1. Background
There is evidence from earliest historical records that examples play a
central role in both the development of mathematics as a discipline and
in the teaching of mathematics. It is not surprising therefore that
examples have found a place in many theories of learning mathematics.
Many would argue that the use of examples is an integral part of the
discipline of mathematics and not just an aid for teaching and
learning. The forum takes as its background both the variety of ways in
which examples are construed within different theories of learning and
the contribution that attention to examples can make to the learning
and teaching processes. Consequently the forum can be seen as
addressing issues at the very heart of mathematics education, both
drawing upon and informing many other research topics. We argue that
paying attention to examples offers both a practically useful and an
important theoretical perspective on the design of teaching activities,
on the appreciation of learners’ experiences and on the
professional development of mathematics teachers.
The importance of these ideas does not actually depend on the framework
used for analysing teachers’ intentions, nor on any terms
used to
describe forms of teaching, such as:
‘analyticinductive’
or ‘syntheticdeductive’,
‘traditional’ or
‘reform’, ‘rotelearning’ or
‘teaching
for understanding’, ‘authentic’ or
‘investigative’. Issues in exemplification are
relevant to
all kinds of engagement with mathematics.
This paper positions exemplification on the research agenda for the
community by giving a historical overview of the way examples have been
seen in mathematics education; an account of associated literature; an
exploration of how exemplification ‘fits’ with
various
perspectives on learning mathematics; accounts of issues relating to
teachers’ and learners’ use of examples; and
directions for
future research.
2. What is a mathematical example?
The word
example
is used in mathematics education in a wide variety of ways. This
section offers a brief overview of the scope of our use of the term and
points to some useful distinctions that can be made between different
uses.
Examples in the form of worked solutions to problems are key features
in virtually any instructional explanation (Leinhardt 2001) and
examples of all kinds are one of the principle devices used to
illustrate and communicate concepts between teachers and learners (e.g.
Bruner et al. 1956, Tall & Vinner 1981, Peled &
Zaslavsky
1997). Diagrams, symbols and reasoning are all treated as particular
yet thought about (by the teacher at least) as general. Examples offer
insight into the nature of mathematics through their use in complex
tasks to demonstrate methods, in concept development to indicate
relationships, and in explanations and proofs. The core issue is
whether learners and teachers are perceiving the same (or indeed any)
generality.
An important pedagogic distinction can be made between examples of a
concept (triangles, integers divisible by 3, polynomials etc.) and
examples of the application of a procedure (finding the area of a
triangle, finding if an integer is exactly divisible by 3, finding the
roots of a polynomial etc). Sowder (1980) tried to avoid this confusion
by distinguishing between ‘examples’ and
‘illustrations’. However, within the category of
‘examples of the application of a procedure’, or
‘illustrations’ we distinguish further between
‘worked(out) examples’, in which the procedure
being
applied is performed by the teacher, textbook author or programmer,
often with some sort of explanation or commentary, and
‘exercises’, where tasks are set for the learner to
complete. The workedout example has been the subject of a body of
research within psychology (e.g. Atkinson et al. 2000, Renkl 2002).
Of course, these distinctions are neither precise nor clear cut. Gray
& Tall (1994) underline the fact that the same notation may be
viewed as signifying a process or an object, so that, for example, a
teacher may offer a representation of the function y = 2x + 3 as an
example of a linear function, but the learner may see it as an example
of a procedure (for drawing a graph from an equation). There is a good
deal of ‘middle ground’ between exercises and
worked
examples, for instance when a teacher ‘leads’ a
class
through the working out of a typical problem using questions and
prompts.
Across these broad categories of form and function of examples there
are three special descriptive labels: ‘generic
example’,
‘counterexample’ and
‘nonexample’. Generic
examples may be examples of concepts or of procedures, or may form the
core of a generic ‘proof’. Counterexamples need a
hypothesis or assertion to counter, but they may do this in the context
of a concept, a procedure or even (part of) an attempted proof.
Nonexamples serve to clarify boundaries and might do so equally for a
concept, for a case where a procedure may not be applied or fails to
produce the desired result or to demonstrate that the conditions on a
theorem are ‘sharp’. In fact all three labels have
to do
with how the person (teacher or learner) perceives the mathematical
object in question, rather than with qualities of the object itself.
The term example here includes anything used as raw material for
generalising, including intuiting relationships and inductive
reasoning; illustrating concepts and principles; indicating a larger
class; motivating; exposing possible variation and change, etc. and
practising technique (Watson & Mason 2002a, 2002b).
Exemplification
is used to describe any situation in which something specific is being
offered to represent a general class to which learners’
attention
is to be drawn. A key feature of examples is that they are chosen from
a range of possibilities (Watson & Mason 2005 p238) and it is
vital
that learners appreciate that range.
3.
Examples
From An Historical Perspective
The whole point of giving worked examples is that learners appreciate
them as generic, and even internalise them as templates so that they
have general tools for solving classes of problems. Unfortunately their
use in lessons is often reduced to the mere practice of sequences of
actions, in contrast to a more investigative approach (Wallis 1682) in
which learners experience the mathematisation of situations as a
practice, and with guidance abstract and reconstruct general
principles themselves.
Whereas mathematical investigations and the use of
‘authentic
or ‘modelling’ approaches appear to be a relatively
recent
pedagogic strategy, there are historical precedents. The earliest
mathematical records (Egyptian papyri, Babylonian tablets and later
copies of lost Chinese manuscripts) all use contextbased problems with
worked solutions to illustrate procedures, or what came to be called
rules and then later algorithms in medieval and renaissance texts. They
sometimes point specifically to a generality with comments such as
‘thus is it done’ or ‘do it
thus’ (Gillings
1972 p232232), and ‘this way you may solve similar
problems’ or ‘by the same method solve all similar
problems’ (Treviso Arithmetic 1478 see Swetz 1987 p151).
By the 16th century European authors of mathematical texts had begun to
justify the presence of examples in their texts, commenting explicitly
on the role that examples play for learners. Girolamo Cardano (1545
see Witmer 1968) used phrases such as:
 We have used a variety of examples so that you may
understand
that the same can be done in other cases and will be able to try them
out for the two rules that follow, even though we will there be content
with only two examples; It must always be observed as a general rule
… ; So let this be an example to you; by this is shown the
modus
operandi in questions of proportion, particularly; in such cases
(Witmer 1968 p3641).
