ORIGINAL ARTICLE Qualities of examples in learning and teaching Anne Watson • Helen Chick Accepted: 5 December 2010 / Published online: 16 December 2010 Ó FIZ Karlsruhe 2010 Abstract In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theo- rising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of ‘‘didactic object’’ (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Nether- lands, pp 191–212, 2002), we talk about ‘‘didacticising’’ an example and observe that the nature of students’ engage- ment is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root ‘‘didact’’. We use the verb ‘‘didacticise’’ but without implying any connection to particular theoretical approa- ches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities. Keywords Examples Didactic object Generalisation Learning from examples 1 Examples in learning and teaching 1.1 The relations between examples and mathematics for learners In her seminal paper, Rissland Michener (1978) examined the role played by examples in understanding mathematics. She described examples as ‘‘illustrative material’’ (p. 362) and highlighted an important dual relation: that examples can be constructed from results and concepts, and in turn examples can motivate concepts and results. Borrowing from Freudenthal’s definition of models-of and models-for (Freudenthal, 1975; cited in van den Heuvel-Panhuizen, 2003), we might view the nature of examples in Rissland Michener’s dual relationship as examples-of—in which the examples are specific instantiations of a previously defined class—and examples-for—in which the examples are the genesis for identifying an as-yet-uncharacterised class. Rissland Michener delineated different roles that examples can play in understanding mathematics. Start-up examples motivate definitions and build a sense of what is going on; reference examples are ‘‘standard cases’’ that link concepts and results, and are returned to again and again; model examples indicate generic cases and can be copied or used to generate specific instances; and, finally, counter-examples sharpen distinctions between, and definitions of, concepts. Lakatos goes further and suggests that counter-examples have historically generated inquiry into new classes of objects (1976), while Goldenberg and Mason (2008) high- light that the difference between example and counter- example depends on one’s attention or emphasis. If attention and emphasis are relevant, then whether an example is ‘‘of’’ some class or actions that are already familiar or ‘‘for’’ the construction of something new depends on the person undertaking the mathematical activity. Thus, a A. Watson (&) Department of Education, University of Oxford, 15 Norham Gardens, Oxford OX2 6PY, UK e-mail: [email protected]H. Chick Melbourne Graduate School of Education, University of Melbourne, Melbourne, Australia 123 ZDM Mathematics Education (2011) 43:283–294 DOI 10.1007/s11858-010-0301-6
Transcript
ORIGINAL ARTICLE
Qualities of examples in learning and teaching
Anne Watson • Helen Chick
Accepted: 5 December 2010 / Published online: 16 December 2010
� FIZ Karlsruhe 2010
Abstract In this paper, we theorise about the different
kinds of relationship between examples and the classes of
mathematical objects that they exemplify as they arise in
mathematical activity and teaching. We ground our theo-
rising in direct experience of creating a polynomial that
fits certain constraints to develop our understanding of
engagement with examples. We then relate insights about
exemplification arising from this experience to a sequence
of lessons. Through these cases, we indicate the variety of
fluent uses of examples made by mathematicians and
experienced teachers. Following Thompson’s concept of
‘‘didactic object’’ (Symbolizing, modeling, and tool use in
mathematics education. Kluwer, Dordrecht, The Nether-
lands, pp 191–212, 2002), we talk about ‘‘didacticising’’ an
example and observe that the nature of students’ engage-
ment is important, as well as the teacher’s intentions and
actions (Thompson avoids using a verb with the root
‘‘didact’’. We use the verb ‘‘didacticise’’ but without
implying any connection to particular theoretical approa-
ches which use the same verb.). The qualities of examples
depend as much on human agency, such as pedagogical
intent or mathematical curiosity or what is noticed, as on
their mathematical relation to generalities.
Keywords Examples � Didactic object � Generalisation �Learning from examples
1 Examples in learning and teaching
1.1 The relations between examples and mathematics
for learners
In her seminal paper, Rissland Michener (1978) examined
the role played by examples in understanding mathematics.
