THE ROLE OF LOGIC IN THE VALIDATION OF
MATHEMATICAL PROOFS -- draft (6.15.96)
Annie and John Selden
Tennessee Technological University
Mathematics Education Resources Co.
Mathematics departments rarely require students to study very much
logic before working with proofs. Normally, the most they will offer
is contained in a small portion of a "bridge" course designed to help
students move from more procedurally-based lower-division courses
(e.g., calculus and differential equations) to more proof-based upper
division courses (e.g., abstract algebra and real analysis). What
accounts for this seeming neglect of an essential ingredient of
deductive reasoning? (Endnote 1) We will suggest a partial answer by
comparing the contents of traditional logic courses with the kinds of
reasoning used in proof *validation*, our name for the process by
which proofs are read and checked.
First, we will discuss the style in which mathematical proofs are
traditionally written and its apparent utility for reducing validation
errors. We will then examine the relationship between the need for
logic in validating proofs and the contents of traditional logic
courses. Some topics emphasized in logic courses do not seem to be
called upon very often during proof validation, whereas other kinds of
reasoning, not often emphasized in such courses, are frequently used.
In addition, the rather automatic way in which logic, such as modus
ponens, needs to be used during proof validation does not appear to be
improved by traditional teaching, which often emphasizes truth tables,
valid arguments, and decontextualized exercises. Finally, we will
illustrate these ideas with a proof validation, in which we explicitly
point out the uses of logic. We will not discuss proof construction,
a much more complex process than validation. However, constructing a
proof includes validating it, and hence, during the validation phase,
calls on the same kinds of reasoning.
Throughout this paper we will refer to a number of ideas from both
cognitive psychology and mathematics education research. We will find
it useful to discuss short-term, long-term, and working memory,
cognitive load, internalized speech and vision, and schemas, as well
as reflection, unpacking the meaning of statements, and the
distinction between procedural and conceptual knowledge.
PROOFS AND VALIDATIONS
In the mathematics education research literature, "proof" has been
used in a variety of ways. The proofs we will discuss here will be
the kind that undergraduate students normally see in their mathematics
texts and classes. These are specialized natural language arguments
which deductively establish the correctness of theorems and are
similar to, although often shorter and more detailed than, the proofs
found in mathematics research journals. Proofs have been discussed
from various points of views -- structure (Leron, 1983), explanatory
power (Hanna, 1989), students' conceptions (Martin and Harel, 1989),
students' reasoning errors (Selden & Selden, 1987), and specialized
formats (Gries & Schneider, 1995; Lamport, 1995). However, all of
these are beyond the scope of this paper, which focuses on proofs as
establishers of the correctness of theorems and the role of logic in
validating proofs.
When a mathematician reads a proof in order to check its correctness,
he/she does something quite different from reading a newspaper or
book. Between the actual reading of the sentences, a mathematician
may assent to claims, draw inferences, ask and answer questions, bring
in outside knowledge, construct visual images, or develop subproofs.
The entire proof may even be reconstructed and supplemented using
one's personal domain specific knowledge, which may itself be altered
during this construction process. Such a personal version of a proof
may be reviewed from beginning to end several times, and some
mathematicians may even produce an abbreviated version which can be
encompassed within a single train of thought, i.e., in a monologue of
internalized speech and vision, uninterrupted by significant attention
to other things.
We call this mental process *validation*. It is a form of reflection
that can be as short as a few minutes or stretch into days or more,
but in general, it is much more complex and detailed than the
corresponding written proof. Although its ostensible purpose is to
establish the correctness of a theorem, it is often used when there is
considerable reason to believe a theorem is correct. In such cases,
mathematicians act as if the theorem were in question. Indeed,
validation appears to be instrumental in mathematicians' learning of
new mathematics. Skill at validation is an implicit part of the
mathematics curriculum, but is rarely explicitly taught. Beginning
undergraduate mathematics students may well be unaware of its
existence and importance.
MEMORY AND VALIDATION
The relationship between validation, especially validation errors, and
the way mathematical proofs are normally written can perhaps best be
viewed by considering the demands on the capacity of short-term
memory, so we begin our analysis with a brief discussion of memory.
Except for processes associated with perception, there are three main
kinds of memory systems -- short-term, long-term, and working
(Baddeley, 1995). The half-life of short-term memory is approximately
fifteen seconds and its capacity is approximately seven chunks of
information (Miller, 1956). These chunks can consist of any
information that can be thought of as a unit, e.g., words, familiar
patterns of chess pieces, or the Pythagorean theorem. One is aware of
the contents of short-term memory and it is readily available for
reasoning. The rest of memory is called long-term and has a very
large capacity. However, one is not aware of its contents and it is
not directly available for reasoning. Information in long-term memory
can be activated, i.e., represented in short-term memory. Finally,
working memory consists of short-term memory, including information
activated from long-term memory, and the reasoning and control
mechanisms that swap information between short-term and long-term
memory.
