Generic examples: Seeing the general in the particular

Abstract

This paper explores some of the ambiguities inherent in the notions of generality and genericity, drawing parallels between natural language and mathematics, and thereby obliquely attacking the entrenched view that mathematics is unambiguous. Alternative ways of construing 2N, for example, suggest approaches to some of the difficulties which students find with an algebraic representation of generality. Examples are given to show that confusion of levels is widespread throughout mathematics, but that the very confusion is a source of richness of meaning.

Full text

JOHN MASON AND DAVID PIMM
GENERIC EXAMPLES: SEEING THE GENERAL IN THE
PARTICULAR
ABSTRACT. This paper explores some of the ambiguities inherent in the notions of
generality and genericity, drawing parallels between natural language and mathematics,
and thereby obliquely attacking the entrenched view that mathematics is unambiguous.
Alternative ways of construing 2N, for example, suggest approaches to some of the
difficulties which students find with an algebraic representation of generality. Examples
are given to show that confusion of levels is widespread throughout mathematics, but
that the very confusion is a source of richness of meaning.
"If you want to soive a problem, first strip the problem of everything that is not essentiaL
Simplify it, specialize it as much as you can without sacrificing its core. Thus it becomes
simple, as simple as it can be made, without losing any of its punch, and then you solve it.
The generalization is a triviality which you don't have to pay much attention to."
Richard Courant, talking about David Hilbert's consciously used principle (1981, p. 161).
Hilbert's approach could be summarized as searching for a generic example,
a specialization which nonetheless speaks the generality. The purpose of this
article is to explore meanings of 'generic' and 'generality' as they are found
both in everyday language, and as they confront students of mathematics. We
start with some everyday occurrences of the idea of generic, leading to the
notion of a generic example of a class, and then move on to explore some
ramifications in elementary algebra.
1. GENERICITY IN NATURAL LANGUAGE
A group of Tel Aviv University scientists who came home from a nearby beach with tar
on their feet decided to do something about it. In the University's biotechnology labs
they developed a petroleum-eating bacterium ....
The scientists are now attempting to isolate similar biologically-produced chemicals to
extend the potential uses further.
"We
see this as a class,"
Rosenberg said, noting how the discovery of penicillin opened
the way to the development of the entire class of antibiotics.
The GUARDIAN 8/1/83
(emphasis added)
UNBRANDED DRUGS SAVE s A YEAR
A four-year project in which doctors and chemists are cooperating to give patients un-
branded drugs instead of expensive brand name equivalents will show that the NHS
could save well over s million a year on only 16 of the 4000 drugs on the market.
Educational Studies in Mathematics
15 (1984) 277-289. 0013-1954/84/0153-0277501.30
9 1984
by D. Reidel Publishing Company.

278 JOHN MASON AND DAVID PIMM
Wherever possible, they agreed, the doctors would use the generic name of prescriptions
(diazepam instead of Valium, for example), and the chemists would dispense an un-
branded version.
The charge of inadequate.., therapeutic effects of generics, so often levelled by the
Association of the British Pharmeceutical Industry and the individual branded drug
manufacturers has not been proved in practice. Only a very small percentage of patients
were found to have a reduced response to generics.., conversely, some patients produce
a markedly better therapeutic response of the generic variety.
The GUARDIAN 15/2/83
A question of concern to the medical profession revolves around whether
prescriptions should bear the class name or a particular brand name. It is
currently unclear whether the doctor is discriminating deliberately among the
various alternatives, particularly since she may not know the names of all
(or even some) of the alternatives. In the last paragraph of the GUARDIAN
report, the medical question of equivalence between members of the class
'diazepam' is raised. 1
Consider the following requests:
"Give me a kleenex"
'~Vhere do you keep your hoover?"
"Can you xerox this for me?"
These last examples are all said many times each day, and in each case the noun
used is intended to refer to a class of objects namely tissues, vacuum cleaners
and photocopiers. Yet the name used is the name of a particular brand, distin-
guished in print but not in speech by the use of a small initial letter for the
generic usage and a capital for the brand name. For various linguistic and
economic reasons, the brand name has come to stand for the whole class. In
general, everyday English words cannot be brand names (see Room, 1982).
The primary reason for the manufacturer's concern at this happening is the
desire to protect the
power
of their product's name - the specificity - intact. If
someone says "I want some kleenex", you don't know for sure (the chemist's
problem of interpreting a doctor's prescription) precisely what is required. It is
not at all incongruous, or evidence of a change of mind, for someone to say
"I want some kleenex. No, not the Kleenex, the Boots."
When we use kleenex as a word for tissue, we are using it as a generic example
of what we want, namely a tissue. The more it is used, the more the word takes
on the meaning of tissue, and the less it is experienced as referring to a specific
brand.