By the late 19th and early 20th century, pedagogic principles become
more and more explicit in some cases, if only to attract teachers to
‘new’ pedagogic approaches. For example a textbook
from
Quebec (MacVicar 1879) claims that:
 The entire drill and discussions [examples] are believed to
be so
arranged, and so thorough and complete, that by passing through them
the pupil cannot fail to acquire such a knowledge of principles and
facts, and to receive such mental discipline, as will prepare him
properly for the study of higher mathematics. (piv)
Some authors scramble different types of problems, or different looking
problems, presumably to engage the learner in recognizing the type,
while others collect exercises according to the technique needed,
perhaps to promote a sense of the general class of which the exercises
are but particulars but more probably to focus on fluency of
performance. For example, the expansion by the schoolmaster Iohn Mellis
(Record 1632) of John Dee’s extension of Robert
Record’s
original arithmetic (Record 1543/1969) offers collections of worked
examples which offer a variety of differences in what is given and what
is sought, so as to draw attention to a wider class of problem type
that can be solved by the same method or ‘Rule’.
The design of sequences of examples is a central issue in
their
instructional use that influences both the inductive and deductive
aspects of learning. For example George Pólya (1962)
provided
long sequences of exercises building up generalisations from a simple
starting idea. He ended one such a chapter with a final task:
 Devise some problems similar to, but different from, the
problems
proposed in this chapter – especially such problems as you
can
solve. (Pólya 1962 p98)
The idea that creating your own examples and questions can aid learning
is not new. Record has his scholar in dialogue with the author
constructing examples, and Cardano invites the reader to construct
their own examples of questions.
Historically there have been two main approaches to the use of
examples, distinguished in the 18th century by the terms analytic and
synthetic. The difference amounted to whether general rules were
presented before or after worked examples (or even not at all). In the
early 19th century Warren Colburn instituted in the USA the inductive
method advocated by Johann Pestalozzi (1801):
 The reasoning used in performing these small examples is
precisely the same as that used upon large ones. And when anyone finds
a difficulty in solving a question, he will remove it much sooner and
much more effectively, by taking a very small example of the same kind,
and observing how he does it, than by [resorting] to a rule. (Colburn
1826 p141142)
Herbert Spencer (1878), developed the ideas further, expecting learners
to infer the general from carefully presented particulars.
 Along with roteteaching, is declining also the
nearlyallied
teaching by rules. The particulars first, and then the generalizations,
is the new method … which, though ‘the reverse of
the
method usually followed, which consists in giving the pupil the rule
first’ is yet proved by experience to be the right one.
Ruleteaching is now condemned as imparting a merely empirical
knowledge – as producing an appearance of understanding
without
the reality. To give the net product of inquiry without the inquiry
that leads to it, is found to be both enervating and inefficient.
General truths to be of due and permanent use, must be earned.
…
While the ruletaught youth is at sea when beyond his rules, the youth
instructed in principles solves a new case as readily as an old one.
(Spencer 1878 p56–7)
Alfred Whitehead summarised this approach as
 To see what is general in what is particular and what is
permanent in what is transitory is the aim of scientific thought.
(Whitehead 1911 p4)
Pólya asserted:
 [in doing mathematics]… we need to
adopt the
inductive attitude [which] requires a ready ascent from observations to
generalizations, and a ready descent from the highest generalizations
to the most concrete observations. (Pólya 1945 p7).
Even more important than the distinction between inductive and
deductive, between ‘general first’ or
‘general
later’, are finer distinctions and hybrid approaches which
will
emerge in later sections. Both inductive and deductive approaches are
compatible with constructive accounts of learning and rely on
exemplification: inductive learning implies that the learner is making
some generalisations about actions or concepts while working with a
range of examples (seeing generality through particulars); deductive
learning implies that the learner is able to make personal sense of a
definition or general principle, and adapt it for current and future
use (seeing particular instances in the general).
Examples can be useful stimuli for prompting selfexplanation leading
to understanding. Cardano acknowledges that sometimes it is too
confusing to state a general method, and suggests that examples provide
explanation. This sentiment is reflected in a wide range of text
authors over the centuries, and by Richard Feynman:
 I can’t understand anything in general unless
I’m
carrying along in my mind a specific example and watching it go
(Feynman 1985 p244).
By contrast, Zazkis (2001) observes that starting with more
complex
problem situations and more complex numbers not only provides an
opportunity for learners to simplify for themselves in order to see
what is going on before returning to the more complex, but also
provides an opportunity for learners to appreciate more fully the range
and scope of generality implied by the particular exemplars.
Furthermore, learners are not deceived by the attraction of doing
simple computations with small numbers rather than attending to
underlying structure.
This survey illustrates a diversity of approaches to examples in
learning and teaching. In some cases the succession of examples is the
important feature of their use. Their explicit and implicit
similarities and differences, the number and variety exhibited, and
their increasing complexity can all be used to promote inductive
learning. In other cases a single example is intended as a generic
placeholder for a completely general expression of a concept, object or
process to support deductive thinking.
4. Examples
From A Theoretical Perspective
Examples play a key role in
various
classes of theories of learning mathematics. Social and psychological
forces and situational peculiarities influence and inform both the
examples and the concept images to which someone has access at any
moment. The notion of a personal example space nicely complements the
notion of a concept image in this respect. Thinking in terms of
variation highlights the importance both of the succession of examples
and the aspects which are varied in that succession in affording
learners access to key features of a concept or technique.
4a The role
of examples in doing mathematics
Various mathematicians
have written
about the importance of examples in appreciating and understanding
mathematical ideas and in solving mathematical problems (e.g.
Pólya, Hilbert, Halmos, Davis, Feynman). Whenever a
mathematician encounters a statement that is not immediately obvious,
the ‘natural’ thing to do is to construct or call
upon an
example so as to see the general through intimate experience of the
particular (Courant 1981). When a conjecture arises, the usual practice
is alternately to seek a counter example and to use an example
perceived generically to see why the conjecture must be true (Davis
& Hersh 1981).
Often
a mathematician will detect and express a structural essence which lies
behind several apparently different situations. Out of this arises a
new unifying concept and an associated collection of definitions and
theorems. Sometimes a particular example will suggest some feature
which can be changed, leading to a richer or more unifying concept, or
at least to an enriched awareness of the class of objects encompassed
by a theory. It is not examples as such which are important to
mathematicians, but what is done with those examples, how they are
probed, generalised, and perceived.
4b The role
of examples in theories of learning mathematics
The importance of
encounters with
examples has been a consistent feature of theories and frameworks for
describing the learning of mathematics This section offers a very brief
overview of different ways in which theories of learning have used
examples.
How people abstract or
extract a
concept from examples has been specifically studied in psychology from
the point of view of how examples and nonexamples influence the
discernment of concepts (e.g. Bruner 1956, Wilson 1986, 1990, Charles
1980, Petty & Jansson 1987). In Artificial Intelligence
attention
on default parameters (expectations and assumptions) for triggering
frames (patterns of behaviour) were used to try to reproduce concept
acquisition (e.g. Winston 1975, Minsky 1975).
