Educ Stud Math
DOI 10.1007/s10649-008-9140-6
Characteristics of teachers’ choice of examples
in and for the mathematics classroom
Iris Zodik & Orit Zaslavsky
# Springer Science + Business Media B.V. 2008
Abstract The main goal of the study reported in our paper is to characterize teachers’
choice of examples in and for the mathematics classroom. Our data is based on 54 lesson
observations of five different teachers. Altogether 15 groups of students were observed, three
seventh grade, six eighth grade, and six ninth grade classes. The classes varied according to
their level—seven classes of top level students and six classes of mixed—average and low
level students. In addition, pre and post lesson interviews with the teachers were conducted,
and their lesson plans were examined. Data analysis was done in an iterative way, and the
categories we explored emerged accordingly. We distinguish between pre-planned and
spontaneous examples, and examine their manifestations, as well as the different kinds of
underlying considerations teachers employ in making their choices, and the kinds of
knowledge they need to draw on. We conclude with a dynamic framework accounting for
teachers’ choices and generation of examples in the course of teaching mathematics.
Keywords Examples . Mathematics teaching practice . Teacher knowledge
1 Examples in mathematics education
Examples are an integral part of mathematical thinking, learning and teaching, particularly
with respect to conceptualization, generalization, abstraction, argumentation, and analogical
thinking. By examples we mean a particular case of a larger class, from which one can
reason and generalize. In our treatment of examples, we also include non-examples, that are
associated with conceptualization and definitions, and serve to highlight critical features of
a concept; as well as counter-examples that are associated with claims and their refutations.
Both non-examples and counter-examples can serve to sharpen distinctions and deepen
understanding of mathematical entities. It should be noted that examples may differ in their
This research was supported by The Israel Science Foundation (grant 834/04, O. Zaslavsky PI).
I. Zodik : O. Zaslavsky (*)
Department of Education in Technology and Science, Technion—Israel Institute of Technology,
Haifa, Israel
e-mail: [email protected]
I. Zodik, O. Zaslavsky
nature and purpose. An example of a concept (e.g., a rational number) is quite different in
nature from an example of how to carry out a procedure (e.g., finding the least common
denominator). Moreover, the purpose for presenting an example may vary. Thus, a teacher
may illustrate how to find a common denominator of two proper fractions for adding
fractions, or s/he may illustrate it as a basis for generalizing the procedure to algebraic
fractions in order to be able to solve more advanced equations (as in Fig. 4).
Studies on how people learn from worked-out examples point to the contribution of
multiple examples, with varying formats (Atkinson, Derry, Renkl and Wortham 2000).
Such examples support the appreciation of deep structures instead of excessive attention to
surface features. Studies dealing with concept formation highlight the role of carefully
selected and sequenced examples and non-examples in supporting the distinction between
critical and non-critical features and the construction of rich concept images and example
spaces (e.g., Vinner 1983; Zaslavsky and Peled 1996; Petty and Jansson 1987; Watson and
Mason 2005). In spite of the critical roles examples play in learning and teaching
mathematics, there are only a small number of studies focusing on teachers’ choice and
treatment of examples. Rowland, Thwaites and Huckstep (2003) identify three types of
novice elementary teachers’ poor choice of examples: choices of instances that obscure the
role of variables (for example, in a coordinate system using points with the same values for
both coordinates); choices of numbers to illustrate a certain arithmetic procedure when
another procedure would be more sensible to perform for the selected numbers (for
example, using 49×4 to illustrate conventional multiplication); and randomly generated
examples when careful choices should be made. These findings concur with the concerns
raised by Ball, Bass, Sleep, and Thames (2005) regarding the knowledge base teachers need
in order to carefully select appropriate examples that are useful “for highlighting salient
mathematical issues” (p. 3). Obviously, the choice of examples in secondary mathematics is
far more complex and involves a wide range of considerations (Zaslavsky and Lavie 2005;
Zodik and Zaslavsky 2007; Zaslavsky and Zodik 2007). This paper provides a close look at
the underlying considerations that experienced secondary school teachers employ.
The specific choice of examples may facilitate or impede students’ learning, thus it presents
the teacher with a challenge, entailing many considerations that should be weighed. Yet,
numerous mathematics teacher education programmes do not explicitly address this issue and do
not systematically prepare prospective teachers to deal with the choice and use of instructional
examples1 in an educated way. Thus, we suggest that the skills required for effective treatment
of examples are crafted mostly through one’s own teaching experience (Leinhardt 1990;
Kennedy 2002). It follows, that there is much to learn in this area from experienced teachers.
Our study proposes to make a step towards learning from experienced teachers—their
strengths and difficulties associated with exemplification in the mathematics classroom.
2 Teachers’ knowledge and its relation to exemplification
Generally, an example should be examined in context. Any example carries some attributes
that are intended to be exemplified and others that are irrelevant. As Rissland (1991) points
out “one can view an example as a set of facts or features viewed through a certain lens”
(p. 190). In Skemp’s (1971) terms, the irrelevant information an example carries can be
We use the term ‘instructional example’, to refer to any example offered by either a teacher or a student
within the context of learning a particular topic.
1
Characteristics of teachers’ choice of examples
regarded as noise—“The greater the noise the harder it is to form a concept” (p. 29). Thus, a
teacher may use a specific example for illustrating certain ideas through his or her lens,
while a student may focus on its irrelevant features. It is the ability to see the general
through the particular that is at the heart of exemplification (Mason and Pimm 1984).
Watson and Mason (2006) suggest the notion of variation in structuring sense-making with
respect to tasks. Basically, they claim that teachers can expose mathematical structure by
varying some of the features of a task while keeping constant others. Consequently, learners
are bound to notice structure and generalize, since they “cannot resist looking for, or
imposing pattern, and hence creating generalizations” (p. 95). This idea is useful when
thinking of exemplification as well, particularly as a means to highlight relevant features.
