chapter 8 the explanatory power of examples in mathematics: … · 2015-07-14 · 8 the explanatory...

Chapter 8The Explanatory Power of Examples inMathematics: Challenges for Teaching

Orit Zaslavsky

The generation or selection of examples is a fundamental partof constructing a good explanation. . . For learning to occur,several examples are needed, not just one; the examples need toencapsulate a range of critical features; and examples need tobe unpacked, with the features that make them an exampleclearly identified.

(Leinhardt, 2001, p. 347)

Instructional Examples in Mathematics Learning and Teaching

Instructional examples are fundamental elements of an explanation, as described byLeinhardt, Zaslavsky, and Stein (1990):

Explanations consist of the orchestrations of demonstrations, analogical representations,and examples. [. . .]. A primary feature of explanations is the use of well-constructed exam-ples, examples that make the point but limit the generalization, examples that are balancedby non- or counter-cases (ibid., p. 6).

I use the term “instructional example,” to refer to an example offered by a teacherwithin the context of learning a particular topic. The important role of instructionalexamples in learning mathematics stems firstly from the central role that exam-ples play in mathematics and mathematical thinking. Examples are an integral partof mathematics and a significant element of expert knowledge (Rissland, 1978).In particular, examples are essential for generalization, abstraction, and analogicalreasoning. Furthermore, from a teaching perspective, there are several pedagogi-cal aspects of the use of instructional examples that highlight the significance andconvey the complexity of this central element of teaching.

O. Zaslavsky (B)Department of Education in Technology & Science, Technion – Israel Institute of Technology,Haifa, Israel; Department of Teaching and Learning, Steinhardt School of Culture, Education andHuman Development, New York Universitye-mail: [email protected]; [email protected]

107M.K. Stein, L. Kucan (eds.), Instructional Explanations in the Disciplines,DOI 10.1007/978-1-4419-0594-9_8, C© Springer Science+Business Media, LLC 2010

108 O. Zaslavsky

According to Watson and Mason (2002) an example is any particular case ofa larger class (idea, concept, technique, etc.), from which students can reason andgeneralize. By and large, an example must be examined in context. Any examplecarries some critical attributes that are intended to be exemplified and others thatare irrelevant. The irrelevant features are what Skemp (1971) considers the noise ofan example. A teacher must be aware that students may not see through an examplewhat it stands for, or what general case it represents, and may be attracted to its“noise.” As Rissland (1991) maintains “one can view an example as a set of facts orfeatures viewed through a certain lens (ibid, p. 190).”

Bills, Dreyfus, Mason, Tsamir, Watson and Zaslavsky (2006) suggests twomain attributes to make an example pedagogically useful. Accordingly, an exampleshould be “transparent” to the learner, that is, make it relatively easy to direct theattention of the target audience to the features that make it exemplary. This notion oftransparency is consistent with Mason & Pimm’s (1984) notion of generic examplesthat are transparent to the general case, allowing one to see the general through theparticular, and with Peled and Zaslavsky (1997) who discuss the explanatory natureof examples.

A “good” instructional example should also foster generalization, that is, itshould highlight the necessary features of an example of the illustrated case and atthe same time point to the arbitrary and changeable features. Examples with someor all of these qualities have the potential to serve as a reference or model example(Rissland, 1978), with which one can reason in other related situations, and can behelpful in clarifying and resolving mathematical subtleties.

Clearly, the extent to which an example is transparent or useful, the way it isinterpreted, and the features that one notices are subjective and context related. Forinstance, in order to exemplify “a function that has a value of −2 when x=3” onecan bring a trivial example such as f(x)=−2 (Hazzan & Zazkis, 1999). Althoughthis example satisfies the required condition, it may be regarded too simple or toonarrow, in the sense that it does not convey the wide range of examples of sucha function, including its mathematical complexity. Thus, while many objects maybe used as an example, it is clear that from a pedagogical perspective some havemore explanatory power than others (Peled & Zaslavsky, 1997), either because theyhighlight the special characteristic of the object or because they show how to buildmany other examples of the focal idea, concept, principle, or procedure.

Peled and Zaslavsky (1997) differentiate three types of examples used by math-ematics teachers, according to their explanatory power: specific, semi-general, andgeneral examples. They maintain that general examples offer explanation and pro-vide insight about a certain phenomenon as well as ideas about how to generatemore examples of this phenomenon. For example, a general example of a pair ofdistinct rectangles with the same diagonal could be the one shown in Fig. 8.1, whilein contrast, Fig. 8.2 provides a sketch of a specific example.

One can easily notice the difference between these two types, in terms of theirgenerality and explanatory power. The first has a stronger explanatory power. As

8 The Explanatory Power of Examples in Mathematics 109

Fig. 8.1 A “general” example of a pair of distinct rectangles with the same diagonal

11

3

9

7

Fig. 8.2 A “specific”example of a pair of distinctrectangles with the samediagonal

shown later, there are instructional situations in which a specific example is moreappropriate than a general one. Interestingly, constructing a specific instructionalexample may be more complicated and demanding than constructing a more generalexample (see Case 5 below).

Another important aspect of the use of examples is the representation of theexample. To illustrate this issue, consider the following examples of a quadraticfunction:

(i) y = (x + 1) (x − 3) ; (ii) y = (x − 1)2 − 4; (iii) y = x2 − 2x − 3.

These are three different representations of the same function. Each example istransparent to some features of the function and opaque with respect to others. Forexample, the first example conveys the roots of the function (−1 and 3); the secondcommunicates straightforwardly the vertex of the parabola (1, −4); and the thirdexample transmits the y-intercept (0, −3). However, these links are not likely to beobvious to the student without some guidance of the teacher. Moreover, it is not clearthat students will consider all three examples as examples of a quadratic function;for instance, in example (i) it is less obvious that there is an exponent of a powerof two; thus, it may not be seen as a quadratic function. A teacher may choose to

110 O. Zaslavsky

deal with only one of the above representations or s/he may use the three differentrepresentations in order to exemplify how algebraic manipulations lead from oneto another, or in order to deal with the notion of equivalent expressions. What astudent will see in each example separately and in the three as a whole dependson the context and classroom activities surrounding these examples. A student whoappreciates the special information entailed in each representation may use theseexamples as reference examples in similar situations, e.g., for investigating otherquadratic functions.

