International Journal of Educational Research 64 (2014) 119–131

Contents lists available at ScienceDirect

International Journal of Educational Research

jo u r nal h o mep age: w ww.els evier .c o m/lo c ate / i jed ur es

Reasoning-and-proving in mathematics textbooks for

prospective elementary teachers

Raven McCrory a,*, Andreas J. Stylianides b,1

a Michigan State University, Department of Teacher Education, 620 Farm Lane #114, Erickson Hall, East Lansing, MI 48824, USAb University of Cambridge, Faculty of Education, 184 Hills Road, Cambridge CB2 8PQ, UK

A R T I C L E I N F O

Article history:

Received 3 December 2012

Received in revised form 9 September 2013

Accepted 17 September 2013

Available online 19 October 2013

Keywords:

Reasoning-and-proving

Textbook analysis

Teacher education

A B S T R A C T

In the United States, elementary teachers (grades 1–5 or 6, ages 6–11 years) typically have

weak knowledge of reasoning-and-proving, and may have few opportunities to learn

about this important activity after they complete their teacher education program. In this

study we explored how reasoning-and-proving is treated in the 16 extant textbooks

written for mathematics courses for future elementary teachers in the United States to

offer insight into the opportunities designed for them to develop knowledge about

reasoning-and-proving. Our findings suggest that reasoning-and-proving is rarely

addressed explicitly in these textbooks. Although many textbooks have one section or

several sections in a single chapter (often the opening chapter) with content about

reasoning-and-proving, references to reasoning-and-proving concepts and methods are

rare outside of these few sections. We discuss methodological issues in studying the

treatment of reasoning-and-proving in textbooks for teachers and implications of our

findings for future research and teacher education.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Over the past decade there has been increased recognition of the importance of reasoning-and-proving in students’mathematical education.2 For example, in the United States, the setting of this paper, policy documents that set curriculumstandards for school mathematics (e.g., National Council of Teachers of Mathematics [NCTM], 2000; National GovernorsAssociation Center for Best Practices & Council of Chief State School Officers, 2010) have made calls for reasoning-and-proving to become central to all students’ mathematical experiences and across all school years.

One reason for the increased emphasis on reasoning-and-proving in school mathematics is its key role inmathematical sense making (e.g., Ball & Bass, 2003; Hanna, 2000; NCTM, 2000). This is consistent with current efforts toorganize classrooms as communities of mathematical discourse in which the validity of ideas rests on reason andargument (e.g., Carpenter, Franke, & Levi, 2003; Lampert, 2001). Although most examples of classrooms organized in

* Corresponding author. Tel.: +1 517 353 8565.

E-mail addresses: mccrory@msu.edu (R. McCrory), as899@cam.ac.uk (A.J. Stylianides).1 Tel.: +44 01223 767550.2 In this paper, we follow Stylianides (2008) in using the hyphenated term reasoning-and-proving to describe the overarching mathematical activity that

encompasses the family of activities broadly related to generating and justifying mathematical generalizations, such as identifying patterns, making

conjectures, and constructing arguments to prove or refute conjectures.

0883-0355/$ – see front matter � 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ijer.2013.09.003

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131120

this way have been taught by exemplary teachers, reasoning-and-proving can be important in all mathematicsclassrooms, regardless of instructional tradition, as long as students are expected to engage in exploring the truth ofmathematical assertions. These assertions may derive from the textbook, the teacher, or students themselves.

Another reason for the increased emphasis on reasoning-and-proving is recognition of the value for children to engage, indevelopmentally appropriate ways, with practices that are honest to what it means to do mathematics in the discipline (e.g.,Lampert, 2001; Stein & Smith, 2011; Stylianides, 2007a). This finds support from the work of educational thinkers such asBruner (1960) who argued that there should be continuity between what a scholar does on the forefront of a discipline andwhat a child does in approaching the discipline for the first time.

