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Reasoning-and-Proving in SchoolMathematics TextbooksGabriel J. Stylianides aa University of Pittsburgh ,Published online: 14 Oct 2009.

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Mathematical Thinking and Learning, 11: 258–288, 2009Copyright © Taylor & Francis Group, LLCISSN: 1098-6065 print / 1532-7833 onlineDOI: 10.1080/10986060903253954

HMTL1098-60651532-7833Mathematical Thinking and Learning, Vol. 11, No. 4, September 2009: pp. 0–0Mathematical Thinking and Learning

Reasoning-and-Proving in School Mathematics Textbooks

Reasoning-And-Proving in Mathematics TextbooksStylianides Gabriel J. StylianidesUniversity of Pittsburgh

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students’mathematical experiences, many students face serious difficulties with this activity. Mathematics text-books can play an important role in students’ opportunities to engage in reasoning-and-proving:research suggests that many decisions that teachers make about what tasks to implement in theirclassrooms and when and how to implement them are mediated by the textbooks they use. Yet, littleis known about how reasoning-and-proving is promoted in school mathematics textbooks. In thisarticle, I present an analytic/methodological approach for the examination of the opportunitiesdesigned in mathematics textbooks for students to engage in reasoning-and-proving. In addition,I exemplify the utility of the approach in an examination of a strategically selected Americanmathematics textbook series. I use the findings from this examination as a context to discuss issuesof textbook design in the domain of reasoning-and-proving that pertain to any textbook series.

INTRODUCTION

The development of proofs in school mathematics has often been treated as a formal process inhigh school geometry, isolated from other related mathematical activities. However, this treat-ment of proof is problematic. A common structure in mathematicians’ practice that culminatesin a proof involves exploring mathematical relationships to identify and arrange significant factsinto meaningful patterns, using the patterns to formulate conjectures, testing the conjecturesagainst new evidence and revising the conjectures to formulate new conjectures that conformwith the evidence, and providing informal arguments that demonstrate the viability of theconjectures (e.g., Lakatos, 1976; Polya, 1954). These activities aid mathematicians in under-standing mathematical relationships, building a foundation for the development of proofs. Thus,by treating proof in school mathematics in isolation from activities that support its development,students are deprived of an important kind of scaffolding that more advanced users of mathematicshave available as they try to make sense of and establish mathematical knowledge.

In this article, I use the hyphenated term reasoning-and-proving (Stylianides, 2008a) todescribe the overarching activity that encompasses the following major activities that are

This article is based in part on the author’s doctoral thesis, which was completed at the University of Michigan underthe supervision of Edward Silver. The author wishes to thank Deborah Ball, Hyman Bass, and Priti Shah for commentsthat influenced the development of the ideas in the thesis. The author wishes to also thank Lyn English and the anony-mous reviewers for useful feedback on earlier versions of the article.

Correspondence should be sent to Gabriel J. Stylianides, University of Pittsburgh, Posvar Hall, Room 5517,Pittsburgh, PA 15260. E-mail: gstylian@pitt.edu

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frequently involved in the process of making sense of and establishing mathematical knowledge:identifying patterns, making conjectures, providing non-proof arguments, and providingproofs. The choice of a hyphenated term to encompass these four activities reflects myintention to view the activities in an integral way. Also, given that the term “reasoning” hasbeen associated with many different aspects of mathematical activity that are not necessarilyrelated to proof (e.g., algebraic reasoning, proportional reasoning), the hyphenated termreasoning-and-proving (RP) clarifies that my focus is on aspects of reasoning related toproof.

Given that RP is central to doing mathematics, many researchers and curriculum frameworksin different countries, especially in the United States, noted that a viable school mathematicscurriculum should make the activities that comprise RP central to all students’ mathematicalexperiences, across all grade levels and content areas (e.g., Ball & Bass, 2003; NCTM, 2000;Schoenfeld, 1994; Yackel & Hanna, 2003). Indicative of the growing focus on making RP activitiescentral to school mathematics is the following excerpt from the Principles and Standards for SchoolMathematics, a curriculum framework released by the National Council of Teachers of Mathe-matics (NCTM, 2000) in the United States.

Instructional programs from prekindergarten through grade 12 should enable all students to—

• recognize reasoning and proof as fundamental aspects of mathematics;• make and investigate mathematical conjectures;• develop and evaluate mathematical arguments and proofs;• select and use various types of reasoning and methods of proof (p. 56).

The authors of the Standards also recommend that proof becomes a “habit of mind” that iscultivated and practiced in various content areas, not just in geometry.

Despite the increased appreciation of the importance of RP in school mathematics, a largebody of research showed that students in different countries face serious difficulties with theconstituent activities of RP, especially the development of proofs (e.g., Harel & Sowder, 1998;Knuth et al., 2002; Küchemann & Hoyles, 2001–03). Also, studies such as the TIMSS (ThirdInternational Mathematics and Science Study) 1995 and 1999 Video Studies (see Hiebert et al.,2003; Manaster, 1998) indicated that in many countries mathematics instruction that fosters RPdeviates from the norm. This state of affairs is problematic: it is unreasonable to expect thatstudents will develop proficiency in RP unless this activity receives systematic attention inmathematics instruction (Yackel & Hanna, 2003).

The studies that examined how mathematics textbooks influence mathematics instructionused different methodological techniques and offered different kinds of evidence, but the bottomline of these studies was that mathematics textbooks have significant influence on students’opportunities to learn mathematics in many classrooms. Specifically, recent studies in theUnited States showed that many of the decisions that teachers make about what mathematics toteach to their students and when and how to teach this mathematics are mediated by the text-books they use (e.g., Porter, 2002; Tarr et al., 2006; Weiss et al., 2003).1 For example, Tarr et al.(2006) concluded the following in a study on the extent of textbook use by 39 middle schoolmathematics teachers utilizing different textbook series.

1I acknowledge that the extent of textbook use by teachers is lower in some other countries such as England.

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[D]istrict-adopted textbook strongly influences both what and how mathematics is taught to middleschool mathematics students. Coupled with the high frequency of textbook use by teachers, thesedata suggest that textbooks likely impact students’ mathematics experience in important ways.Moreover, the type of mathematics textbook being used seemed to matter little in the extent to whichit serves as resource for students and teachers (p. 200; italics in original).

Although mathematics textbooks can play an important role in the opportunities that studentshave to engage in RP, to date, we lack knowledge of how RP is promoted in contemporarymathematics textbook series. In this article, I present an analytic framework and a methodologicalapproach of how to use this framework in order to examine the opportunities designed in mathe-matics textbooks for students to engage in RP.2 Also, I present findings I obtained from applyingthe analytic/methodological approach to an analysis of the Connected Mathematics Project(CMP) (Lappan et al., 1998/2004), which is the most popular reform-based middle school math-ematics textbook series in the United States (U.S. Department of Education, 2000).3 The findingsI report in this article complement the findings from the analysis of CMP I reported in Stylianides(2007c). Specifically, in Stylianides (2007c) I examined the guidance that CMP teacher editionsoffer to teachers in order to implement in their classrooms proof opportunities designed in CMP.

ELABORATION ON THE SCOPE

Research on RP has tended to examine the teaching and learning of RP in classroom settings(e.g., Ball & Bass, 2003; Lampert, 1992; Stylianides, 2007a) or to elicit students’ and teachers’conceptions about RP (e.g., Harel & Sowder, 1998; Knuth, 2002; Küchemann & Hoyles, 2001–03;Martin & Harel, 1989; Stylianides et al., 2007). Undoubtedly, these are important and necessaryresearch strands, but our understanding of the findings from these strands is constrained by ourlack of knowledge of what is available for teachers and their students in the mathematics text-books they use for their work on RP. This lack of knowledge is, to some extent, the consequenceof the lack of an analytic/methodological approach that can be used by researchers to map theRP opportunities designed for students in mathematics textbooks.

In this article, I propose one such analytic/methodological approach and I illustrate its utilityin an analysis of a strategically selected American textbook series (i.e., CMP). In addition, I usethe findings from my analysis of CMP as a context to discuss issues of textbook design in thedomain of RP that pertain to any textbook series. Most important, I discuss different designmodels for the distribution of RP opportunities in mathematics textbooks across grade levels andcontent areas. These models are singled out for discussion from numerous possible models

2I talk about opportunities designed for (as opposed to opportunities offered to) students in mathematics textbooks inorder to (1) indicate my focus on the written curriculum, i.e., what is included in the students’ textbooks and the teachers’editions of a textbook series; and (2) emphasize that there may be differences between the written and implemented curriculum,whereby implemented curriculum I mean the way in which the written curriculum is enacted in the classroom.

3CMP is currently undergoing revision. Parts of the new version of CMP are now available. All the analysis anddiscussion in this paper refers to the first iteration of CMP. Also, I use the phrase “reform-based textbook series” to referto textbook series that were designed to embody the recommendations of the NCTM (1989, 2000) Standards. I use thephrase “conventional textbook series” to refer to textbook series that do not associate their existence with the Standards.Although many of the conventional textbook series have been updated to address the NCTM Standards, their updatinghas been primarily one of retrofitting as opposed to complete redesign.

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because they seem to receive better support from existing research and theory (e.g., Balacheff,1988; Dewey, 1903; Schwab, 1978; Stylianides & Stylianides, 2008). Future research needs toexamine the implications of these and other possible models for the teaching and learning of RPin classrooms, thereby offering insight into the relative value of each of them.