In terms of irrelevant features, although commonly used, in the above exam-ples of quadratic functions, it is irrelevant what symbols we use, i.e., we couldchange x to t and y to f(t). Yet, a student may regard x and y as mandatory sym-bols for representing a quadratic function. Another irrelevant feature is the factthat in all three representations all the numbers are integers. A student may con-sider this a relevant feature, unless s/he is exposed to a richer “example space”(Zaslavsky & Peled, 1996; Watson & Mason, 2002). Mason and Pimm (1984)warn about a mismatch that often occurs between the teacher’s intention and stu-dents’ interpretations. Thus, an example that is meant to demonstrate a general caseor principle may be perceived by the learners as a specific instance, ignoring itsgenerality.

In addition, one may generalize and think that for any quadratic function all threerepresentations exist, while the first one depends on whether the specific quadraticfunction has real roots. Hence, the specific elements and representation of an exam-ple or set of examples, and the respective focus of attention facilitated by the teacher,have bearing on what students notice, and consequently, on their mathematicalunderstanding. Thus, the role of the teacher is to offer learning opportunities thatinvolve a large enough variety of “useful examples” to address the diverse needsand characteristics of the learners.

It follows that the use of examples is a significant and complex terrain.Apparently, teachers are not used to articulate their considerations, not to mentionsharing and debating surrounding the issue of exemplification.

In spite of the critical roles examples play in learning and teaching mathematics,there is only a small number of studies focusing on teachers’ choice and treatment ofexamples. Rowland, Thwaites, and Huckstep (2003) identify three types of elemen-tary teachers’ poor choice of examples: choices of instances that obscure the roleof variables, choices of numbers that are used to illustrate a certain arithmetic pro-cedure when another procedure would be more sensible to perform for the selectednumbers, and randomly generated examples when careful choices should be made.

Rowland et al.’s (2003) findings concur with the concerns raised by Ball, Bass,Sleep, and Thames (2005) regarding the knowledge base teachers need in order tocarefully select appropriate examples that are useful for highlighting salient mathe-matical issues. Not surprisingly, the choice of examples in secondary mathematicsis far more complex and involves a wider range of considerations (Zaslavsky andLavie, 2005; Zodik and Zaslavsky, 2007, 2008).


In this chapter, I offer ways of examining instructional examples in mathe-matics from two perspectives: (1) their explanatory power and (2) the demandsthey present on teachers. I am aware that the explanatory power of an exampleis “in the eyes of the beholder,” and as discussed above, one cannot automati-cally assume that the teacher’s intention in offering a particular example will beperceived as expected. This applies for any example. However, I maintain thatthere is great value in analyzing examples in terms of their potential explanatorypower.

As to the demand on teachers, as illustrated below, generating an appropriateexample for a given purpose is often an art, or a problem-solving process. I use anumber of cases to unpack and highlight these two aspects of instructional examplesin mathematics, namely, their explanatory power and the challenge of coming upwith appropriate ones. The cases I discuss address main themes in teaching math-ematics, all related to explanations: (1) conveying generality and invariance; (2)explaining and justifying notations and conventions; (3) resolving uncertainty (orestablishing the status of (pupils’) conjectures or assertions); and (4) connectingmathematical concepts to real-life experiences. In addition, I examine the chal-lenges of example generation, with a focus on unexpected difficulties in generatingan instructional example with certain constraints. The issue of the “correctness” ofan example is also discussed.

All the cases presented in this chapter evolved from actual observations ofclassroom situations or carefully designed workshops with experienced and highlyreputable secondary mathematics teachers or with prominent mathematics educa-tors (i.e., researchers in mathematics education who teach in teacher educationprograms). The workshops drew on real classroom events (some described in theliterature and some from my own work on examples, as well as the work of Zodikand Zaslavsky, 2008). The work with teachers involved ongoing reflective accountsthat often included an iterative process of designing – analyzing – re-designingexperiences that engaged participants in dealing with instructional examples inmathematics and probing for their thinking and guiding principles (similar to theprocesses described in Zaslavsky, 2008). My overarching claim is that there is muchmore to examples than meets the eye. It is a complex and fascinating domain toexplore. I see a great challenge providing teachers with experiences that preparethem for such demands.

As mentioned above, the chapter is organized around illustrative cases that tosome extent share the features of the cases in Stein, Smith, Henningsen, and Silver(2000). These cases serve as “meta-examples”; they may not all be generic meta-examples, but they are at least existential examples. They reflect genuine practice,thinking, and concerns surrounding instructional examples. My intention is that thereader will be able to see through and beyond them to more general issues, ignoringthe “noise” attached to them. To assist in capturing the issues these cases are meantto convey, I offer my own lens by addressing, for each case, the question: “what isthis case an example of?”.

112 O. Zaslavsky

Cases Illustrating the Challenges Entailed in Teacher’s Choiceand Use of Instructional Examples in Mathematics

Conveying Generality and Invariance with Examples(Case 1)

Zaslavsky, Harel, & Manaster (2006) describe an eighth-grade teacher, who chose aset of examples that build on students’ knowledge of how to calculate the area of arectangle and lead them to a way of calculating the area of a triangle.

According to their report (ibid), the teacher began the lesson by putting on theboard three examples (Fig. 8.3), in order to move from a rectangle and its area cal-culation – already familiar to her students – to a right-angled triangle that is clearlyhalf of the rectangle, to a seemingly more general triangle. She chose to keep thegiven measurements constant. This allowed a better focus on the varying elements,e.g., the type of figure, the connection between a side and its corresponding height.