The importance of reasoning-and-proving in school mathematics raises, then, issues about opportunities offered toprospective elementary teachers, in university mathematics courses required for teacher preparation, to learn about thismathematical activity. Investigating these opportunities is crucial in light of (1) the difficulties that many elementaryteachers face with reasoning-and-proving (e.g., Martin & Harel, 1989; Simon & Blume, 1996); (2) the research findingthat students’ opportunities to learn are dependent on the quality of their teachers’ knowledge (e.g., Hill, Rowan, & Ball,2005); and (3) the recent research findings about teacher guides that offer inadequate support to teachers forsupporting their students’ learning of reasoning-and-proving (Stylianides, 2007b).

In this paper, we contribute to this body of research and explore the following question:

In textbooks for undergraduate mathematics courses for prospective elementary teachers in the United States, whatopportunities are designed for them to learn about reasoning-and-proving?

Our purpose is to understand, from the perspective of the future teachers and their instructors, what a textbookcan provide about reasoning-and-proving. We use the term ‘‘designed’’ and say ‘‘can provide’’ to emphasize that, nomatter what the content of the textbook is, it does not determine what students learn or what the instructorteaches. The textbook is a resource for, but not a determinant of, student learning. We take the perspective of seeking toexplore what students could learn from the textbook if they used it for studying issues and concepts of reasoning-and-proving, and what instructors might teach if they based their course primarily on the textbook. Our assumptionis that these two overlap to the extent that both students and instructors rely on the textbook for defining thecourse.

Although we focus on textbooks written for use in the United States, the methodological issues we faced would likelyapply to similar textbook analyses in other countries. Also, the issues raised by the findings of our examination can offer abasis for reflection on relevant issues in other countries where (to the best of our knowledge) no similar analyses have beenconducted.

2. Method

The data for this paper derive from analysis of 16 current mathematics textbooks in print in the U.S., for prospectiveelementary (grades 1–5 or 6, ages 6–11 years) teachers in the U.S. (a list of these textbooks is in Appendix A). The firststep in our analysis was to decide how to locate reasoning-and-proving in these texts. Unlike topics such as congruenceor regular polygons, reasoning-and-proving is a mathematical activity that spans mathematical topics and may appearin multiple places in textbooks. Taking the perspective of the students and instructors as users of the textbook, wefocused our analysis on locations in textbooks where authors explicitly addressed issues related to reasoning-and-proving. That is, we sought reasoning-and-proving in the textbook without reading every chapter and section,imagining a student who wants to recall a specific idea mentioned in class or study for a test (asking a question such as,‘‘what does it mean to do an indirect proof?’’), or an instructor who is basing the sequencing and flow of the course onthe textbook. This turned our attention to the published maps of each textbook; specifically, tables of contents andindexes.

This approach supports a consistent analysis of textbooks. In addition, it allows us to take the perspective of thestudent aiming to study reasoning-and-proving using the textbook, and of an instructor planning a course andinterested in focusing on reasoning-and-proving. In each case, these users might look for explicit references toreasoning-and-proving within the textbook. Without such links, a student would need to search page by page, orremember something from earlier uses of the textbook. Similarly, an instructor who aimed to prepare lessons building acoherent approach to reasoning-and-proving while using the textbook as a primary source might also rely on theauthor’s pointers to reasoning-and-proving as for content, organization, and trajectory. In short, the table of contentsand index of a textbook give users a means for studying, learning, or teaching particular content. The disadvantage ofthis approach is that it may miss some instances of reasoning-and-proving that are not referenced in either the table ofcontents or the index. Since this limitation reproduces the problem a student or an instructor would have, our method islegitimate for our purposes.