She described examples as ‘‘illustrative material’’ (p. 362)
and highlighted an important dual relation: that examples can
be constructed from results and concepts, and in turn
examples can motivate concepts and results. Borrowing
from Freudenthal’s definition of models-of and models-for
(Freudenthal, 1975; cited in van den Heuvel-Panhuizen,
2003), we might view the nature of examples in Rissland
Michener’s dual relationship as examples-of—in which the
examples are specific instantiations of a previously defined
class—and examples-for—in which the examples are the
genesis for identifying an as-yet-uncharacterised class.
Rissland Michener delineated different roles that examples
can play in understanding mathematics. Start-up examples
motivate definitions and build a sense of what is going on;
reference examples are ‘‘standard cases’’ that link concepts
and results, and are returned to again and again; model
examples indicate generic cases and can be copied or used to
generate specific instances; and, finally, counter-examples
sharpen distinctions between, and definitions of, concepts.
Lakatos goes further and suggests that counter-examples
have historically generated inquiry into new classes of
objects (1976), while Goldenberg and Mason (2008) high-
light that the difference between example and counter-
example depends on one’s attention or emphasis. If attention
and emphasis are relevant, then whether an example is ‘‘of’’
some class or actions that are already familiar or ‘‘for’’
the construction of something new depends on the
person undertaking the mathematical activity. Thus, a
edge—one of our characteristics as learners—influenced
our choice. Where we differed was in the approaches that
we took to the possible parameters. To demonstrate the
subtleties of our use of examples, we give a detailed
exposition.
One approach treated the phrase ‘‘and one other place’’
as defining a fixed but unknown third zero at the point
(m,0), thus making m a parameter for the problem. With
three given zeros, and the class of functions restricted to
cubics, the factor theorem implies a function of the form:
f ðxÞ ¼ kðx� 0Þðx� 2Þðx� mÞ ¼ kxðx� 2Þðx� mÞ:
In the absence of other constraints, k is free to vary,
giving a family of cubics with zeros at x = 0, x = 2 and
x = m. However, Task 1 imposes an extra constraint that
impacts on k: the function passes through (3,3). This
implies
3 ¼ k � 3� ð3� 2Þð3� mÞ ¼ 3kð3� mÞ:
Solving for k, which is constrained by the four points,
gives
k ¼ 1
3� mand hence f ðxÞ ¼ x
3� mðx� 2Þðx� mÞ:
This represents a family of cubics, fixed by the two
given zeros and (3,3), and governed by the location of the
third unspecified zero, determined by the parameter m.
The other approach recognised that there were two
degrees of freedom associated with the problem: the position
of the third zero and the steepness of the cubic. The resulting
form of the function, f(x) = kx(x - 2)(x - m), was as
obtained earlier, as was the use of (3,3) to yield 3 = 3k(3 -
m). However, this formalisation was understood differently
as it was then solved for m in terms of k, to give
m ¼ 3� 1
k:
The difference between these approaches lies in the
interpretation of what is ‘‘known’’ and what is determined
‘‘in terms of’’. In this second case, the ‘‘steepness’’, k, was
regarded as fixed, with the third zero (the ‘‘one other
place’’ of the original problem) being determined by it.
On seeing each other’s solutions, we immediately began
discussing the role of parameters and how our interpreta-
tions of the problem had led to differing treatments. In fact,
parameters are at the heart of this problem, and this task
could be an example for understanding their nature and
how they differ from variables. To understand our
approaches, we unpicked the role of parameters. They are
the quantities that structure a particular object. In many
cases, they might be viewed as ‘‘unknown knowns’’; they
are treated as if we know their value, and yet we do not,
which allows us to wonder what happens when they vary,
and to imagine them to be fixed or variable according to
purpose. Thus, when considering functions, there are two
kinds of variation a learner might be attending to: the
variation of variables, and the variation and invariance of
parameters. The first is characteristic of an example; the
second is characteristic of the class.
It is, of course, important to note that we brought con-
siderable mathematical experience to tackling Task 1. Our
previous experiences meant that not only we were com-
fortable with the forms and properties of polynomial
functions, but also we could manage the ‘‘unknownness’’
of the third zero. Finally, our approaches were algebraic,
although we both had images of cubic graphs in our minds.