GENERIC EXAMPLES
279
2. GENERICITY IN MATHEMATICS
The same thing happens frequently in mathematics. For example, the words
'rectangle' and 'square' admit at least two uses. Most teachers wish to subsume
'square' under 'rectangle', in the sense that a square is a particular kind of
rectangle, but most pupils want rectangles and squares to be different, indeed
distinct (possibly because they are different words?). Use of the term 'oblong'
for a non-square rectangle often causes dissension among teachers on the
grounds that it is not a 'proper', that is 'mathematical', term.
In the case of fractions, the symbol 2/3 is used to refer to both the fraction
and to the rational number, exactly analogous to the use of kleenex and
hoover, but without even the benefit of capitals to distinguish the different
meanings in print. The inherent ambiguity leads to questions such as:
What is the numerator of the rational number 2/3?
The question seems a little peculiar at first, and it is hard not to give the
answer 2. However the question refers to the
rational
number 2/3, which is
the same as the rational number 4/6, since for instance, both are placed at
the same place on a number line. What then
is
the numerator?
There are difficulties in interpreting 'numerator', when 2/3 is used as a name
for the whole class, namely
...-6/-9,-4[-6,-2/-3,
2/3, 4/6, 6/9,...
or
{2t/3t:
t a non-zero integer}.
It might be feasible to describe the numerator of the rational number 2/3 as 2t.
'Numerator' and 'denominator' are terms describing features of the
notation
for fractions and have nothing to do with properties of the numbers them-
selves. Neither are fractional, decimal or binary "numbers" particular types
of numbers. Algorithms for operations on equivalence classes tend to be given
in'terms Of operations on the representatives (see Stewart and Tall, 1979, for
a detailed discussion of this common mathematical procedure). Similarly an
integer does not have digits, but its representation in a given base does. By way
of contrast,
When 5 is written in base two it is 101. is it a prime in base two?
The property of being a prime is independent of which base is used to repre-
sent the number, because primality is a property of the number and not of
the name. (Further examples of mathematical confusions of this sort are
collected in Adda, 1982.) A similar confusion is due to the propensity of
mathematicians to embed one collection of objects (here fractions) in larger

280 JOHN MASON AND DAVID PIMM
collections, and then to assume that everyone really knows what is going on, so
an identical name can be used without confusion (The connection with meta-
phor is explored in Pimm, 1981).
3. INTRODUCTORY ALGEBRA
Real confusion arises when symbols, particularly single letters, are introduced
to stand for numbers in beginning algebra (see for example Hart, 1981, or
Booth, 1981, 1983.)Consider the two statements:
The square of an even number is even.
and The sum of any two even numbers is even.
On encountering the first statement, someone writes down 2N.
What does 2N stand for? What IS N?
There are at least two possible responses. On the one hand, 2N is standing for
any (or is it all) even number(s). It is a shorthand for, or a name for
{2N: N a whole number}.
On the other hand 2N is a particular, but not specific, even number. By this it
is intended to imply that 2N is actually an even number in the same way that
18 is an even number, and not a set of numbers. It
is particular
in that it IS an
even number, but not
specific
in that it is NOT a specific even number like 2,
4, or 6. There seems to be a dual perception invoked by the use of the symbols
2N, and it is these shifts of perception which may lie at the heart of some
algebraic difficulties.
To begin to unfold some of the confusions, notice the language of the two
statements, and in particular the use of articles. The first uses 'an' and the
second uses 'any'. Most mathematicians would make little if any distinction
between them, because both evoke the mathematician's sense of generality,
of generalization about numbers.
There are four words which require clarification by being compared:
specific
particular
generic
and general.
Leaving aside the question of 'particular', we can distinguish the other three
as follows.