To begin this process, we developed a list of terms that would be logical indicators of reasoning-and-proving. Theseterms related broadly to the two major activities comprising reasoning-and-proving (Stylianides, 2008), namely,generating mathematical generalizations and justifying mathematical generalizations. Under generating mathematicalgeneralizations we had the terms conjecture, generalization, and inductive reasoning (or simply reasoning). Underjustifying mathematical generalizations we had: (1) terms that relate to the development of mathematical arguments:

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131 121

assumption, compound or conditional statements (including specifically contrapositive and converse), deductive and indirect

reasoning (or simply reasoning), definition, modus ponens (or law of detachment) and modus tollens (or contrapositive); (2)terms that relate to different functions that an argument or a proof can serve: explanation, falsification (or refutation),

justification (or verification); and (3) terms that relate to different proof methods: contradiction, counterexample, indirect

proof, mathematical induction, reductio ad absurdum, as well as the term proof itself (or proving). In the tables of contents,we scrutinized the titles of all chapters and sections (including sub-sections), looking for terms that indicated places inthe textbooks likely to include content about reasoning-and-proving. In every case, we used the most detailed version ofthe table of contents available.

We used this method not to argue that any specific approach to including reasoning-and-proving in the table ofcontents is best, but rather to investigate whether and how representation of relevant terms in the table of contents isindicative of the role of reasoning-and-proving in the text and of the kind of support given to students (and instructors)to find content about reasoning-and-proving. Note that, to be consistent across textbooks, we analyzed only studenteditions.

In our examination of indexes, we considered all of the terms that we listed earlier, thus, we conjectured, increasing thechances of finding reasoning-and-proving addressed explicitly in the textbooks. We do not suggest that any given termshould be included. Rather, we use their presence as a way to find relevant content, and we suggest that without references toat least some of these terms, students (and their instructors) may have difficulty finding content about reasoning-and-proving in a textbook even if such content is actually there. Also, we consider the total number of pages indexed by a term tobe an indication of the emphasis placed on that term specifically, and reasoning-and-proving generally, especially if thatnumber of pages is viewed in relation to the total number of pages in the textbook and the overall pages referenced by any ofthe reasoning-and-proving terms.

Once we identified the relevant pages through the tables of contents and indexes, we inspected those pages to understandwhat the textbooks offer and how they are similar and different. In particular, we asked the following questions:

1. I

s the referenced page about reasoning-and-proving? 2. O n the referenced page, how is the term used or presented? 3. D oes the referenced page indicate how the term is connected to, or used in, other parts of the textbook?

Question 1 is not easily answered. For our analysis, we took a qualitative approach, seeking evidence that the textbookused the term or addressed the topic explicitly, not merely including them, but using them and/or explaining them in thecontext of reasoning-and-proving. For example, an indexed reference to the term ‘‘proof’’ could point to a page with thefollowing sentence, and nothing more:

‘‘Throughout the book, we use proof as a primary method of argument.’’

This sentence does not provide the student, or the instructor, with content from which they could learn reasoning-and-proving. In other cases, the term ‘‘proof’’ in the index could point to a definition of proof or an actual proof of a theorem.Although we did not quantify these differences, we wrote memos describing the role of reasoning-and-proving found onindexed pages. We used these memos to compare and contrast textbooks, aiming to develop qualitative understanding ofsimilarities and differences found in this corpus of textbooks.

Question 2 (On the referenced page, how is the term used or presented?) calls for a qualitative analysis, since we did not startthe analysis with a list of expected uses or presentations of terms. In the results that follow, we describe instances of use andpresentation, and, in some cases, report patterns that emerged across textbooks.

Question 3 (Does the referenced page indicate how the term is connected to, or used in, other parts of the textbook?) isrelatively easy to assess. Here we look for language that references other instances of the concept or term, or connects it toother parts of the textbook.

In cases where we found no references in the table of contents or in the index, we investigated the textbook furtherto appraise whether there was an explanation for the lack of explicit references. Our method was to simply peruse thesetextbooks (n = 2) to develop a possible explanation. These two cases are explained in Section 3.1.5.