In analyzing validations, which are trains of thought rather than
simple incidences of recall or inference, we will also discuss two
subcategories of long-term memory. We will call memory that lasts a
number of years, one's *knowledge base*. In the case of validation,
this is one's knowledge of mathematics and its conventions, as well as
of logic and reasoning. On the other hand, the more ephemeral,
partially activated memory arising from a validation itself, we call
*local memory*. It might include that a proof started with "Let x be
a real number" or that one has used the triangle inequality in
justifying some part of the proof. One is not continually aware of
this local memory. It is not part of working memory, but is very
easily accessible to it. Such local memory grows and persists during
a validation, but may well be forgotten when it is no longer useful.
THE STYLE IN WHICH PROOFS ARE WRITTEN
The style in which proofs are written greatly influences validation,
increasing the reliability of proofs, which are mathematicians' form
of natural language argument, and are. as in other disciplines, the
ultimate arbitrator for rationally establishing (public) knowledge.
Alternative, simplified, "clearer" styles, such at the two-column
proofs of high school geometry have been devised, and it seems
plausible that such approaches might produce more student successes
and generally improve reasoning ability. However, using such an
alternative style, students often come to view the form of proofs,
rather than their substance as what counts (Schoenfeld, 1988).
In practice, even geometry students who are successful in making
two-column proofs may not connect proofs with reasoning in other
settings. A hint of this problem came from a colleague who, in
teaching a noncredit college geometry course, reported than one of his
students incredulously asked, "You mean proofs can contain words?"
More detailed analyses have revealed successful geometry students who
fail to see the utility of deductive arguments in making geometric
constructions, even when a relevant proof, from which one could
extract the proper construction, remained in full view on the
blackboard (Schoenfeld, 1985, pp. 355-357). Another "clearer" style
with more reasons, consistent symbols, and semi-fixed format has been
developed for use in discrete mathematics by Gries and Schneider
(1995a, b).
Although the style in which proofs are written has evolved within the
mathematics community mostly without plan or design, it is very robust
and has a number of identifiable features. Perhaps the clearest of
these is that proofs are not reports of the proving process. No one
publishes in a proof that he/she originally thought, say, that
commutativity would play a key role, but subsequently discovered it
led nowhere, or that he/she had an inspiration regarding a new
concept, such as quaternions, while alighting from a bus. Such
observations, while interesting, are left to historical accounts and
works like Hadamard's on the psychology of mathematical invention
(1945).
Indeed, we will argue that, in contrast to reporting the proving
process, the main distinguishing features of proofs all tend to
minimize the probability of validation errors (by experts). As an
inescapable consequence, the validation process may take longer, the
domain knowledge required may be increased, and novice validators may
be perplexed.
Another global feature of proofs is their terseness. There is little
of the redundancy that enhances understanding in everyday verbal
communication. One does not say essentially the same thing in several
ways. For example, while a local restaurant might post a sign saying
"Our soups are prepared fresh each and every day," in a proof
involving a homomorphism, f, that has just been shown to be an
isomorphism, one does not normally remind readers that this means that
f is one-to-one and has zero kernel. Such terseness is not a matter
of oversight or ignorance of the needs of one's readers. Proofs are
written for an idealized reader -- mathematicians know from their own
experience that differing validators have differing knowledge with
which to validate a given proof.
Few people would write in such terse prose were their primary
intention to sway as many readers as possible. This concise style has
the effect of placing much of the locus of control with the validator,
instead of the writer of a proof. While validators know they can
supplement proofs, they may well resist omitting or altering
significant parts of them (concerned they might overlook something
important). Thus, a terse style allows a validator's supplemental
proofs to "fit" closely with his/her own knowledge base and avoids
mentally cycling through unnecessary or distracting material -- all of
which helps minimize demands on the validator's working memory and the
probability of validator error.
Here we are suggesting only an effect of a terse style, not that this
effect is a conscious goal of authors, who may think more about
avoiding publication of "obvious" material and journal page
limitations. In addition, terseness is a matter of balance -- no
proof at all would shift all control (and the burden of constructing
the proof) to the validator, which is hardly desirable.
However abstract a proof's subject, the style of its writing tends to
be "concrete", that is, wherever possible quantified statements are
avoided through use of universal instantiation. For example, one
would avoid including "For every positive number x, f(x) < 0" in a
proof. Instead, one might first write "Let x be a positive number"
and later observe that "f(x) < 0."
The quantified version is precise and may well represent the
information the author of a proof has in mind. However, from the
perspective of a validator, it is likely to (momentarily) require more
of the limited resources of working memory than either of its
replacements. While a portion of propositional calculus (e.g., modus
ponens) appears to be immediately accessible to everyone's working
memory, this appears to be less so for predicate calculus. For
example, beginning university students often lack an understanding of
the relationship between negation and quantifiers, thinking that "All
x are not y" is the negation of "All x are y" (Endnote 2).
It has been noted that the ability to draw correct inferences in
predicate calculus can be improved by the mere suggestion that one
examine arguments in terms of single instances of the variables
involved (Stenning and Oaksford, 1993). The properties of one
instance can readily be held in short-term memory. However, more than
one such instance may be required in a given argument and this can
place considerable burden on working memory, and hence, increase the
probability of error.