GENERIC EXAMPLES 281
specific generic general
6 6 can be 2
." . ~ ~ ~ ~
9 o o
2N 3
We are led to several different expressions:
THE even number 6,
AN even number (like) 6,
ANY even number (like) 6,
2N
THE even number 2N
AN even number 2N
ANY even number 2N
The use of 'the' emphasizes the definiterxess, the specificness of 6. Attention
is drawn to the number 6 as 6, while by contrast, the use of 'an' suggests
indefiniteness. It is not 6 as such that is of interest, but some quality possessed
by 6. The use of 'any' suggests perhaps even more generality. The phrase
ANY even number 6
is meaningless. It requires the 'like' in order to make sense, whereas the phrase
with 'an' can just about survive without it. Contrast the questions:
Is there any even number which is prime?
Is any even number prime?
The first question seems fine, and might be uttered in despair at finding many
examples of odd primes but no even ones, and an answer of 'yes, 2' is appro-
priate. But the second can be interpreted in two conflicting ways.
Is any (one specific) even number prime?
Is any (at all, i.e., every) even number prime?
The answers are of course 'yes' and 'no' respectively.
Mathematicians tend to use 'any' to mean 'every', and occasionally their
meaning conflicts with ordinary usage. For example, an Open University
Foundation course assignment reads: W stands for the set of all 2 x 2 matrices
a--2 2--a
2
with a E R, a =~ 2. For any matrix A in W, show that A 2 = A.

282 JOHN MASON AND DAVID PIMM
Six out of 13 submitted answers from one tutorial group had chosen a
particular matrix for A (each student a different one) and derived the result.
When asked about this one student commented "Well it said show it for any,
so I just picked one. I thought that was what they meant".
Thornton (1970) also explored this confusion and Tall (1977) has dis-
cussed first-year undergraduates' interpretations of 'some' and 'all' and
discovered that for many students 'some' implied 'not all', so that a statement
such as
'Some rationals are reals'
is judged to be false. Further discussion of this whole problem area can be
found in Hawkins (1978). 4
Earlier 2N was referred to as being an even number, but some care needs to
be exercised here. Strictly speaking, 2N is not to be found in any list of even
numbers, no matter how far extended, despite it being commonplace to write:
2, 4, 6 ..... 2N,...
which suggests the contrary. One function of its inclusion is to render un-
ambiguous precisely which sequence is intended, since
2,4,6 ....
can be continued in many ways, as Wittgenstein (1967) was keen to point out.
In what sense then is 2N an even number? Perhaps it
represents
an even
number, or perhaps all even numbers. We mentioned earlier in the context of
kleenex, that 2N can be seen as the name of a number, and not the number
itself. Instead of being strict and saying that the symbols 2N never appear in a
list of even numbers, we could try out the idea that 2N is the name of an even
number.
What seems to be going on is that the mathematician, interested in general
statements which apply to a wide class of objects, avoids the semantic-
philosophical difficulties, and treats 2N as if it were an even number. Put
another way, the symbols 2N cause connections to be made with {2N: N a
number} in the mind. The 2N is like a reminder or signal of the 2N inside the
set description {2N: N a number}. When set brackets are actually written,
the 2N appearing inside is not itself a number, but an instruction to carry
out a calculation, and a criterion for membership (in formal logical terms, a
predicate). Thus 2N is a template for identifying even numbers, a condition
for admission to the ranks of the chosen. Thinking back again to the kleenex-
type of examples (notice the generic use of kleenex here to stand for all our
examples, and the widespread use of the suffix '-type' to turn a specific into a