The range of possibilities for what is indexed or referenced suggests both a strength and a weakness of our method. Thestrength is that we analyzed each textbook using a similar method, from the perspective of a student using the textbook toget help with reasoning-and-proving, or an instructor aiming to design a course or lesson that emphasized reasoning-and-proving based on the textbook. This gives a perspective about students’ opportunities to learn from and with the textbookirrespective of what goes on in the classroom; and a perspective about what an instructor could use (and what might needto be supplemented). The weakness is that the method may not give a complete picture of instances of reasoning-and-proving in the textbook. In addition, we make no claim about how the instructor actually uses the textbook in practice or ofhow the textbook, or any course using the textbook, actually impacts students. For these reasons, our analysis is notintended to quantify or assess these textbooks individually; instead, we provide descriptions of characteristics of thiscorpus of textbooks based on this method of analysis, and of similarities and differences across the textbooks. Finally wediscuss the emerging profile of the corpus of textbooks, aiming to connect our findings to current thinking about reasoning-and-proving.

Table 1

Reasoning-and-proving in tables of contents.

Included in: Chapter with reasoning or

proof/proving in the title

Chapter on

logic

Chapter on problem

solving

Different

chapter/section

Not explicitly

included

Bassarear (2012) U

Beckmann (2011) U

Bennett and Nelson (2012) U

Billstein et al. (2013) U

Brumbaugh et al. (2005) U

Darken (2007) U U

Jensen (2003) U

Jones et al. (2000) U

Long and DeTemple (2009) U U

Masingila et al. (2006) U

Musser et al. (2004) U

O’Daffer et al. (2008) U

Parker and Baldridge (2004, 2006) U U

Sonnabend (2010) U

Sowder et al. (2010)

Wheeler and Wheeler (2009) U U

Totals 6 2 4 6 2

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131122

3. Results

3.1. Reasoning-and-proving in tables of contents

The findings from analysis of the table of contents show that textbooks vary in the location of table-of-contents referencesto reasoning-and-proving. We identified five different categories of textbooks with respect to this analysis.3 Table 1 showsthe results for each textbook in the corpus of 16.

The categories are:

1. C

3

tex

hapter with reasoning or proof/proving in the title

2. C hapter on logic 3. C hapter on problem solving 4. R easoning-and-proving in a different chapter, not included in 1–3 above 5. N o table-of-contents level reference to reasoning-and-proving

In what follows we discuss each category separately and present few illustrative examples.

3.1.1. Chapter with reasoning or proof/proving in the title

Six textbooks (Bennett, Burton, & Nelson, 2012; Brumbaugh, Moch, & Wilkinson, 2005; Darken, 2007; Parker &Baldridge, 2004; Sonnabend, 2010; Sowder, Sowder, & Nickerson, 2010) have a chapter with a title that indicatesattention to reasoning-and-proving. The chapter titles included Sets, Functions, and Reasoning (Bennett), Deductive

Reasoning and the Real Numbers (Darken). Each of the six used either ‘‘reasoning’’ or ‘‘proof’’ in these chapter titles. Thechapters in this category typically treated reasoning-and-proving as a topic, explaining logical statements, providingexamples of formal or informal proofs, addressing deductive and/or inductive reasoning, definitions, and counter-examples. The exception to this is Sowder, which includes reasoning in the title of every section of the textbook, butdoes not indicate any particular subsection where reasoning-and-proving is the focus. Excluding the Sowder textbook,sections on reasoning-and-proving in these chapters ranged from 8 pages (Brumbaugh) to 29 pages (both Parker andSonnabend).

3.1.2. Chapter on logic

Two textbooks (Musser, Burger, & Peterson, 2010; Wheeler & Wheeler, 2009) include issues related to reasoning-and-proving in chapters on logic. Both of these chapters are expositions of logic that cover truth tables, Venn diagrams, and typesof logical statements. Musser makes reference to mathematical proofs and includes several deductive proofs. Wheeler, onthe other hand, includes only logical arguments that use symbols or linguistic examples.

For simplicity, in the first citation of each textbook, we provide a complete reference. Thereafter, we use only the first author’s name to designate a

tbook, followed by the date in cases where there is more than one volume by the same author.