Genetic
epistemology (Piaget 1970, see also Evans 1973) assumes that
individuals actively try to make sense of their world of experience,
supported by social groupings (Confrey 1991) in which they find
themselves. It underpins many current theories of mathematics learning,
by assuming the impact of new examples on existing mental schema
through assimilation and accommodation. Piaget’s notion of
reflective abstraction (Dubinsky 1991) implies experiences and actions
performed by the learner through which abstraction is possible.
Building
on Piaget’s notion of schema, Skemp (1969) wrote about the
learning of mathematical concepts through abstraction from examples,
which meant that the teachers’ choice of which examples to
present to pupils was crucial. His advice on this topic includes
consideration of noise, that is the conspicuous attributes of the
example which are not essential to the concept, and of nonexamples,
which might be used to draw attention to the distinction between
essential and nonessential attributes of the concept and hence to
refine its boundaries.
Once a
concept is formed, later
examples can be assimilated into that concept (Skemp 1979) and a more
sophisticated concept image can be formed (Tall and Vinner 1981).
Vinner (1983, 1991) describes a gap between learners’ concept
image and the concept definition: concept images can be founded on too
limited an exploration of the examples encountered so that features of
the examples which are not part of the concept are retained in the
concept image, a process recognised and elaborated on by Fischbein
(1987) as figural concepts. Concept images are therefore often limited
to domains with which learners are most familiar and so may be too
limited to be useful. A considerable part of research results on wrong,
alternative and partial conceptions can be convincingly interpreted in
this way. Thus improving learners’ conceptions amounts to
reducing the gap between their concept images and the concept
definition. Tall and Vinner point to the importance of the
examples in closing this gap.
Thorndike et al. (1924)
followed a
behaviourist line in using examples as stimuli to provoke learning
responses, and Gagné (1985) developed this into a hierarchy
of
behaviours of increasing complexity. Dienes (1960) used cleverly
constructed games and structured situations as examples of mathematical
structures in which to immerse learners so that they would experience
examples of sophisticated mathematical concepts through their own
direct experience. Others follow historical precedents in trying to
describe what it is like for learners to make sense of new concepts
(Davis 1984) and worked examples (Anthony 1994).
Marton and colleagues
(Marton &
Booth 1997, Marton & Tsui 2004 ) developed the notion of varied
examples as a way to encounter concepts noting that what is needed is
variation in a few different aspects closely juxtaposed in time so that
the learner is aware of that variation as variation. Marton even
formulates a definition of learning as becoming aware of one or more
dimensions of variation which an example could exhibit. Since teacher
and learner may not appreciate the same dimensions of variation, Watson
& Mason (2005) expanded this to appreciating a particular
concept
as being aware of dimensions of possible variation and with each
dimension, a range of permissible change within which an object remains
an example of the concept.
Recent articulations
which connect
the genesis of mathematical knowledge with the processes of coming to
know also clarify the central role of examples as the raw material for
generalizing processes and conceptualizing new objects. Sfard (1991)
follows Freudenthal (1983) in seeing learners moving from an
operational to a structural understanding of concepts through a process
of interiorisation and condensation leading to reification.
Interiorisation and condensation are slow, gradual processes, taking
place over time and through repeated encounters with examples. Dubinsky
and his colleagues (see Asiala et al. 1996) have introduced a theory of
the development of mathematical knowledge at undergraduate level which
they call APOS theory (actions, processes, objects, schemas). Again the
theory predicts that encounters with examples will be part of the
process by which learners will move from action to process and then to
object conceptions. The Pirie & Kieren (1994) onion model of
the
growth of understanding focuses on image construction and folding back
between states, yet still recognizes that it is direct experience of
examples which contribute to the formation of personal images and
knowings.
Another aspect of the
relationship
between examples and concepts or processes centres on the notion of
generic example, or prototype. A generic example:
involves making
explicit the reasons
for the truth of an assertion by means of operations or transformations
on an object that is not there in its own right, but as a
characteristic representative of the class. (Balacheff 1988, p. 219)
Freudenthal (1983)
describes
examples with this potential as paradigms. A strand of psychological
research beginning with Rosch (1975) has explored how these prototypes
(representatives of categories) are used in reasoning. Hershkowitz
(1990) drew attention to the tendency to reason from prototypes rather
than definitions in mathematics, and the errors that this kind of
reasoning can produce. Often learners' concept image is largely
determined by a limited number of prototype examples (e.g. Schwarz
& Hershkowitz 1999) so it is important to go beyond prototypes
using nontypical examples to push toward and beyond the boundary of
what is permitted by the definition, becoming aware of that boundary
during the process (the range of permissible change). Approaches to
helping learners expand their reasoning beyond prototypes have been
described in a number of specific areas of mathematics.
Dreyfus
(1991) discusses the role of examples in abstraction, and in particular
the different uses that might be made by learners of single examples
and collections of examples. He suggests that, for a relatively
sophisticated mathematical learner, a definition and a single example
may be sufficient, whereas less experienced learners may need large
numbers of carefully selected examples before they can abstract the
properties of the concept.
4c
The theory of personal example spaces
The collection of
examples to which
a learner has access at any moment, and the richness of interconnection
between those examples (their accessible example space) plays a major
role in what sense learners can make of the tasks they are set, the
activities they engage in, and how they construe what the teachertext
says and does. Zaslavsky & Peled (1996) point to the possible
effects of limited examplespaces accessible to teachers with respect
to a binary operation on their ability to generate examples of binary
operations that are commutative but not associative or vice versa.
Watson and Mason (2005)
formulated
the notion of a personal example space as a tool for helping learners
and teachers become more aware of the potential and limitations of
experience with examples. They identify two principles:
 Learning
mathematics consists
of exploring, rearranging, gaining fluency with and extending your
example spaces, and the relationships between and within them.

Experiencing
extensions of
your example spaces (if sensitively guided) contributes to flexibility
of thinking and empowers the appreciation and adoption of new concepts.
A personal
example space is what
is accessible in response to a particular situation, to particular
prompts and propensities. Example spaces are not just lists, but have
internal idiosyncratic structure in terms of how the members and
classes in the space are interrelated. Their contents and structures
are individual and situational; similarly structured spaces can be
accessed in different ways, a notable difference being between
algebraic and geometric approaches. Example spaces can be explored or
extended by searching for situationallypeculiar examples as doorways
to new classes; by being given further constraints in order to focus on
particular characteristics of examples; by changing a closed response
into an open response; by glimpsing the infinity of a class represented
by a particular.
4d Summary
While there is a long
history of
attention to the provision of suitable examples intended to indicate
the salient features which make examples exemplary, recent developments
indicate that social and psychological forces and peculiarities play a
central role in both the personal example space to which learners have
access and the concept image which they develop. Particular attention
needs to be paid to the succession of examples and both the dimensions
of possible variation and their associated ranges of permissible change
to which learners are afforded access.