Three aspects of teacher knowledge strongly relate to exemplification in mathematics
education: knowledge of mathematics, knowledge of students’ learning, and pedagogical
content knowledge (concurring with Shulman 1986, 1987, and Harel, in press). The quality
of a teacher’s mathematical knowledge affects what is taught and how it is taught. With
respect to exemplification, the mathematical aspect of an example has to do with satisfying
certain mathematical conditions depending on the concept or principle it is meant to
illustrate. Knowledge of students’ learning refers to teachers’ understanding of how
students come to know and how their existing knowledge affects their construction of new
knowledge. It also relates to the teacher’s sensitivity to students’ strengths and weaknesses,
and with respect to examples—to teachers’ awareness of the consequences of students’
over-generalizing or under-generalizing from examples, and students’ possible tendency to
notice irrelevant features of an example instead of attending to its critical features.
Pedagogical content knowledge has to do with transforming mathematics into means by
which its learning can be facilitated; this includes “ways of representing and formulating
the subject that make it comprehensible to others” (Shulman 1986, p. 9). Clearly, examples
are inseparable from their representations, and indeed they are meant to help make
mathematics comprehensible to learners.
A central part of teachers’ work involves decision-making; some parts can be done in
advance with careful planning while others are done in real time “on one’s feet” in response
to classroom situations, many of which occur unexpectedly. Mason and Spence (1999)
elaborate on the latter treating it as another significant aspect of teachers’ knowledge, which
they term knowing to act in the moment. This aspect of knowing relates to teachers’ creative
ways of responding to actual (often unexpected) classroom situations that require an
immediate action on the part of the teacher; it heavily relies on teachers’ increasing
awareness and ongoing reflection. Choosing and generating examples for teaching often
requires in-the-moment decisions in response to classroom interactions. This kind of acting
in the moment can also be examined in terms of teachers’ flexibility, for example, as done
in the work of Leikin and Dinur (2007) in which they identified different patterns of teacher
flexibility in the context of problem solving in the classroom. Simon (1995) connects
teachers’ knowledge, their pre-planning of a lesson, and the actual classroom interactions
that often call for spontaneous actions, in his Mathematics Teaching Cycle. His model
accounts for teachers’ ongoing learning through their practice.
In our study we use the above constructs for examining teachers’ choice of examples. We
examine their underlying considerations with respect to the knowledge on which they draw, and
highlight their knowledge-in-action and accessibility to their personal example spaces. Inspired
by Simon (1995), we offer a framework for thinking about teachers’ choices of examples as
part of a Mathematics Example-Related Teaching Cycle (see Fig. 7 later in this paper).
As discussed above, there is a growing interest in articulating the mathematical
knowledge needed for teaching (Ball et al. 2005). At the same time, increasing attention is
I. Zodik, O. Zaslavsky
given to the critical and multifaceted roles of examples in learning and teaching
mathematics (Zazkis and Chernoff 2008; Zaslavsky and Zodik 2007; Bills, Dreyfus,
Mason, Tsamir, Watson and Zaslavsky 2006; Watson and Mason 2005). Our study lies at
the intersection of these two significant themes. We focus on a somewhat under-researched
aspect of teacher practice and knowledge, that is, their knowledge and use of examples for
teaching mathematics. We address the challenge of learning from secondary experienced
teacher practice in order to characterize this aspect of their practice, on the one hand, and to
offer a sound basis for designing teacher education programs that better prepare secondary
mathematics teachers for judicious choice and use of examples.
3 The study
Goal The study reported here is part of a larger research programme addressing the role
and nature of examples in mathematical thinking, learning and teaching. The main goal of
this part of the study is to characterize mathematics teachers’ underlying consideration in
their choice and generation of examples for teaching secondary mathematics. In particular,
the aim is to identify and characterize in-the-moment decisions teachers take in the
classroom in response to their interactions with students that lead to modification or
construction of new examples.
Participants The study can be seen as a collection of five interrelated case studies of
teachers (Stake 2000). The main participants of the study were five experienced secondary
teachers (with at least 10 years of mathematics teaching), who taught classes of different
grade level and achievement level. Altogether 15 groups of students participated in the
study: three seventh grade, six eighth grade, and six ninth grade classes. The classes varied
according to their level—seven classes of top level students and six classes of average and
low level students.
Data Sources The data collected included 54 observed and documented lessons of five
different teachers accompanied by field notes and audio transcripts. Following Wiersma’s
(2000) recommendation that “Much of the qualitative research involves observation by
multiple observers as at least part of the data collection” (p. 211), in our study 22 of the
classroom observations were conducted by two researchers. In order to enhance internal
reliability (Wiersma 2000) data was triangulated by other sources, which included pre and
post lesson interviews with the observed teachers. In addition, we collected several related
documents (e.g., students’ worksheets, teachers’ notes and lessons plans) and the researcher
managed a research journal.
We observed both randomly and carefully selected mathematics lessons of five
experienced secondary mathematics teachers. By ‘carefully selected’ classroom observations we refer to lessons which teachers invited us to observe on the basis of their personal
views of good ways to use examples in the classroom; these lessons were considered by the
teachers as their ‘best cases’, that is, lessons which illustrate a particularly good way of
choice and treatment of examples in the classroom. The ‘carefully selected’ observations
were conducted with each teacher towards the end of the study, after a number of randomly
selected observations were performed. By this we were able to get access to the more or
less regular practice of the teacher, without disclosing what we were looking for; since our
goal was to learn from experienced teachers, we also wanted to include lessons that
reflected what they thought were manifestations of “good practice” with respect to
Characteristics of teachers’ choice of examples
exemplification. By observing both kinds of lessons, randomly vs. carefully selected, we
hoped to get a rich picture of teachers’ choice and treatment of examples. For the carefully
selected lessons, we conducted pre and post lesson interviews with the teachers, in order to
prompt them about their views of and underlying considerations in choosing what they
considered “good” examples. In all other lessons we just had an informal conversation
before and after the lesson regarding their overall lesson plan, with no specific attention to
examples. This was done in order to reduce the influence interviews are known to have on
the interviewee.