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131 123

3.1.3. Chapter on problem solving

Four textbooks (Billstein, Liebeskind, & Lott, 2013; Long, DeTemple, & Millman, 2012; Masingila, Lester, & Raymond,2006; Wheeler, op. cit.) have sections on issues related to reasoning-and-proving in chapters on problem solving. Thesesections closely resemble in both length and content the six textbooks of the first type. They cover topics such as logicalstatements, truth tables, inductive and deductive reasoning, and range from 7 to 11 pages. Only one of the four – Masingila –actually includes a proof in the chapter, proving two theorems: (1) the sum of two even numbers is even and (2) that thereare infinitely many natural numbers.

3.1.4. Reasoning-and-proving in a different chapter

Six textbooks cover reasoning-and-proving in a different chapter (Bassarear, 2012; Beckmann, 2011; Darken, op. cit.;Long, op. cit.; O’Daffer, Charles, Cooney, Dossey, & Schielack, 2008; Parker & Baldridge, 2006). Here, we found the greatestvariation across textbooks in what was included and how it was presented. Although three of these textbooks (Bassarear,Long, O’Daffer) featured content similar to what was found in the first three categories above (e.g., logic, Venn diagrams,direct exposition about deductive and inductive reasoning, one or more example proofs), three textbooks (Beckmann,Darken, Parker) provide extensive and qualitatively different exposition of ideas related to reasoning-and-proving. Thesethree are discussed in more detail below.

Beckmann has a section titled Explaining Solutions in which she uses the NCTM (2000) Reasoning and Proof Standard andgives detail about creating and writing a mathematical explanation. In a section titled Using Moving and Additivity Principles

to Prove the Pythagorean Theorem, she defines proof as follows:

Fig. 1. ‘

From P

Proof is required to know for sure that a statement really is true. A proof is a thorough, precise, logical explanation forwhy a statement is true, based on assumptions or facts that we already know or assume to be true. . .. Proofs are one ofthe important aspects of this book, too, even if we don’t usually call our explanations proofs. (p. 546, emphasis inoriginal)

Darken treats reasoning-and-proving as a strand throughout the textbook, including a section at the end of every chaptercalling attention to whether and how the NCTM Reasoning and Proof Standard is addressed. The textbook includes a sectionon proving the Pythagorean theorem in a chapter on measurement, in which the author gives a thorough explanation of theproof.

In Volume II (2006), Parker has a chapter titled Deductive Geometry that begins with an introduction to proof: ‘‘Ingeometry, a written logical argument is called a proof’’ (p. 71). In a section in this chapter titled Unknown Angle Proofs, theauthors explain the difference between a ‘‘teacher’s solution’’ and a proof, and give nine examples of teacher’s solutions andproofs. They explain why proofs are important, and ‘‘should be fun’’; and they give a graphic of the features of an ‘‘ElementaryProof’’ reproduced (with permission) here in Fig. 1. This exposition illustrates explicit attention to reasoning-and-proving:not only providing a mathematical argument, but also calling attention to the steps, reasons, and presentation of theargument.

In particular, Fig. 1, in conjunction with text on the nearby pages of the Parker (2006) volume, calls attention to statingwhat is given; stating what is being proved; using an illustration that is complete, clearly labeled, and connected to both thestatement of the problem and the proof; and providing reasons or facts that support steps of the proof. Although thisillustration is in the geometry volume of the textbook, Parker provides a similar illustration in the first volume for what hecalls a teacher’s solution to any mathematical problem. A teacher’s solution does not reach the level of proof, but it is a guide toreasoning and problem solving using diagrams, knowns, and logic.

‘Elementary Proof’’.

arker and Baldridge (2006, p. 77).

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131124

3.1.5. No table-of-contents level reference to reasoning-and-proving

Two textbooks (Jensen, 2003; Jones, Lopez, & Price, 2000) do not cover reasoning-and-proving in a way that is visible inthe table of contents. Jones is a paperback textbook unlike most of the others: it has no index and is considerably shorter. Ourhypothesized explanation for Jones’ lack of table-of-contents references to reasoning-and-proving is that it is generally a lesscomprehensive and less-indexed text than the others. We did find one instance of proof in the chapter on geometry; therecould be others that we did not find.

