood the statement to be proven, and questions asking to
state explicitly the chosen proof methodology.
We now compare and contrast the teaching methodologies that are
most effective for the objectives of knowledge and of applica-
tion. Suitable methods for the presentation and the acquisition
of knowledge are lectures, printed material, and the like; these
methods are usually preferred because of their simplicity
[Bloom56]. The most effective way to achieve the application ob-
jectives is the method of guided discovery, that is, by proceed-
ing a step at a time and demonstrating the consequences that a
particular decision, method, or strategy would bring about [Kato-
na67]. The methods of lecturing and guided discovery can be com-
pared as follows. Lecturing, as we remarked above, is a perfect-
ly suitable way to achieve objectives related solely with
knowledge, but it was found that the pure demonstration of
several solutions of similar problems is the least effective
teaching method to pursue problem solving objectives. On the oth-
er hand, guided discovery is the most effective teaching method
in this situation. Finally, problem solving without assistance
is often too difficult for many subjects [Katona67], and it is
not recommended for pursuing application objectives.
3.3. Analysis
At this level, the student should be able to identify and distin-
guish conclusions from supporting statements, relevant from ex-
traneous material, necessary steps from corollaries, factual from
normative statements, logical fallacies in a proof, questionable
term usages, unstated assumptions, and the purpose of a proof
passage or a proposition. The student should also be able to re-
late analogies and differences among proofs and proof techniques.
She should also be able to identify the fundamental part of a
proof from the technical details and grasp the structure of a
proof. She should be able to outline the main points of a proof
and correlate them. Finally, she should acquire facility in
reading a well-structured paper, as for example those published
in journals.
3.4. Synthesis
An educational objective for this level is the ``ability to make
mathematical discoveries and generalizations'' [Bloom56]. When
synthesis is the main objective, the educational approach should
be as non-direct and stimulating as possible, the reason being
that at this level the greatest amount of intellectual growth
should occur. In particular, the student should feel free from
tension and from pressure to adopt a particular viewpoint, and
should be free to set his own objectives [Bloom56].
3.5. Evaluation
At this level, the students should be able to judge the value,
significance, and relevance of a proof. He should be able to sum-
marizes proofs and whole lines of thought, to compare different
proof techniques, and to discuss the application of proof
methods.
4. Conclusions
In this paper, we have given a detailed taxonomy of partial ob-
jectives implied by the necessity of teaching how to prove a
statement in a mathematically rigorous way. We found that some
objectives are surprisingly legitimate and fall in established
teaching methodologies --- these include the rote learning of
proofs, the ability of interpreting mathematical statements, the
analysis of proofs and proof techniques, and the ability of read-
ing the current literature. We also discovered that different
objectives require very different teaching methodology; in par-
ticular, application and synthetic abilities cannot be effective-
ly taught by lecture methods commonly used to pursue knowledge
objectives.
Acknowledgments
We would like to thank Ann Yasuhara for many comments.
References
[Bloom5] Benjamin S. Bloom, editor. Taxonomy of Educational Objectives,
volume 1. McKay, New York, 1956.
[Gagne85] Robert M. Gagne'. The Conditions of Learning and Theory of
Instruction. Holt, Rinehart and Winston, New York,
4th edition, 1985.
[Katona67] George Katona. Organizing and Memorizing. Studies in the
Psychology of Learning and Teaching. Hafner, New York, 1967.
[Rosen95a] Kenneth H. Rosen. Discrete Mathematics and Its Applications.
McGraw-Hill, NY, third edition.
[Rosen95b] Kenneth H. Rosen. Instructor's Resource Guide to Discrete
Mathematics and Its Applications. McGraw-Hill, NY, third edition.
Appendix: An Example
One of the main conclusions of this paper is that the same topic
could be presented in very different ways, depending on the in-
tended educational objectives. In this section, we will illus-
trate how the same theorem and proof can be approached in quite
different ways according to the educational goals.
In the first levels of the taxonomy, the educational approach is
very well-defined, but, as we move on to subsequent objectives,
the teaching strategy becomes less and less rigid, and in the
more advanced levels, the approach is highly dependent on the
context of instruction and on the students' reaction. Therefore,
only the first four levels (knowledge through analysis) will be
considered in this example, as it is much more difficult to con-
fine the other levels in only one case study.
We will use for our example the Pascal's identity and its proof.