4 Data analysis
In our research, we had two preliminary stages in analyzing teachers’ choices of examples,
corresponding to two dimensions: Conditions for an example (that is, whether a particular
object/case qualified as an example according to our criteria), and mathematical correctness
(that is, whether the example satisfied what it was intended/supposed to from a
mathematical perspective). This enabled us to focus on the collection of mathematically
correct examples for further analysis and characterization according to other categories that
emerged as we repeatedly looked into the data, in the spirit of the grounded theory approach
(Strauss and Corbin 1998). We report on these categories in the findings section. One of the
insights we gained as we examined the data relates to the unit of analysis. It turned out that
it made a lot of sense to analyze the underlying considerations that led to a particular
example, rather than to try to characterize an example in itself. These considerations
reflected not only the mathematics, but also the pedagogy that was employed, including the
teacher’s goals, available tools, interpretation of the situation, etc.
In order to provide a stronger and unbiased picture of the data, and to better understand
the phenomena that were examined, we used some simple statistics following Miles and
Huberman (1987) and Wiersma (2000). There is no claim about generalizing beyond the
scope of the study. However, given the limited number of studies that examined teachers’
use of examples in and for their classrooms, this analysis may help form some hypotheses
for future research.
For internal consistency we followed Wiersma (2000), who claims that “If two or more
researchers independently analyze the same data and arrive at similar conclusions, this is
strong evidence for internal consistency” (p. 211). Thus, for each stage that required some
sort of coding according to a classification system that we applied, we had two researchers
code independently at least 15% of the relevant data. In all cases we got at least 90%
agreement, with no discussions between these researchers. In addition, in cases of that were
vague, the validity was enhanced by stimulated recall interviews with some of the teachers,
in which they were asked to reason about their choice of examples and react to the
researchers’ interpretations.
4.1 Identifying examples
The guiding question for this part of the analysis was, for any given case that was observed
in the lesson, “Is it an example?”. Theoretically, every mathematical object can be seen as
an example, that is, as a particular case of a larger class. We take the stand that for a
mathematical (or any) object to become an example of something, there should be some
mental interaction between the person and the object that registers in the eyes of the person
as an example of a larger class. Thus, in our study we included just the cases for which we
I. Zodik, O. Zaslavsky
had evidence that the person who proposed or discussed a particular case, thought of it and
treated it as an example of a larger class, in a way that we could answer the question: What
(for him/her) is this an example of? Thus, a practice exercise, such as solving an equation,
was not regarded an example unless the teacher or students explicitly referred to it as an
example of a more general class of equations, or as an illustration of a more general
procedure. This is not to ignore the fact that any exercise could have been thought of or
implicitly treated as an example without our noticing it. Yet, we took the stand that by our
approach, we may be taking the risk of leaving out some relevant cases that were hard to
identify, however, we were able to justify that the collection that we further analyzed
consisted only of elements that were observed to be treated to some extent by someone in the
classroom as examples. Altogether, there were only a small number of possibly missed cases.
We turn to an example of a case that was identified as an example:
The drawing in Fig. 1 was offered by a teacher in response to a student’s query regarding
the possibility of a triangle having two obtuse angles. The teacher drew two obtuse angles
with a common side, calling students’ attention to the construction process, which
illustrated why the sides of the two obtuse angles do not intersect in a way that “closes it to
a triangle”. The teacher and the students treated this as an example of two obtuse angles
sharing a common side.
This part of the analysis was based on explicit classroom utterances, as well as pre and
post lesson conversations and interviews with the teachers. The rest of the analysis was
done just for the cases that were identified as examples.
4.2 Determining mathematical correctness
This part of the analysis examined the extent to which an example is mathematically
“correct”, similar to the way correctness was determined in Zaslavsky and Peled (1996). In
part, this category can be considered rather “objective” although it is context based. The
correctness was determined with respect to the claims made regarding the example.
Generally, there are three types of possible “incorrectness” of a mathematical example: One
has to do with whether the case that is treated as an example of a more general class in fact
satisfies the necessary conditions to qualify as such example. For instance, treating the
absolute value function as an example of an everywhere differentiable function would be
considered incorrect, or maintaining that 0:333 is an example of an irrational number is also
incorrect. The second type has to do with counter-examples. Treating an example as a
counter-example for a particular claim or conjecture when it does not logically contradict
the claim is also considered mathematically incorrect. For instance, treating the following
binary operation a*b=ab as a counter-example to the false claim that any commutative
operation is also associative, or regarding the function f(x)=x3+1 as a counter-example to
the claim that any odd function is monotonic, is considered mathematically incorrect. A
third type of mathematical incorrectness is manifested in treating a non-existing case as if it
were a possible example. For instance, Fig. 2 gives an example of an (non-existent)
isosceles triangle that illustrates the third type of incorrectness.
Fig. 1 A teacher’s example illustrating why a triangle cannot have
more than one obtuse angle
Characteristics of teachers’ choice of examples
Fig. 2 An example of a
“non-existing example” of an
isosceles triangle
6
6
12
5 Findings
In all our observations, we identified altogether 604 teacher-generated2 examples and only
35 student-generated examples. This in itself suggests that the lessons observed were rather
teacher-centered. Our findings refer only to the teacher-generated examples. Note that
within a particular teacher some lessons were richer in examples than others (e.g., it seems
that teachers use more examples for introducing a new topic than in an “ordinary” lesson).