Jensen is also unusual in this corpus of textbooks, having more proofs than any other of the textbooks. There are nochapter headings that include any of the words in Table 1, but the entire textbook is organized by proving mathematicaltheorems, preceded by axioms and definitions. In the preface, Jensen writes:

The presentation of material in this book is in the old-fashioned style of definition, theorem, proof used in Euclid’sElements. . . . The first step towards understanding a concept is to have a clear definition of it. . .. Having chosenthe definition, we then prove as theorems the equivalence of alternative definitions. . .. The mention of proofsscares mathematics students at all levels. But a proof is nothing but an explanation for why a theorem is true, andeverybody wants to know that. Students’ antipathy to proofs comes from their experience of proofs which don’texplain anything to them. . .. A proof is also an exercise in problem solving. The problem is, ‘why is this theoremtrue?’ (p. viii–ix)

Proof is interwoven throughout the textbook, and there is no chapter or section set aside to explain the nature of proof,methods of proving, logic, or other aspects of reasoning-and-proving.

3.2. Reasoning-and-proving in indexes

In Table 2, we indicate the presence of references to terms associated with reasoning-and-proving in indexes of thetextbooks. None of the textbooks indexed the terms falsification, justification, mathematical induction, reductio ad absurdum,refutation, or verification, and those terms are not shown on the table. Most often, the references in the indexes of thetextbooks point to the chapters described above in which issues of reasoning-and-proving are presented.

Although the textbooks tend to be quite long (mean = 654.5 pages, median = 731.5 pages), the indexed terms appear toreceive relatively little attention in the textbooks. Specifically, of the whole collection of 24 terms related to reasoning-and-proving that had index entries, few appear in each textbook’s index (mean = 7.5 terms, median = 8.0 terms). Furthermore,each indexed term appears in a relatively small number of textbooks (mean = 6.3 textbooks, median = 5.5 textbooks) and istreated in a relatively small number of pages in each textbook with a few exceptions to this pattern (notably the term‘‘reasoning’’) that account for a rather large difference between the mean and median number of pages (mean = 6.7 pages,median = 2.0 pages).

Some textbooks treat terms that appear in Table 2, but either do not name them according to their formal names orinclude them much less frequently in the index than in the textbook itself. An example of the former is found inO’Daffer: Although they cover explicitly the essence of the laws of detachment and contrapositive, they do not namethem as such (they call them Logic Rule A and Logic Rule B, respectively, p. 27) and therefore one does not find theseterms in the index. An example of the latter is found in Beckmann: She devotes half of her first chapter (Problem Solving)to ‘‘Explaining Solutions’’ and writes explicitly about what constitutes a good explanation in mathematics and how todevelop a good explanation. She writes about assumptions and provides a list of criteria for good explanations.Throughout the textbook, she gives explicit examples of good explanations, and she uses the term explanation in lieu ofproof, as she explains in her definition of proof on page 546 (the relevant quotation was offered in Section 3.1.4). Butproof is indexed only once and explanation five times.

In the following sections, we present our findings of the content of the textbooks referenced by the terms we selectedrelated to reasoning-and-proving, and we offer illustrative examples.

3.2.1. Assumption

Two textbooks (Brumbaugh; Darken) have assumption in the index, but neither of them explains the role of assumptionsin reasoning-and-proving. In Brumbaugh, the index refers to the chapter on statistics where statistical assumptions areexplained. In Darken, on the pages indexed, the only use of the word assumption is in the sentence, ‘‘You spotted a patternand made an assumption that the pattern would continue’’ (p. 10). In the pages following this sentence, many examples aregiven of faulty conclusions based on incorrect assumptions, but the word assumption is not used again and the importance ofassumptions is not discussed.