Unquantified sentences such as "Let x be a positive number" and "f(x)
< 0" can be regarded as about a single arbitrary, but fixed object so
only one's propositional calculus abilities need be invoked when
drawing inferences. Thus, a concrete style of proof reduces demands
on a validator's (and a prover's) working memory; thereby, reducing
the chance of processing errors.
Another stylistic feature of proofs is the insistence that the symbols
correspond in a one-to-one way with the objects under consideration.
That is, if one lets x represent a number (even an arbitrary, but
fixed one), one does not later let x represent another number, or even
let y also represent the first number. This also extends across
domains for which differences in context would allow differing
meanings of a symbol to be understood. That is, one does not normally
let X represent both a topological space and its dimension, even
though the differing contexts of spaces and numbers would allow the
meaning to be discerned.
This restriction might be thought to apply so broadly to discourse as
not to be a distinguishing characteristic of proofs. However, we have
observed that novice undergraduate proof writers often violate it,
suggesting it is not generally followed outside of mathematics. The
benefit of this feature of proofs is not to the prover for whom it
amounts to a restriction, but to the validator. If there is a chance
that a symbol might refer to two distinct objects, then a validator
would require additional information on this at each occurrence of the
symbol. To constantly retain this information in short-term memory
would be a considerable burden, and thus, increase the probability of
error. If, on the other hand, this information were contained within
local memory, a validator would not be continually aware of it and
would have to recall it when needed. This additional activity would
also draw on the resources of working memory and increase the
probability of error. In either case, ignoring this stylistic feature
would tend to increase validator errors.
In general discourse, there are a number of characteristics of
arguments which, while not essential, tend to make them more widely
understood or convincing. These include the tendency to provide
detailed explanations or supporting evidence and the inclusion of
definitions and rewordings (a form of redundancy) for purposes of
clarity or completeness. However, such features are usually reduced
to a minimum or avoided entirely in proofs.
Assertions in proofs, as in other forms of argument, can be followed
by reasons or explanations, but in proofs, these are usually kept very
brief, perhaps only the mentioning a theorem or definition. Even much
of the detailed reasoning leading to a (sub-)claim may be omitted in
proofs. This is not a matter of oversight nor of attributing
unrealistic knowledge or sophistication to validators. Often authors
write out the details for themselves and then remove them from the
final published proof. Similarly, overviews, which in other forms of
argument can support understanding, are reduced to a minimum in
proofs. Typically, these consist only in remarking that a proof is by
contradiction or involves cases. Also, questions can sometimes be
used to focus a reader's attention on important points, thereby making
a general argument more convincing or memorable. However, this way of
writing (and teaching) is forgone in proofs, even proofs in expository
or pedagogical texts.
Definitions, especially unfamiliar ones, are sometimes included in
general arguments, and students who are new to proving theorems tend
to insert them in their proofs. However, general mathematical
definitions (i.e., those useful beyond the confines of a single proof)
are almost never quoted within proofs. For example, if one has just
proved that f is a homomorphism, one would not also ask readers to
recall that a function g is a homomorphism if and only if for every x
and y, g(x + y) = g(x) + g(y). This is also true for definitions
introduced for the first time, and hence, not familiar to readers.
Furthermore, general arguments sometimes include the entirety of a
logical argument, such as one employing modus ponens, perhaps to make
them more convincing (or inescapable). In a mathematical setting, for
example, this would amount to saying, "We know differentiable
functions are continuous and f(x) is differentiable, so f(x) must be
continuous," rather than simply asserting "f(x) is continuous" and
leaving affirmation to the validator.
Finally, it appears that most people depend heavily on redundancy for
comprehension of everyday text. However, in proofs this is avoided.
Even the symbols in proofs may not explicitly represent the complexity
of the structures to which they refer. For example, in dealing with
parametric equations, one can write "x, y" rather than "x(t), y(t),"
and in referring to a group, one normally writes "G," rather than "(G,
+)."
The above features, while seeming to enhance general arguments, are
avoided or minimized in proofs. Their main purpose is not explanation
or enhancement of readers' understanding (Endnote 3). Such a
minimalist style does, however, tend to reduce the moment-to-moment
load on working memory. One effect of avoiding everything inessential
to a proof is that an expert validator need not devote working memory
to constantly deciding what is or is not important to check --
everything is important. Another effect is to enhance a validator's
control over his/her chunking. Were inessential material included in
proofs, a validator might feel compelled to read it, even on multiple
passes through a proof, making it more difficult to grasp the essence
of the proof. But if the same information were chunked in a
validator's local memory, it would be readily accessible when needed.
While a validator may require considerable effort to get such
information into local memory initially, this would occur prior to,
and not interfere with working memory while, reviewing the entire
proof.