3

6

3

6

3

6

Fig. 8.3 A teacher’s initialset of examples

Then, with the “help” of some students who she invited to the board, she movedfrom one case to the next, adding auxiliary segments (the dashed segments inFig. 8.4) and building on the previous one.

3

6½(3 × 6) = 9

3

6

3

63 × 6 = 18

Fig. 8.4 Building gradually from one example to its subsequent

In the third case – the more general triangle – it is not obvious how to calcu-late its area and how to build on the previous example. The teacher pointed to thetwo right-angled triangles into which the height divides the triangle, and helped thestudents notice that embedded in the drawing are two cases that are just like theprevious case. She repeated the method of “completing” a right-angled triangle upto a rectangle that is twice the area of the triangle. In order to continue in this direc-tion and calculate the area of each sub-rectangle, as done in the middle example, itis necessary to know the lengths of the sides of each one. Thus, it appears that inthe third example (see Fig. 8.4) the lengths of the sides of each sub-rectangle aremissing.


The teacher turned to the students and asked them for suggestions how to “splitthe 6 up”. They were expected to choose two measurements that added up to 6in order to fill what seemed missing in the example. They chose to “split” it into“2 and 4”, and once these measurements were determined, the calculation becamestraightforward.

Figure 8.5 depicts the stages the class went through to calculate the area of thetriangle. It clearly depended on the choice of 2 and 4.

3

6½ (3 × 2) = 3 ½ (3 × 4) = 6

3 + 6 = 9

2 4

Fig. 8.5 Deciding at randomhow to “split” the side oflength 6

What Is Case 1 an Example of?

Case 1 is an example of a sequence of examples for which one provides an explana-tory basis for the next. It leads from a specific case of a right-angled triangle to amore general case of a triangle that is not necessarily a right-angled triangle.

It is also an example of an attempt to convey generality by a random choice ofthe specifics of an example. The choice of 2 and 4 as the measurements of the twoparts of the side of the triangle was actually arbitrary. The students obviously reliedon the drawing and estimated that the left part is shorter than half the side of length6 and the right part is longer than its half. Tending toward whole numbers, theynaturally picked 2 and 4. Therefore, in this sense, the choice was not done totallyat random. However, this approach could have been reinforced to reach a moresound generalization and a better sense of one of the “big ideas” in mathematics –invariance.

Constructing a set of examples by controlling variation and keeping a core ofelements constant may be helpful in moving from one case to another and allowingfocus on those that change. Along this line, a more powerful set of examples wouldbe one that better deals with the general case of a triangle (even if still restricted toacute-angled triangles, it could set the grounds for dealing in a similar way with aset of obtuse-angled triangles, where instead of adding two area measurement youneed to subtract one from the other).

This would be equivalent to asking the students to suggest alternative ways tosplit the (6-unit length) side of the triangle. By actually repeating the same reasoningand calculation procedure as in Fig. 8.5 for each case in Fig. 8.6 (as suggested byHarel, 2008) the generality would probably be made more transparent, and the sensethat the area of the triangle is invariant under change of location of its vertex along a

114 O. Zaslavsky

Fig. 8.6 A range of instances that captures the invariance of area of a triangle

line parallel to its side could be developed. It would also allow connection betweenthe distributive law and this invariance.

To convey more of the complex web of considerations that a teacher needsto make in choosing or generating explanatory instructional examples, note thatalthough a random choice of specifics of an example could be powerful in manycases, randomness has its limitations. In Case 1, the choice of lengths of 3 and 6is not as “generic” as, for instance, 8 and 5. In the procedure that is illustrated inFig. 8.5, a student is more likely to attend to irrelevant features of the specifics (asin Mason and Pimm, 1984); this case may lead some students to over-generalize themere coincidence that the area of one triangle is 3 as is the length of the height, andthe area of the second triangle if 6 as is the length of the side. When a large varietyof such examples are encountered, this kind of over-generalization is less likely tooccur.

This case can be regarded to a certain extent as a worked example, whichLeinhardt (2001) considers key features in virtually any instructional explanation.

Another word of caution with respect to random choice of instructional examplesstems from the studies of Zodik and Zaslavsky (2007) and Rowland et al. (2003). Intheir work, they identify several cases where a random choice impedes the purposeof the example and limits its explanatory power. There are many cases in which acareful choice of examples is needed. Case 2 that follows indicates how subtle thechoice may become.

Explaining and Justifying Notations and Conventions (Case 2)

In school mathematics, we introduce students to several mathematical conventionsand notations. Many seem quite arbitrary and convey a rigid conception of the dis-cipline of mathematics. In many cases, there is a reason for such conventions that isnot always obvious to the students. The “big idea” of an agreed upon notation is acommunicative one: we want to avoid ambiguity and make sure that when using acertain notation it is well defined and it clearly indicates to what it refers. Thus, it isimportant to find examples for which this really matters.

For example, the common notation of a polygon requires listing its vertexes con-sistently either clockwise or counterclockwise. This requirement offers a degreeof freedom, yet has its restrictions. Why do we insist on this? Why not name itby listing its vertexes in any order that suits us? A group of mathematics educa-tors examined this question for quadrangles. Very soon they realized that a random


Four vertexes of aquadrangle

Quadrangle no. 1.1 Shape no. 1.2 Shape no. 1.3

A

C

D

B

A

C

D

B

A

C

D

B

A

C

D

B

A→B→C→D→A A→B→D→C→A A→C→B→D→A

Fig. 8.7 A specific choice of 4 points that determine a unique quadrangle

choice of four vertexes will not be equally helpful in explaining this convention.They wanted to find an example for which unless we follow this convention it willnot be clear to which quadrangle we refer.

Figure 8.7 is an example of a set of four specific points, A, B, C, and D, thatdetermine one and only one quadrangle with these vertexes (Quadrangle 1.1). Notethat the order in which the vertexes are listed determines how they are connected.