What was exemplified to us in Task 1? Pedagogically,
we could see it was an example of a learner-construction
task that might develop learners’ understanding of such
curves and might challenge their treatment of variables and
unknowns, as will be discussed shortly. However, when we
did the task ourselves, we were not doing it as learners, but
as curious mathematicians: we wanted to know what this
family of cubics looks like. Finding one example was less
interesting to us than learning about the whole class, and
what does any class of cubics look like when we know two
roots and one other point?
Our personal characteristics determined our choices: in
Goldenberg’s terms, our familiarity with the domain
informed our decision to work algebraically; our under-
standing of the language ‘‘find the equation’’ as ‘‘find a
general equation’’ was due to being familiar with extending
the meaning of ‘‘find’’ to mean ‘‘find a class’’; our expec-
tations that there would be a wide general class to explore
led us to constrain ourselves to cubics. Our different con-
figurations of past mathematical work led to the focus on
different parameters. Perhaps, a difference between our
approach and a novice approach would be that we were
aware of defining the domain and level of generality. We
were also aware of constructing a specific exploratory
Qualities of examples in learning and teaching 287
123
purpose for the example as set: to treat it as an example-for
generating a class of functions, with the intention of then
exploring the appearance of the class using graphing soft-
ware and possibly raising new question by changing the
givens. Because of our experiences with it, Task 1 also
became an example-for discussing parameters. We did not
bother to construct an example-of until we needed one for
illustrative purposes to communicate (see Fig. 2).
2.2 Pedagogical issues
The task, as written, appears to assume certain knowledge,
although perhaps not as much as we employed. What might
happen, then, in a classroom where some of this knowl-
edge—particularly the use of parameters—is not as
familiar? Imagine giving Task 1 to students with less, but
still some, knowledge of polynomials and the factor theo-
rem. In this case, what happens and what might be exem-
plified will depend on the students’ experiences, and on the
teacher’s pedagogical intent and implementation. Chick
(1988) writes about students’ responses to a similar ques-
tion; what she found, together with our own knowledge of
teaching and students, helps us engage in a thought
experiment about how this task might be used by students.
• Students might sketch a cubic passing through the
given points and an arbitrary point on the x-axis chosen
to be the ‘‘one other place’’ where the curve crosses the
axis. Comparing differences among these examples-of
a cubic sketched by students could highlight the effects
Fig. 2 Some functions
satisfying Task 1: passing
through (0,0), (2,0), (3,3), and
(m,0). The parametrised cubic
function can be set up in
Geogebra or TI-Nspire, with the
parameter m as a ‘‘slider’’ that
can be varied to obtain members
of the family of functions
288 A. Watson, H. Chick
123
of the choice of the third zero and lead to thinking
about the relationship between the third point and
possible shapes (examples-of become examples-for).
• Everyone in the class might agree to have the same
point as the third zero. It should be evident that many
different curves can be drawn through the four points.
These examples-of curves satisfying the constraints
might then become examples-for discussing what
classes of curves will work, whether or not they can
be expressed algebraically, what is the ‘‘simplest’’
function that passes through the four points, and how
many such ‘‘simple’’ functions there are. If continuous,
what turning points does it have to have? These require
considerable pedagogical decision-making to get fur-
ther than merely a collection of wiggles.
• Students familiar with the factor theorem might express
possible functions in polynomial form by writing
y = f(x) = (x - 0)(x - 2) …, with an awareness that
this is incomplete. There may be varied understandings
of what impact different choices will have on the
resulting function and how to express these. For
example, a student might assume a particular value
for the third zero, say x = -4. With f(x) = x(x - 2)
(x ? 4) as a tentative candidate function, the student
might substitute x = 3 to obtain y = f(3) = 21, which
is 18 more than the desired value of y = 3. The
function may then be posited as f(x) = x(x - 2)
(x ? 4) - 18, without realising that this function no
longer has zeros in the requisite places (cf. Chick,
1988). These examples-of (incorrect) functions may
become examples-for building principles for determin-
ing functions that satisfy certain conditions.