3.2.2. Logical statements: compound statements, conditional statements, converse, law of detachment (modus ponens)

Twelve of the textbooks include references to one or more types of logical statements in their chapters on reasoning orlogic. These terms are not referenced outside of those sections in any of the textbooks.

3.2.3. Contradiction (or proof by), indirect proof, indirect reasoning, contrapositive (modus tollens), reductio ad absurdum

Five textbooks talk about indirect proof (Bassarear; Brumbaugh), contradiction (Brumbaugh), or proof by contradiction(Beckmann; Long; Parker, 2004). On the pages referenced by the index, they mention this concept in the following contexts:

Table 2

Terms Associated with Reasoning-and-Proving in the Indexes.a

Book

lengthb

Index Glossary Assumption Compound

Statements

Conditional

Statements

Conjecture Contradiction Contrapositive Converse Counterexample Deductive

reasoning

Total unique number of pages indexed by the term

Bassarear (2012) 657 Y N 5 2 2 2 7

Beckmann (2011) 704 Y N 1

Bennett and

Nelson (2012)

813 Y N 7 1 2 2 2 12

Billstein

et al. (2007)

920 Y N 3 6 2 2 4 1 4

Brumbaugh

et al. (2005)

230 Y N 1 2 5 1

Darken (2007) 838 Y N 1 3 1 2 2 1 38

Jensen (2003) 378 Y N 1

Jones et al. (2000) 306 N N

Long and

DeTemple (2012)

825 Y Yd 2 1* 1 2 3

Masingila

et al. (2006)

356 Ye Y 1* 1* 1* 1 1 1*

Musser et al. (2011) 944 Y N 1 1 2 1 2 7f

O’Daffer et al. (2008) 853 Y Yg g 8 2 3 3* 16*

Parker and Baldridge

(2004, 2006)h

441 Yh N 2

Sonnabend (2010) 759 Y N 2 1 2 3 13

Sowder et al. (2010) 791 Y Y 1* 1* 1* 7

Wheeler and

Wheeler (2009)

657 Y N 1 7 3 1 1 30i

Total books indexing

the term

2 4 10 8 4 9 10 10 11

R.

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esearch

64

(20

14

) 1

19

–1

31

1

25

Book

lengthb

Index Glossary Definition Detachment,

law of

Explanation Generalization Indirect

proof

Indirect

reasoning

Inductive

reasoning

Proof Reasoning

Total unique number of pages indexed by the term

Bassarear (2012) 657 Y N 1 1 3 3 9 13

Beckmann (2011) 704 Y N 5 1

Bennett and

Nelson (2012)

813 Y N 1 1 4 102c

Billstein et al. (2007) 920 Y N 1 2 2 3 3 30

Brumbaugh

et al. (2005)

230 Y N 9

Darken (2007) 838 Y N 1 8 1

Jensen (2003) 378 Y N

Jones et al. (2000) 306 N N

Long and

DeTemple (2012)

825 Y Yd 1 1 18 75d

Masingila et al. (2006) 356 Ye Y 1* 1 *

Musser et al. (2011) 944 Y N 71f 5 1 9

O’Daffer et al. (2008) 853 Y Yg 15* 13* 24

Parker and Baldridge

(2004, 2006)h

441 Yh N 1 4

Sonnabend (2010) 759 Y N 2 10 11

Sowder et al. (2010) 791 Y Y 2 2*

Wheeler and

Wheeler (2009)