Although the proof is rather short, it is complicate enough to
cause students' uneasiness [Rosen95b]. Also, Pascal's identity
is a very important combinatorial equality, and its proof is the
prototypical examples of a combinatorial proof. Usually Pascal's
identity is introduced in a basic course in discrete mathematics
or combinatorics after the sum and product rules, the definition
of permutations, combinations, and binomial coefficients; we will
assume that the students are familiar with all these concept. In
the discussion of the proof we will use the following notation by
now standard in combinatorics: [n] is the set {1, 2, ... , n} and
C([n],k) is the collection of all subsets of [n] of size k.
If the educational objective is the knowledge of the result and
of the proof, then the instructor could expound the following ar-
gument, adapted from [Rosen95a].
Theorem(Pascal's Identity) Let n and k be positive integers with
n greater than or equal to k. Then C(n + 1,k) = C(n,k - 1) +
C(n,k).
Proof. Let S be a set containing n+1 elements, a in S, and T = S
- - {a}. There are C(n + 1,k) subsets of S containing k elements.
However, a subset of S with k elements either contains a together
with k-1 elements of T, or contains k elements of T and does not
contain a. Since there are C(n,k - 1) subsets of k - 1 elements
of T, there are C(n,k - 1) subsets of k elements of S that con-
tain a. And there are C(n,k) subsets of k elements of S that do
not contain a, since there are C(n,k) subsets of k elements of
T. Consequently, C(n + 1,k) = C(n,k - 1) + C(n,k). QED
Then, in a test addressing the knowledge objective, the students
will be required to repeat the proof as it is. The next level is
the comprehension objectives. When the instructors wishes to test
for the comprehension of the proof, he could for example ask the
following questions:
1. List C([3],2), C([2],1) and C([2],2). Is it true that
C([3],2) = C([2],1) U C([2],2)? Is it true that
C(3,2) = C(2,1) U C(2,2)?
2. Recall that binomial coefficients count the number of subsets
that can be extracted from a certain universe and that have a
certain size.
What does C(n,k-1) count in the proof of Pascal's identity?
What does C(n,k) count? What does C(n+1,k) count?
3. How is a chosen among all elements of S? Why is it always possible
to choose such an a?
4. Where is the sum principle used? How do we know that the sets are
disjoint?
5. Where is the multiplication principle used?
6. What is wrong in the following argument: "The subsets of S of
size k containing a are subsets of S and have k elements,
so there are C(n+1,k) of them"?
7. Show that if n is a positive integer, then C(2n,2) = 2 C(n,2) + n^2
with a combinatorial argument.
Some of the questions above deserve some comments. The two ques-
tions about the sum and product rule belong to the comprehension
objective as long as the verbal phrasing of the two principles
has been emphasized. Otherwise, these questions are probably too
hard for the comprehension objective and belong more properly to
the analysis objective - as we noted in the introduction the tax-
onomy objectives are sometimes partly overlapping. Analogously,
question 7 would appear to belong to the application objective,
but moves to the comprehension objective because of the similari-
ty of the answer with the proof of Pascal's identity.
Suppose now that Pascal's identity is exploited to pursue appli-
cation objectives. We assume that students have already mastered
the main ideas pertaining combinatorial proofs, as well as other
proof techniques, and we want to use the method of guided
discovery. Then, the student is given only the statement of the
theorem, and asked to prove it, without suggesting that a com-
binatorial proof is the best way to attack the problem. After a
while, if the student did not obtain the solution, the best hint
is probably to demonstrate that an algebraic proof leads quickly
to quite complicate calculations. Now, the student has to come
up with another proof method. If he tries induction, we could
point out that singling out one element of S is in fact a form
of induction. The student and the instructor then continue alter-
nating ideas and hints, until the final proof is obtained. The
interaction between the student and the instructor is usually su-
perior to the lecturing method. The interactive method is also
superior to the following common strategy: the instructor ex-
pounds several combinatorial proof, and then asks the student to
obtain an original combinatorial proof of a new result without
assistance. Unfortunately, if the new proof differs significant-
ly from those in the student's repertoire, many subjects will
find the assignment too difficult; but if the proof closely fol-
lows other known proofs, then the assignment does not qualify for
the pursuit of application objectives. However, most students
should be able to carry out proofs without assistance once the
method of guided discovery has been used by the instructor.
Eventually, at the analysis level, the students should be able
for example to relate analogies and differences between the proof
of Pascal's identity and other known combinatorial proofs, and
the difference and analogies between the proof of Pascal's iden-
tity and a proof by induction.
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