5. Examples
From A Teacher's Perspective
The treatment of examples presents the teacher with a
complex
challenge, entailing many competing features to be weighed and
balanced, especially since the specific choice of and manner working
with examples may facilitate or impede learning. Note that the aspects
mentioned here are interrelated, not disjoint.
5a Examples as tools for communication and explanation
Examples are a communication device that is fundamental to
explanations and mathematical discourse (Leinhardt 2001). The art of
constructing an explanation for teaching is a highly demanding task
(Ball 1988; Kinach 2002a, 2002b), as described by Leinhardt et al.
(1990):
 Explanations consist of the orchestrations of
demonstrations,
analogical representations, and examples. […]. A primary
feature
of explanations is the use of wellconstructed examples, examples that
make the point but limit the generalization, examples that are balanced
by non or countercases. (ibid p6).
Leinhardt & Schwarz (1997) claim that when teaching metaskills
 The purpose of an instructional explanation is to teach,
specifically to teach in the context of a meaningful question, one
deserving an explanation. (ibid p399).
That is to say that the meaningful question, the example, plays a key
role in the instructional explanation.
Peled & Zaslavsky (1997) distinguish between three types of
counterexamples suggested by mathematics teachers, according to their
explanatory power: specific, semigeneral and general examples. They
assert that general (counter)examples explain and give insight
regarding the reason why a specific conjecture is not true and
strategies to produce more counterexamples.
The conjecture that two rectangles with the same
diagonal
must be congruent, is false. The diagram (taken from Peled &
Zaslavsky 1997) can be regarded as a general counter example because it
communicates an explanation of why the conjecture is false without
reference to particular values. Furthermore, inherent to this example
is the notion that there are an infinite number of different rectangles
with the same diagonal. 

With respect to communication, a teacher must take into
consideration that an example does not always fulfil its intended
purpose (Bills 1996; Bills & Rowland 1999). Mason &
Pimm (1984)
suggest that a generic example that is meant to demonstrate a general
case or principle may be perceived by the learners as a specific
instance, overlooking its generality. What an example exemplifies
depends on context as well as perceiver.
Attributes which make an example ‘useful’ include:
 Transparency: making it relatively easy to direct the
attention of the target audience to the features that make it
exemplary.
 Generalisability: the scope for generalisation afforded by
the
example or set of examples, in terms of what is necessary to be an
example, and what is arbitrary and changeable.
Examples with some or all of these qualities have the potential of
serving as a reference or model example (RisslandMichener 1978), with
which one can reason in other related situations, and can be helpful in
clarifying and resolving mathematical subtleties. Clearly, the extent
to which an example is transparent or useful is subjective. Thus, the
role of the teacher is to offer learning opportunities that involve a
large variety of 'useful examples' (yet not too large a
variety
that might be confusing) to address the diverse needs and
characteristics of the learners.
To illustrate some of the distinctions mentioned so far, consider the
following examples of a quadratic function (these examples and the
subsequent elaboration appear in Zaslavsky & Lavie 2005,
submitted):
These are three different representations of the same function. Each
example is more transparent about some features of the function and
more opaque with respect to others (e.g., roots; position of the vertex
and minimum value; yintercept). However, these links are not likely to
be obvious to the learner without some guidance on how to read or
interpret the expressions. Moreover, it is not even clear that learners
will consider all three as acceptable examples of a quadratic function,
since, for example the power of two is less obvious in the factored
form, and a quadratic may have been defined to look like the third
expression. A teacher may choose to deal with only one of the above
representations, or s/he may use the three different representations in
order to exemplify how algebraic manipulations lead from one to
another, or in order to deal with the notion of equivalent expressions.
Each different representation communicates different meanings and
affords different mathematical engagement, but there are further
possible differences in perception. What a learner will see in each
example separately and in the three as a whole depends on the context
and classroom activities surrounding these examples, and her own
previous experience and disposition. A learner who appreciates the
special information entailed in each representation may be informed by
them to be alert to their differing qualities in the future, even to
the extent of effectively using them as reference examples or reference
forms when investigating other (quadratic) functions.
To an expert there are some irrelevant features, such as the use of
particular letters yet, a learner may regard x and y as mandatory
symbols for representing a quadratic function. Another irrelevant
feature is the fact that in all three representations all the numbers
are integers. A learner may implicitly consider this to be a relevant
feature, unless s/he is exposed to a richer examplespace. Learners may
also generalise and think that all three representations can be used
for any quadratic function.
None of these considerations need be conscious; even the learner who is
not deliberately making sense of what is offered is still becoming
familiar with a particular range of examples which create a sense of
normality. Hence, the specific elements and representation of examples,
and the respective focus of attention facilitated by the teacher, have
bearing on what learners notice, and consequently, on their
mathematical understanding. Paul Goldenberg (personal communication)
pointed out that sometimes an example can be too specific to be useful;
learners and teachers need to be aware that the shift to seeing
examples as ‘representative and therefore
arbitrary’ is
nontrivial and may need classroom discussion.
5b.
Uses of examples for teaching
Some authors have categorised examples according to the use for which
they are particularly suited. Notable amongst these, RisslandMichener
(1978) distinguishes four types of examples (not necessarily disjoint),
which have epistemological significance: startup (which help motivate
basic definitions and results, and set up intuitions in a new subject),
reference (which are used as standard instances of a concept or a
result, model, and counterexample and referred to repeatedly in the
development of theory), model (which are paradigmatic, generic
examples) and counterexamples (which demonstrate
that a
conjecture is false and are used to show the importance of assumptions
or conditions in theorems, definitions and techniques).
Rowland and Zaslavsky (2005) distinguish between providing examples of
something as raw material for inductive reasoning, as particular
instances of a generality, and providing an environment for practice.
For example, in order to teach subtraction by decomposition, a teacher
might work through say, 6238 in column format; for practice a
collection of wellchosen subtly varying particular cases might be set
as an exercise. In the case of concepts, the role of examples is to
facilitate abstraction. Once a set of examples has been unified by the
formation of a concept, subsequent examples can be assimilated by the
concept
Another kind of use of examples in teaching, more often called
‘exercises’, is illustrative and practiceoriented.
For us,
exercises are examples, selected from and indicative of a class of
possible such examples. Typically, having learned a procedure (e.g. to
add 9, to find equivalent fractions, to solve an equation), the learner
rehearses it on several such ‘exercise’ examples.
This is
first in order to assist retention of the procedure by repetition, then
later to develop fluency with it (Rowland & Zaslavsky 2005).
When
the teacher repeatedly demonstrates how to perform on these practice
exercises, the learning mechanism that is facilitated may share some
characteristics of the learning from workedout examples (see section
6b).