There appear to be three main impediments to an (expert) validator's
reliably determining the correctness of a proof: lack of domain
knowledge (including logic), domain misconceptions, and working memory
overload. Lack of specific domain knowledge, e.g., not knowing or not
thinking of using the triangle inequality, is easily detectable during
validation, which can be suspended while the validator locates or
derives such information. This causes no threat to the integrity of
proofs and does not influence the style in which proofs are written.
Misconceptions, e.g., believing all continuous functions are
differentiable, are not easily detectable by a single validator,
although two validators who disagree on the correctness of a proof
could bring these to light. The most grievous misconceptions are
those shared by the prover and a validator, but nothing about the
style in which proofs are written would enable these to be discovered.
Thus, the only effect upon the integrity of proofs grounded in the
style of their writing comes from working memory overload, and we have
argued above that proofs are written so as to minimize this. This is
done, even at the expense of making validations longer and proofs less
persuasive or less widely understood (to naive readers).
BEGINNING LOGIC COURSES AND VALIDATION
Perhaps the first thing to notice about beginning logic courses is
that they are not essential to mathematics. Many mathematicians have
started their work with proofs without ever having taken a logic
course and without any familiarity with symbolic logic. Such
mathematicians must have first validated proofs using only their
natural, everyday reasoning ability, which was then perfected through
critical use. This natural reasoning or logical ability is, in some
degree, universal, i.e., everyone seems to correctly use modus ponens,
universal instantiation, and a few other aspects of propositional and
predicate calculus. However, this pervasive natural ability by no
means includes all, or even most of, propositional and predicate
calculus. For example, many people do not consider modus tollens
arguments as valid (48% in a study by Rips, 1994, p. 177), confuse the
conditional with the biconditional, and incorrectly negate multiply
quantified statements. Calculus teachers and textbook writers
routinely warn students that the theorem, "If a series converges, then
the limit of the n-Th. term is zero," provides a test for divergence,
not for convergence (Stewart, 1991, p. 581), usually to no avail.
This untrained, natural ability has been studied extensively by
cognitive psychologists (Rips, 1994; Garnham & Oakhill, 1994),
but not much seems to be known about the way it is perfected
through critical use or formal courses. It would be informative
to see the research techniques of mathematics education
applied to the learning of logic (Endnote 4).
TOPICS NOT MUCH USED IN VALIDATION OF PROOFS
Much of what is in beginning logic courses is little used during the
validation of proofs, e.g., truth tables and Venn diagrams are not
directly used. While logical equivalences, implications, and the
validity of arguments from propositional calculus are all used in
proofs, only the simpler forms occur very often. Complex logical
argument can be broken down into simpler, more familiar arguments
leading from premises to conclusion, and mathematical proofs tend to
do this -- consequently, validators rarely confront great logical
complexity. Thus, what validators need most is to be very familiar
with the simpler aspects of propositional calculus (somewhat beyond
what people can normally do without training). While beginning logic
students are occasionally asked to analyze Lewis Carroll type
"nonsense" arguments involving a half dozen or so premises (Endnote
5), real expertise at handling this sort of logical complexity is not
called for in mathematical proofs, despite their length and domain
specific complexity.
While some predicate calculus is needed to validate proofs, only the
simpler aspects of it occur -- those which tend to avoid quantifiers.
For example, if a theorem is about all integers, the proof may begin,
"Let n be an integer," after which the argument is about a fixed, but
arbitrary n. However, predicate calculus is essential in establishing
the link between the statement of a theorem and its proof and in
bringing information from (outside) definitions and theorems into a
proof.
The probability that a particular argument form, e.g., DeMorgan's
laws, will occur in any randomly selected proof appears to be quite
small. It would be instructive to know which aspects of logic courses
occur most often in proofs that undergraduate students normally
encounter.
TOPICS UNDEREMPHASIZED IN BEGINNING LOGIC COURSES
There are several kinds of logic or logic-related topics, which though
useful in validation, are not emphasized in traditional logic courses,
perhaps because these are seen as unimportant or transparent. We will
discuss substitution, interpreting the logical structure of informally
written statements, applying theorems and definitions to situations in
proofs, understanding the language of proofs, and recognizing logical
structures in the context of mathematics. We suspect all of these are
difficult for students just beginning their work with proofs and are
amenable to explicit instruction (Endnote 6).
Most college students have no problem substituting "3" for the "x" in
x + 2 = 5 and one might hope that more complex substitutions were
equally transparent for them. After all, the constellation of
concepts consisting of substitution, variables, and quantifiers is
used by quite young children when understanding the natural language
sentence, "You will all get a piece of candy." If arithmetic teachers
regularly took a small amount of time away from concrete calculation
to build on these ideas, students might gradually come to see
variables as functioning like pronouns in English sentence, with
similar substitution properties. For example, children could be
asked, "Which of the numbers 7, 16, 26, 13, is greater than 5? All,
none, some?" Apparently, this does not happen. The concept of
variable appears to be first introduced in algebra, in a way
disconnected from previous experience.