Important to note is that I did not develop the analytic/methodological approach for CMP orfor exclusive use in the American context; the application of the approach to analyze the afore-mentioned textbook series should be viewed as an exemplification of the approach and of itspotential to be used more broadly. As I explain in the last section, the analytic/methodologicalapproach can find application in examinations of the place of RP in other textbooks series.

The choice of CMP was strategic for two main reasons. First, CMP, being a reform-basedtextbook series, it aims (and claims) to be aligned with the recommendations set forth by theNCTM (1989, 2000) Standards, including the recommendation for making RP central throughoutgrade levels and content areas. Given that the major goal of the article was to develop and test ananalytic/methodological approach for examining the place of RP in mathematics textbooks, itwas important to choose a textbook series that had the potential to provide a rich context forsuch an examination. Second, there are findings available from longitudinal examinations of thedevelopment of CMP students’ understanding of RP (Knuth et al., 2002; Knuth & Sutherland,2004) that provide a useful context as I discuss my findings and the different design models.

In this textbook analysis, I was concerned not only with the extent to which RP opportunitieswere designed in CMP but also with whether and how the distribution of these opportunitiesvaried by content area (algebra, geometry, and number theory) and grade level (grades 6 to 8).My decision to examine the distribution of RP opportunities across content areas and gradelevels in CMP and to discuss different possible models for the distribution of RP tasks acrosscontent areas and grade levels in any textbook series is important for various reasons.

The first reason relates to the distribution of RP opportunities across content areas. AsI mentioned earlier, many researchers and curriculum frameworks agree that RP should be cen-tral to all content areas of the school mathematics curriculum. But, what does this recommenda-tion mean in practical terms? For example, does it mean that there should be non-statisticallysignificant differences among the number of RP opportunities across different content areas?A long historical tradition in school mathematics in the United States and elsewhere has associ-ated RP with Euclidean geometry (see, e.g., Stylianides, 2008b, for a review of the place ofproof in the American school mathematics curriculum the last century). Does this imply thatgeometry in the middle grades should also be privileged over other content areas in terms of thenumber of RP opportunities that is devoted to it? The second reason relates to the distribution ofRP opportunities across grade levels. Recent longitudinal studies on CMP and other students’understanding of various aspects of RP indicated only modest improvements of students’ under-standing of RP with age (Knuth et al., 2002; Knuth & Sutherland, 2004; Küchemann & Hoyles,2001–03). How do/should mathematics textbooks deal with the development of students’ RPabilities across grade levels?

ANALYTIC FRAMEWORK

I used the conceptualization of RP I outlined earlier to develop an analytic framework that I usedin the textbook analysis. An important advantage of the framework is that it integrates major

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activities related to engagement in proof and, thus, allows examination of this engagement in abroader context. The framework unfolds along two inter-related dimensions (see Table 1).

Dimension 1 uses (1) the notion of “making mathematical generalizations”—the transportingof mathematical relations from given sets to new sets for which the original sets are subsets(Polya, 1954)—to capture two of the activities that comprise RP (“identifying a pattern” and“making a conjecture”), and (2) the notion of “providing support to mathematical claims” tocapture the other two activities that comprise RP (“providing a proof” and “providing a non-proof argument”). Dimension 1 includes also a further breakdown of some of the RP activities tocapture important distinctions.

Dimension 2 is complementary to Dimension 1 and concerns the purposes (functions) thatpatterns, conjectures, and proofs may serve in students’ engagement in RP. The examination ofthese purposes is important for several reasons. For example, in mathematics, one importantconnection among patterns, conjectures, and proofs is that patterns can generate conjectures,which in turn can give rise to the development of proofs (e.g., Polya, 1954). To what extent arethe structures “pattern → conjecture” and “conjecture → proof” found in mathematics text-books? In addition, numerous researchers (e.g., de Villiers, 1998; Yackel & Hanna, 2003) calledfor an emphasis on the explanatory purpose of proof in school mathematics and argued againstthe traditional emphasis on the verification purpose of proof. Do mathematics textbooks reflectthis emphasis on explanatory proofs in school mathematics?

Next I elaborate on the two dimensions of the analytic framework. In Stylianides (2008a), Ipresented an analytic framework of RP that included three components (mathematical, psy-chological, and pedagogical) and that aimed to serve as a useful platform for conducting dif-ferent kinds of investigations with a focus on RP: textbook analyses, classroom-basedexaminations, analyses of professional development sessions, etc. Therefore, the presentationof the framework in Stylianides (2008a) was generic and did not address some important

TABLE 1 The Analytic Framework

Reasoning-and-Proving

Making Mathematical GeneralizationsProviding Support to Mathematical

Claims

Identifying a Pattern Making a Conjecture Providing a Proof

Providing a Non-proof Argument

Dimension 1 Components and Subcomponents of Reasoning-and-proving

• Plausible Pattern• Definite Pattern

• Conjecture • Generic Example• Demonstration

• Empirical Argument

• Rationale

Dimension 2 Purposes of Pattern, Conjecture, and Proof

• Conjecture Precursor• Conjecture Non-Precursor

• Proof Precursor• Proof Non-Precursor

• Explanation• Verification• Falsification• Generation of

New Knowledge

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issues that are particular to any specific kind of investigation. Dimension 1 of the framework Ipresent in this article corresponds with the mathematical component of the framework inStylianides (2008a). My description of Dimension 1 includes new examples and elaboratesparts of the framework that particularly pertain to a textbook analysis. Dimension 2 is pre-sented here for the first time.

Dimension 1: Components and Subcomponents of Reasoning-and-Proving

Identifying a Pattern

In this framework, a pattern is defined as a general mathematical relation that fits agiven set of data. This notion of pattern transcends content areas and representationalforms (e.g., algebraic, pictorial).4 The framework distinguishes between two kinds of pat-terns—“definite” and “plausible”—according to whether the patterns are uniquely deter-mined. In Stylianides (2008a), I elaborated on the theoretical grounding of definite andplausible patterns in the notions of structural and empirical generalizations (Bills &Rowland, 1999; Küchemann & Hoyles, 2009). Next I define and exemplify definite andplausible patterns.

In definite patterns, it is possible mathematically for an expert (given the information in atask5) to provide conclusive evidence for the selection of a specific pattern. In plausible pat-terns, it is not possible mathematically for an expert to provide conclusive evidence for theselection of a specific pattern over other patterns that also fit the data; the selection of a patternmay represent the simplest or most evident relation that fits the data. Example 1 illustrates thedistinction between definite and plausible patterns.

Example 1The table shows how two variables relate. Find a pattern in the table and use the pattern to

complete the missing entries.

One pattern (expressed algebraically) that fits the data in the table is y = 2x. Based on thispattern, the missing entries are 8 and 16. Another pattern that fits the data is y = 2x. According tothis pattern, the missing entries are 6 and 8. Yet, another pattern that fits the data is that y equals2 if x is odd and 4 if otherwise. According to this pattern, the missing entries are 2 and 4. Thethree pairs (8, 16), (6, 8), and (2, 4) are illustrative of an infinite number of possible solutionsthat fit the given set of data. From a mathematical standpoint, any answer to the task could becorrect. The pattern is not uniquely determined and, thus, is a plausible pattern. How could thetask in Example 1 be modified so that the pattern would become definite? One way to do this is

4The notion of “pattern” in this framework includes also statements that describe covariation between structures,properties, or variables, i.e., it includes patterns that describe relationships between different patterns. A more refinedapproach in the treatment of pattern in the framework would be to distinguish between patterns that relate differentpatterns and patterns that do not. The former category can be more challenging for students because it requires coordina-tion of multiple structures, properties, or variables and, thus, may require special attention in instruction.

5See in the method section for how I use the term “task.”

x 1 2 3 4

y 2 4

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to place the task in a situation that implies a particular structure. For example, the situationdescribed in part A of Example 4 (see Appendix) defines uniquely the pattern y = 2x, where y isthe number of ballots and x is the number of cuts.

The previous discussion shows that whether a pattern is definite or plausible influences thekind of justification one might offer for why the pattern holds. In definite patterns one can offerconclusive evidence for the pattern, whereas in plausible patterns this is not possible. Yet, thisdoes not mean that plausible patterns cannot engage students in productive justifications ofmathematical generalizations. For example, the students can assume a particular structure thattransforms a plausible pattern into definite and then use the assumed structure to provide conclu-sive evidence for the specific pattern that fits the data.

Making a Conjecture

In this framework, a conjecture is defined as a reasoned hypothesis about a general mathe-matical relation based on incomplete evidence. The term “reasoned” highlights the non-arbitrarycharacter of the hypothesis. The term “hypothesis” indicates a level of uncertainty about thetruth of a conjecture and denotes that further action is needed for its acceptance or rejection(Cañadas & Castro, 2005; Reid, 2002; Arzarello et al., 1998).

Although the activities of making conjectures and identifying patterns are parts of the moregeneral activity of making mathematical generalizations, there are two major differencesbetween them. First, in conjecturing the solver formulates a hypothesis with a domain of refer-ence that extends beyond the domain of cases that gave rise to it. This is not always the case inpattern identification (Reid, 2002). Second, in conjecturing the solver sets forth a hypothesisthat, although is accompanied by an expression of conviction about its truth, is not considered tobe true or false and is subject to testing. As Harel and Sowder (2007) put it, “[a] conjecture is anassertion made by an individual who is uncertain of its truth” (p. 808). Therefore, the very natureof conjecture calls for further examination. In pattern identification, however, once the solverfinds a relation that fits a given set of data, the solver presents this relation in a way that does notnecessarily communicate possible doubt about its truth. To illustrate these two distinctionsconsider the following example from CMP.