• Students with a better understanding of the structural
relationships might take an algebraic approach similar
to that taken by the authors, in which case, the task
might become an example-for discussing the role of
parameters.
• Students might try to use the general polynomial form
of a cubic y = f(x) = ax3 ? bx2 ? cx ? d (cf. Chick,
1988). The two zeros and the point (3,3) are enough to
generate two equations in three unknowns. Those who
can deal with the arbitrariness of the third zero might
pick (m,0) as the fourth point on the function, and
obtain a third equation: 0 = m3a ? m2b ? mc. For
many, this may be an uncomfortable moment: m is an
‘‘unknown known’’, while a, b, and c are the givens for
which they must solve. This distinction is difficult for
students and again allows parameters to become the
focus of what this task might exemplify.
Establishing the general role of the parameters in
defining the family is, to our minds, difficult and might be
facilitated through the pedagogical implementation of
appropriate technology (see Fig. 2).
Any combination of the approaches suggested above
could arise in a classroom, ultimately leading to a collec-
tion of examples of functions meeting the requirements of
the task. The actual combination would depend on famil-
iarity, expectations, and language of pedagogic interven-
tion, as we have said before, but it also appears to depend
on fluency with different representations and availability of
technological tools. We also want to be more specific about
‘‘familiarity’’. Having used this thought experiment to
reflect on our own work on the task, it is clear to us that our
habits of attention were crucial too. We came to the task
attending to parameters; others might come to the task
attending to curve shapes, or covariation. The resulting
examples or families can exemplify a number of important
mathematical principles. Some possibilities are listed in
Table 1.
The realisation of any of the exemplifying affordances
in Table 1 depends on the mathematical backgrounds and
expectations of those who attempt the task, the way the
examples are didacticised by the teacher, available tech-
nology, and also on the actions undertaken by learners
(construct, compare, attend in certain ways). Given the vast
scope of this task for exemplification, the choices made
about how to use the examples as didactic objects
Table 1 Mathematical
principles that can be
exemplified by Task 1
Principle How exemplified
There are infinitely many possible functions Graphically or algebraically; explicitly or
‘‘sketched’’
There is a unique cubic through 4 points Construct algebraically with a specific third zero
Role of parameters CAS software
Different parametrisations will be related Contrast solutions using different
parametrisations
A function is a mathematical object Compare different functions, highlighting
structural properties
Different representations of functions afford attention
to different properties
Varying the representation and foregrounding the
effect of variation
Qualities of examples in learning and teaching 289
123
fundamentally depend on a teacher’s own view of the task,
his/her evaluation of what it is perceived to exemplify, and
his/her decisions about what it is important to emphasise
through such examples.
The task affords at least three layers of exemplification:
1. There are the individual examples-of generated in
doing the task (i.e., functions that satisfy the
constraints).
2. The individual examples can become examples-for
representing a family of cubics in which the role of
parameters can be highlighted, and then cubics can be
seen as the simplest examples of the range of
polynomial functions that might be used.
3. Task 1 itself can be generalised to prompt learners to
construct objects and then explore superordinate
classes of polynomials determined by different-sized
sets of given points.
This experience leads us to realise that, in addition to
situational givens described earlier, technologies and rep-
resentations contribute to the didacticisation of examples,
which includes indications of appropriate dimensions of
variation. Unless a teacher deliberately guides attention,
tool use, and discussion in some way, it is unlikely that
novices would be able to handle the different kinds of
variation of variables and parameters. However, we are not
losing sight of the fact that when we did the task, we acted
on the examples to make them ‘‘an example of something’’.
A teacher can facilitate this in the didacticisation process,
but the learners will also, as we did, imbue the task with
personal purpose.