Perhaps partly as a result of this background, beginning calculus
students often have difficulty with long or multi-level substitutions
such as those required to calculate the derivative of a particular
function at a point using the definition. When applying complex
theorems to their own research, mathematicians sometimes make explicit
substitutions for each occurrence of a variable in order to ensure
that they have grasped the exact meaning. We suspect that
mathematicians make such substitutions in several steps, keeping track
of them in local memory, aided by few written symbols. However, many
beginning students may lack such procedural skills or the tendency to
use them, and hence, introduce errors by attempting to carry out
complex multi-level substitutions in working memory. Whatever the
origin of students' difficulties with substitution, they could
probably benefit from additional explicit instruction in a variety of
multi-level substitution problems (Endnote 7).
Another logical difficulty students have when validating proofs is
interpreting the logical structure of theorems and definitions written
in the informal style of most textbooks. Theorems and definitions can
be expressed formally in natural language, explicitly naming all
variables and employing the usual logical connectives and quantifiers
(and, or, not, if-then, if-and-and-only-if, for all, there exists).
However, theorems are often expressed less formally, that is, their
wording departs significantly from this more formal style. For
example, "f + g is continuous at a point, provided f and g are" is
informally stated. In contrast, "For all functions f, all functions
g, and all real numbers a, if f is continuous at a and g is continuous
at a, then f + g is continuous at a" is its more formal counterpart.
Because theorems and definitions are so often expressed informally,
there are probably some cognitive benefits to the practice. Indeed,
we suspect one reason authors favor an informal style is to make
theorems and definitions more "memorable," more easily remembered and
recalled. However, formally stated theorems can be relatively easily
linked to what we call a *proof framework*, i.e., the portion of a
proof that can be written without knowledge of the specific
mathematical concepts involved and which determines whether a proof
actually proves the theorem it claims to prove, rather than some other
theorem. For example, the theorem mentioned above has the following
simple proof framework: Proof: Let f and g be functions and let a be a
number. Suppose f and g are continuous at a. . . . Thus, f + g is
continuous at a. QED.
Since a proof can contain subproofs, it may also contain nested proof
framework structures. However, there is evidence that mid-level
mathematics majors are unable to reliably unpack the logical structure
of informally stated calculus theorems and definitions. Thus, it is
unlikely that they can reliably recognize proof frameworks while
validating proofs (Selden & Selden, 1995). This suggests that
explicit instruction, both in unpacking the logical structure of
informally written statements and in the recognition and construction
of proof frameworks would be very useful.
Proof frameworks, however, are not the only link between the (largely
propositional calculus) language of proofs and the predicate calculus
language of (outside) definitions and theorems. In validating a
proof, one applies theorems and definitions to the mathematical
situations arising within a proof. We found that many mid-level
undergraduate mathematics majors, in a "bridge course" designed to
introduce proofs and mathematical reasoning, were unable to reliably
translate informally written mathematical statements into their
predicate calculus equivalents -- a process we call *unpacking*, i.e.,
associating with an informally worded mathematical statement a
logically equivalent formal statement, including all those logical
features that are often understood by convention, rather than
explicitly expressed. Just 8.5% of the 94 unpacking attempts we
collected from six small sections of a "bridge course" were
successful. For example, given the following (incorrect, but
meaningful) mathematical statement: For a < b, there is a c so that
f(c) = y whenever f(a) < y and y < f(b), students wrote such things
as: For all a < b, there exists c such that for all f and for all y,
f(a) < y and y < f(b) implies f(c) = y. Note that if f were a
continuous function, this statement would be the Intermediate Value
Theorem as stated in many calculus texts. However, the interchange of
universal and existential quantifiers radically alters the meaning of
the original statement. It now asserts the rather bizarre idea that
in every interval (a,b) there is a peculiar number c, at which each
nonconstant f takes on all values y between f(a) and f(b). [See
Selden & Selden, 1995, pp. 136-139.] This and other examples like it
suggest that mid-level math majors are unable to understand and
correctly apply quantified statements in context, and thus explicit
instruction in their application would probably be useful. One might
ask students of a particular theorem, "What does this theorem say
here?", about various mathematical situations.
Also, the language of proofs, and in particular, the meanings
of a few words, might receive some explicit instruction,
just as the meaning of "or" and "if and only if" are explicitly
taught in logic. For example, "let" can be used in at least
three distinct ways in proofs. In proving "For all numbers e . . . ",
a proof might include "Let e be a number." In that case one
needs to be sure e has not occurred earlier in the proof and that
e is not restricted in some way. In proving, " . . . there is a d
. . .", a proof might contain "Let d = x/2." Here also d should
not have appeared earlier in the proof, but in contrast to the
previous use of "let," d must arise from the previous portion of the
proof and either be explicitly described or shown to exist. It would
be clearer to say, "Set d = x/2," but that is not common usage. The
third use of "let" occurs in the proofs of theorems containing
conditionals, e.g., " . . . if f is continuous, then . . .". Here
"Let f be continuous" means "Suppose f is continuous." In this case,
f can, indeed should have, occurred previously in the statement of the
theorem or its proof. Furthermore, suppositions can be made about f
more than once, as for example, when subsequently, an additional
property about f is to be proved by contradiction. These three uses
of "let," for universal instantiation, assignment, and supposition,
are not, to our knowledge, often explained to mathematics students,
although we make a practice of doing so in "bridge courses" (Endnote
8).