GENERIC EXAMPLES 283
generic class name), it seems that names of things are used to evoke recollec-
tion of (images of) objects or experiences that they name, a form of visual
onomatopoeia.
A more careful philosophical examination of the question of whether 2N
is an even number, suggests that 2N is literally only marks on paper, and as
such is neither specific, general nor particular. It all has to do with perception.
Coming to grips with algebra entails explicit development of changing per-
ceptions connected with generality, with multiplicity of meaning. (See Mason
et al.,
1983 for a description of various contributory routes to a sense of
algebra as expression of generality.)
It seems sensible to admit 2N as a non-specific even number, and to empha-
size the perception of it as a single, but indefinite, even number, a perception
for which the word 'particular' seems appropriate. Thus 2N comes out as a
particular, non-specific even number, s
Have all difficulties been overcome by admitting the required multiplicity
of perception, analogous to the use of kleenex, and of 2/3? Not at all! We
began with two statements involving even numbers, but have looked only at
the first. The second was
The sum of any two even numbers is even.
Students newly exposed to algebraic notation frequently respond to requests
for a proof of this statement by:
Let 2N be any even number.
Then 2N + 2N = 4N which is even.
The 'any' is still causing some difficulty. It is very reasonable to argue that
since 2N stands for ANY even number, the sum of ANY two even numbers
can be represented by 2N + 2N. What seems to be missing is an awareness
of 2N as a PARTICULAR, but non-specific even number. The 'any' has two
interpretations as mentioned earlier, and it is an essential part of the per-
ception of generality that the 'every' interpretation is used. Thus using 2N
and 2N is using the same particular even number twice, and not ANY two
particular even numbers.
Transition to a representation of every even number, in the form of 2N
is not easy. Diagrams are a very useful intermediary. There seems to be con-
siderable evidence that some version of Pythagoras' theorem was known to
the Babylonians (Neugebauer, 1969) and to the Chinese (Swetz and Kao,
1977) before Pythagoras, and that early 'proofs' were of the same pictorial
nature as the following proof that the sum of two even numbers is always
even:

284 JOHN MASON AND DAVID PIMM
+ =
It serves to remind us of an image or perception of even numbers as numbers
which can be displayed as two matching rows of dots. Since in both numbers
the dots pair up, so too will they in the sum, formed by amalgamating the dots.
Yet some students might see this as proof by example (namely that since
14 + 24 = 38, the result holds generally), a topic to which we shall return later.
This theme is explored in detail in Vinner (1983) where he provides pupils
with a choice of three proofs of a result, which we would classify as specific,
generic and general respectively. The generic proof, although given in terms of
a particular number, nowhere relies on any specific properties of that number.
4. PARAMETERS
There is a further aspect of generality which needs considering and that is the
ambiguous nature of parameters. Students who are working on quadratic
equations are expected,eventually to treat
ax 2 + bx + c
as if it were a specific example of a quadratic. Thus a typical examination
question reads:
Find where the graph ofy =
ax 2 + bx + c
crosses the x-axis.
The coefficients are fixed but unspecified, and so capable of manipulation,.
and x is supposed to be found in terms of them. The answer is particular in
the sense that there is a correct formula. 'The' graph implies that there is
but one graph, and that the generic picture has a > 0 and real, unequal roots.
When quadratics are first encountered, x 2 acts as a generic quadratic and
so is studied in detail. Mathematicians sometimes speak of x 2 as being the
same as any quadratic up to scaling, translation and reflection, thereby being
careful to indicate the scope of generality being captured by the example; in
other words the scope of its genericity. Later in a student's experience with
quadratics, a, b and e are expected to be seen as standing for any real number.
Then figure and ground are reversed by the request to
"hold x fixed, e.g., put x = Xo, and let a vary - what happens to...".
Notice how the subscript helps to make the x particular. If you have trouble
seeing 2N as a particular even number, try 2No.