3 Case two: decimals, percentages, and fractions
Our second case examines the didacticisation process more
closely by considering how an experienced teacher used
examples with her students over two lessons addressing
decimals, percentages, and fractions. The lessons were
selected from a bank of videos that had been accumulated
for professional development purposes. These lessons were
chosen because the teacher used several sets of examples
and had discussed her pedagogical intentions and decisions
with us. Hence, they provided raw material for the con-
tinuation of our theoretical work on relations between
examples and exemplified classes.
Zara was an experienced secondary teacher close to
retirement. Her school was recently designated a ‘‘spe-
cialist college for mathematics’’. She and her students were
used to being filmed and she had in the past filmed teachers
herself for research and development purposes.
We observed two lessons in which she was introducing
conversion among decimals, percentages, and fractions to
an all-attainment class of 11- and 12-year-olds whose prior
knowledge of these was varied. In the first lesson, her aim
was to develop a continuous linear image of number. We
watched a video of this lesson, focusing on her actions and
public statements, to identify what examples were offered
to the whole class, and how they were offered. We knew
from conversations that Zara was aware of the importance
of example choice, and our aim in observing Zara was to
learn more about variety in example use. We shall report
on three episodes from the lesson, and a related part of a
subsequent lesson, and then show how these prepare stu-
dents for different kinds of action. Our purpose is not to
identify things she could have done differently in a critical
way, but to analyse, from the evidence available to us, the
relations between her example provision and students’ use
of the examples.
3.1 Episode 1
Zara asked students to ‘‘Write down as many sets of three
numbers as you can that add up to 4’’. After a few minutes,
she asked for their suggestions and wrote them on the
board. These were the first three sets:
3 1=2 1=21:5 0:5 2
3 1 0
Another student offered ‘‘googolplex to the power of
four divided by googolplex to the power of three’’ and Zara
engaged in a brief discussion to reformulate the suggestion,
and reminded them that they were only allowed to use
‘‘add’’. She praised the whole class for using a broad
meaning of number that included fractions, decimals, zero,
and negatives.
3.2 Episode 2
Pairs of students were given a metre stick and a lump of
modelling plastic, which they had to roll out into a snake
1 m long. She asked them to find out anything interesting
about fractions and numbers using these two materials.
She asked them if they had any ideas about why she had
given them the metre stick. One student replied that it had
100 cm marked on it. Zara said that the stick is ‘‘one
whole’’ and the snake is also ‘‘one whole’’ and then wrote
‘‘1/100 = 0.01’’ on the board. The public statement of the
task was very open, but she went round the class helping
them decide what to do and discussing with them the
relations between fractions of the 1 metre snake and
decimal readings on the stick. About 10 min after they
had rolled out the modelling plastic, she stopped the class
and asked students whether these were the same or
different:
290 A. Watson, H. Chick
123
0:5 0:50
Several students replied that they were the same and
gave explanations using the metre stick as ‘‘one whole’’
and 0.50 as indicating 50 cm.
3.3 Episode 3
Zara pointed to the ‘‘1/100 = 0.01’’ and asked what 1 cm
is as a fraction of a metre. She then asked students to find
out what they could from the metre stick and from cutting
their snakes into one half, one-third, one quarter, one-fifth,
and one-eighth. Towards the end of the lesson, she spent
10 min reviewing what students had found out. She said it
was very important that they should have a picture of how
decimals and fractions match ‘‘in your mind’’. She then
wrote � = 0.5 and asked for the decimal equivalent of a
quarter. A student replied and she wrote � = 0.25. She
asked a particular student for ‘‘a third’’ and the student
replied ‘‘zero point three three’’. When she asked for an
eighth, two students almost simultaneously called out
‘‘nought point eight’’. Her response to the first one was
‘‘Did you do it? Did you write it in your sheet?’’ She then
said: ‘‘This needs to be in your mind … picture where it
is’’. Another student said: ‘‘Zero point one two five’’. At
that point a bell sounded and some students began to pack
away, but she said, ‘‘We are not going until you can match
some fractions and decimals’’ and asked for one-fifth, and
then one-tenth, which were answered correctly. The lesson
was now over time, and she began to have difficulty
keeping the students’ attention. At the end of the lesson,
this list was on the board:
1
2¼ 0:5
1
4¼ 0:25
1
3¼ 0:33 ½sic�
1
8¼ 0:125
1
5¼ 0:2
1
10¼ 0:1:
3.4 Episode 4
In the next lesson with this group, Zara moved away from a
linear model of number and used some commercial soft-
ware that gives manipulable images of coins with their
monetary, fraction, decimal, and percentage equivalents on
the interactive white board. This software drew on
students’ everyday knowledge of money and used indi-
vidual coins to represent elementary objects that could be
combined to make other sums of money. Because of the
partial isomorphisms between money and decimal number,
Zara could pose questions starting from any given, and then
ask students to work at the board and ‘‘fill in the blanks’’
for the other cells, such as:
50p 12
0:5 50%
The software allowed these four cells to be revealed in
any order. Both Zara and the software emphasised that the
cells were always expressing fractions ‘‘of a pound’’.