Example 2 (from CMP unit titled Kaleidoscopes, Hubcaps, and Mirrors, p. 26)Terrapin Crafts wants to rent a space of 12 square yards.

A. Use 12 square tiles to represent the 12 square yards. Find all the possible ways the TerrapinCrafts owner can arrange the squares. Copy each rectangle you make onto grid paper, and label itwith its dimensions (length and width).

B. How are the rectangles you found and the factors of 12 related?

For the purposes of this illustration I focus on part B. Let us suppose that the solver noticedthat, “the factors of 12 are the dimensions of the rectangles found.” This pattern is limited tothe specific domain of factors of 12. Also, it is presented in a way that does not communicateany doubt about its truth. Let us assume now that there was a part C asking the following:“What can you say about the relation between the factors of numbers and the dimensions ofthe rectangles you can make with each number?” If the solver said that, “the factors of anumber are the dimensions of the rectangles you can make,” this would again count as a

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pattern. This new pattern, however, has a domain of reference that extends beyond particularcases and is more general than that in part B. If the solver said instead, “I expect that the fac-tors of a number will be the dimensions of the rectangles you can make,” this new responsewould count as a conjecture, for it is phrased as a hypothesis that suggests a need for furtherinvestigation.

Providing a Proof

In this framework, the definition of proof is compatible with the conceptualization of proofdetailed in Stylianides (2007a). A proof is defined as a valid argument based on accepted truthsfor or against a mathematical claim that makes explicit reference to “key” accepted truths that ituses. The term “argument” denotes a connected sequence of assertions. The term “valid” indi-cates that these assertions are connected by means of accepted canons of correct inference(modus ponens, modus tollens, etc.). The term “valid” should be understood in the context ofwhat is typically agreed on currently in mathematics. Of course, this is not to say that this termhas universal meaning in mathematics nowadays, but it is beyond the scope of this paper to elab-orate on this issue. The term “accepted truths” is used broadly to include the axioms, theorems,definitions, and statements that a particular community may take as shared at a given time.Which accepted truths are “key” and, thus, should be explicitly referenced in a proof is adelicate issue. Yet, a response to this issue needs to consider the audience for the proof, which inthe case of a textbook analysis I consider to be the (hypothetical) classroom community thatimplements the textbooks series. As one makes decisions about which accepted truths should bereferenced in a textbook argument in order for it to qualify as a proof, it is important to considerwhat preceded the focal argument in the textbook series and what was the students’ expectedknowledge and understanding at that point. In the method, I explain how I determined students’expected knowledge and understanding as I was coding textbook tasks. In my discussion of the“non-proof argument” category of the framework I revisit the issue of what can count as a “key”accepted truth and provide examples.

The framework distinguishes between two kinds of proof: generic examples and demonstra-tions. A generic example is a proof that uses a particular case seen as representative of thegeneral case (Balacheff, 1988; Mason & Pimm, 1984; this is also similar to Harel and Sowder’s,1998, “transformational proof” and Movshovitz-Hadar’s, 1988, “transparent pseudo-proof”).Generic examples are important because they can provide students with a powerful and easilyreached means of conviction and explanation and can allow students to prove mathematicalclaims even when they lack mathematical language to express their proofs in more sophisticatedways.

A demonstration is a proof that does not rely on the “representativeness” of a particular case(this is similar to Harel and Sowder’s, 1998, “axiomatic proof” and Balacheff’s, 1988, “thoughtexperiment”). Valid arguments by counterexample, contradiction, reductio ad absurdum,mathematical induction, contraposition, and exhaustion are examples of demonstrations. I donot associate demonstrations with a particular representational form. For example, the followingpictorial and algebraic arguments for the claim “the sum of any two consecutive odd naturalnumbers is a multiple of 4” could qualify as demonstrations (assuming that the argumentsaddressed appropriately contextual features of the focal classroom community, e.g., whatcounted as an accepted truth within the community).

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Pictorial proofFirst we define “odd” and “multiple of 4” using pictures and then we prove the conjecture.

Algebraic proofOdd natural numbers are the numbers of the form 2n + 1, when n is a whole number.Multiples of 4 are the numbers of the form 4k, where k is an integer.Sum of two consecutive odd natural numbers = (2m + 1) + (2m + 1 + 2) = 4m + 4 = 4 • (m + 1) =

4n, where m is a whole number and n is an integer. Thus, the sum of two consecutive odd naturalnumbers is a multiple of 4.

Providing a Non-Proof Argument

In this framework, a non-proof argument is defined as an argument for or against a mathe-matical claim that does not qualify as a proof. I clarify that circular or non-genuine responses(i.e., responses that show minimal engagement, irrelevant responses, or responses that are poten-tially relevant but their relevance is not made evident by the solver) do not count as non-proofarguments. The framework distinguishes between two kinds of non-proof arguments: empiricalarguments and rationales.

An empirical argument is an argument that purports to show the truth of a mathematicalclaim by validating the claim in a proper subset of all the possible cases covered by the claim(this is similar to Balacheff’s, 1988, “naïve empiricism” and Harel and Sowder’s, 1998, “empir-ical justification”). Students’ engagement in empirical arguments, which are invalid, is likely toreinforce the common misconception that examples can prove general mathematical claims(e.g., Knuth et al., 2002; Küchemann & Hoyles, 2001-03; see Harel & Sowder, 2007, for areview).

The notion of rationale was introduced in the framework to capture valid arguments for oragainst mathematical claims that do not qualify as proofs. Specifically, an argument counts as arationale instead of a proof if it does not make explicit reference to key accepted truths that ituses (in the context of a particular community where these truths can be considered as key), or if

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it uses statements that do not belong to the set of accepted truths of a particular community (inthis case the argument is incomplete because it omits necessary steps). For example, in thecontext of an elementary school classroom community where the students have just beenintroduced to definitions of odd numbers and multiples of 4, the pictorial argument that I pre-sented earlier would count as a rationale if it made no explicit reference to the definitions itused. The underlying assumption is that in this context the definitions are key accepted truthsthat cannot be left implicit in an argument that meets the standard of proof. Of course, this isnot to say that proofs need to make explicit reference to every accepted truth they use. Forexample, the pictorial proof above was using but made no explicit reference to the transitivityof equality. Requiring explicit reference to such intuitively obvious properties before argu-ments could qualify as proofs would shift the emphasis of developing a proof from sense-making to carrying out ritual procedures. For another example of a rationale, consider asituation where a student is trying to prove the same claim about the sum of two consecutiveodd natural numbers using the following statement, which does not belong to the set ofaccepted truths of the student’s classroom community: “the sum of any two consecutive evennatural numbers is a multiple of 4 with 2 left over.” The following argument would count,then, as a rationale: “I can get any two consecutive odd natural numbers by subtracting onefrom each of two consecutive even natural numbers. Therefore, (odd) + (consecutive odd) =(even – 1) + (consecutive even – 1) = (even + consecutive even) – 2 = (multiple of 4 + 2) – 2= multiple of 4.”

To conclude, in contrast to students’ engagement in empirical arguments, students’ engage-ment in rationales is desirable for at least three reasons:

1. Unlike empirical arguments, rationales do not support the development of inaccurateunderstandings of proof that would have later on to be addressed by instruction.

2. Being valid but not as developed as proofs, rationales offer a good choice of an argumentwhen the production of a proof is impractical (e.g., due to time constraints), impossible(e.g., due to conceptual barriers), or undesirable (e.g., due to the focus of activity beingon a concept other than the justification of a claim).

3. Being valid but less developed arguments than proofs, rationales can be more easilyaccessible to students than proofs and, thus, have the potential to serve as transitionalstage between empirical arguments and proofs.

The third reason points to a possible hierarchy in terms of the level of mathematical sophisti-cation of the different kinds of argument in the analytic framework: empirical arguments, ration-ales, generic examples, and demonstrations. Later in the paper I discuss such a hierarchy.

Dimension 2: Purposes of Pattern, Conjecture, and Proof

Purposes of Pattern

As I discussed earlier, a major role that patterns play in mathematics is to lead to the develop-ment of conjectures. This observation gives rise to the distinction of whether students haveopportunities to identify patterns that forerun the generation of conjectures associated with thesepatterns (conjecture precursors) or not (conjecture non-precursors). A pattern can serve the

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purpose of a conjecture non-precursor when it does not have the potential to be extended to aconjecture, or when the solver does not investigate/is not expected to investigate the pattern fur-ther to make a conjecture.

To illustrate these two purposes of pattern let us consider a student who added four pairs ofmultiples of three and noticed the pattern that the sum in all cases was a multiple of three. If thestudent pursued this pattern further to make the conjecture that “the sum of any two multiples ofthree is a multiple of three,” then we say that the pattern served the purpose of a conjectureprecursor. If the student did not pursue the pattern further (e.g., because the student was notasked to do so in the context of the particular task or subsequent tasks in the textbook), then wesay that the pattern served the purpose of a conjecture non-precursor.