GENERIC EXAMPLES. 285
Suddenly a has attained the status of a parameter. In what ways do para-
meters differ from variables? Is it a question of perception?
5. FURTHER EXTENSIONS: GENERIC EXAMPLES
If it seems from the discussion so far that the difficulties with generality are
confined to beginning algebra, and that once the algebraic hurdle has been
surmounted then the student finds life plain sailing, consider the following
observation of Desmond MacHale (1980, p. 752). In order to convince students
that the converse of a theorem is false, or that the conditions of a theorem
cannot be weakened in any way, a counter-example is usually offered. MacHale
complains that there is a paucity of examples actually provided for students.
For example,
the function x -+ Ixl
is often the only example of a continuous but non-differentiable function
presented (with the possible exception of functions so complicated that no
student believes that they are really even functions!). What is happening is that
the lecturer, in presenting the example, is seeing it as generic. It indicates a
whole class of functions
x -+ klx + al + C
at the very least, just as Rosenberg
perceived his one oil-cleaning bacterium as but one of a large class. The students
however are concentrating on the particular example. They see, not a class of
functions, but a single function. MacHale provides eleven other contexts in
which a standard counter-example is usually the only one offered, which could
be seen as generic by the lecturer, but is often not seen that way by students.
We suggest that students are frequently uncertain as to the role and nature
of such a counterexample. Not only is it particular rather than general, but in
the main they are not clear about the general statement it is defeating. Seeing
one strange example often has little effect, and students are here in good
company. It is often claimed that just one counterexample disproves a con-
jecture, yet the history of mathematics provides evidence that this is false.
Research mathematicians do not actually operate that way. A
single
counter-
example to a putative theorem is- often incorporated into the statement of the
theorem as the sole exception. Another approach is to alter one of the defi-
nitions which lie behind the theorem, in other words to refuse to accord the
example the status of counterexample. Lakatos (1976) provides a graphic
description of the various mathematical strategies for dealing with counter-
examples (see also Pimm, 1982). Thus students subconsciously exclude the
specific function x -+ Ix[ from the scope of the theorem, or even deny that it
deserves to be called a function at all. Their intuitions are therefore not in

286 JOHN MASON AND DAVID PIMM
need of adjusting, and the whole point of the exercise is lost. Due to the air of
artificiality which accompanies specifically contrived counterexamples, many
students may take this view.
Such strategies are employed at all mathematical levels. The following
example arose in an interview with 12 yr. old Christopher. He was asked to
work on testing generalizations about the area and perimeter of rectangles,
such as:
For any rectangle there is one with the same area and a larger
perimeter.
He was invited to come up with a similar generalization. On the basis of three
examples, he proposed the result:
"The area is always less than the perimeter".
Another example was tried which didn't work, so an extra clause was tacked
on: "when the area and perimeter are less than 20"
in order to exclude this newcomer from consideration. See Pimm (1983) for
further discussion.
So much for counterexamples, ts the situation any different for ordinary
or confirming examples? If examples are always examples
of
something, and
counterexamples counter
to
something, how can students become aware of
the 'something' which is being exemplified? A partial answer is provided by
Michener (1978), but one important aspect is the relation between generality
and specificity.
A teacher having written an example of a technique or theory on the board,
is seeing the generality embodied in the example, and may well never think
of indicating the scope Of the example, nor of stressing the parts that need
to be stressed in order to appreciate the exampleness. However, the pupils
have far less experience, even with a particular instance of the situation under
discussion (and may well be unaware there are others) which as a consequence,
absorbs all their attention. The pupils may see only the particular (which is
possibly for them still quite general, i.e., not mastered). As a result they often
try to 'learn the example'.
We earlier raised the question of proof by example in the context of proof
by generic example. Despite what is commonly claimed, there are many
situations redolent of this. Instances include integration by partial fractions
and decomposition of permutations into disjoint cycles or into transpositions,
each suitably chosen to illustrate the range of possibilities. The sign of the
derivative on an entire intel~!al is found by checking it only at one point.