Encouragingly, there were just 12 mathematically incorrect examples. Of the incorrect
examples, six were in geometry and six in algebra. There were three “non-existing”
examples, and nine examples that did not meet the necessary conditions. Eleven of these
examples were acknowledged or modified in the course of the lesson, as a result of
interactions with students. Altogether, teachers used 18 counter-examples, all of which were
treated logically appropriately.
Throughout the classroom observations it became clear that some of the examples
teachers use are planned in advance (i.e., pre-planned) while others are chosen or
constructed on their feet either in response to classroom interactions or as part of the way
they planned to conduct the lesson. Some teachers plan in advance rather generally what
type of examples they will use and just determine possible dimensions and range of change
within which to select the specifics, but generate the specific examples in the classroom,
either on their own or by involving the students in the process.
We turn to a description of features of two main kinds of examples that we identified:
pre-planned vs. spontaneous examples, and present teachers’ considerations in choosing
these kinds of examples.
5.1 Pre-planned vs. spontaneous examples
As mentioned above, we examined the examples that a teacher chose to use in the classroom
in terms of the amount of pre-planning underlying his or her choices. On the one hand, a
teacher could carefully plan and choose which examples to use, as part of the plan of a
lesson. On the other hand, a teacher may face a completely new and unfamiliar classroom
situation that calls for in-the-moment generation or choice of an appropriate example.
Pre-planned examples are examples for which there was some evidence indicating that
the teacher thought of them in advance and intended to incorporate them in the lesson.
Thus, they appeared in the teachers’ planning notes, worksheets they prepared for their
students, textbooks on which they based the lesson, or could be inferred from teachers’
utterances and actions. In cases for which there was no sufficient evidence, we prompted
teachers in our conversations with them in an attempt to find out when and how they came
up with the examples they used.
We consider a spontaneous example to be an example for which there is evidence that
choosing it involved to some extent in-the-moment decision making. This includes cases
In this section we consider “teacher-generated examples” any example selected and presented by the
teacher, even if it was taken from textbooks or other sources, with no actual generation on the part of the
teacher.
2
I. Zodik, O. Zaslavsky
when a choice of an unplanned example was made (even if it were a familiar one for the
teacher), or in-the-moment construction—either of a completely new (to the teacher)
example or a modification of an example that had been introduced (by the teacher or the
students) in the classroom. The evidence supporting the claim that a certain example is
spontaneous was based on teachers’ utterances in and outside the classroom, the length of
time devoted to generating the example, their hesitations and body expressions. Utterances
that indicated in-the-moment acting included expressions like: “I’m trying to construct a
simple example but it’s not working”, or “I just chose these numbers now without giving
them more thought”. There were cases where a student asked the teacher: “Are you
inventing the example right now?” and she confirmed this.
Teachers’ spontaneous examples were often generated in response to students’ queries or
claims. One case of a spontaneous example was seen in a situation where a teacher had a
clear plan for the lesson, but no specific examples. She constructed the examples on her feet
in front of the students step by step, correcting some steps (by erasing parts that did not fit
her intentions), until she reached the desirable examples that fit her plan. This was a lesson
on algebraic expressions. The teacher wanted to illustrate that a complicated algebraic
fraction could be equivalent to a number, say 13. She deliberately wanted to conceal the
intended result (which she considered surprising) and attempted to generate a lengthy and
4 2 3
b c 4a3 b
compound case, yielding: 3a36a
7 b3 c3 . It took some iteration until she reached this specific
example that met her purpose.
Another example of a spontaneous example is a counter-example that a teacher
generated in a geometry lesson in response to a student’s unexpected claim that if in a
quadrangle there are two opposite right angles it is a kite. In order to convince the student
that this claim was false the teacher generated on her feet a (dynamic) counter-example with
an explanatory dimension (in terms of Peled and Zaslavsky 1997) by taking two right
angles and moving them around until they intersected as shown in Fig. 3.
Of the teacher-generated examples that were observed in this study, there were 317 preplanned and 287 spontaneous examples, that is, close to half were spontaneous. We are
aware that it is extremely difficult and highly ambitious to clearly distinguish between preplanned and spontaneously constructed or chosen examples, yet we feel that this distinction
is helpful in making sense of teachers’ choice of examples. Thus, we acknowledge that we
may be biased towards attributing to examples used by experienced teachers more planning
than actually was involved in their choices; what appears to be pre-planned could in fact be
generated instantly. Consequently, it may be that some examples that we considered preplanned could have some spontaneous elements in them, to which we were not aware.
However, the many obvious spontaneous examples that we were able to identify were
insightful sources for examining and following in real time how teachers act with respect to
Fig. 3 A spontaneous example
of a quadrangle with two opposite
right angles that is not a kite
Characteristics of teachers’ choice of examples
examples. These are occasions where teachers manifest their knowing to act in-the-moment
(Mason and Spence 1999).
In general, teachers’ pre-planned examples drew more on their available resources,
mainly textbooks. There was a case in which a teacher claimed that her choice of the
textbook for a particular group of average and low level students was based on the way
examples are structured there (i.e., gradually, from “easy” to “difficult”). Some of the
teachers’ spontaneous examples were rather immediate, that is, seemed to be easily accessible
from their example spaces. However, those that took longer to come up with and often needed
a number of iterations until the teacher was satisfied that they served their purpose indicated
either remote accessibility to the teacher’s example space or a need to generate examples that
were new to the teacher. These “moments” can be seen as learning opportunities for the
teachers (Zodik and Zaslavsky 2007); their learning could result in a richer example space.