Purposes of Conjecture

The previous discussion suggests that a major role that conjectures play in mathematics is tolead to the development of proofs. This observation gives rise to the distinction of whetherstudents have opportunities to make conjectures that forerun the development of proofs associ-ated with these conjectures (proof precursors) or not (proof non-precursors). A conjecture canserve the purpose of a proof non-precursor (1) when the solver does not/is not expected to inves-tigate the conjecture further to produce a proof or (2) when the solver investigates the conjecturebut develops an argument that does not qualify as a proof.

Let us consider again the previous example with the addition of two multiples of three and letus assume also that the student developed the conjecture that, “the sum of any two multiples ofthree is a multiple of three.” If the student did not pursue the conjecture further to prove it (e.g.,because the student was not asked to do so in the context of the particular task or subsequenttasks in the textbook), then we say that the conjecture served the purpose of a proof non-precur-sor. The conjecture would again serve the purpose of a proof non-precursor if the student madean attempt to prove the conjecture but developed a non-proof argument. If the student developedan argument that qualified as a proof, then we say that the conjecture served the purpose of aproof precursor.

I clarify that my focus on the aforementioned purposes that patterns and conjectures canserve does not imply that the relation between patterns, conjectures, and proofs is always linear“pattern → conjecture → proof.” Sometimes, for example, it is possible to formulate a conjec-ture without first identifying a pattern (e.g., formulate a conjecture deductively from an alreadyestablished theorem) (see Arzarello et al., 1998, for discussion on why conjecturing is notconfined to inductive reasoning). Also, the process of constructing a proof can follow a zig-zagpath between attempts to generate arguments and criticisms of these attempts (e.g., through theconstruction of counterexamples) that may lead to identification of new patterns and formulationof new conjectures (Arzarello et al., 1998; Lakatos, 1976; Lampert, 1992; Reid, 2002).

Purposes of Proof

In this framework, proofs can serve one or more of the following purposes: explanation, ver-ification, falsification, and generation of new knowledge. In mathematics, proofs serve also anumber of other important purposes, which are not included in the framework for differentreasons. For example, the purpose of communication—“the transmission of mathematical

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knowledge” (de Villiers, 1999, p. 5)—was not included in the framework because it was hard todefine it in a clear and operational way for the needs of a textbook analysis. Another purpose ofproof that is missing from the framework is systematization—“the organization of variousresults into a deductive system of axioms, major concepts and theorems” (ibid). This purposewas not included because, as de Villiers (1998) noted, “in mathematics [ . . . ] the systematiza-tion purpose of proof comes to the fore only at a very advanced stage” (p. 390).

Next I define and discuss the purposes of proof listed in the framework:

1. Explanation, when the proof provides insight into why a claim is true (Bell, 1976; deVilliers, 1999; Hanna, 1990) or false.

2. Verification, when it establishes the truth of a given claim (Bell, 1976; de Villiers, 1999).Direct methods of proof and proof by contradiction fit under the verification purpose ofproof. I use “proof by contradiction” as the method of verifying the truth of a given claimby the supposition that it is false and the subsequent drawing of a conclusion that contra-dicts a claim that is proved or assumed to be true.

3. Falsification, when it establishes the falseness of a given claim. This purpose of proofhas not been previously identified in the literature as a separate purpose of proof, in partbecause proof has often been defined as an argument for the truth of a mathematicalclaim. Yet, based on my definition of proof, a proof can also be an argument against aclaim. So, according to this definition, reductio ad absurdum and proof by counterexam-ple relate to the purpose of falsification and cannot be captured under verification. I use“reductio ad absurdum” as the proof method that proceeds by stating a claim and thenshowing that it results in a contradiction, thus demonstrating the claim to be false.

4. Generation of new knowledge, when it contributes to the development of new results(Davis & Hersh, 1981; de Villiers, 1999; similar also to Lakatos’, 1976, “proof-gener-ated” notion). The phrase “new results” is used to describe products that solvers in a par-ticular community add to their knowledge base as a result of constructing a proof.

The purposes of proof described above (to be illustrated later) have not attracted equalattention in school mathematics. Traditionally the verification purpose of proof prevailed. Indic-ative are the results of Knuth’s (2002) study with 16 inservice secondary school mathematicsteachers about their conceptions of proof. One of the questions that Knuth asked the teacherswas the following: What purpose does proof serve in mathematics? Although all teachers in thesample endorsed verification, none of them mentioned explanation. Yet, in recent years, therehas been a call from many researchers for more emphasis on the explanatory purpose of proof inschool mathematics (e.g., de Villiers, 1998; Hanna, 1990). For example, de Villiers (1998)argued against the traditional emphasis on verification by pointing out the mismatch that thisemphasis creates between school mathematics and the discipline of mathematics.

METHOD

Sample

CMP has a total of 24 units, 8 in each grade level. The sample includes all the units in algebra(six units), number theory (one unit), and geometry (five units). From the 12 units in the sample,

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3 units come from sixth grade, 3 from seventh grade, and 6 from eighth grade (see Stylianides,2005, for more information about these units).

CMP includes both student and teacher support materials and is organized around problemsolving activities. I focused my analysis on the algebra, number theory, and geometry unitsmainly because these three content areas are important in the middle grades and are likely toconcentrate the bulk of the textbook opportunities designed for students to engage in RP. Toidentify the CMP units associated with these three content areas, I used the textbook authors’perception about which of the four broad strands that CMP addresses (number, geometry,algebra, statistics and probability) is primarily being promoted in each unit.6 This decisioncreated two complications. The first is that under “geometry” the authors also include themeasurement units. The second is that under “number” the authors, besides number theory, alsoinclude number and number relationships, as well as computation and estimation. To deal withthese complications, I examined the contents of each candidate unit (see Stylianides, 2005, formore detail).

Design

Defining the Unit of Analysis

Different textbook series follow different marking systems for their exercises, problems,and activities. For example, (American) reform-based mathematics textbook series mark asone exercise, problem, or activity what conventional textbook series would typically markas many different ones. It was important to define the unit of analysis so that the same unitcould be applied in analyses of textbook series that follow different marking systems. Thiswould allow comparisons between the results of this study and future studies that would usethe same analytic/methodological approach to investigate the place of RP in other textbookseries. Given these considerations, I used the task as the unit of analysis; by task I meanany exercise, problem, activity, or parts thereof that have a separate marker in the students’textbook.

Producing the Analytic Framework

The process I followed to produce the analytic framework was comprised of four stages. Inthe first stage, I produced a preliminary framework by studying related literature and the natureof RP in the discipline of mathematics. Given that RP is a fundamental mathematical activity, itwas important to develop a framework that would be consistent with important aspects of thenature of RP in mathematics.

In the second stage, I made an initial test and validation of the framework. Specifically,I tested whether the framework would be applicable to examining both reform-based and con-ventional textbook series by analyzing, in addition to CMP tasks, tasks from MathematicsApplications and Connections (MAC) (Collins et al., 1993), which is the most popular conven-tional middle school mathematics textbook series in the United States (U.S. Department of

6In the cover page of each CMP unit, the textbook authors denote which of the four strands they consider to beprimarily promoted in that unit.

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Education, 2000). Developing an analytic tool applicable across different kinds of textbookseries would enhance its importance. Through solving a broad range of tasks from various gradelevels and content areas of the two textbook series, I tested the extent to which the frameworkcould adequately capture the RP activity involved in these tasks. In this analysis, I followed aconstant comparative method (Glaser & Strauss, 1967): I was comparing the definitions of exist-ing framework categories to see whether the features of a task indicated mismatches that couldlead to generation of new categories or adjustment of existing categories to include those fea-tures. For example, this analysis led to the addition of the falsification purpose of proof:although proofs that established the falseness of given claims were rare in the CMP tasks Itested, the falsification purpose of proof was not uncommon in the MAC tasks.

In the third stage, two mathematics educators and two research mathematicians evaluated,respectively, the framework’s potential to capture important aspects of the textbook opportuni-ties designed for students to engage in RP and the quality of its definitions. Using theirfeedback, I revised the framework accordingly. For example, the mathematicians’ suggestionshelped me sharpen the definitions of definite and plausible patterns and make clearer the distinc-tions between patterns and conjectures.

In the fourth stage, I tested the applicability of the revised framework by analyzing moretasks from CMP and MAC in different content areas and grade levels. This analysis led only tominor refinements of the definitions of some of the framework (sub)categories.

Coding

The textbook analysis involved (1) identification of the tasks in the sample that designedopportunities for students to engage in at least one of the four major activities that comprise RP(Dimension 1 of the analytic framework), and (2) coding of these tasks using the differentframework subcategories (Dimensions 1 and 2 of the analytic framework). To decide about howto code each task, I used the definitions of the different framework subcategories. Multiplecoding of tasks was permitted to capture as much as possible of their complexity.

An important issue I faced in the analysis was how to make decisions about what opportuni-ties each task was designing for students. To code the tasks in both Dimensions 1 and 2 of theframework, I developed a way to make reasonable inferences about the expected formulation ofthe tasks—the path students were anticipated to follow in solving the tasks, as this was reflectedin the written curriculum. Specifically, I determined the expected formulation of tasks bysolving them in the order they appeared in the textbook series and by considering together thefollowing three factors: (1) the approach suggested by the students’ textbook, (2) the approachsuggested in the teachers’ edition, and (3) the students’ expected knowledge and understandingwhen encountering a given task. I determined the latter by looking at what preceded the giventask in the students’ textbook and considering it to be known to the students (e.g., Whattheorems, definitions, mathematical conventions, and methods were the students expected toknow up to that point?).