Hejný (2005) notes that the focus of attention needs to be
not
only on what can be generalised from one example, but also on a
structured set of tasks which may direct learners to find a general or
abstract idea. For example, he suggests helping learners in
primary grades discover a formula for the area of a triangle, by
offering them a rich problem situation, from which a general
relationship can be induced.
Divide a given rectangle ABCD by a segment EF to make
two
rectangles AEFD and EBCF. These rectangles are divided by diagonals AF
and BF into four rightangled triangles. Consider eight shapes: five
triangles: AEF, AFD, EBF, BCF, ABF and three rectangles: ABCD, AEFD,
EBCF.
Given the area of two of these shapes, find the areas of all the others;
Given the length of three segments from the following: AE, EB, AB, DF,
FC, DC, AD, EF, BC find the areas of all the triangles.
What do you have to know to find the area of triangle ABF?


Most of the studies that deal with sets of example suggest
that
the specific sequence of examples has an impact on learning. In
particular, it is recommended to combine examples and nonexamples
within a sequence of examples, in order to draw attention to the
critical features of the relevant examples. There is an argument for
examples to be ‘graded’, so that learners
experience
success with routine examples before trying more challenging ones.
However, it should be noted that sequencing examples from
‘easy’ to ‘difficult’ is not
always effective
(Tsamir 2003). Exercises designed for fluency are likely to be
differently structured to exercises designed to promote or provoke
generalisation (Watson & Mason 2006).
Leron (2005) uses the term
generic
proof to
refer to what MovshovitzHadar (1988) calls a
transparent proof or
pseudo proof.
Leron illustrates generic proofs with reasoning to justify the fact
that every permutation can be decomposed as the product of disjoint
cycles. As a simpler example consider the proof that the sum of two odd
numbers is an even number. One can use two
‘general’ odd
numbers that are not special in any obvious way, e.g. 137 and 2451, and
present a ‘proof’ for these two numbers,
137 + 2451= (136 + 1) +
(2452 – 1) = 136 + 2452.
This form of presentation can be read generically as justification that
the sum of any two odd numbers is equal to the sum of two even numbers
and hence even. However, learner attention has to be directed
appropriately in order to have this effect. The specific choice of
examples together with the transparency with respect to the main ideas
of the proof both play an important role.
Finally, examples (or exercise examples) can be used for assessment of
learners’ performance and understanding in a broad sense. The
more conventional way would be to present learners with examples of
problems or mathematical objects and ask them to follow certain
instructions (e.g., solve the problem, compare the objects etc.). In
this, the teacher assumes that these examples are cases of a more
general class of problems or objects, and considers learners’
performance with these examples as a representation of their knowledge.
In a way, several researchers use carefully selected examples to
investigate learners’ schemes (e.g., Dreyfus & Tsamir
2004;
Peled & AwawdyShahbari 2003). Section 7 elaborates on
researchers’ use of examples.
Another approach that some teachers (as well as researchers) use for
revealing learners’ conceptions and ways of thinking is by
asking
learners to generate their own examples of problems and of objects
(e.g. van den HeuvelPanhuizen et al. 1995, Zaslavsky 1997, Hazzan
& Zazkis 1999, Watson & Mason 2005).
5c.
Teachers’ choice of examples
Research on teachers’ choice of examples is rather scarce.
Ball
et al. (2005) maintain that a significant kind of mathematical
knowledge for teaching involves specific choices of examples, that is,
considering what numbers are strategic to use in an example. Similarly,
Rowland & Zaslavsky (2005) note that the choice of 6238 in
column
format to teach subtraction by decomposition is not a random choice:
the digits are all chosen with care because constructing examples is
not an arbitrary matter, though there is usually some latitude in the
choice of effective examples. The 8 could have been a 9; on the other
hand, it could not have been a 2. It could have been a 4, say, but
arguably the choice of 4 is pedagogically less effective than 8 or 9,
because subtracting 4 from 12 would lead some pupils to engage in
fingercounting, distracting them from the procedure they are meant to
be learning. Attending to the range of change of digits that is
permissible without changing the learners’ experience (Watson
& Mason 2005) is essential in choosing instructional examples.
Novice teachers’ poor choices of examples have been
documented by
Rowland et al. (2003) who considered the way in which student teachers
give evidence of their subject knowledge in their teaching of
mathematics to primary school children, one aspect being the choice of
examples. The authors present instances of choices which, in their
words, ‘obscured the role of the variable’ (p244):
reading
a clock face set at half past the hour by using the example of half
past six; using as the first example to illustrate the addition of nine
by adding 10 and subtracting one, adding nine to nine itself. Often the
unintentionally ‘special’ nature of an example can
mislead
learners.
In selecting instructional examples it is important to take into
account learners’ preconceptions and prior experience. In
particular, careful construction of examples could enable teachers to
identify and help learners cope with the effect of previous knowledge
and existing schemes (implicit models) on the construction of new
knowledge. Research findings on learning could serve as a
rich
source for teachers’ selection of effective examples for this
purpose. For example, Peled & AwawdyShahbari (2003) suggest
asking
learners to compare carefully selected pairs of decimal or common
fractions, in order to identify the implicit models by which they
operate. An effective example for decimal fractions would be to ask
learners which number is bigger: 2.8 or 2.85. Some learners claim that
2.8 is bigger “because tenths are bigger than
hundredths”.
Similarly, in comparing and , some will
say
that is larger because fifths are larger than
sixths,
because they focus on the size of the fractional part and ignore the
number of parts. Similarly, the study by Tsamir & Tirosh (1999)
regarding learners’ tendencies to address inclusion
considerations when dealing with comparisons of infinite sets informed
the choice of examples Tsamir and Dreyfus subsequently presented to
learners (Tsamir & Dreyfus 2002).
In secondary school the considerations in selecting specific examples
seem to be far more complex than in primary school. Zaslavsky and Lavie
(2005, submitted) and Zaslavsky and Zodik (in progress) discuss
teachers’ considerations underlying their choice of examples.
Issues that came up in their study include: the tension between the
teachers’ desire to construct ‘reallife’
examples
and the mathematics accuracy they feel they are
‘sacrificing’ when doing so; the dual message of
randomly
selected examples since the randomness may convey the generality of the
case, however it may also yield impossibilities or inadequate
instances; the visual entailments of examples in geometry, and the
ambiguity regarding what visual information may be induced and what
should not. A classic instance is that when a
‘general’
triangle is sketched, some learners rely on the relative magnitude of
length of its sides, leading to examiners asserting with every diagram
‘not drawn to scale’.
5d.
Summary of teacher perspective
The use of examples in the classroom is an essential but complex
terrain. It involves careful choices of specific examples which
facilitate the directing of attention appropriately so as to explain
and to induce generalisations. Desirable choice of examples depends on
many factors, such as the teaching goals and teachers’
awareness
of their learners’ preconceptions and dispositions.
It has been proposed (e.g., Tall & Vinner 1981, Chi et al.