We turn now to context or "situatedness." Beginning logic courses
often seem to present logic very abstractly, in essence as a form of
algebra, with examples becoming a kind of applied mathematics.
Students are asked to convert natural language mathematical situations
into symbolic form, analyze them, and then convert these back again to
get the result. This kind of application does occur in validations,
although often the links to symbolic logic are not so explicit. For
example, a validator may be required to decide whether an assertion is
warranted by recognizing it as the conclusion of a valid argument
(written in natural language) with premises that are scattered through
the previous portion of the proof. This is quite different from the
abstract, symbolic setting of many exercises in beginning logic
courses. Since there is evidence that the application of knowledge
can be remarkably dependent on the situations in which it is learned
(Endnote 9), the reasoning involved in validation may depend as much
on students' common sense background as on logic courses. Thus, it
would probably be useful to teach logic in more realistic contexts,
e.g., those provided by fragments of proofs. For example, students
might be asked to select and justify one of several possible
assertions that might follow from a proof fragment in which the
premises of a valid argument are embedded.
WAYS OF KNOWING LOGIC
Since the 1980s, research in mathematics education has concentrated on
how students develop conceptual knowledge, rather than procedural
knowledge -- it has focused on understanding, rather than performance.
However, in complex trains of thought such as validations, both
conceptual and procedural knowledge appear to be inextricably
integrated into cognition. In particular, in assenting to an assertion
in a proof, a validator may simply recognize that it is the
consequence of a familiar embedded logical argument, such as modus
ponens. In that case, the validator's only conscious response may be
to say "OK." That is, whatever the validator does is below the
conscious level of internalized speech or vision. One could think of
the validator as activating a small *schema* corresponding to modus
ponens, i.e., filling in the blank spaces between the connectives that
form the pattern of modus ponens (Endnote 10). There may be other
ways to view such automated responses of the validator, but in any
case, he/she is unaware of them in his/her conscious mind. Thus, the
validator has no opportunity to check them, say, in the way a
sequences of statements occurring in internalized speech might be
broken into parts, rehearsed slowly and checked individually.
However, the validator's conscious, automated response, "OK," may
leave working memory largely free for other tasks.
This situation is somewhat analogous to that of an elementary student
who simply "knows" that "8 times 7 is 56," rather than having to think
"7 times 7 is 49 and 49 plus one more 7 is 56." This kind of
procedural knowledge is very efficient, but if the student thought "8
times 7 is 57" and suspected an error, the only way to resolve this,
would be to switch to a more conceptual approach.
Since, in validations, some logic appears to be applied in this
procedural, automated way, it should be beneficial to teach a small
part of beginning logic for procedural (as well as conceptual)
knowledge, i.e., using drill and practice, especially in appropriate
(mathematical) contexts.
A SAMPLE VALIDATION (Endnote 11)
The following example illustrates some of the ideas discussed in this
paper. For this purpose, we have selected a familiar calculus
theorem, but are not suggesting what follows should be used in
teaching calculus. Nor is this example meant to illustrate how a
theorem or its proof might be invented since it does not consider such
things as intuition, insight, visualization, false starts, or
cognitive strategies.
Although the theorem we have selected is similar to Leron's Theorem
2.2 (1983), our purpose is different. Leron presents the argument in
a nontraditional, top-down, structured way for pedagogical purposes in
order to facilitate students in grasping its basic structure, as well
as the relationship between relevant mathematical concepts. We, on
the other hand, give a traditional linear proof which facilitates the
validation process.
Validations can be done in various ways -- they depend heavily upon
the individual validator, his/her domain knowledge, and his/her
knowledge of validation procedure.
We present a plausible, but imaginary, transcript detailing the main
questions, answers, and comments of our (hypothetical) validator.
This is followed by a (short) analysis of the logic needed for this
validation. The transcript is written entirely as internalized
speech, but an actual validator might also draw some pictures, think
out loud, or visualize portions of the proof. For reference, we have
numbered the sentences in the proof, using brackets to enclose both
reference numbers and our comments.
Theorem. f + g is continuous at a point, provided f and g are.
Proof: [1] Let a be a number and let f and g be functions continuous
at a . [2] Let e be a number greater than 0. [3] Note that e/2 is
greater than 0. [4] Now because f is continuous at a , there is a d1
greater than 0, such that for any x1 , if | x1 - a | < d1, then | f
(x1 ) - f (a) | < e/2. [5] Also there is a d2 greater than 0, such
that for any x2 , if | x2 - a | < d2, then | g (x2) - g (a) | < e/2.