Our findings point to ways in which teachers’ choices of examples, both pre-planned
and spontaneous, are guided by some underlying principles or considerations. These
considerations interact with their example spaces and determine the actual choice or
generation of “new” examples.
5.2 Teachers’ considerations in choosing examples
Based on our observations and conversations with the teachers, we identified a number of
interrelated and somewhat overlapping principles that guide them in choosing and/or
generating examples. Some considerations reflect pedagogical content knowledge, and
some indicate sensitivity to students. In all cases, their choice and generation of examples
relied to a large extent on sound mathematical knowledge of the relevant topics.
There were several examples for which more than one consideration was employed. On
the other hand, there were cases consisting of a sequence of examples for which the same
consideration was employed. Thus, altogether we identified 474 explicitly expressed
considerations, and provide details on how they were distributed according to the categories
that we present. We also provide information on the distribution of these considerations
between spontaneous and pre-planned examples.
Interestingly, all five teachers whom we observed claimed that they had never articulated
how to select and generate examples—not throughout their years of pre-service and inservice education nor with colleagues in their school or other forms of professional
communications. Moreover, they had never explicitly thought about these issues. Thus, by
asking them to reflect on their work and examine what principles or rules of thumb guide
them, they became more aware of their planning and in-the-moment actions.
We turn to the description of the types of considerations teachers employed in selecting
or generating instructional examples. The first four were particularly common. In all but
one category (4), these principles were employed for both pre-planned as well as for
spontaneous examples. We present them according to a decreasing level of frequency.
5.2.1 “Start with a simple or familiar case”
This approach has some resemblance to Bills and Bills’ (2005) findings. It was expressed
by utterances such as: “I won’t start with a ‘tilted’ right triangle, but rather with a simple
case where the sides of the right angle are vertical and horizontal. I’ll move to the other
cases later” [T1]; “I begin teaching the Square Root with simple and familiar cases, like
square roots of 9, 16, 25, …; thus at first they [the students] can find the square root
immediately without the need to use their calculator” [T2].
I. Zodik, O. Zaslavsky
Some teachers consider sequences of examples, and construct them gradually by a
certain criterion. Accordingly, the sequence may “vary just one element at a time” or
gradually increase in its level of complexity or difficulty. Figure 4 provides a sequence of
three systems of equations with fractional algebraic expressions that were selected by a
teacher [T3] applying her principle of going from simple to more complicated. The notes on
the right hand side are her explanations for this choice.
This category was the most common one. We identified 193 instances, of which 59%
(114) were related to spontaneous examples and 41% (79) to pre-planned.
5.2.2 “Attend to students’ errors”
Teachers often build examples according to common errors they know students make, or
other difficulties students encounter. Some of the expressions they used to explain this
approach were: “I didn’t just choose this, I thought what difficulties they [the students]
might have” [T3]; “I use students’ error, that is they think that when cancelling a fraction if
‘everything cancels’ then it equals zero when it really equals one” [T1]; Accordingly, she
4 3 2
b c 3a2 b4 c
chose to use the following example: 4a
6a2 b4 c2 2a4 b3 c helping the students realize that it is
equivalent to 1 and not to 0.
pffiffiffiffiffi
Other cases similar to the latter were exhibited in examples such as 50 which a teacher
[T2] stated that she included in the collection of examples she wanted her students to
encounter when learning
about
pffiffiffiffiffi p
ffiffiffiffiffiffiffiffiffiffiffisquare roots. She did this in order to draw their attention
that even though 50 ¼ 2 25 the result is not 10; based on her experience, she realized
that students
tempted
to induce that it is equal p
toffiffiffiffiffi
10 usingpthe
pare
ffiffiffiffiffi often p
ffiffiffiffiffi
ffiffiffiffiffi following fallacious
reasoning: 50 ¼ 2 p25
¼
2
5
¼
10,
while
in
fact
50
¼
6
2
25. Similarly,
a choice
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffi
2 þ 32 6¼
of
an
example
like
25
may
serve
to
draw
attention
to
the
fact
that
4
pffiffiffiffiffi pffiffiffiffiffi
42 þ 32 :
Another example of how a teacher [T1] takes into account students’ possible errors, was
observed in a geometry lesson in which a teacher introduced the terms corresponding and
x
+
3 2
x− y
1)
2
2)
3)
y
{
{
=2
−
x+4
2 ( x − 3)
5
+
( 6 ,0 )
Example 1: Simple expressions in the
numerator; Easy to find the appropriate
least common multiple (LCM) of the
denominators.
( 10 , 4 )
Example 2: Not as simple – there is a
need to use the distributive law; and
finding the LCM is more complicated.
( 7 ,3 )
Example 3: A different representation
of the fractions as well as choice of
signs that present a difficulty.
=1
y−2
=
4x − 5
7
4
14
3 ( 2 y − x ) − 2 y + 14 = 0
3
2
( x − y) − ( x − 2 ) = 4
2
5
3( x − 2) − 5 y
Fig. 4 A teacher’s choice of and reasoning about a gradual sequence of examples
Characteristics of teachers’ choice of examples
alternate angles between a pair of lines and a line intersecting them both. Based on her
prior experience, she knew that students tend to think that these terms are used only when
the pair of lines is a pair of parallel lines. Thus, she deliberately chose to illustrate how to
identify corresponding and alternate angles in several cases not only when the lines appear
to be parallel but also where the lines are clearly not parallel.
In another case, a teacher [T1] identified in the course of the lesson a student’s difficulty
in determining whether the order of the equations in a system of two equations they were
solving matters, and decided to illustrate that the solution is invariant with respect to the
order in which the equations appear (i.e., that it is equally possible to “subtract” the
equation that appears on the top from the one on the bottom). She chose to move from
the common algebraic representation in which indicating the order of equations is
inevitable, to a graphical representation of a simpler system of equations. In the example
she chose, the two equations were represented by two intersecting straight lines, where no
order was embedded in the representation. Through this example the teacher was able to
draw the student’s attention to the above invariance about which he was puzzled. The
student’s reactions indicated that the teacher’s choice was helpful, and that he and his peers
were able to make the necessary connections and build on them.