In brief, the analysis of the textbooks was a hypothetical enterprise: What opportunities toengage in RP would students have if their classroom community covered serially all the parts intheir textbooks and in the ways designed in the written curriculum? This approach to thetextbook analysis does not imply a particular perspective on how teachers should implementtextbook tasks in their classrooms. Rather, the approach aims to offer a meaningful and reliable

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way to draw inferences about the expected formulation of tasks, and also a way that is consistentwith what is included in the written curriculum. Below is an example of how I coded a CMP task(see Appendix for more examples).

Example 3 (adapted from CMP unit titled Prime Time, p. 29)Make a conjecture about whether the sum of two even numbers will be even or odd. Then justify

your answer.

Regarding Dimension 1, the task was triple-coded as “identifying a pattern (definite pattern),”“making a conjecture,” and “providing a proof (demonstration).” This is because in the expectedformulation of the task the students were anticipated to (1) examine a few cases and notice thedefinite pattern that the sum they got in all cases was an even number; (2) use the pattern to for-mulate the conjecture that “the sum of any two even numbers is even”; and (3) use their defini-tion of even numbers as rectangles with a height of two square tiles to provide the followingdemonstration: “The sum of two even numbers is even because we can combine two rectangleswith height of two square tiles (i.e., two even numbers) to get another rectangle with height oftwo square tiles.” One could argue that sixth-graders should be expected to produce a more for-mally presented argument for the claim, such as an algebraic proof. This could be possible if thestudents were introduced to an algebraic definition of even numbers, but this was not expectedto happen up to this point in CMP. Nevertheless, the argument the students were expected toproduce covered adequately the general case without relying on examples of any kind and, thus,qualified as a demonstration.

Regarding Dimension 2, the pattern served the purpose of a “conjecture precursor”because the students were expected to use the pattern to make a conjecture. The conjectureserved the purpose of a “proof precursor” because the students were expected to use theconjecture to develop a proof. Finally, the proof served the purposes of (1) “explanation”because it provided insight into why the conjecture was true; and (2) “generation of newknowledge” because, up to that point in the textbook series, the students were not expectedto know that the sum of two even numbers was even. The proof did not serve the purposesof verification or falsification because the claim to be proved was not given to the students;rather, the students were expected to formulate the claim as a result of their engagementwith the task.

I tested the inter-rater agreement of the coding scheme by comparing my codes with thecodes of a second rater and calculating three reliability values. The first two values werebased on a subsample of tasks that offered the opportunity for different kinds of codes. Thefirst reliability value related to our decisions on whether each task in the subsample wasdesigned to provide students an opportunity to engage in RP. The inter-rater agreement was92.7%; kappa statistic = .9 (Siegel & Castellan, 1988). The second reliability value related toour decisions on how to sort the RP tasks to the seven subcategories of the four categories ofDimension 1. The inter-rater agreement was 88.5%. The third reliability value related to thepurposes proofs served in CMP and involved a stratified sample from different grade levelsand content areas of more than 15% of the tasks that were coded as proofs.7 The inter-rateragreement was 92.6%.

7It was not necessary to calculate separately the inter-rater agreement for the purposes of patterns and conjecturesbecause this inter-rater agreement was embedded in the second reliability value.

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Limitations

One limitation of the study related to the restriction of the sample to one textbook series. Arelated limitation is that the analysis focused on content areas I hypothesized would concentratethe bulk of the textbook opportunities designed for students to engage in RP, but this hypothesiscan be refuted by a comprehensive analysis of the textbook series.

Other limitations relate to different aspects of engagement in RP, which are not currentlycaptured by the analytic framework. Examples of such aspects include: What opportunities aredesigned for students to understand the zig-zag path between attempts to generate valid argu-ments and criticisms of these attempts that can lead to identification of new conjectures? Do stu-dents receive consistent epistemological messages about RP from a textbook series (similar toRaman’s, 2004, textbook analysis on the notion of continuity)? What is the role of mathematicallanguage in the RP tasks included in a textbook series? For example, what is the role of impera-tives used to signal mathematical action within tasks (e.g., show, explain, give evidence, con-vince, prove)?

Another limitation related to my assumption during the analysis of tasks was that studentshad all prerequisite knowledge that was covered in the textbook series. This assumption cannotapply in actual classroom settings, but it offered a meaningful way for me to code tasks in areliable and consistent way that also made justice to the written curriculum.

A related limitation was that the study did not provide insight into the implemented curricu-lum on RP. It is likely that some of the textbook tasks that were coded as designing opportunitiesfor students to engage in RP would not qualify as such in an analysis of classroom practice thatimplemented those tasks, and vice versa. Also, although some RP activities may receive a smallrelative emphasis in a textbook series, this does not imply that the students in a classroom thatimplements the textbook series will have fewer opportunities to develop ability in these activi-ties as compared with other activities that receive more emphasis in the textbook series: teachersmay implement higher percentages of tasks from the seemingly underemphasized activities. Thelatter statement is consistent with the findings from Tarr et al. (2006). Specifically, theseresearchers found that the implemented curriculum for 39 middle school teachers correspondedwith 60% to 70% of the written curriculum. From the 30% to 40% of the textbook content thatthe teachers omitted, Number and Operations comprised only 11 to 17%, “a remarkably low fig-ure given the heavy emphasis on number in the written curricula” (p. 196).

RESULTS

Reasoning-and-Proving Tasks

The analysis included 4855 tasks. About 40% of these tasks (1852 tasks) were designed to offerstudents at least one opportunity for RP; I refer to these tasks as reasoning-and-proving tasks(RPTs). More than 50% of the 4855 tasks (2726 tasks) did not design any opportunities forstudents to engage in RP; I refer to these tasks as non-(reasoning-and-proving) tasks (n-RPTs).For the remaining 277 tasks, there was not enough information in the textbook series to codethem as RPTs or n-RPTs. For this reason these 277 tasks were eliminated from the analysis.Therefore, the findings I report here are based on a sample of 4578 tasks.

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Regarding the ways in which the RPTs were distributed across the different framework(sub)categories within each CMP unit in the sample, the reader is referred to Stylianides (2005).Given that CMP is used in this article as a context for illustrating the utility of the analytic/methodological approach and for discussing broader issues related to textbook design in thedomain of RP, I only mention important findings from the analysis.

In more than half the CMP units in the sample the percentage of RPTs exceeded 40%. Theseunits spanned different grade levels and content areas. Only in a seventh-grade algebra unit wasthe percentage of RPTs below 30% (26%). Sixty two percent of the RPTs in the sample weredesigning opportunities for students to give rationales, 18% were designing opportunities forstudents to identify definite patterns and 6% to identify plausible patterns, and 12% to providedemonstrations. There was a significant variation in the percentages of the RPTs in each unitthat were designing opportunities for students to provide demonstrations. These percentagesranged from 1% in a seventh-grade algebra unit to 32% in an eighth-grade algebra unit. Genericexamples were virtually non-existent in the sample (0%), and the percentages of the RPTs codedas empirical arguments or conjectures were 2% and 3%, respectively.

Important to note is that Prime Time (sixth-grade number theory), which is the first CMP unitteachers are expected to implement in their classrooms, was the richest unit in the sampleregarding the RP opportunities designed for students (the RPTs in this unit were 49%). Also, theproportion of RPTs that were designing opportunities for students to provide demonstrations inPrime Time was more than twice the sample average (28% versus 12%), the proportion ofconjectures was more than three times the sample average (10% versus 3%), the proportion of def-inite patterns was well above the sample average (22% versus 18%), and the proportion of empiri-cal arguments was the lowest in all units (0% for Prime Time versus 2% for all other units).

Table 2 summarizes the distribution by percent of RPTs in different grade levels and contentareas. I ran chi-square tests to check for significant differences in the distribution of differentkinds of RPTs across grade levels and content areas. The actual numbers associated with thesetests can be found in Stylianides (2005). Next I summarize important outcomes from the tests.

Regarding grade-level comparisons, there was evidence to indicate that (1) the examinedseventh-grade units were more likely to have fewer RPTs than the examined units in the othertwo grades; (2) the examined sixth-grade units were more likely to design opportunities forconjectures and proofs than were tasks in the examined units in the other two grades; and (3) theexamined sixth-grade units were more likely to design opportunities for rationales than weretasks in the examined units in seventh-grade and less likely to design opportunities for rationalesthan were tasks in the examined eighth-grade units. Finally, there was no significant differencein the distribution of tasks that were designing opportunities for students to identify patterns orempirical arguments in different grade levels.

Regarding content area comparisons, there was evidence to indicate that (1) the numbertheory unit was more likely to have more RPTs than the units in the other two content areas;(2) tasks in the geometry units were less likely to design opportunities for patterns than weretasks in the units in the other two content areas; (3) tasks in the geometry units were morelikely to design opportunities for proofs than were tasks in the algebra units and less likely todesign opportunities for proofs than were tasks in the number theory unit; and (4) tasks in thenumber theory unit were more likely to design opportunities for rationales than were tasks inthe geometry units and less likely to design opportunities for rationales than were tasks in thealgebra units.

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Purposes of Patterns, Conjectures, and Proofs

Purposes of Patterns and Conjectures

Analysis of the purposes patterns and conjectures serve in a mathematics textbook series canreveal whether there are opportunities designed for students to understand an important connec-tion that exists in mathematics among patterns, conjectures, and proofs: patterns can lead to thegeneration of conjectures, which, in turn, can lead to the development of proofs. If a patternserved the purpose of a conjecture precursor, then I counted this as an opportunity designed forstudents to understand the relation I just described between patterns and conjectures. Similarly,if a conjecture served the purpose of a proof precursor, then I counted this as an opportunitydesigned for students to understand the relation I described previously between conjectures andproofs.