Students first had to predict what the cells would contain
for 20p and 10p coins. Zara asked ‘‘Why?’’ for each
selection. She then asked the students to work individually
to produce the same four representations for other sums of
money, and asked two students to reveal, on the board,
those they found hard. The 1p and 5p values were selected
by students and Zara discussed these with the whole class,
ending up with:
1p 1100
0:01 1%5p 1
200:05 5%
When Zara asked how a student had arrived at 1/20, the
reply was that ‘‘there are twenty lots of five in a pound’’,
and when she asked about ‘‘0.05’’ the reply was that it was
‘‘five times 0.01’’.
She then illustrated 60p as three 20p coins, and 75p as a
group of 50p, 20p and 5p coins, and asked the students to
work on these. After a short while answers were given and
written up:
20p 15
60p 35
0:6 60%
One student called out that the fraction for 60p could be
6/10 and Zara agreed. Multiplying 0.1 by six, or 0.2 by
three, were other methods students reported using. Then
she recorded
75p 3=4 0:75 75%
and a student explained � by saying that ‘‘there are four
lots of 25p in a pound, and 75p is three of them’’.
3.5 Analysis of the affordances of the example sets
In the first episode, Zara was aware of the limited meaning
students in lower secondary school can give to ‘‘number’’
and needed them to have a different example space on
which to draw for this lesson. In workshops with teachers
and teacher educators, we find the same limitation, and our
experience with such tasks suggests that her students were
well-used to extending their idea of number to the whole
class of reals, as they knew it. Zara told us afterwards that
Qualities of examples in learning and teaching 291
123
she hoped students would go on to use a visual image of a
number line to relate fractions, decimals, and percentages.
They would hence have access to broader generalisations.
These learner generated sets were examples-of sets of three
numbers that add to four, but Zara’s deeper role for them
was as examples-for access to a larger class of numbers.
In the second episode, Zara was explicit about some
things but not others as she set up a task comparing lengths
on a metre stick with lengths of modelling plastic. She
fixed the notion of ‘‘one whole’’ and showed that this was a
special decision by asking students why, and they respon-
ded with their realisation that it was something to do with
100. Because some of the students would already know that
� = 0.5 (and maybe a few other standard decimals), they
could have seen ‘‘1/100 = 0.01’’ as an example of the kind
of equivalence they already knew something about. Dis-
playing it on the board allowed it to function as an
example-of a class of relations, and also as a special ele-
mentary case from which other examples could be con-
structed (an example-for). The next example, equating 0.5
and 0.50, is also indicative of a class, but this time not a
class of relations but a class of equivalences, namely that it
does not matter how many zeroes are on the end in a
decimal representation. However, it is not clear whether
students understood this to be a reference example-for a
rule of equivalence or a special case. If the latter, perhaps
pedagogical implementation (e.g., use of only a single
instantiation) and characteristics of the learners restricted
the example’s power.