We identified 85 instances in this category, of which 79% (67) were related to
spontaneous examples and 21% (18) to pre-planned ones.
5.2.3 “Draw attention to relevant features”
This concern has to do with a deliberate attempt to reduce “the noise” of an example, that
is, to try to avoid cases that are special in ways that could distract attention from the more
general case. This type of consideration was expressed, for example, by a teacher [T1] who
wanted to illustrate the Pythagorean Theorem, giving examples of right triangles with the
measurements of two sides, and asking the students to calculate the hypotenuse. The
teacher gave first the pair 3, 4, then moved to 6, 8 (note that the second pair is a multiple of
the first). She then deliberately gave 5, 12, in order to “break the pattern” of pairs of
numbers that are multiples of the first pair, because “even students in a low level grade
immediately notice such patterns and may be distracted by them”.
Another way teachers try to draw students’ attention to relevant features, in addition to
breaking an irrelevant pattern, can be seen in their attempts to apply principles similar to
what Watson and Mason (2006) consider
structured
Thus, a teacher [T5]
variation.
deliberately moved from dealing with x2 x < 20 tox2 þ x < 20; or another teacher
[T1] gave a sequence of linear functions for an exploration task: f ð xÞ ¼ x þ 5; f ð xÞ ¼
2x þ 5; f ð xÞ ¼ 3x þ 5, varying one thing at a time (the coefficient of x). Then she
decided to “break the pattern” and added f ð xÞ ¼ x2 þ 5 to the sequence, changing the
degree of the polynomial (from 1 to 2), keeping the free term constant.
We identified 68 instances in this category, of which 57% (39) were related to
spontaneous examples and 43% (29) to pre-planned ones.
5.2.4 “Convey generality by “random” choice”
Choosing specific examples “at random” could be useful in conveying the idea of
generality (Zaslavsky, Harel and Manaster 2006). However, as Rowland et al. (2003)
indicate, it is critical to distinguish between cases in which it is appropriate to choose
examples at random and those that could be misleading or missing the point. This category
of considerations was manifested in both types of cases—some for which “random choice”
I. Zodik, O. Zaslavsky
was helpful and some for which such approach was not. It should be noted that these
choices were not done strictly at random, e.g. by tossing dice (as in Rowland et al. 2003).
Some were manifested as a quick selection of what first comes to mind from a large set of
possibilities. In other cases, teachers asked randomly selected students to provide the
specifics for an example.
Some teachers exhibited awareness of occasions when it is appropriate and valuable to
create examples by randomly selecting the specifics, namely, the numbers. This type of
consideration was associated solely with spontaneous examples, as the teacher made this
process transparent to the students. In many such cases, the teacher had in mind the range of
possible numbers, or according to Goldenberg and Mason (this issue) this could be
considered the range of permissible change. It should be noted, that teachers’ range of
permissible change included not only mathematical but also pedagogical constraints.
This kind of approach was observed in a seventh grade lesson in which the teacher [T5]
introduced
the notion of a rational number and wroteon the board its common definition:
Q ¼ Any number of the form mn ; m 2 Z; n 2 Z; n 6¼ 0 . She wanted to illustrate for many
different numbers how they satisfy the definition. Thus, she asked her students to suggest
various numbers within a certain range of possibilities. To ensure a wide range of
“different” numbers and at the same give a sense of randomness, she asked for several
decimal numbers between 0 and 1, and then she asked for several improper fractions, and
so on. By this she was able to show that most of the numbers that the students were familiar
with at that stage were rational numbers, and at the same time she illustrated the different
ways to represent them as a quotient of two integers.
Another case in which a teacher conveyed generality with the use of randomly selected
numbers from a range of possibilities can be seen in the following classroom event. A
teacher [T4] wanted to illustrate the theorem about the measure of an external angle of a
triangle (Fig. 5). For this he sketched a triangle on the board and first asked the students to
choose any vertex and an adjunct side; he then extended the chosen side beyond the chosen
vertex, pointing to the external angle that became obvious. This was to emphasize that
whatever they do for one external angle would equally apply for the others. After this stage
he asked them to suggest measurements for the two interior angles with the remaining two
vertices. Within the range of possible choices they suggested 42° and 73°. Now they could
calculate the measurement of the external angle and notice that it was the sum 42°+73°.
Note that the numbers 42 and 73 gave a sense of random numbers.
A
β
A
β
A
α
73o
C
B
C
B
42o
C
B
Step 1:
Step 2:
Step 3:
Draws a "general"
triangle
Marks an external angle
"at random"
Assigns measures
"at random"
Fig. 5 A teacher’s illustration of “random” choice of angles and measures
Characteristics of teachers’ choice of examples
A few random choices were not as helpful. For example, a teacher [T4] wanted to
illustrate how to apply the Viète formulae3 that deal with the sum and the product of the
roots of a quadratic equation in a ninth grade algebra course where the students had no prior
experience with or knowledge of complex numbers that are not real (Zaslavsky and Zodik
2007). In such a situation, we consider the following example, 2x2 þ 4x þ 5 ¼ 0 that was
based to some extent on a random choice of integer coefficients, to be a poor choice
(similar to the views expressed by Rowland et al. 2003), since at this point the students lack
the tools and knowledge to deal meaningfully with the square root of a negative number,
and all they can do, if at all, are meaningless manipulations in applying the formulae. In
Rowland’s term (this issue) this would not be considered a judicious choice for this
particular situation; it can be seen as a random choice of numbers, when carefully chosen
numbers are more appropriate (Rowland et al. 2003).