The majority of patterns in the sample served the purpose of conjecture non-precursors,with plausible patterns having higher proportions of conjecture non-precursors than definitepatterns (97% and 88%, respectively). A chi-square test showed evidence that there was ahigher proportion of conjecture precursor definite patterns than conjecture precursor plausiblepatterns.

TABLE 2Distribution by Percent (Rounded to the Nearest Integer) of Reasoning-and-proving Tasks in

Different Grade Levels and Content Areas

Reasoning-and-proving (sub)category

Grade Level Content Area

Sixthb (44) Seventhb (33) Eighthb (43) Algebrab (40) Geometryb (39)Number

Theoryb (49)

Identifying a PatternPlausiblea 1 9 6 9 1 1Definitea 17 17 19 19 17 22Totalc 11 9 11 11 9 12

Making a ConjectureConjecturea 7 2 2 2 4 10Totalc 3 1 1 1 2 5

Providing a ProofGenerica 0 0 0 0 0 1Demonstrationa 18 9 11 6 19 28Totalc 8 3 5 3 7 14

Providing a Non-proof ArgumentEmpiricala 3 1 2 2 3 0Rationalea 54 65 63 66 57 50Totalc 25 22 28 27 24 24

aThe values in this row show the percentage of tasks in each grade level or content area that were designed to offerstudents at least one opportunity to engage in the activity indicated, calculated in relation to the total number of reasoning-and-proving tasks in the grade level or content area.

bThe value in parenthesis shows the percentage of the reasoning-and-proving tasks in the grade level or content area.cThe values in this row show what percentage of the total number of tasks in the grade level or content area the tasks

in the cells above correspond to.

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Most conjectures in the sample served the purpose of proof non-precursors (70%). ABinomial test showed that there was significant difference between the proportions of proofprecursors and proof non-precursors.

Purposes of Proofs

Because only one task in the sample was coded as a generic example, I focused on the pur-poses of proof in the demonstration subcategory. Most demonstrations in the sample were foundto serve multiple purposes. Almost all served the purpose of explanation (94%), and about threeout of five (61%) contributed to the generation of new knowledge. About one out of six (17%)demonstrations served the purpose of verification. Only 3% of the demonstrations were coded asfalsifications.

DISCUSSION

In this section, I use the findings from the CMP analysis as a context to discuss issues oftextbook design in the domain of RP. The section is organized into two parts. The first partconsiders key findings from the CMP analysis to explore the question of what can be said aboutthe opportunities designed for students to engage in RP. Although the findings are specific toCMP, the discussion is broader and pertains to any textbook series.

The second part emanates from an important finding from the comparisons of the differentCMP units in the sample; namely, that the first unit in the textbook series was the richest unit interms of the RP opportunities designed for students. This finding raises some important issuesrelated to the distribution of RPTs in any textbook series: Does it matter how students’ opportu-nities for RP are allocated in a textbook series, or is what really matters to have these opportuni-ties in the textbook series? And, if it matters how the RP opportunities are allocated in atextbook series, what are some principles that textbook authors might follow in distributingthese opportunities across grade levels and content areas? As a field, we do not yet have answersto these questions, but we have a good understanding of several factors that are important toconsider in addressing the questions. In this part, I discuss one possible model for the distribu-tion of different kinds of RPTs across grade levels and two possible models for the distributionof different kinds of RPTs across content areas. Also, I compare these models with thosereflected in the way RPTs were distributed across grade levels and content areas in CMP. Thiscomparison aims to offer an image of how one could map the findings from a textbook analysisto the models.

Key Findings from the CMP Analysis

Reasoning-and-Proving Tasks

Four notable findings from the percentages of different kinds of RPTs in the sample are thelow percentages of tasks that were designed to engage students in empirical arguments, conjec-tures, and generic examples; and the high percentage of tasks that were designed to engage stu-dents in rationales. I use the phrase “low/high percentages of tasks” in relative terms,

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considering the percentages of tasks that were designed to engage students in different RP activ-ities. No claims are made about “ideal or appropriate percentages” for the distribution of differ-ent kinds of RPTs in textbooks.

As I explained earlier, students’ engagement in providing empirical arguments is likely toreinforce their common misconception that checking examples constitutes a proof for a generalmathematical claim (e.g., Knuth et al., 2002; Küchemann & Hoyles, 2001–03). Thus, I consideran advantage of CMP and any other textbook series to have a low percentage of tasks designedto engage students in empirical arguments.

Conjectures, contrary to empirical arguments, emphasize the need for providing proofsbecause they are statements that call for further examination (Arzarello et al., 1998; Harel &Sowder, 1998, 2007; Reid, 2002). Thus, the low percentage of tasks in the sample that weredesigned to engage students in making conjectures does not seem to be as desirable as the lowpercentage of tasks that were designed to engage students in empirical arguments.

Students’ engagement in providing rationales is desirable for the three main reasonsI discussed earlier. The second reason was that rationales offer a good choice of an argumentwhen the production of a proof is impractical, impossible, or undesirable. Yet, although thehigh percentage of rationales in the sample seems to be an advantage, before one can concludethat a high percentage of rationales in a textbook series is appropriate, one needs to considerthe following: Should some of these rationales be developed further to engage students inproofs? The analytic framework does not provide the information necessary to answer thisquestion.

The almost complete absence of generic examples from the sample was not desirable.Generic examples are important, especially in the middle grades, because they have the potentialto serve as a transitional stage between non-proof arguments, which are likely to prevail inelementary school mathematics textbooks, and demonstrations, which are more sophisticatedarguments than generic examples and are likely to prevail in high school mathematics textbooks.

Next I discuss how the findings from the studies by Knuth and his colleagues on CMPstudents’ understanding of proof (Knuth et al., 2002; Knuth & Sutherland, 2004) might connectto the findings from my textbook analysis. It is clear that with lack of data on how CMP wasimplemented in the classrooms of the students who participated in the Knuth studies, no definiteconclusions can be drawn in regard to explaining these students’ limited understanding of RP.Yet, the available data support important hypotheses on this issue that can guide future researchon the relations among the textbooks used, teaching, and student learning in the domain of RP.

The finding from the Knuth studies that older students showed a better understanding of thedeductive process might relate to the accumulated opportunities these students had to engagewith proofs throughout the years, and to the developmental psychology finding that students’reasoning resources develop with age (see Stylianides and Stylianides, 2008, for review anddiscussion of this psychology research). One could hypothesize that the finding that students ofall three grades demonstrated an overwhelming reliance on the use of examples to justify thetruth of mathematical claims would imply large numbers of CMP tasks that were designed toengage students in empirical arguments. This hypothesis is not supported by my textbook analy-sis: only a small percentage of the tasks in the sample were designed to engage students inempirical arguments.

Yet, there are many other hypotheses that explain the propensity of these students for empiri-cal arguments. An important hypothesis is that many of the CMP tasks that were designed to

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engage students in proofs ended up being implemented as empirical arguments in these students’classrooms. This hypothesis finds support in the limited guidance that CMP offered to teachers,in the teachers’ edition, for implementing with their students the proof opportunities designed inthe algebra, number theory, and geometry units (Stylianides, 2007c). Specifically, in 90% of theproof tasks in these units the guidance offered to teachers was limited to possible solution(s) tothe tasks. Only in 10% of the proof tasks in the examined units were there forms of guidance tothe teacher that included, in addition to possible solution(s) to the tasks, one or more of thefollowing three features: explanations about why students’ engagement in a proof task mattered,cautious points on how to manage student approaches to a proof task, and discussions thatsupported teachers’ knowledge of proof. These additional forms of guidance are important forthe fidelity of implementation of proof tasks in the classroom for at least three reasons: (1)teachers often have a limited understanding of proof (e.g., Knuth, 2002; Martin & Harel, 1989);(2) middle school teachers often lack images of what it means to promote proof in the class-room, presumably due to the long historical tradition that has restricted proof in school mathe-matics to the high school grades; and (3) proof tasks are hard to implement in the classroom notonly because they are cognitively demanding but also because students face serious difficultieswith them (e.g., Knuth et al., 2002; Küchemann & Hoyles, 2001–03).

Purposes of Patterns and Conjectures

The majority of patterns in the sample were coded as conjecture non-precursors. This was notsurprising given that a much larger proportion of tasks in the sample were designed to engagestudents in patterns than in conjectures. Also, the finding that the proportion of conjectureprecursor definite patterns was higher than that of conjecture precursor plausible patterns wasexpected because it is more meaningful to make conjectures based on uniquely determinedpatterns than based on non-uniquely determined patterns.

Most conjectures in the sample did not lead to the development of proofs so they were codedas conjecture non-precursors. This finding was surprising because a significantly larger propor-tion of tasks in the sample were designed to engage students in proofs than in conjectures.Students would likely be more motivated to engage in proofs of conjectures they developed thanto engage in proofs of mathematical claims provided to them by the textbook (see, e.g., Balacheff,1988).