[6] Let d be the smaller of d1 and d2. [7] Note that d is greater
than 0. [8] Let x be a number. [9] Suppose | x - a | < d. [10] Then
| x - a | < d1, so | f (x) - f (a) | < e/2. [11] Also | x - a | < d2,
so | g (x) - g (a) | < e/2. [12] Now | (f (x) + g (x) ) - ( f (a) + g
(a) )| = | ( f (x) - f (a) ) + ( g (x) - g (a) )| (< or =) | f (x) - f
(a) | + | g (x) - g (a) | < e/2 + e/2 = e . [13] Thus | ( f (x) + g
(x) ) - ( f (a) + g (a) )| < e. [14] Therefore f + g is continuous at
a . QED
Our (hypothetical) transcript follows.
[1] "OK, a , f , and g can be introduced this way because they haven't
been used before -- this is the first line of the proof. Why is the
argument starting with something about f and g when the theorem starts
with f + g ? Does the argument really prove the theorem? The theorem
really means: For every number a and every function f and every
function g , if f and g are continuous at a , then f + g is continuous
at a . [This is an instance of unpacking the original informal
statement into a more formal one.] So the proof should start with an
arbitrary a , f , and g and assume f and g are continuous. That
amounts to [1] . After some argument, the proof should end with f + g
is continuous at a, which turns out to be the last line, [14] . OK,
if the argument checks out, the theorem is true." [That is, a proof
framework obtained from the statement of the theorem agrees with the
one embedded in this argument].
[2] "OK, e hasn't come up before, so it's permissible to let it
represent any number."
[3] [This comes from [2] and a bit of domain knowledge.] "Half a
positive number is positive. So [3] is OK, but why say it?" [The
answer to this would be helpful but is not essential to the
validation.]
[4] "This is supposed to come from ' f continuous at a' which means:
For every e greater than 0 there is a d greater than 0 so that for all
x , if | x - a | < d, then | f (x) - f (a) | < e. Here the symbols
may mean something different from those in the proof. [This comes
from domain knowledge, perhaps with some unpacking.] This says 'for
every e' so this part can be dropped with e replaced by any positive
number. So e/2 can replace e . Also, the x can be renamed as x1 and
the d as d1 throughout. That gives me [4] .
[5] "There is no 'because' here. But, it looks like [4] . So I can
start with the definition of ' f is continuous at a' and change 'f '
to 'g'. Just as before in 'e greater than 0', I can replace e by e/2.
Also the x2 can replace x and d2 can replace d. So [5] is OK."
[6] "Now 'd' has not been used before in the proof, so it can
represent any number made out of numbers already there." [It will
turn out later that d has to be treated this way, and cannot be
arbitrary like e.]
[7] [This comes from some domain knowledge.] "The smaller of two
positive numbers is positive."
[8] "OK, 'x' hasn't been used yet so it can represent any number."
[9] "This inequality is a statement because x , a , and d already
represent numbers and one can suppose any statement. But why?" [It
turns out this supposition is part of proving f + g is continuous at a
. Knowing that would help the validator keep track of the argument,
but it is not essential to the validation.]
[10] "The first part of this assertion follows from [9] and [6] [and
simple domain knowledge -- if a quantity is less than the smaller of
two numbers, then it is less than either number separately.] The
second part comes from the first part and the implication in [4] ,
replacing 'x1' by 'x' . This is permissible because [4] said 'for any
x1'. " [In addition, [10] is using modus ponens.]
[11] "The first part comes from [9] and [6] , just as in [10] . The
second part comes from the first part and [5] , just as in [10] ."
[12] "The first equality comes from (a + b) - (c + d) = (a - c) + (b -
d). [Some domain knowledge.] The inequality comes from the triangle
inequality, | p + q | (< or =) | p | + | q | , where 'f (x) - f (a)'
replaces 'p' and 'g (x) - g (a)' replaces 'q'. Next the strict
inequality comes from the last parts of [10] and [11] [and some domain
knowledge, namely, a + b < c + d can be inferred from a < c and b < d
]. The final equality is just arithmetic."
[13] "This follows from [12] [and several instances of the transitive
property for real numbers, e.g., a (< or =) b and b < c implies a < c
]."
[14] "This must depend on the definition of continuous at a , applied
to f + g . That is, for every e , there is a d greater than 0, such
that for every x , if | x - a | < d , then | ( f + g )(x) - ( f + g
)(a) | < e.
"Now I need to check that the definition of continuity is satisfied
and that amounts to proving a theorem. [The validator, in effect,
next constructs a proof framework.] Let e be an arbitrary real number
greater than 0. That's [2] of the proof. Let d be a real number
greater than 0, which can depend on d but not on x . That's [6] and
[7] of the proof. Let x be an arbitrary real number. That's [8] ,
which comes after [6] , so d doesn't depend on x . Suppose | x - a |
< d. That's [9] . Then, after some argument has been given, | ( f +
g) (x) - ( f + g)(a) | < e . This doesn't look like [13] , but it
means the same thing because ( f + g)(x) means f (x) + g (x) and ( f +
g)(a) means f (a) + g (a) . [Here the validator does some notational
unpacking.]
"So [14] is OK."
"Now the entire argument is OK, and at the beginning, I checked
that if the argument was OK, then it would prove the theorem." [The
validator decided, as part of checking [1] , that the proof framework
of the argument was appropriate for the statement of the theorem.]