1989,
Chapman 1997) that the key feature of learning is not what is presented
but rather what is encoded in the learner's mind, what is constructed
by the learner, what practices are internalised.
6. Examples From A Learner's Perspective
The crucial factors for appreciating and assimilating
concepts,
and for learning techniques are the form, format and timing of examples
encountered, and experience of ways of working with and on examples.
When invited to construct their own examples, learners both extend and
enrich their personal example space, but also reveal something of the
sophistication of their awareness of the concept or technique.
6a.
Concept formation
Davis (1984) described mathematical objects emerging from specific
experiences:
When a procedure is first being learned, one experiences it almost one
step at time; the overall patterns and continuity and flow of the
entire activity are not perceived. But as the procedure is practiced,
the procedure itself becomes an entity  it becomes a thing.
[…]
The procedure, formerly only a thing to be done  a verb  has now
become an object of scrutiny and analysis; it is now, in this sense, a
noun. (p2930, ibid).
In the process of concept formation, the operational conception
(focussing on the process) is often first to develop, gradually moving
towards a structural approach (focusing on the object) (Rumelhart
1989). Gray and Tall (1994) use the example ‘2 + 3’
to
illustrate how a symbol sequence or expression may be conceived either
as aa process (add) or a concept (sum). A learner might perceive an
example either as a process, or as an object, or both (proceptually).
For example, if a learner’s only experience of equations is
of
being shown how to solve them, with the language only of
‘doing’, then it is unlikely that a conceptual
understanding will be formed easily.
Charles (1980) argues that while for ‘easy’
concepts a
sequence of examples from which to generalise may be sufficient, for
more ‘difficult’ concepts nonexamples are also
necessary
to delineate the boundaries of the concept. Wilson (1986) points out
that learners can be distracted by irrelevant aspects of examples, so
the presence of nonexamples provides more information about what is,
and is not, included in a definition. Since examples are far more
effective than formal definitions in appreciating concept (Vinner
1991), learning might be enhanced by contact with a rich variety of
examples and nonexamples. Paul Goldenberg (private communication)
observed that there is a big difference between noticing for oneself a
salient feature in a collection of examples and then naming it, and
being given a new word followed by a sequence of objects which are
supposed to illustrate its meaning.
How rich and in what variety needs careful study however. Bell (1976)
reported that school learners often do not recognize the significance
of counterexamples and would not necessarily alter their conjectures or
proofs if a counterexample did crop up, and this is reflected in the
observation that undergraduates also tend to monsterbar (MacHale 1980)
rather than modify their concept image. It is fairly obvious that a
limited experience of examples and nonexamples may lead to a
restricted concept image, but it is also the case that limiting
mathematics to sequences of examples ‘to be done’,
rather
than sets of examples to be understood, may induce learners to focus on
completing their tasks rather than on making sense of the tasks as a
whole (Watson & Mason 2006). A succession of examples does not
add
up to an experience of succession. Not attending to the whole may
result in an overly restricted understanding of the nature of
mathematics.
6b.
Learning from workedout examples
Several studies point to the contribution of workedout examples for
learning to solve mathematical problems (e. g. Reed et al. 1985;
Reimann & Schult 1996; Sweller & Cooper 1985). However,
providing workedout examples with no further explanations or other
conceptual support is usually insufficient. Learners often regard such
examples as specific (restricted) patterns which do not seem applicable
to them when solving problems that require a slight deviation from the
solution presented in the workedout examples (Reed et al. 1985, Chi et
al. 1989). Note however that the immensely insightful mathematician
Ramanujan was, while a student, able to treat a book of summarised
generalities as a sequence of particular examples!
Watson & Mason (2002a, 2002b) suggest that workedout examples
might even inhibit learners' ability to generalise apart from
recognition of the syntactical template. One explanation of this
phenomena was given by Reimann & Schult (1996), based on
Artificial
Intelligence literature. They claim that the information captured and
attention drawn in workedout examples is mostly the solution steps,
which limit matching and modification processes. Furthermore, Reimann
& Schult (ibid) assert that it is important to specify in a
workedout example the steps that were taken and the reasons for taking
them, that is, how attention is directed. This is consistent with the
findings of Chi et al. (1989) and Renkl (2002) who emphasise the
importance of learners’ selfexplanation of the workedout
example, and also with the work of Eley & Cameron (1993) who
found
that learners considered an explanation to be better if it included the
‘trigger’ for each step. Workedout examples may
enhance
learners' learning, and in particular their problem solving
performance, but only if they are used in ways which encourage
explanation and reasoning.
Much of the research in this area has been directed towards a view of
learning as measurable by performance of techniques and solution of
word problems, rather than of learning as conceptual understanding or
mathematical enquiry. The role of workedout examples in conceptual
understanding deserves further research.
6c.
The role of examples in mathematical reasoning and problem solving
Examples can play a role in facilitating nonroutine problem solving, a
process in which reasoning about the situation allows the learner to
apply and adapt sequences of techniques whose purposes need to be
understood. If this is seen as a process of applying known techniques,
the relevant workedout examples which the learner has experienced need
to be sufficiently different, and sufficiently explained, for the
purpose of the techniques used to be understood. If, on the other hand,
problemsolving is seen as a process of modelling a situation and
tackling it heuristically, a learner needs to have some knowledge of
similar situations in order to be successful.
One of the main processes of reasoning about novel situations is
reasoning by appealing to similarity (Rumelhart 1989). Rumelhart refers
to a continuum, moving from ‘remembering’ a
suitable
example to ‘analogical reasoning’. Another central
kind of
mathematical reasoning that necessitates generation of examples is
proving by refutation. Addressing learners' difficulties in producing
and using appropriate counterexamples is another challenge for
teachers' use of examples (Zaslavsky & Peled 1996; Zaslavsky
&
Ron 1998). Pólya (1945, 1962) elaborates on the processes of
inductive (examplebased) reasoning, generalization, and analogical
reasoning, all of which greatly depend on examples.
It seems that all learners who are even only partially engaged try to
generalise from sequences of examples, implicitly or explicitly, and
that this is done by the natural process of discerning differences and
similarities in what is available to be perceived. What they choose to
stress and ignore, and what they ‘get from it’ is
highly
variable. Discerning invariance and variation explains many standard
misconceptions in mathematics: learners generalise inappropriately, but
in ways which can be seen to be the products of mathematical reasoning,
given their experience. Thus learners are always engaged in
mathematical reasoning whenever they are exposed to a set of examples
of anything, although this may not be recognised or made explicit.
There are many unresolved issues. For example, Hejny (personal
communication) questions whether ‘natural’
generalisation
is always the same kind of process, or whether it differs according to
whether one is encountering a concept, a process, etc..