As a field, we do not yet know whether there is a critical number of opportunities thatstudents need to receive in order to understand that in mathematics an important connectionamong patterns, conjectures, and proofs is that patterns can lead to the formulation of conjec-tures, which, in turn, can lead to the development of proofs. Yet, it is well known that themore opportunities students have to make connections among different mathematical ideasthe more likely students are to develop a deep understanding of them (e.g., Hiebert &Carpenter, 1992; Romberg & Kaput, 1999). Therefore, tasks that design opportunities forstudents to make connections among patterns, conjectures, and proofs are likely to helpstudents develop a deeper understanding of these activities. Also, such tasks are likely toengage students in mathematical activity related to RP that is authentic in its reflection of thewider mathematical culture (Lampert, 1992) because, for example, a mathematician does notmake a conjecture for the sake of a conjecture but makes a conjecture to designate a claim thatis worth proving.

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Purposes of Proofs

Almost all proofs in the sample served the purpose of explanation, whereas only one out ofsix proofs served the purpose of verification. These findings suggest that the examined CMPunits addressed the call of many researchers who stressed the need for an emphasis on theexplanatory purpose of proof in the school mathematics curriculum and argued against the tradi-tional emphasis on the verification purpose of proof (e.g., de Villiers, 1998; Hanna, 1990).

The high percentage of proofs in the sample that served the purpose of generation of newknowledge suggests good potential of the proof tasks in the examined units to motivate students’engagement in proofs: students are more likely to see the need for proof when they explore mathe-matical claims whose truth value is unknown to them. According to Balacheff (1988), “a contextpromoting awareness of the need for proof [is one that] holds some risk linked to uncertainty, andtherefore something to gain by entering a proving process” (p. 285; italics added). A mathematicalclaim of unknown (to the students) truth value can satisfy the condition of uncertainty; establishingthe truth value of the claim via the development of a proof can satisfy the condition of gain.

Another finding that is worth attention is that only a small percentage of proofs in the samplewere intended to establish the falseness of given mathematical claims. Yet, the knowledge ofhow to refute a claim in school mathematics is important, especially in classroom communitieswhere disagreements occur frequently and have to be resolved on mathematical grounds ratherthan by appeal to the authority of the teacher or the textbook (see, e.g., Ball & Bass, 2003; Reid,2002; Lampert, 1992).

Possible Models for the Distribution of Reasoning-and-Proving Tasks Across Grade Levels and Content Areas

In my discussion of the three possible models, I do not discuss percentages but rather general trendsfor the distribution of different kinds of RPTs in mathematics textbooks (discussion of percentageswould be premature considering the little we know from existing research and theory). The threemodels have not been the focus of discussion in prior research. I have singled them out for discussionfrom numerous possible models because they seem to receive better support from existing researchand theory (e.g., Balacheff, 1988; Dewey, 1903; Schwab, 1978; Stylianides & Stylianides, 2008).

One Possible Model for Distribution of Tasks Across Grade Levels

To manage the complexities of developing a possible model for the distribution of differentkinds of RPTs across grade levels in middle school mathematics textbooks, I focus only on thefour kinds of argument under “providing support to mathematical claims” in the analytic frame-work (i.e., empirical arguments, rationales, generic examples, and demonstrations). In addition,I set as the ultimate goal of the model to support the development of students’ ability to producemore (mathematically) sophisticated arguments as they progress through the grades. In addition,I base the model on the following premises.

1. The total number of tasks that a mathematics textbook series can include to engagestudents in the four kinds of argument is fixed, and

2. this number of tasks is distributed equally among the different grade levels.

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The important question that the model needs to address is how to distribute the available tasks inthe four kinds of argument across the different grades.

In order to develop this model, I organized the four different kinds of argument in a hier-archy based on their level of sophistication. A close examination of the definitions of thefour kinds of argument in the description of the analytic framework gives rise to the hierar-chy in Figure 1.

Empirical arguments, being invalid arguments, are in the bottom of the hierarchy. Rationales,being valid arguments but not as developed as proofs, are listed in the space between empiricalarguments and proofs. Demonstrations are at the top of the hierarchy because, contrary togeneric examples, they do not rely on the representativeness of a particular case.

Next I consider separately each of the four kinds of argument in the hierarchy and discusswhat the model proposes for them. The discussion is based on research I reviewed in previoussections of the article and, to avoid repetition, I do not repeat discussion of this research here.

Empirical arguments have no place in the model. As I already explained, empiricalarguments are likely to reinforce students’ common misconception that insufficient empiricalevidence is proof and so they should be avoided. This perspective is consistent with Dewey’s(1903) suggestion that whatever the preliminary approach to learning is it should not inculcate“mental habits and preconceptions which have later on to be bodily displaced or rooted up inorder to secure proper comprehension of the subject” (p. 217).

Rationales have an important role to play in the model for the three reasons I discussedearlier. Yet, in alignment with my conceptualization of rationales as a possible transitionalstage between empirical arguments and proofs and the goal of the model to support the devel-opment of students’ ability to produce more sophisticated arguments as they progress throughthe grades, the number of rationales in the proposed model decreases with grade levels. As Iexplain below, the number of proofs in the model increases with grade levels and so the num-ber of rationales had to decrease accordingly (see the first premise on which the model isbased).

The findings from developmental psychology research that students’ capabilities indeductive reasoning and proof develop with age (see Stylianides & Stylianides, 2008),8 give

8With this statement I do not suggest that students’ capabilities in deductive reasoning and proof develop only withage. The teacher has an important role to play in this development. Mathematics education research offers evidence that,in supportive classroom environments, even elementary students are able to successfully engage in deductive reasoningand other forms of reasoning involved in the development of proofs (see, e.g., Ball & Bass, 2003; Lampert, 1992; Reid,2002; Stylianides, 2007a; see also earlier research into young children’s abilities to reason, e.g., Donaldson, 1978;Gattegno, 1987).

FIGURE 1 A hierarchy of arguments based on their level of mathematicalsophistication.

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rise to the hypothesis that students are likely to benefit less if a disproportionally large per-centage of their opportunities to engage in proofs in the middle grades comes at a stagewhen their capabilities in deductive reasoning and proof are, from a psychological stand-point, not as highly developed. A second argument in favor of this hypothesis relates toissues of coherence in students’ school mathematical experiences (see “continuity princi-ple” in Stylianides, 2007b). Middle grades, as the predecessors of high school grades, areexpected to progressively prepare students for the proofs that are likely to prevail in highschool: if the bulk of students’ proof opportunities is found in the lower middle gradesrather than in the upper middle grades, this can create a discontinuity in students’ experi-ences with proof from middle school to high school. Therefore, in the model, the textbookopportunities for students to engage with proofs in the middle grades increase across gradelevels. Also, the percentage of proofs treated as demonstrations (as opposed to genericexamples) increases across grades in the model because older students are better equipped(both in terms of their cognitive development and their accumulated knowledge resources)to deal with such arguments.

To sum up, the model proposes the following statements for the four different kinds ofargument in the hierarchy.

1. No empirical arguments in middle school mathematics textbook series.2. Rationales are important across all grades, but their number should decrease across the

grades.3. The number of proofs should increase across the grades.4. The relative percentage of demonstrations as compared to the percentage of generic

examples should increase across the grades.

The examined CMP units seem to fulfill to great extent statement 1 in the model summary.The high percentage of rationales in the sample seems to fulfill also the first part of statement2. CMP does not fully fulfill the second part of statement 2 because tasks in the examinedsixth-grade units were more likely to design opportunities for rationales than were tasks in theexamined seventh-grade units. The almost complete absence of generic examples in the sam-ple does not leave much space for discussion for statement 4. The results from the grade levelcomparisons in CMP do not seem to be aligned with statement 3: the examined units in sixthgrade were more likely to have more proof tasks than the examined units in the other twogrades.

Two Possible Models for Distribution of Tasks Across Content Areas

Existing theory and research seem to provide support to two different models for the distribu-tion of different kinds of RPTs across content areas. A third possible model could emerge fromthe blending of the two models, but space constraints do not allow me to discuss this third modelhere.

First model. The first model is based on the premise that it is easier in certain content areasthan in others to design opportunities for students to engage in particular kinds of RPTs. Thus,an uneven distribution (in terms of statistical significance) of different kinds of RPTs acrosscontent areas is both expected and desirable.

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The focal textbook series seemed to be aligned with this model. My analysis showed thatthe geometry units were more likely to have a higher proportion of proof tasks than the alge-bra units and a lower proportion of proof tasks than the number theory unit. The higher pro-portion of proof tasks in number theory compared with geometry was surprising: the longhistorical tradition that associated proof with high school courses on Euclidean geometry cre-ated an expectation that there would be more proof tasks in middle school geometry in orderto better prepare students for their encounters with proof in high school geometry. As Iexplain below, there are some good ways in which one can (retrospectively) explain this unex-pected finding. Yet, this finding provides a good example of how the present textbook analy-sis can help challenge existing assumptions about how RPTs should be distributed intextbooks.

A possible explanation for why number theory had more proof tasks than the other twocontent areas has to do with students’ level of understanding of definitional concepts in differ-ent content areas. Number theory concepts (even numbers, factors, multiples, etc.) are acces-sible to even young children and attract a lot of attention by elementary and middle schooltextbooks. Therefore, an important component of the necessary foundation for formulatingproofs, namely, definitions, is presumably well developed in middle grades’ number theory,but this does not seem to be so much the case in the other two content areas because the sys-tematic development of algebra and geometry does not usually begin before the middlegrades. Another possible explanation for having more proof tasks in number theory than inalgebra and geometry has to do with that many results in number theory are understandable toeven young children, thus stimulating their curiosity to engage in explorations that naturallyraise the need for proof (see, e.g., Balacheff, 1988; Ball & Bass, 2003). In middle schoolgeometry, for example, it is not easy to find properties that are not evident as being true orfalse through suitable, simple to perform drawings. The asymmetrical representation of prooftasks in algebra and geometry might have historical origins: the deductive aspects of arith-metic began to be stressed in the teaching of algebra in the early 1800s, whereas geometry hasalways been taught according to the Euclidean tradition established in 300 B.C. (Davis &Hersh, 1981).