This completes the validation.
The proof in this example is fairly detailed -- some might say overly
so. Many proofs omit lines like [1] which serve to "introduce" a , f
, and g . An expert validator would not need [1]. The importance of
[1] however is that it emphasizes the need to check that a , f , and g
are arbitrary. If this were not the case -- say f somehow depended on
g or a -- then an expert validator would notice the error even though
[1] were missing. That is, he/she would act as if [1] were there
should arbitrariness fail, but might not even notice its absence
otherwise.
All or part of [12] might also be omitted from the proof. In that
case, the validator might add the omitted portions to his/her own
expanded version of the proof.
An actual validator might well carry out the validation process in
this example much more quickly than the transcript can be read. On
the other hand, if [12] were omitted from the proof, a relatively
inexperienced student might take some time to realize that the key to
proving [13] was the triangle inequality. Furthermore, if the need to
insert an argument to justify [13] were not noticed until [14] was
checked, another problem might arise. The checking of [14] is
sufficiently complex that stopping for some time to justify [13] might
overload working memory and cause part of the checking of [14] to be
neglected. To guard against this sort of potential error, some
validators make multiple passes through a proof.
The proof we have discussed here is short and "structurally simple,"
consisting only of a brief proof within a proof. Many proofs are much
longer and structurally more complex, and hence, require more
complicated validations.
Let us now examine the above validation for uses of logic or logic-
related topics. For example, in [1], a validator must know which of
the three uses of "let" is intended, unpack the informally written
statement of the theorem to sort out the quantifiers, the premises,
and the conclusion, construct a proof framework, and check that the
first and last lines of the proof agree with this framework. These
are all things not taught in traditional logic courses. In addition,
as the validation proceeds, there are a number of instances of
substitution combined with application of the definition of
continuous, e.g., in [4], [5], [10], [12], and [14] This is also not
taught in traditional logic courses. The argument is largely concrete
-- it deals with arbitrary, but fixed instances of a, f, g, x, and
e>0. The logic used seems confined to universal instantiation, e.g.
in [1], [2], and [8], and modus ponens, e.g., in [10], [11], and [13].
ENDNOTES
1. It appears to be widely believed among mathematicians that logic
courses, per se, are of little benefit in helping students learn to
understand and create proofs. Research in mathematics education has
not directly examined the accuracy of this belief, but it is
consistent with Nisbett's, et al's work (1987) on the effects of logic
courses.
2. This difficulty with negation of universal quantifiers seems
common in our culture. A recent ad on television proclaimed, "All
people are not created equal," presumably intending the negation of
"All people are created equal."
3. Some mathematics education researchers have noted that the style
in which many proofs are written is ill-suited for pedagogical
purposes. They prefer "proofs that explain" over "proofs that merely
prove" (Hanna, 1989).
4. Mathematics education researchers might find the work of diSessa
(1993) on p-prims a useful guide here. He and other physics education
researchers has often been concerned with altering students'
pre-existing conceptions of the world (those originating in everyday
experience) and bringing them into alignment with current scientific
theory -- a situation that seems relevant here. To date, much of
mathematics education research has been devoted to studying the
construction of concepts, like function, limit, or integral, which
have not spontaneously arisen from students' everyday experiences.
5. An example is the following:
If he goes to a party, he does not fail to brush his teeth. To look
fascinating it is necessary to be tidy. If he is an opium eater, then
he has no self-command. If he brushes his hair, he looks fascinating.
He wears while kid gloves only if he goes to a party. Having no
self-command is sufficient to make one untidy. Therefore,
____________________ .
6. By "explicit instruction," we mean a variety of instructional
techniques including explorational activities and group work, as well
as the more traditional lecturing and homework exercises. We are
suggesting certain topics, now omitted or underemphasized, should be
part of the explicit, rather than the implicit curriculum. Most
students need help learning these topics, or even, knowing they are
important. Unfortunately, many of these have been considered part of
"mathematical maturity" in the past.
7. Epp (1990) has several sections on substitution into combinatorial
formulas, but much less than we think students need.
8. Some "bridge course" texts do list some of the various ways the
conditional (and biconditional) can be expressed in mathematics, e.g.,
if p, then q; p implies q; q, provided p; q, whenever p, p is
sufficient for q; q is necessary for p (Kurtz, 1992).
9. Within mathematics education, a growing number of researchers
consider all learning as situated, i.e., concepts and procedures are
inexorably linked to the context in which they are learned (Brown,
Collins, & Duguid, 1989) .
10. Schemas have also been called *scripts* or *frames* and the
blanks spaces referred to as *variables* or *slots*. They have been
described as unconscious cognitive structures that underlie knowledge
and skills. Schemas are active processors of knowledge that have been
likened to small theories, procedures, and parsers. They are used in
perception, text comprehension, memory, learning, and problem solving
(Rumelhart, 1980).
11. This theorem and validation appear in Appendix 1 (Selden &
Selden, 1995).
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