In this case it really does not matter how we “name” it. There are eight acceptableways to name a quadrangle with these vertexes: clockwise – ABCD, BCDA, CDAB,and DABC; or counterclockwise – ADCB, DCBA, CBAD, and BADC; however,for this particular example, violating the clockwise–counterclockwise conventiondoes not create ambiguity regarding the designated quadrangle. For instance, ABDCviolates the convention but its corresponding shape (no. 1.2 in Figure 8.7) violatesthe definition of a quadrangle. Thus, ABDC does not designate a quadrangle, so noambiguity is caused.

ABDC corresponds to Shape 1.2, while ACBD corresponds to Shape 1.3. Bothdo not satisfy the definition of a quadrangle, thus raising the question: so why fuss?If we restrict ourselves to notations of quadrangles, non-quadrangles do not count,and so Quadrangle 1.1 could be denoted by its four vertexes regardless of the orderin which they are listed.

After much contemplation the group came up with another set of four points asin Fig. 8.8, which determines three different quadrangles; thus, they considered it arather good explanatory example, since here the “name” must uniquely correspond

A C

D

B

A C

D

B

A

C

D

B

A C

D

B

Four vertexes of a quadrangle

Quadrangle no. 2.1 Quadrangle no. 2.2 Quadrangle no. 2.3

A→B→D→C→A A→C→B→D→A A→B→C→D→A

Fig. 8.8 A specific choice of 4 points that determine three different quadrangles that seemcongruent

116 O. Zaslavsky

to one of the three quadrangles. Thus, thanks to the agreed-upon convention, ABCDrefers only to Quadrangle no. 2.3, ABDC refers to Quadrangle 2.1 and ACBD toQuadrangle 2.2.

When presenting this example at an in-service workshop with secondary mathe-matics teachers, most of the teachers were overwhelmed – they had never managedto convince their students of the necessity of this convention – and instantly feltthey now had a tool with which to explain and convince (themselves as well as theirstudents). However, one teacher claimed that this was not a convincing example.She argued that this particular choice of points is symmetrical; therefore, all threequadrangles are congruent, so in a sense there is just one quadrangle. Thus, for herthe example in Fig. 8.8 was not the most effective in convincing why the conventionis necessary.

A C

D

B

A C

D

B

A C

D

B

A C

D

B

Quadrangle no. 3.1

A→B→D→C→A A→C→B→D→A A→B→C→D→A

Quadrangle no. 3.2 Quadrangle no. 3.3 Four vertexes of aquadrangle

Fig. 8.9 A specific choice of 4 points that determine three distinct quadrangles

This argument led to a generation of an example that was agreed to be of astronger explanatory power (Fig. 8.9): it is a case of four points that determine threedifferent non-congruent quadrangles. Thus, unless we strictly follow the convention,it will not be possible to avoid ambiguity with respect to the specific quadran-gle in question. For instance, without the convention, how would we know whichQuadrangle we mean by ABCD?

Thanks to the well-defined convention, Quadrangle 3.1 is ABDC, Quadrangle 3.2is ACBD, and Quadrangle 3.3 is ABCD. There is still a degree of freedom for eachnotation, and an alternative notation can be used provided the order is maintained(e.g., Quadrangle 3.1 may also be denoted as BCDA or CDBA, but not as ADCB).


Case 2 is an example of the potential power of examples to justify mathematicalconventions. Its explanatory power rests on its potential in convincing that withoutthe convention ambiguity may arise and impede (mathematical) communication.

It also illustrates the subtleties and iterative nature of a judicious choice of aninstructional example that is neither commonly found in textbooks nor addressed inteacher education settings. This process involves problematizing the situation, set-ting an explicit goal that addresses this problem, and checking each example in light


of this goal. The goal reflects a sound understanding of big ideas in mathematics.Achieving it involves thinking “out of the box” – in this case, moving from rathercommon convex quadrangles to concave.

Case 2 is also a manifestation of a necessity-based approach to learning math-ematics (Harel, 2008). Even if this particular convention is not a central one(compared, for instance, to the order of executing arithmetic operations), it con-veys a desired “explanation-based” mindset that drives a teacher to constantly dealwith the natural question of “why?”. As reflected in Case 2, such explanations oftenrely on convincing examples.

Establishing the Status of Pupils’ Conjectures and Assertions(Case 3)

Bishop (1976) begins his paper with a classroom event, which he experienced asa teacher and invites the reader to think of how s/he would deal with it. It goes asfollows:

Teacher: Give me a fraction which lies between 1

2 and

3

4

Pupil; 2

3

Teacher: How do you know that 2

3 lies between

1

2 and

3

4?

Pupil: Because the 2 is between the 1 and the 3, and the 3 is between the 2 and the 4

How would you deal with that response? Bishop, 1976, p. 41

A group of secondary experienced and highly reputable mathematics teachersdiscussed this case. None of them had the prior knowledge regarding the validity ofthe pupil’s claim, although their initial gut feeling was that it could not be true forall cases; thus it is not a valid argument.

They began by examining several examples. Note that generating an example inthis context requires a choice of a pair of fractions, and a choice of another fractionfor which two properties are checked: Does it lie between the two fractions? Is itsnominator between the nominators of the two fractions? Is its denominator betweenthe denominators of the two fractions?

All in all, the teachers examined 12 examples until they reached a warrantedconsensus:

118 O. Zaslavsky

(1) 3/4 as a fraction that lies between 2/3 and 4/5, and indeed the 3 is between the 2and the 4, and the 4 is between the 3 and the 5; thus this is a supporting example(i.e., an example that satisfies the pupil’s claim).

(2) 2/3 as a fraction that lies between 1/2 and 4/5, and indeed the 2 is between the 1and the 4, and the 3 is between the 2 and the 5; thus, this is another supportingexample.

(3) 3/4 as a fraction that lies between 1/2 and 4/5, and indeed the 3 is between the 1and the 4, and the 4 is between the 2 and the 5; strangely – another supportingexample.