Although proof tasks seemed to be underemphasized in the algebra units of the focaltextbook series, tasks that designed opportunities for students to engage in patterns received a lotmore attention in these units than in the number theory and geometry units. This finding wasexpected. One explanation for it is that the topics usually covered in middle school algebra(solution of polynomial equations, basic properties of functions and graphs, etc.) supportnaturally the design of “real-world” tasks that involve pattern identification.

Second model. The second model for the distribution of RPTs across content areas claimsthat a balanced distribution (i.e., no statistically significant differences) of different kinds ofRPTs in a textbook series is desirable.

The following question about an extreme case scenario provides an argument in favor ofthis model: What would be the unintended message sent to students and teachers if almost allof the proof opportunities in a textbook series resided, for example, in geometry? Students andteachers would possibly be led to believe that in mathematics proofs have a central role toplay only in geometry. In fact, the “new math” movement caused many secondary schoolteachers to develop this belief. According to Davis and Hersh (1981), “as late as the 1950s

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one heard statements from secondary school teachers, reeling under the impact of the ‘newmath,’ to the effect that they had always thought that geometry had ‘proof’ while arithmeticdid not” (p. 7).

Another argument for a balanced distribution of the different kinds of RPTs across contentareas is that in the discipline of mathematics RP activities are central to mathematicians’ work inall content areas. Given that it is hard to make a case that specific RP activities prevail in certainareas of the discipline and given also theoretical ideas that the school curriculum should repre-sent, in an undistorted way, the structure of the disciplines (Schwab, 1978), one may say that thedifferent content areas in a textbook series should contain comparable numbers of opportunitiesfor students to engage in different kinds of RPTs.

IMPLICATIONS OF THE ANALYTIC/METHODOLOGICAL APPROACH

In this article, I proposed an analytic/methodological approach for the examination of the oppor-tunities designed in mathematics textbooks for students to engage in RP and I illustrated itsmerit in an analysis of a strategically selected American textbook series. In addition, I used thefindings from the textbook analysis as a context to discuss issues of textbook design in thedomain of RP that pertain to any textbook series.

The analytic/methodological approach can serve as a useful tool to study the relationsbetween the written and the implemented curriculum in the domain of RP. This is an importantpotential contribution to research because, as Stein et al. (2007) suggested, the field is currentlylacking tools that can support inquiry on the relations between the written and the implementedcurriculum in particular areas. Such an inquiry in the domain of RP could address questions likethe following: To what extent is teachers’ implementation of RPTs in their classrooms alignedwith what is designed in the written curriculum? How do the design features of differenttextbook series (e.g., the ways they distribute RPTs across grade levels and content areas) relateto the enactment of RPTs in classrooms? A textbook analysis such as the one discussed in thisarticle can offer a good picture of the place of RP in the written curriculum. To study the placeof RP in the implemented curriculum one could use the same analytic framework and follow asimilar methodological approach. Yet, instead of making inferences about the “expected formu-lation of the tasks” in order to code them, one can use the “actual formulation of the tasks,” thatis, how the tasks played out in the classroom.

Another important implication of the analytic/methodological approach is that it can serveas a basis for examining the place of RP in other content areas (probability, measurement,etc.), school levels (both elementary and high school), and kinds of school mathematics text-book series (conventional in addition to reform-based). Regarding the latter, a comparativestudy on the place of RP in different textbook series can be useful for various reasons: (1) bycomparing different textbook series against the same analytic/methodological approach, wecan gain a better understanding of the endeavors of each series to develop students’ RP capa-bilities; and (2) the findings of such a study can challenge existing notions of what studentscan be expected to learn about RP at different grade levels or challenge popular assumptionsfor how different kinds of RPTs should be distributed in textbooks (e.g., more proof tasks ingeometry). Yet, it is likely that parts of the analytic framework will need to be slightlymodified in analyses of non-middle school mathematics textbook series in order to capture

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particular characteristics of these school levels. For example, in an analysis of high schoolmathematics textbooks it would be meaningful to further code demonstrations into subcatego-ries that correspond to different methods of proof (proof by contradiction, mathematicalinduction, etc.).

Finally, the methodology I followed to develop and implement the analytic framework wouldbe useful (if not equally applicable) to textbook analyses on other mathematical activities thatare complex and are considered to be critical for student learning, such as problem solving.Prerequisite steps to such an analysis, however, are (1) the “decomposition” of problem solving,for example, into its constituent activities; and (2) the formulation of definitions for these activi-ties that are, at once, honest to the nature of problem solving in mathematics and operational in atextbook analysis.

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APPENDIX

More Examples of How CMP Tasks were Coded(for additional examples, refer to Stylianides, 2005)

Example 4 (adapted from CMP unit titled Growing, Growing, Growing, p. 6)

Each part of this example would be coded as a separate task. Part A would be coded as non-(reasoning-and-proving), because it does not design an opportunity for students to engage in any ofthe activities that comprise reasoning-and-proving. Part B would be coded as identifying a pattern(definite pattern). The pattern serves the purpose of a conjecture non-precursor, because the studentsare not expected in this or subsequent tasks to make a conjecture out of the pattern they will identify.

Problem: A. Cut a sheet of paper in half. Then stack the ballots and cut them in half. Continue to stack the

ballots you get each time and cut them in half. Count the ballots after each cut. Make a tableto show the number of ballots after 1 cut, 2 cuts, 3 cuts, and so on.

B. Look for a pattern in the way the number of ballots changes with each cut. Use your observa-tions to extend your table to show the number of ballots for up to 10 cuts.

Commentary (not appearing in the teachers’ edition): In this example, the situation in the task determines uniquely the pattern that needs to be

chosen. The students, in order to extend the table to show the number of ballots for up to 10 cuts,would have to multiply each time the previous number of ballots by 2.

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REASONING-AND-PROVING IN MATHEMATICS TEXTBOOKS 287

Example 5 (from CMP unit titled Kaleidoscopes, Hubcaps, and Mirrors, p. 67)

In this example, I focus on part c. Part c would be double-coded as identifying a pattern(definite pattern) and making a conjecture. The pattern serves the purpose of a conjectureprecursor because it gives rise to a conjecture, and the conjecture serves the purpose of a proofnon-precursor because the students are not expected in this or subsequent tasks to develop aproof in order to verify or refute their conjecture.

Problem:a. Describe the relationship between pairs of opposite sides and between pairs of opposite

angles for parallelogram ABCD.

b. Suppose the translation T1 matches point A with point B, and translation T2 matches point Awith point D.i. Where is the image of point A under the combination T1 * T2?ii. Where is the image of point A under the combination T2 * T1?

c. What conjecture would you make about commutativity of translations based on your answersto part b?

Commentary (not appearing in the teachers’ edition):In part c, the students are expected to notice the pattern that their answers to part b are the

same, and then to make the conjecture that the * operation is commutative for this set oftranslations.

Example 6 (adapted from CMP unit titled Thinking with Mathematical Models, p. 10)

The task in this example designs an opportunity for providing an empirical argument.Problem: A group in Ms. Hollister’s class finds that the equation model y = 5.5x fits their

bridge-thickness data [y corresponds to the breaking weight in pennies and x to the thicknessof the bridge in layers of paper strip]. The group conjectures that if they repeat the experimentwith cardboard, the relationship will be modeled by y = 4(5.5x).Is y = 5.5(4x) equivalent to y = 4(5.5x)? Is y =22x equivalent to y = 4(5.5x)? Give evidence to

support your answers.

Expected Formulation (Teachers’ Edition, p. 10):Yes, these three equations are all equivalent. The table below confirms this.

x 0 1 2 3 4 5

y = 5.5(4x) 0 22 44 66 88 110y = 4(5.5x) 0 22 44 66 88 110y = 22x 0 22 44 66 88 110

B

C

A

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288 STYLIANIDES

Commentary (not appearing in the teachers’ edition):In this example, the students are expected to confirm that the three equations are equivalent

to each other by examining six particular cases (for x = 0, 1, 2, 3, 4, 5).

Example 7 (adapted from CMP unit titled Looking for Pythagoras, p. 44)

In this example I focus on part b, which would be coded as providing a rationale.Problem: Square ABCD has sides of length 1. Draw a diagonal, dividing the square into two

triangles. Cut out the square and fold it along the diagonal.

a. How do the two triangles compare?b. What are the measures of the angles of one of the triangles? Explain how you found each

measure.

Expected Formulation for Part b (Teachers’ Edition, p. 44):The angle measures in each triangle are 45°, 45°, and 90°. The diagonal line divides the

corner angles into two equal angles, so the smaller angles must each be half of 90°, or 45°.

Commentary (not appearing in the teachers’ edition):Part b would be coded as a rationale and not as a proof, because the expected formulation

does not explain why the diagonal line divides the corner angles into two equal parts. Is itbecause the two triangles are congruent and the measures of corresponding angles are equal?

A B

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