We identified 68 instances in this category, all of which were related to spontaneous
examples.
5.2.5 “Include uncommon cases”
By uncommon cases we mean either cases that are rather exceptional within mathematics or
cases which are under-represented in teaching mathematics. The former includes, for
example, the special statusp
offfiffiffi 1 and 0 as the onlypnumbers
that are invariant under integer
ffiffiffi
)ffiffiffi 1 ¼p1.
A
powers, that is, 02 ¼ 0 ) 0 ¼ 0; or 12 ¼ 1 p
ffiffiffi number of teachers pointed to
this property when explaining their choice of 0 and 1 when teaching the definition of a
square root (in seventh grade).
Other considerations of this type indicated awareness to non prototypical examples that
are often overlooked. Thus, one of the teachers [T3] explained why she chose to present a
concave kite when introducing the notion of a kite. She sketched on the board a kite that
looked concave and asked the students to check whether they could apply the definition
they had of a kite to this case.
We identified 40 instances in this category, of which 22% (9) were related to
spontaneous examples and 78% (31) to pre-planned ones.
5.2.6 “Keep unnecessary work to minimum”
This approach was expressed by utterances such as: “It’s a pity to have the class spend too
much time on the technical work instead of the essence” [T5]. This was manifested in the
choice of 17 in order to illustrate the period of a rational (decimal) number. The teacher
1
1
or 19
, because 17 had a long enough period, and it
explicitly said that she chose 17 instead of 17
1
1
or 19
that are much
was unnecessary to add the work needed for obtaining the periods of 17
longer.
Another manifestation of this approach can be seen in teachers’ decisions to highlight the
relevant part of an example and not go into extra details, e.g., an example of a problem was
presented to illustrate a general solution strategy without completing all the calculations.
We identified 20 instances in this category, of which 25% (5) were related to
spontaneous examples and 75% (15) to pre-planned ones.
3
According to the Viète formula, the roots x1 and x2 of a quadratic equation a x2 þ b x þ c ¼ 0 satisfy
the following conditions: x1 þ x2 ¼ ba ; x1 x2 ¼ ac ðx1 ; x2 2 CÞ.
I. Zodik, O. Zaslavsky
Altogether, 64% (302) considerations were associated with spontaneous examples and
36% (172) with pre-planned examples. This difference in itself does not say much, because
for many of the pre-planned examples we did not have access to the underlying
considerations that led the teachers to their selection or generation. However, within this
limitation, we can still observe some trends (Fig. 6). A look at the above six categories,
indicates that categories 2 and 4 are more frequent for spontaneous examples while
categories 5 and 6 are more common in pre-planned examples. With respect to category 2,
this difference may reflect the fact that teachers attend to students’ difficulties mainly as
they become aware of them in the course of the lesson, and not as much in advance. In
addition, the nature of demonstrating random choices (category 4) has to do with in-themoment acting. Considerations of types 5 and 6 require much more planning, in identifying
and searching for special cases (category 5) and analysing the amount of work needed for
each choice (category 6). Interestingly, choosing to begin with the simple or familiar
(category 1) is the leading type of consideration; it was the most frequent not only in its
total number of occurrences but also within the spontaneous examples (38% of 302) as well
as within the pre-planned examples (46% of 172).
The above considerations reflect a broad knowledge base on which teachers draw. Some
are closely connected to mathematical knowledge, e.g.: the logical aspects of a counterexample (Fig. 3); the degree of freedom or range of permissible change in constructing
triangles; some number properties such as the length of the period of a decimal fraction; the
conditions for which a quadratic equations has real roots. Other considerations draw a lot on
content pedagogical knowledge, e.g.: highlighting relevant features by “breaking a pattern”
and reducing the “noise” of an example; constructing gradual sequences of examples; using
explicit random choices of numbers to convey generality. Finally, there were many
considerations that reflected teachers’ tendency to take into account students’ ways of
thinking when selecting examples, which is a manifestation of their knowledge of and
sensitivity to students, e.g.: building examples that address students’ known errors and
overgeneralizations; moving from simple and familiar to more complicated and less familiar.
5.3 The dynamics of teachers’ choices of examples in the course of teaching
As mentioned earlier, we find Simon’s (1995) Mathematics Teaching Cycle helpful in
thinking about the different roles and stages of use of spontaneous and pre-planned
examples. It should be noted that in reality teachers’ considerations are not employed in
Start with a Attend to
simple or
students’
familiar case errors
N=474
Draw
Convey
Include Keep needless
attention to generality uncommon “work” to
minimum
relevant by “random” cases
choice
features
Fig. 6 Distribution of number of considerations between pre-planned examples and spontaneous examples
(in percentage)
Characteristics of teachers’ choice of examples
isolation. There is often a dynamic sequence of events, triggering different considerations
within a learning trajectory. As shown above, a lot of the actual choices teachers make,
within the guiding principles they employ, rely on their knowledge base, or in this context
on their accessible example spaces. Thus, for instance, when they run short of accessible
examples they need to generate new examples that eventually may become part of a richer
example space they hold (for an elaboration on the notion of example spaces and their
accessibility see Goldenberg and Mason, this issue).
Inspired by Simon’s (1995) teaching cycle, the diagram in Fig. 7 is an attempt to capture
the dynamics of mathematics teachers’ choice and generation of examples in the course of
their teaching, as reflected by our findings.
The teaching cycle depicted in Fig. 7 focuses on the locations of examples in the course of
teaching mathematics, with respect to teacher knowledge, the planning stage, and the actual
lesson. The interplay between these elements is indicated by various arrows. In terms of
teacher knowledge we see teachers’ example spaces as well as various resources (mainly
textbooks) as the main but not sole sources for teachers’ choices of examples. While
textbooks serve mostly in the planning stage, example spaces serve in both the planning and
the actual implementing stages. The actual choice is guided by the different types of
considerations teachers attend to, according to their personal inclinations and judgements.