These examples reinforced a sense that the pupil may be right. One of the teachersexpressed this feeling in the following words: “If you can’t find a counter-exampleeasily the claim is probably right.” However, they continued checking and lookingfor additional supporting or refuting examples. The next three were as follows:

(4) 2/4 is a fraction that does not lie between 1/2 and 4/5, although the 2 is betweenthe 1 and the 4 and the 4 is between the 2 and the 5. This example seemed tocontradict the pupil’s claim; however, it was treated as a (degenerate) specialcase; since 2/4 = 1/2, it did not seem to violate the general assertion.

(5) 3/3 is a fraction that does not lie between1/2 and 4/5, although the 3 is betweenthe 1 and the 4, and the 3 is between the 2 and the 5. However, for similarreasons this was treated as another special case; since 3/3 = 1, it did not seemto violate the general assertion.

(6) 3/4 is a fraction that does not lie between 1/3 and 5/10, although the 3 is betweenthe 1 and the 5, and the 4 is between the 3 and the 10. However, this againappeared as a special case; since 5/10 = 1/2, it did not seem to violate thegeneral assertion.

They now decided to approach the problem more systematically. They articulatedthe implied pupil’s claim to say: “Given two fractions, a/b and c/d, the fraction k/nlies between them if the k is between the a and the c, and the n is between the band the d.” Thus, the teachers chose two fractions – one for a/b and the other forc/d, kept them fixed, and began checking and listing some examples of fractions thatwork and some that do not, as in Fig. 8.10.

At this point, the entire group was convinced that the pupil’s assertion did nothold for all fractions. They were so preoccupied with figuring out for themselvesthe status of this assertion that they did not attend to the original question that wasposed to them, namely, “How would you deal with that response?” Bishop (1976)describes his way of dealing with this response as “buying time,” until he managedto think on his feet and come up with a counter-example.


Case 3 is an example of a classroom situation that calls for in-the-moment deci-sion. It is a case where the teacher is uncertain regarding the validity of a student’s


The fixed fractions were chosen: 12

and 57

Examples that support the pupil's

claim Examples that violate the pupil's claim

23

, because 1 2 52 3 7

< <

35

, because 1 3 52 5 7

< <

46

, because 1 4 52 6 7< <

25

, because it does not lie between 12

and 57

34


and 57

45


and 57

Fig. 8.10 A systematic sequence of examples for a fixed pair of fractions

assertion, thus raising a genuine need to generate various examples in search forevidence and conviction. It shows how the process of moving from a sense that theconjecture is true to a conviction that it is false depends on the specific examplesunder investigation. This case is also a not so common case – for which there aremany supporting examples as well as many counter-examples. In a way, this casereflects a typical intellectual need for example-based reasoning (Rissland, 1991).It is also an example of a sort of Lakatos (1976) style dialog involving “monsterbarring”; some of the examples that violated the pupil’s assertion were treated asextreme or special cases that do not count. It follows that Case 3 is also an exampleof how the process of example generation may serve to resolve teachers’ uncertaintywith respect to whether a conjecture is true or not. A similar situation could be rathereasily orchestrated in a real classroom.

Case 3 highlights the challenge and demands that teachers face with respectto choice of and inference from examples as well as the significance of teachers’subject matter knowledge in being able to act in the moment and come up withappropriate examples (Mason & Spence, 1999). Actually, as explained below, thepupil’s assertion would be valid if he meant the nominator lies exactly in the mid-dle between the two nominators, and the denominator lies exactly in the middlebetween the two denominators. This knowledge and understanding on the part of theteacher would change dramatically his or her ability to choose appropriate exam-ples. Moreover, as illustrated below, the analysis of case 3 suggests a significantinterplay between examples and (visual) representations, and the explanatory powerof a visual representation of a particular example.

Bishop’s pupil’s assertion relates to the mediant property of fractions: The medi-ant of two fractions a/b and c/d (for which a, b, c, d are positive integers) is(a+c)/(b+d). That is to say, the numerator and denominator of the mediant are

120 O. Zaslavsky

(a)2

1

4 6 10

1

5

6

3

5O x

y

(b)

O

1

3

4

2

2 4 63

y

x

4+

⎫⎪⎪⎬⎪⎪⎭

14

56

610

35

12

34

23

46

3+

Fig. 8.11 Visual representations of the pupil’s response that convey an explanatory feature to thegeneralizable elements of his method

the sums of the numerators and denominators of the given fractions, respectively.Interestingly, as illustrated in Fig. 8.11, if a/b<c/d then a/b<(a+c)/(b+d)<c/d. Thisis a valid way to construct a fraction that lies between two given fractions – atask that the teacher posed to the pupils without realizing this connection. In fact,the pupil’s assertion is valid if you restrict it to the (arithmetic) mean, and can

be formulated as follows: if a/b<c/d then ab <

a+c2

b+d2

< cd . Knowledge of this

property of fractions would help dealing with the classroom event and comingup with examples that shed light on the affordances and limitations of the pupil’sgeneralization.

One of the main properties of a mean (of any kind) is that it is an intermediatevalue. So this provides an explanation why the pupil’s strategy worked. He took aspecial intermediate value: the (arithmetic) mean.

As Arcavi (2003) suggests, we can represent the fraction 1/2 by the point (2,1)in a Cartesian coordinate system. Thus, the fraction 1/2 is the slope of the line thatconnects the origin O with the point (2,1). Similarly, the fraction 3/4 is the slope ofthe line that connects the origin O with the point (4,3). Actually, in this representa-tion all equivalent fractions lie on the same line and correspond to the same slope.For example, 1/2 and 2/4 are on the same line. If you continue the line with slope1/2 you will see that it passes through the point (4,2). The slope corresponds to theangle between the line and the x-axis; the larger the fraction, the larger the angle andthe line’s slope.

This representation has an explanatory power for why the pupil’s suggestionworks if you construct a new fraction from the two given ones by taking the means

of the nominators and denominators, respectively; indeed, 23 = 1+3

22+4

2lies between

1/2 and 3/4. This method will always work. The line of slope 2/3 is the diagonal ofthe parallelogram, the vertexes of which are the origin O, (2,1), (4,3), (6,4). (Notethat (6,4) is (2+4,1+3); this relationship guarantees that it is the 4th vertex of theparallelogram determined by the origin and the two given fractions).