In the third episode, Zara listed the fractions she was
particularly interested in hearing about, and included
fractions related by halving, a fraction whose decimal
recurs, and two that can be confused with each other
through not understanding the reciprocal: � = 0.5 and
1/5 = 0.2. Her aim, she told us, was for the students to
relate everything to the number line so that they abandoned
‘‘food’’ images of fractions such as cakes and pizzas. She
was explicit with them about making ‘‘the match’’—a
connection between fractions and decimals—via the linear
model. It seemed she meant these examples to be exam-
ples-for showing students the relationship between deci-
mals and the number line, and the materials to provide
them with a model of it, but the specific examples that she
chose were examples-of the relationship.
Students’ responses, however, suggest that something
else was happening. Whereas we can imagine students
cutting a 50 cm snake in half and reading off ‘‘0.25’’ from
the stick, to read ‘‘0.33’’ is harder to believe, when one
considers the plausible error bounds involved in measuring.
Furthermore, the student who gave ‘‘0.125’’ for one-eighth
could not have been reporting a reading a length of mod-
elling plastic from the stick, given the level of accuracy
required. Perhaps, these were students who already knew
the relation and had edited their readings to make the right
answer.
More interesting were the two (and maybe more) stu-
dents who believed that one-eighth is ‘‘nought point
eight’’—a result that could not have come from measuring
an eighth of the whole snake. This was the common error
which she had designed the task to address, the kind of
intuitive error relating to understanding the reciprocal (cf.
‘‘reciprocal thinking’’ in Stacey & Steinle, 1998). Zara’s
response to ‘‘0.8’’ was to ask them if they had actually
made and measured this length. From this interchange, she
reported later, she found that a few students were assuming
an answer, and then rolling their modelling plastic into
thinner lengths to reach this result. In other words, they did
not see the task as about matching lengths and reading off
from the stick, but about assuming a relation—based in this
case on the common reciprocal error—and making their
lengths match. For students to believe the 0.8 result, they
must have generalised from some relation in which digits
do match, or be enacting some kind of meaning-free
manipulation. It is interesting to note that the ‘‘0.8’’ answer
followed from the ‘‘0.33’’ answer, which may have rein-
forced a direct relation between the digits of the denomi-
nator and the decimal digits. The final fraction of the
lesson, 1/10, would not have dispelled that myth, nor would
the example of 1/100 that was on the board. The latter
appears to have been used incorrectly as a template rather
than as an element with which to build. Here, some stu-
dents’ expectations and experience (Goldenberg, 2005)
may have reduced the effectiveness of the examples.
Zara reported being disappointed in the lesson; she had
hoped that students would adopt the linear image as a
reference for meaning, but found that the affordances of the
material allowed something different to happen that had, if
anything, confirmed the misconception that decimals had to
contain the same digits as the fraction notation. Further-
more, some students had relied on prior knowledge and
therefore may have missed the experience of matching that
she had designed. She resolved to use paper strips next time
so that no one could stretch the materials to fit their pre-
conceptions. Her reaction at the time prompted her to
introduce a new model using the coin software in the next
lesson.
Zara’s aim in episode 4 was to utilise knowledge they
already had and to try a ‘‘fresh start’’ after the difficulties of
the previous lesson. The software she chose gave her
complete control over the examples she used, and she
started with coins whose proportion of a pound she judged
to be fairly easy to express and explain. She tried to
establish (she told us afterwards) the language pattern
‘‘There are five 20ps in one pound’’ which we observed
students using later in the lesson. The lesson proceeded to
consider the coins students had found hard to express—1p
292 A. Watson, H. Chick
123
and 5p—but Zara had been aware that these were likely to
be problematic. She worked publicly with these, putting 1p
first, so that 5p could be seen to be a multiple of 1p. This
mirrored the use of 1 cm in the previous lesson: an ele-
mentary example from which others could be constructed.
The process of multiplicative construction was repeated at
the end of the lesson, when ‘‘20p’’ was written as a pre-
cursor to working on representations of 60p. Zara had
illustrated 60p using three 20p coins, and ‘‘times 20p by
three’’ was offered as a method by students. 75p was
offered additively as 50p ? 20p ? 5p, but the multiplica-
tive method for arriving at 3/4 was praised by Zara.