Novices and expert mathematicians alike depend on experiences with a
single rich generic example, or else, as with most novices, numerous
examples, in order to get some intuition about the situation and then
try to generalise and reason from them. (Bills & Rowland 1999,
Zaslavsky & Lavie 2005). This mixture of logicalbased
reasoning
(using deductive mechanisms) and examplebased reasoning (Lakatos 1976)
characterises mathematical competence at every level.
Weber & Alcock (2004, 2005) documented how undergraduates
learning
to prove use examples in reasoning and constructing proofs. They
recognised that professional mathematicians switch fluently between
examples (specific cases) and formal definitions, so they asked how
learners make the transition to this fluency, if this shift has not
been made explicit for them. They found that example use for such
learners is often illustrative and empirical rather than general and
deductive. Where their reasoning failed, they were more likely to
selfcorrect errors to do with the individual example than errors to do
with the underlying rationality. Alcock & Weber (in press) then
distinguished between two learners who used a referential approach to
proof and a syntactical approach. The learner who used referential
approach rejected examples as a tool for developing structural
understanding and may have needed help in describing examples more
formally, to see how doing so might offer the structure for a formal
proof. The learner who approached the task of proof construction as if
it were solely a manipulative exercise might have benefited from using
specific examples to give her work some meaning, but selfgeneration of
appropriate examples is not trivial for learners who are unused to
doing so.
6d.
The role of learner generated examples in learning
Learning is an activity which requires initiative and intention.
Getting learners to construct their own examples proves to be a highly
effective strategy for transferring initiative from the teacher to the
learner (e.g. Zaslavsky 1995, Niemi 1996, Dahlberg & Housman
1997,
Hazzan & Zazkis 1999, Zazkis 2001, Watson & Mason 2005).
The current shift from teachercentred to learnercentred pedagogical
environments in order to foster mathematical classroom discourse, fits
with encouraging learners to construct their own examples, which in
turn enables teachers to detect the kinds of understandings reflected
by learners' examples (e.g. Watson & Mason 2005, and as
suggested
by Zaslavsky 1995). Creation of an example is a complex task that calls
upon conceptual links among concepts (Hazzan & Zazkis 1999).
Dahlberg & Housman (1997) showed that learners who generated
examples as a strategy of learning were more likely to understand new
concepts. 'Give an example of …' tasks prove very useful in
assessing learners’ understanding (Niemi 1996).
When learners have been asked to create their own examples, they
experience the discovery, construction or assembly of a space of
objects together with their relationships. Whereas RisslandMichener
(1978) saw example spaces as canonically objective, construction is
often idiosyncratic, combining modifications of conventional and
familiar objects to construct new objects, to recognise new
relationships, and to enjoy new meanings and personal understandings.
Easilyavailable canonical spaces, such as those teachers and textbooks
commonly use, form suitable starting points for further extension, just
as in any learning the learner can only start from what is already
known, which may be a proper subset of what is relevant. In other
words, through construction, learners become aware of
dimensions
of possible variation and corresponding ranges of permissible change
within a dimension, with which they can extend their example spaces.
From a mathematical perspective it may be possible for an expert to see
a large potential space of examples, or at least to have past
experience of a large space, but what comes to mind in the moment may
only be fragments of that potential. Spaces are often dominated by
strong images, some of which may be almost universal. What is
accessible in one situation may not be so readily accessible in
another. The experience of constructing examples for oneself can
contribute to increased sensitivity in future, triggering richer
example spaces.
6e.
Summary of learner perspective
Examples play a crucial role in learning about mathematical concepts,
techniques, reasoning, and in the development of mathematical
competence. However, learners may not perceive and use examples in the
ways intended by teachers or textbooks especially if underlying
generalities and reasoning are not made explicit. The relationship
between examples, pedagogy and learning is underresearched, but it is
known that learners can make inappropriate generalisations from sets of
examples, or fail to make any conceptual inferences at all if the focus
is only on performance of techniques. The nature and sequence of
examples, nonexamples and counterexamples has a critical influence of
what opportunities learners are afforded, but even more critical are
the practices into which learners are inducted for working with and on
examples.
The relationship between examples and logical deduction in
proof, or analogical reasoning in problem solving, cannot be assumed to
be assimilated or even accommodated by learners without explicit
support and provocation. It is valuable for learners to create their
own examples, since this process requires complex engagement with
concepts and mathematical structures.
Learners naturally perceive variation and invariance in what they
experience, and make generalisations from this activity, developing
example spaces whose contents may be triggered in future situations.
How these contents are structured and interrelated is the outcome of
past experience and with ways of working with examples.
7.
Examples From A Researcher's Perspective
From a researcher’s perspective the role of
examples in
mathematics education research concerns choices based instructional
design, in research on learning, and the role of case studies,
considered as research examples, in theory development in mathematics
education. The three points will be illustrated by means of examples
from a research project, in which they are prominent without, however,
being explicit.
7a.
Researchbased design
Research findings depend critically on specific properties of examples
just as much as teaching and learning. For example, in the study by
Dreyfus & Tsamir (2004); and Tsamir & Dreyfus (2002,
2005),
which deals with the comparison of the cardinalities of infinite sets,
the task set initially was to compare the numbers of elements in the
set of natural numbers with the number of elements in the set of
perfect squares. Two representations were used: numeric and geometric.
In the numeric representation, the sets were represented on three
cards:
 Card A {1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, …}
 Card B {1, 4, 9, 16, 25, 36.
49, 64 81, 100, 121, 144, 169, 196, 225, …}
Card M was identical to Card A. The inclusion relationship was
highlighted by asking learners to choose and mark the perfect squares
on Card M. The geometric representation used squares and the
correspondence between side length and area (see Tsamir &
Dreyfus
2002, for a detailed description).
The examples in this first task were chosen with attention to research
findings (Tsamir & Tirosh 1999) regarding learners’
tendencies to think in terms of inclusion when presented with
a
numeric representation of the task, and to identify the onetoone
correspondence in reaction to the geometric representation of the same
task. Consequently, learners may be expected to reach contradictory
answers.
After several more tasks using either or both representations as well
as algebraic correspondence rules between the sets, the task in the
third session was to compare the set A of natural numbers to a set V,
which was given numerically as {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55,
66, 78, ...} (Tsamir & Dreyfus 2005). Here the algebraic rule
of
the onetoone correspondence is not easily apparent, nor is the
establishment of such a correspondence geometrically. Without adequate
preparation, a learner could thus be expected to use only inclusion
considerations.
When designing a sequence of tasks, whether for the purpose of teaching
or research, the characteristics of each specific example need to be
taken into consideration. These characteristics include the different
representations in which the example can be cast, and whether the
example triggers certain types of reasoning, such as analogy or
cognitive conflict. Whereas in teaching not all examples will usually
be determined ahead of time since inspired and creative teaching
involves sensitivity to the flow of events and on the spot decisions by
the teacher, in research, on the other hand, researchers usually do
plan all examples in advance; nevertheless, decisions to add or omit
examples in a specific stage of the research may be made on the basis
of the analysis of previous stages or exigencies in the moment.