Connecting Mathematical Concepts to Real-Life Experiences(Case 4)

This case is a fair description of an actual classroom event that was a trigger for aworkshop with a group of prominent secondary mathematics teachers. It is based onthe work of Zaslavsky and Lavie (2005).

An eighth-grade teacher whom we observed in a lesson that aimed at introducingthe notion of the slope of a (linear) function in its qualitative sense rather than as aspecific measure decided to draw on students’ real-world experiences. For this, shechose the mountain metaphor, and sketched the following example of two mountainsM1 and M2 (Fig. 8.12):

M1

M2

Fig. 8.12 The initial example for introducing the notion of slop – two mountains with differentheights

Her intention was to draw students’ attention to the differences between the twomountains, by focusing on their relative difference in terms of their “steepness.” Infact, all the students agreed that mountain M1 was “steeper” than M2. However, oneof them gave his reason for this assertion, by explaining that M1 is higher than M2

(this was a manifestation of a well-known (mis)-conception of students, confusingheight for slope (Leinhardt et al., 1990), of which apparently the teacher was notaware. However, as a response to the student’s claim, she immediately “corrected”the example and drew a different one (Fig. 8.13), highlighting that they now havethe same height, yet M1 is steeper than M2.

M2 M1Fig. 8.13 A modifiedexample for introducing thenotion of slop – twomountains with the sameheights

She went further and drew two “steps”, for which the horizontal sides were equalin length, in order to give the students a measurable way to look at and compare therelative degrees of “steepness,” as illustrated in Fig. 8.14.

She then went back to deal with linear functions and their graphical representa-tions and used a similar drawing to highlight the visual aspects of slope (Fig. 8.15),that is, the ratio between the “rise” and the “run.” Without defining slope, they wereable to discuss the relative “steepness” of the two graphs, by comparing the “rise”for a fixed “run.”

122 O. Zaslavsky

M2 M1Fig. 8.14 Adding “steps” forcomparing the relativesteepness of the twomountains

Fig. 8.15 Connecting themeasure of relative steepnessof the mountains to linearfunctions – keeping the “run”fixed and examining the“rise”


This case illustrates that “exampleness” is in the eyes of the beholder. It beginswith a manifestation of the gap that may occur between a teacher’s intentions andwhat students actually notice. Teachers are not always aware of such discrepancies,mainly because the focal location of attention is not always explicitly expressed.In this case, the student’s explanation revealed what he was attending to and theinterpretation he attributed to it.

Case 4 also demonstrates a learning opportunity for the teacher. The student’sreaction drew her attention to the limitation of her original example (Fig. 8.10) andled her to improve it (Fig. 8.13). Moreover, the state of awareness led the teacher tofurther considerations that are reflected in the additional visual aid that she added(Fig. 8.14), in anticipation that this would help her students focus on the relevantfeatures of the example, which reflect the main idea that she was trying to high-light through it. In an interview that followed, it appeared that this classroom eventwould affect the teacher’s future choice of examples. Thus, this episode can alsobe seen as a glimpse into the way knowledge of and about instructional exam-ples in mathematics is crafted in the course of teacher practice (Leinhardt, 1990;Kennedy, 2002).

This classroom event is also an example of a case that elicited a rich discus-sion among teachers who considered the merits and limitations of such “real life”example for their own classrooms. The discussion focused on the idea of usingmountains as examples to set the grounds for learning about the slope of a linearfunction.

Although there was a consensus, that learning mathematics should relate to stu-dents’ informal knowledge and out-of-school experiences, some teachers objected


to the use of mountains in this context. They argued that the graph representing themountain (which resembles a parabola) does not have a constant slope, so com-paring the degree of steepness of the two mountains is not straightforward. Theseteachers felt it would be misleading to connect the notion of slope of a linear func-tion that is constant for any point on its graph to the steepness of a mountain. Theysuggested replacing the metaphor of mountains by cable cars or pyramids. In short,constructing instructional examples that connect to familiar context and map well tothe mathematical concepts they are supposed to illustrate relies on a web of complexand often competing considerations.

The Challenge of Constructing Examples with Given Constraints(Case 5)

In their study on counter-examples, Peled and Zaslavsky (1997) analyzed examplesthat mathematics teachers gave of two non-congruent rectangles that have diagonalsof equal length. Looking back at the data from that study, it is striking that thespecific examples that were proposed were not the kind one would expect as anexample of a rectangle. While it is likely that a specific example of a rectanglewould have its two sides’ measurements, e.g., in Fig. 8.16, what teachers proposedwere examples of pairs of rectangles with measurements of one side and a diagonal,as in Fig. 8.17.

3

8

4

9

Fig. 8.16 Typical specific examples of rectangles

2

8

4

8

Fig. 8.17 An example of two non-congruent rectangles with diagonals of equal length

This can partially be attributed to the request to focus on the equal-length diag-onals. However, there is more to it. In order to better understand this phenomenon,the following task was given to a group of highly prominent mathematics educators:

124 O. Zaslavsky

Suppose you wanted to design a hands-on activity for your students, to helpthem realize that two non-congruent rectangles could have equal-length diag-onals, by constructing two different rectangles, on a grid paper (15×20), andmeasuring or comparing their diagonals.

What example would you use for this purpose?

Note: To accurately construct a rectangle on such a grid paper, it would behelpful to offer the students the measures of the “length” and “width” inintegers, between 1 and 15 or 20.