GENERIC EXAMPLES 287
Behind these examples are local-to-global theorems (see Fowler (1973) for
development and exploitation of this idea in elementary analysis) guaranteeing
the invariance of the specific result and attesting that any example can be
taken as generic. Might not this look to students as proof by example, and
confirm their prior impressions? Is it not in fact proof by example?
A result proved to be true for equivalence classes is illustrated on particular
elements. Yet why is this not the same as
7 is related to 4 and 4 is related to 6 so
7 is related to 6
as a sufficient proof of transitivity of a relation R? Many students see these as
the same.
Teachers always have ulterior motives; their attention is directed at several
different levels in preparing for and conducting classes. Seldom, if ever, are the
teachers in the classroom simply to be with the children. When their own
attention is directed at a level of generality, despite the fact that they are
ostensibly working with particular examples, it is not surprising that their
pupils' attention is on the particular and that many remain blissfully unaware
of the general.
In this article we have tried to bring out some of the inherent difficulties
in mathematical expressions of generality and their relation to the particular.
A generic example is an actual example, but one presented in such a way as
to bring out its intended role as the carrier of the general. This is done by
means of stressing and ignoring various key features, of attempting to structure
one's perception of it. Different ways of seeing lead to different ways of
knowing. Unfortunately it is almost impossible to tell whether someone is
stressing and ignoring in the same way as you are.
In conclusion, we offer the following questions which seem to us difficult,
but worthy of considerable attention.
Questions
How can you expose the genericity of an example to someone who sees only
its specificity? Apart from stressing and ignoring, and repeating the general
statement over and over, how can the necessary act of perception, of seeing
the general in the particular, be fostered?
How can you discern the extent of the generality perceived by someone else
when looking at a particular example together?
Why do we offer students examples in class, and what are they supposed to

288 JOHN MASON AND DAVID PIMM
make of them? If examples are always examples
of
something, how can
students become aware of that which the examples are supposed to be ex-
emplifying?
NOTES
1 Our deliberate use of 'she' in this paragraph produces a further instance of generic
ambiguity. The possible incongruity which can arise from using 'she' coutd be due to
the (currently challenged) perception of 'he', but not 'she', as generic for the class of
humans. fman~
man woman
he she
his hers
This phenomenon, an element of a class being used to name the class as well, is precisely
the focus of our attention. Linguists, who have a name for everything, refer to the relation
'X is a specific example of Y' as 'X is a hyponym of Y'. One of the central concerns of
this paper car therefore be succinctly described as reflexive hyponymy.
2 6, when seen in the context of even numbers, CAN be seen as a generic even number,
but this requires stressing some of its features (its evenness) and ignoring other features
(that it is a product of two primes, that it is divisible by 3 .... ). It is almost impossible to
tell whether someone else is stressing and ignoring the same way that you are, without
recourse either to algebra or considerable discussion. This important observation will be
expanded shortly.
3 2N cannot be seen as a generic even number if it is not accepted as an even number
at all, because generic examples have first and foremost to be examples. However the
symbols 2N are extremely helpful because they stress precisely the qualities which make
2N generic of even numbers, and ignore (by hiding inside the N) all irrelevant features.
It is possible to rewrite 6 as 2 • 3 to emphasize this particular quality.
4 The following extract from a fictional conversation between the philosophers B. Russell
and G. E. Moore in BEYOND THE FRINGE indicates that mathematicians are not alone.
Philosophers too have struggled with definite and indefinite articles:
tn there was Moore, seated by the fire with a basket upon his knees.
"Moore", I said, "have you any apples in that basket?"
"No", he replied, and smiled seraphically as was his wont.
I decided to try a different logical tack.
"Moore", I said, "do you then have some apples in that basket?"
"No", he said, once again.
Now I was in a logical cleft stick, so to speak, and had but one way out.
"Moore", I said, "do you then have apples in that basket?"
"Yes", he replied, and from that day forth we remained the very closest of
friends. cited in Fowler (1973, p. 117)
s Compare this with the style of joke which begins
There was
this
woman
or
A certain
young man was...

GENERIC EXAMPLES 289
in which the purpose of the underlined definiteness is obscure. The purpose is not very
clear either of the American legal practice whereby arrest warrants can be issued in the
name of John Doe, whereas in England they can be left blank.
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