When analysing these lessons, we were struck by the
contrast between the continuous linear image of number
afforded in the first and the discrete image afforded in
the second. We could also see that the second afforded
articulation of multiplicative relations between the reals,
and multiple isomorphisms between four representations,
where the first lesson had offered a relation between
fraction and decimal notation, and a focus on actions of
‘‘cutting’’ and ‘‘adding to’’. The different models had
structured different relations to be inferred from the
examples. Nevertheless, the use of money in her second
lesson played a similar role to the metre stick and the
modelling plastic: the isomorphisms there and the
examples-of specific relationships (e.g., between 5p, 0.05
and 1/20) are also examples-for recognising the con-
nections between fractions and decimals and for ways of
obtaining the relationships (multiplicatively, additively,
etc.)
Reflection on the roles of examples in these lessons
shows that Zara used:
• Extending a class beyond obvious examples by asking
students to construct several cases that fulfil a
constraint;
• Indicating and naming types of example that constitute
a class, where there is the danger of assuming that a
subclass represents the whole class;
• Individual examples and sets of examples which
indicate a relation between classes, via a particular
layout inviting structural inductive reasoning;
• Examples which provide elementary cases from which
others in a class can be built or generated;
• Examples which express equivalence (same thing,
different representations);
• Formatted references to use as templates when dealing
with other class members;
• Sets of examples that span the possibilities in a class,
the subtypes, and also can be used later as raw material
to identify relations within the class;
• Examples in which superficial (possibly incorrect)
relations can be inferred from appearance;
• Individual and sets of examples indicating a relation
between classes through a particular layout; and
• Examples as situations in which to develop language
patterns suitable for thinking about fractions of a whole.
4 Examples and examplehood
Our analysis of our own mathematical work showed that an
example can exemplify many different mathematical ideas,
depending on learners’ experience and choice of actions. It
suggested that what was exemplified depended on choice
of dimensions of variation, choice of technologies, and
imbuing a task with purpose. The role of the teacher in
determining the intent and pedagogical implementation is
also likely to be critical. One aspect of this is guiding how
learners need to act on examples (construct, compare,
attend in new ways) in order to achieve analysis, general-
isation and abstraction.
Our analysis of Zara’s use of examples suggests two
kinds of example use not described in our earlier exposi-
tion: examples to be used to build other examples and
examples that afford a shift of focus. In Zara’s teaching, the
latter were examples that afforded multiplicative instead of
additive reasoning. It also suggests a further element of
didacticisation, namely the use of layout or particular
representations to draw attention to patterns and relations.
Another feature of Zara’s teaching was that her use of
examples did not always relate to a superordinate class for
generalisation or abstraction, but sometimes was intended
to direct learners’ attention analytically to a property or
relation within an example, such as equivalence, or a new
feature to pay attention to.
Her use of examples was fluent and flexible, adapted to
learners’ needs as she perceived them, so her practice
provides a wide repertoire of possible pedagogical pur-
pose in example use. It also challenges whether our dis-
tinction between ‘‘examples-of’’ and ‘‘examples-for’’ is
meaningful from a pedagogic perspective. Not only do
some examples appear to serve both purposes, but also
the uses that students made of examples were not always
as intended. In the lessons, there were incidents where it
seemed students used an example not for meaning-making
or for building-with, but merely for copying its visual
features, yet Zara’s purpose was for it to be an element
for building, or an instantiation of a relation. From Zara’s
comments to us, we know she always had a purpose for
her example use, but only a few times in the lessons did
she make that purpose explicit. Learning to learn math-
ematics includes learning what to do with examples, so
that—like we did in Task 1—learners can choose from a
range of ways to engage.
Qualities of examples in learning and teaching 293
123
The relation between an example and examplehood is,
as we have shown, dependent on human agency: how the
teacher didacticises the example and how the learner
engages with it. The pedagogic task is to align the learners’
engagement with the teacher’s intentions. In this paper, we
have illustrated and drawn attention to the variety of fluent
uses of examples made by experienced mathematicians and
experienced teachers, and hence indicated how careful and
knowledgeable teachers need to be to bring about such
alignment.
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