This task proved rather demanding. Aparently, finding two specific rectangleswith integer measurement and equal-length diagonals is a non-trivial task even forexpert mathematics educators, and perhaps more so for mathematics teachers. It isbased on the Pythagoras theorem: The relation between the lengths of the sides of arectangle – a and b, and its diagonal – c, satisfies the following equality: a2+b2=c2.In other words, the solution to the task is four integers, a, b, m, n, between 1 and 20,that satisfy the following equality: a2+b2 = m2+n2. Here are a few of the responseswe received:

One approach was by systematic trial and error: “If we determine one rectangle,for example, with sides of lengths 1 and 8, thus the diagonal is the square root of thesum 12+82 which equals 65; now we need to find another pair of integers for whichthe sum of their squares also equals 65. We begin by subtracting integer squaresfrom 65: 65–4, 65–9, 65–16, and so on, until we reach a integer square. This occursat 65–16=49, so since 49 is an integer square, a second rectangle can be obtained,of lengths 4 and 7. Indeed,

√12 + 82 = √

42 + 72, so the students could be askedto construct, on their grid paper, two distinct rectangles, one of measures 1×8 andthe other of measures 4×7. By comparing the lengths of their diagonals, they cansee that they are of equal length.”

Another systematic search for such pairs of integers was suggested, as illustratedin Table 8.1.

This method provides several specific examples of pairs of rectangles. The fol-lowing pairs of rectangles have the same diagonal, and can be accurately sketchedon a 15×20 grid paper: They are given by their side lengths: (1,7) and (5,5);(1,8) and (4,7); (2,9) and (6,7); (7,9) and (3,11); (2,11) and (5,10); or (1,12)and (8,9).

Another response to this example-generation problem was as follows: “I recalledthat a prime number is the sum of two squares if and only if its remainder in thedivision by 4 is 1. Then I recalled that it is possible to have two representations,for example for numbers that are a product of two primes whose remainder in thedivision by 4 is 1. The smallest are 5 and 13. Their product is 65. Then I had tofind the integers a, b, m, n. I noticed that 65=64+1 and that 64=82, so 8 and 1 aretwo of these numbers. I had to find the others. At this point I tried and I found that49+16=65, and that the other numbers are 7 and 4.”


Table 8.1 A systematic search for examples of two pairs of integers for which the sums of theirsquares are equal

+ 12 22 32 42 52 62 72 82 92 102 112 122

12 2 5 10 17 26 37 50 65 82 101 122 145

22 8 13 20 29 40 53 68 85 104 125 148

32 18 25 34 45 58 73 90 109 130 153

42 32 41 52 65 80 97 116 137 160

52 50 61 74 89 106 125 146 169

62 72 85 100 117 136 157 180

72 98 113 130 149 170 193

82 128 145 164 185 208

92 162 181 202 225

To give the reader a sense of how far this task went, here is another solution thatwas sent to me: “I asked a number theorist about this. He said that there are manyexamples and that there is a theory which gives, in a number of cases, the numberof solutions in the integers of pairs, (a,b) with a2+b2=N, for the same integer N.Here’s a specific solution: Pick any four distinct positive integers a, b, c, d. Then:(ac–bd)2+(ad+bc)2 and (ac+bd)2+(ad–bc)2 are both sums of squares and are bothequal to the same integer N=(a2+b2)(c2+d2). If you choose a=1, b=2, c=3, d=4,then you get the two pairs of lengths of the sides of the rectangle, (5,10) and (2,11).

For both the diagonal length is√

125(√

52 + 102 = √22 + 112

). So an example

for the students could be to draw on a grid paper two rectangles: one with sides 5and 10, and the other with sides 2 and 11.”


Case 5 is an example of the mathematical challenge of generating instructionalexamples, even for elementary pupils. The intended task for the pupils is a sim-ple and straightforward one, but finding specific measurements that allow pupils toexplore such relationships without tedious calculations is extremely demanding onbehalf of the teacher.

This clearly was a genuine problem-solving situation for the mathematics educa-tors. As demonstrated above, it was rich in the possible approaches to the problem –some relied on sophisticated number theory and others mainly on the knowledgeof the Pythagorean relationship. This experience was helpful in conveying thatgenerating (instructional) examples in mathematics is an art.

126 O. Zaslavsky

Concluding Remarks

As demonstrated throughout this chapter, the task of choosing an example toillustrate a mathematical idea is a non-trivial one. The choice of an example forteaching is often a trade-off between one limitation and another. Choosing instruc-tional examples entails many complex and even competing considerations, someof which can be made in advance, and others that only come up during theactual teaching. Many considerations require sound curricular and mathematicalknowledge.

Zodik and Zaslavsky (2008) add another dimension to this complexity by rais-ing the issue of correctness of an example. In their study, they identified three typesof “incorrectnesss” with respect to teachers’ treatment of mathematical examples:The first has to do with whether the case that is treated as an example of a moregeneral class in fact satisfies the necessary conditions to qualify as such example,e.g., treating 0.333̇ as an example of an irrational number. The second type has to dowith counter-examples. Treating an example as a counter-example for a particularclaim or conjecture when it does not logically contradict the claim is mathematicallyincorrect, e.g., bringing the example of the following binary operation a ∗ b=ab

as a counter-example to the false claim that any commutative operation is alsoassociative. A third type of mathematical incorrectness is manifested in treating anon-existing case as if it were a possible example, e.g., bringing the supposed tri-angle in Fig. 8.18 as an example of an isosceles triangle illustrates the third type ofincorrectness, since, contrary to this “example,” the sum of the lengths of any twosides of a triangle is always larger than the length of the third side.

5 5

10

Fig. 8.18 An “example” of anon-existing isoscelestriangle

The choice of examples presents the teacher with a challenging responsibility,especially since the specific choice of and treatment of examples may facilitateor impede learning (Zaslavsky & Zodik, 2007). The knowledge teachers need forjudicious construction and choice of mathematical examples is a special kind ofknowledge that can be seen both as core knowledge needed for teaching and as adriving force for enhancing teachers’ knowledge (Zodik and Zaslavsky, 2009). Itbuilds on and enhances teachers’ knowledge of pedagogy, mathematics, and studentepistemology. In Ball, Thames, & Phelps’ (2008) terms, it encompasses knowledgeof content and students and knowledge of content and teaching, as well as “pure”content knowledge unique to the work of teaching.