7b.
Research on learning
Learners’ abstract mathematical constructs usually emerge
from
their occupation with specific cases, i.e. examples. This becomes
particularly clear in the research mentioned above, which analyzes the
case of one learner, Ben, addressing the comparison of powers of
infinite sets. How and what exactly learners may or may not learn from
examples only becomes clear after detailed, careful and controlled
observation, and analysis of the observations, by researchers.
As an example, consider what Ben did (not) learn from the two tasks
presented above. When presented with the first task, Ben claimed, as
expected, that the number of elements in set A was larger than the
number of elements in set B, explaining that “set B is
actually
part and I mean REALLY part of set A”, and that “it
is easy
to notice that the further I go [in set B] the larger the
intervals”. Over the next two sessions, Ben gained insight
into
the problematic aspects of using inclusion and correctly solved this
and all other tasks presented to him by using onetoone mappings
between infinite sets in numeric/algebraic and geometric
representations. He reached what the researchers interpreted as
consolidated indepth constructs allowing him to solve such tasks, and
it seems that this was on the basis of a carefully designed sequence of
tasks. For example, with respect to the comparison of set A above with
the set of natural numbers greater than 2, he explained:
 “The two extra, unmatched elements stand out and
trigger
the conclusion that here we have infinity and here infinity plus two,
which SEEMS larger. Instead of matching numbers at the same ORDINAL
place [pause]. I mean, assuming that if for each place n there is one
and only one element in each the two sets, then they go on hand in
hand, corresponding, and extra elements are just in our imagination.
The infinite nature makes it possible that no matter which number you
chose in one of the sets, at the same ordinal place there is a matching
specific number placed in the other set. It cannot be that the numbers
in the second set are finished and cannot provide a matching element,
because the set is infinite, and this behavior of plus two goes on,
like, forever.”
In the third session, Ben was asked to compare the sets A and V (see
above). This example, which was intended to introduce more challenging
tasks, turned out to provide the researchers with insight into the
complexity of what had been interpreted as Ben’s consolidated
knowledge about the comparison of countable infinite sets. For over 20
minutes, Ben assiduously tried to establish, geometrically or
algebraically, a onetoone correspondence between A and V. He even
noticed that there is a onetoone correspondence between set A and the
set of differences between successive elements of V. But then he ended
up concluding,
 “The differences between successive elements get
larger and
larger. Wow! REALLY larger. I see. Set V consists of fewer elements.
REALLY fewer.”
Even insistent questioning by the interviewer did not sway his opinion.
The interviewer remarked:
 “You once told me that using inclusion and
correspondence
leads to contradiction. And then you read that only equivalence
correspondence should be used for comparing infinite sets.
Right?”
To this, Ben replied that yes, indeed, using inclusion and onetoone
correspondence may lead to a contradiction, and that he had not used
inclusion except to prove that there exists no onetoone
correspondence.
Based as it was on careful choice of a sequence of examples, this
research has advanced our understanding of the important
characteristics of consolidation (Dreyfus & Tsamir 2004).
Equally
interestingly, the choice of the introductory example to the third
session also turned out to have an important, though unplanned role in
the research because it led to modification of our conception of
consolidation.
Research on learning is necessarily based on examples because all
learning is either fundamentally based on examples, or at least
strongly supported by examples. The choice of examples thus influences
research on learning, and possibly research results. Are such research
results reliable? Not quite. An example was found where Ben’s
supposedly consolidated knowledge broke down. Without this example,
conclusions about Ben's consolidation of knowledge about the comparison
of infinite sets would have been exaggerated.
There are two ways researchers can counterbalance this influence of
examples: One is to be acutely aware of it, and attempt to analyze it,
thus recognizing the influence, and the possible ensuing limitations of
any specific piece of research; and the other is to carry out several
parallel research studies using different sets of examples, the subject
of the next subsection.
7c.
Theory building
It is generally agreed that theory building is one of the aims of
research. In mathematics education, researchers' theoretical constructs
about X (e.g. a specific learning process such as consolidating) tend
to emerge from observation of a few, sometimes of a single example of
X, combined with theoretical reflection on X. The small number of
examples is a necessary limitation, due to the fact that examples are
often “large” in the sense that they may require
weeks of
detailed observations and subsequent painstaking analysis of the
observations.
Research on constructing and consolidating knowledge is a case in
point. Learners can be given opportunities for constructing knowledge
– but they cannot be forced to construct; researchers thus
provide learners with opportunities, and hope they can observe what
they are looking for. Consolidating recently constructed knowledge, by
definition, is an ongoing process, that may last hours or years.
Dreyfus & Tsamir (2004) have proposed characteristics of
consolidation on the basis of a single, albeit detailed and very
carefully analyzed, but still only a single example, namely the example
of Ben constructing and consolidating his knowledge about the
comparison of infinite sets.
In a similar vein, the entire ‘RBC theory’ made up
of
Recognizing, Buildingwith and Constructing (Hershkowitz et al. 2001),
within which the consolidation research is located, has been proposed
on the basis of a single example, a 9th grade learner learning about
rate of change as a function. Again, one example has served to propose
an entire theory. Subsequently, the same and other researchers have
shown that the theory is applicable to many other contexts, possibly
after suitable modification. The theory has thus been strengthened and
validated. It is important to stress that this validation is based on
examples as well. In this sense, examples play a central and crucial
role in the establishment of theory, the other basic element of theory
building being theoretical reflection.
7d.
Summary of research perspective
The choice of examples, and their sequencing, is crucial in
instruction. Examples may be chosen for using specific representations
and they may be sequenced to go from easy to difficult for triggering
analogy, or from difficult to easy for triggering cognitive conflict
(Tsamir 2003). Consequently, research on learning mathematics is
necessarily based on examples as well, and the choice of mathematical
examples may influence research results. Researchers can counterbalance
this influence by being aware of it, by taking it into account when
drawing conclusions, and by carrying out parallel research studies
using different sets of examples.
Moreover, there is a second level of example use in research. A
research study, such as the one about Ben, may itself serve as an
example that forms the basis for theory building. Additional examples
of research studies are a tool for validating the theory.
8. For Further Research
Particular attention needs to be paid to
 the sequencing and timing of a succession of examples, and
both
the dimensions of possible variation and their associated ranges of
permissible change to which learners are afforded access.
 ways of directing learner attention so as to perceive
exemplariness;
 ways of drawing teachers’ attention to the
importance of the choices of examples they make when working with
learners;
 the role of workedout examples in concept formation;
 ways of directing learner attention so that sets of
exercises are pedagogically effective.