Apparently, a major part of teachers’ planning of their lessons has to do with selecting or
generating examples. In the course of the lesson, classroom events often consist of teacher
moves and interactions with students (in addition to student interactions in which the teacher
is not involved). These events often raise the need for the teacher to act in-the-moment and
find an appropriate example to address the need that arises. In our study, in some cases this
Fig. 7 Mathematics examplerelated teaching cycle
Teacher Knowledge
Teacher Resources
EXAMPLE-SPACE
TEXTBOOKS
LESSON PLANNING
PRE-PLANNED
EXAMPLES
ACTUAL LESSON
Classroom Event
Teacher Moves
Interactions of and
with Students
In-the-Moment Acting
SPONTANEOUS EXAMPLES
I. Zodik, O. Zaslavsky
was done instantly, indicating the teacher’s drawing on his or her accessible example space,
while in several cases such a need called for a less immediate response. In the less immediate
responses, apparently teachers drew also on other sources of knowledge, including guiding
principles and rules of thumb which they assume. It seems that a rich and accessible example
space may support teacher in-the-moment choices and increase their flexibility.
The unexpected events in which a teacher is faced with a need for additional and different
examples could turn into a learning experience of the teacher, first in terms of developing
awareness to such needs, and then in terms of the actual examples or ways of producing
them. Thus, as we see in Simon’s mathematics teaching cycle, teachers learn through their
teaching. Figure 7 highlights the learning that occurs through example selection and
generation. Apparently, this kind of experience activates and enriches existing example spaces.
6 Discussion
The findings of this study provide insight into the kind of knowledge teachers craft in the
course of their teaching experience. As noted earlier, none of the teachers who participated
in the study had encountered any formal training that explicitly addressed mathematical
exemplification. The remarkable articulations of how they choose examples for their
teaching point to principled learning that occurs through practice.
We argue that teachers’ spontaneous examples rely on their accessible (mathematical and
instructional) example spaces that are to a large extent situated (Watson and Mason 2005,
p.76). We suggest distinguishing between an easily accessible example space (that includes
examples that have become immediate and automatic), and a more remotely accessible
example space that becomes accessible through analytical thinking and self monitoring. One
can also think of a teacher’s example space as including a collection of examples with labels
attached to them. A label refers to the class of cases which is represented by an example and
the context in which it would be useful to use. An expansion of an example space may mean
not only additional examples, but also additional labels to already existing examples. For
example, a teacher may think of the function f(x)=x3 as a good example of an odd function
and use it frequently when discussing with students the concept of an odd function.
However, if at a certain point the teacher realizes that this could also serve as a good
example of a function that has a tangent that intersects its graph, then this example will get
another label and could become accessible for wider range of contexts and purposes.
It should be noted that teachers’ crafted principles concur with common approaches to
teaching in particular, the need to attend to students’ difficulties is a common principle.
However, how to do so through examples is crafted over years of experience. Teachers also
addressed the issue of variation, using their own wording, in order to draw attention to
relevant features; their realization is closely related to aspects that Watson and Mason
(2006) discuss.
The findings point to the special role and nature of randomly selected examples. On the
one hand, teachers recognize the strength of randomness in conveying generality, as
manifested also in Zaslavsky et al. (2006). On the other hand, there are cases for which
randomly selected specifics miss the point or even mis-lead (Rowland et al. 2003).
We identified two main reasons for teachers’ need to create examples spontaneously ‘on
their feet’: (1) as a response to students’ utterances, such as a falsifiable claim; (2) as a
response to their own recognition of certain limitations of a pre-planned example in the
course of the lesson (e.g., the teacher may realize that broader range of examples of the
same kind are needed).
Characteristics of teachers’ choice of examples
For instance, all the counter-examples that we observed were spontaneously constructed
by the teacher in response to a student’s unexpected invalid conjecture/statement. Thus, all
the cases in which a teacher dealt with a counter-example were spontaneous, as a result of a
classroom situation that called for it. Interestingly, in all the classroom observations we
conducted, there was not one instance in which a teacher pre-planned to deliberately
incorporate a counter-example in the lesson.
The numerous episodes we observed form a rich source of cases that we are beginning to
adapt for teacher education programs, both pre-service and in-service. This can be useful in
providing systematic learning opportunities for teachers in order to facilitate both practical
and theoretical knowledge of treatment of instructional examples. Along this line, it is
recommended that teachers encounter experiences in generating examples spontaneously
for (real or hypothetical) classroom situations and reflect on them. Our experience with inservice and pre-service teachers indicates that such encounters provide rich and powerful
learning opportunities that lead to teachers’ deeper understanding of mathematics (e.g., of
certain mathematical concepts), an expansion of their personal example spaces, and their
awareness to different aspects of creating and choosing examples in mathematics.
Our study raises the need for further research. One direction would be exploring how the
findings of teachers’ example-related practice could be used for teacher education programs.
How can craft knowledge of experienced teachers be disseminated to prospective teachers?
Our study was conducted in rather traditional teacher-centered classrooms. It would be
interesting to examine teachers’ and students’ choices and use of examples in other types of
classrooms (e.g., explorative, students-centered).
It seems worthwhile to examine more closely the mechanism by which example spaces
expand and how examples are encoded in teachers’ example spaces. What makes an
example more accessible for a particular context?
The rich collection of observations that include numerous example-related classroom
events could be turned into simulations of practice based cases (in the spirit of Stein, Smith,
Henningsen, & Silver 2000) and serve for teacher education and professional development
activities.
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