In this chapter I tried to unpack and capture some of the ingredients of math-ematics teacher thinking, knowledge, and practice surrounding the art of craftinginstructional examples of explanatory power. The cases I presented and analyzedmay be considered meta-examples – some specific and some more general – ofwhat Leinhardt (2001) refers to in the following passage:


In developing or selecting an example, teachers are faced with difficult tasks. They mustunderstand the critical features that they need to explicate. These features may be criticalbecause they are important within the subject matter domain or because they are key to thestudents’ understandings. The teacher needs to be aware of the purposes that the examplemay help to serve: Can the example exemplify the way a principle is to be applied, theway new ideas connect to the older ones, or the ways in which the question can be prob-lematized? Finally, the teacher needs to have the skills to refine and extend examples posedby the students themselves. Examples can fail because they are irrelevant, because theyare confusing, or because they themselves are so complex that untangling them leads theinstructional explanation astray and the point is lost. (ibid., p. 348).

References

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. EducationalStudies in Mathematics, 52, 215–241.

Ball, D., Bass, H., Sleep, L., & Thames, M. (2005). A theory of mathematical knowledge forteaching. Paper presented at a Work-Session at ICMI-Study15: The Professional Educationand Development of Teachers of Mathematics, Brazil, 15–21 May 2005.

Ball, D.L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes itspecial? Journal of Teacher Education, 59(5), 389–407.

Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplificationin mathematics education. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.),Proceedings of the 30th conference of the international group for the psychology of mathemat-ics education (Vol.1, pp. 126–154).

Bishop, A. J. (1976). Decision-making, the intervening variable. Educational Studies inMathematics, 7(1/2), 41–47.

Hazzan, O., & Zazkis, R. (1999). A Perspective on “give and example” tasks as opportunities toconstruct links among mathematical concepts. Focus on Learning Problems in Mathematics,21(4), 1–14.

Harel, G. (2008). DNR perspective on mathematics curriculum and instruction, Part II. Zentralblattfuer Didaktik der Mathematik, 40, 893–907.

Kennedy, M. M. (2002). Knowledge and teaching [1]. Teachers and Teaching: Theory andPractice, 8, 355–370.

Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge:Cambridge University Press.

Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location forcontrast. In V. Richardson (Ed.). Handbook of research on teaching (4th ed., pp. 333–357).Washington, DC: American Educational Research Association.

Leinhardt, G. (1990). Capturing craft knowledge in teaching. Educational Researcher, 19(2),18–25.

Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks,learning, and teaching. Review of Educational Research, 60(1), 1–64.

Mason, J. & Pimm, D. (1984). Generic examples: Seeing the general in the particular. EducationalStudies in Mathematics 15(3), 277–290.

Mason, J., Spence, M. (1999). Beyond mere knowledge of mathematics: The importance ofknowing-to act in the moment. Educational Studies in Mathematics, 38(1–3), 135–161.

Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that(also) explain. FOCUS on Learning Problems in Mathematics, 19(3), 49–61.

Rissland, E. L. (1978). Understanding understanding mathematics. Cognitive Science, 2(4),361–383.

Rissland, E. L. (1991). Example-based reasoning. In J. F. Voss, D. N. Parkins, & J. W. Segal (Eds.),Informal reasoning in education (pp. 187–208). Hillsdale, NJ: Lawrence Erlbaum Associates.

128 O. Zaslavsky

Rowland, T., Thwaites, A., & Huckstep, P. (2003). Novices’ choice of examples in the teachingof elementary mathematics. In A. Rogerson (Ed.), Proceedings of the international conferenceon the decidable and the undecidable in mathematics education (pp. 242–245) Brno, CzechRepublic.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: TeachersCollege Press.

Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth, UK: PenguinBooks, Ltd.

Watson, A., & Mason, J. (2002). Student-Generated Examples in the Learning of Mathematics.Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237–249.

Zaslavsky, O. (2008). Meeting the challenges of mathematics teacher education through designand use of tasks that facilitate teacher learning. In B. Jaworski & T. Wood (Vol. Eds.), Themathematics teacher educator as a developing professional, Vol. 4, of T. Wood (Series Ed.),The international handbook of mathematics teacher education (pp. 93–114). Rotterdam, TheNetherlands: Sense Publishers.

Zaslavsky, O., Harel, G., & Manaster, A. (2006). A teacher’s treatment of examples as reflec-tion of her knowledge-base. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková(Eds.), Proceedings of the 30th conference of the international group for the psychology ofmathematics education (Vol. 5, pp. 457–464).

Zaslavsky, O., & Lavie, O. (2005). Teachers’ use of instructional examples. Paper presentedat the 15th Study Conference of the International Commission on Mathematical Instruction(ICMI), on the Professional Education and Development of Teachers of Mathematics. Águasde Lindóia, Brazil.

Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teach-ers and student–teachers: The case of binary operation. Journal for Research in MathematicsEducation, 27(1), 67–78.

Zaslavsky, O., & Zodik, I. (2007). Mathematics teachers’ choices of examples that potentiallysupport or impede learning. Research in Mathematics Education, 9, 143–155.

Zodik, I., & Zaslavsky, O. (2007). Is a visual example in geometry always helpful? In J-H. Woo, H-C. Lew, K-S. Park, & D-Y. Seo (Eds.), Proceedings of the 31st conference of the internationalgroup for the psychology of mathematics education (Vol. 4, pp. 265–272).

Zodik, I., & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for themathematics classroom. Educational Studies in Mathematics, 69, 165–182

Zodik, I., & Zaslavsky, O. (2009). Teachers’ treatment of examples as learning opportunities. InTzekaki, M., Kaldrimidou, M., & Sakonidis, C. (Eds.). Proceedings of the 33rd Conferenceof the International Group for the Psychology of Mathematics Education, v.5, pp. 425–432.Thessaloniki, Greece: PME.

chapter 8 the explanatory power of examples in mathematics: … · 2015-07-14 · 8 the explanatory...

Documents