657 Y N 5 21i

Total books

indexing the term

5 3 2 5 1 3 11 5 8

* Indicates that the term is included in the glossary.a The terms falsification, justification, mathematical induction, reductio ad absurdum, refutation, and verification are not included on the table because none of the books indexed them.b Book length excludes answers to selected problems, appendices, indexes, and any templates at the end of the book for use in activities.c Bennett and Nelson includes sections titled ‘‘Reasoning and Problem Solving’’ as part of problem sets at the end of each section. These are all indexed under ‘‘reasoning’’. The problems in these sections often

include worked out examples of reasoning and illustrations of one or more of the Polya strategies.d Long and DeTemple has a ‘‘Mathematical Lexicon’’ that explains word origins. Fifty of the pages indexed by ‘‘reasoning’’ point to the sections on algebraic reasoning; fifteen of the pages point to the sections on

proportional reasoning.e For one volume only.f Musser et al. entry for ‘‘deductive reasoning’’ refers the reader to ‘‘direct reasoning’’. Those pages are included here. Entries under ‘‘definition’’ include references to 110 terms on 117 pages. 36 of those pages

include multiple references, resulting in 71 separate pages referenced under definition.g O’Daffer et al. glossary is titled ‘‘Glossary of Geometric Terms’’. The book includes a page indexed by ‘‘assuming the converse’’.h The two volumes of Parker and Baldridge are 226 and 215 pages respectively. The second volume does not include an index.i The index entry for ‘‘deductive reasoning’’ in Wheeler & Wheeler points to ‘‘logic’’. All the pages indexed by these two terms are included in this total. The 21 references listed under ‘‘reasoning’’ are indexed by

the term ‘‘Reasoning and proof standard’’.

Table 2 (Continued)

R.

McC

rory

, A

.J. Sty

lian

ides

/ In

terna

tion

al

Jou

rna

l o

f E

du

catio

na

l R

esearch

64

(20

14

) 1

19

–1

31

12

6

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131 127

� t

o show that ‘‘you can’t divide by zero’’ and also the primes in the Sieve of Eratosthenes (Bassarear); � t o prove that H3 is irrational (Beckmann, p. 685); � c ontradiction as a synonym for counterexample and indirect proof as eliminating contradictory answers in a multiple

choice test (Brumbaugh, pp. 213, 217);

� a s a logical rule and to prove that H2 is irrational (Long, pp. 64, 428); � t o show that there are infinitely many prime numbers, that H2 is irrational, and that if a whole number is not a square, then

its square root is irrational (Parker, 2004, pp. 123, 224–225).

Four other textbooks (Billstein; Masingila; Musser; Sonnabend) have in their index the term indirect reasoning, which istypically used to describe the kind of reasoning applied in indirect proofs. For example, Billstein introduces the use of indirectreasoning as a problem-solving strategy and describes it as follows:

A different type of reasoning, indirect reasoning, uses a form of argument called modus tollens. For example, consider the

following true statements:

If a figure is a square, then it is a rectangle.

The figure is not a rectangle

The conclusion is that the figure cannot be a square. Modus tollens can be interpreted as follows:

If we have a conditional accepted as true, and we know the conclusion is false, then the hypothesis must be false. (Billstein,p. 61, emphasis in original)

Similar quotes can be found elsewhere (e.g., Musser, p. 152). These sections first define and then illustrate what the termmeans. The explanation follows, rather than precedes, the elaboration of this reasoning type. Parker (2004), for example,notes the following about proof by contradiction, after they have used it to prove the statement that there are infinitely manyprime numbers:

This method of proof is called proof by contradiction. Proofs using this technique boil down to the following logic:

1. Either a statement or its negation must be true—but not both. For instance, the negation of ‘‘There is an infinite number of

primes’’ is ‘‘There is a finite number of primes’’; one and only one of these statements is true.

2. We assume that the negation is true and show how that assumption leads to a logical contradiction. In the above proof,

the assumption that the number of primes is finite was shown to be incompatible with a known fact (Lemma 3.2).

3. We conclude that our assumption is wrong, so the original statement is true.

Proof by contradiction is a powerful logical technique. (Parker, 2004, p. 123)

It is not always the case, however, that the authors index their use of a particular method. For example, Long proves theirrationality of H2 using proof by contradiction (pp. 428–429) but that page number is not included in the index entry forproof by contradiction, even though they explicitly say on p. 429 that this is the proof method they are using. In othertextbooks, the concept is presented, but is not indexed or is indexed with a different term. For example, Bennett includes thelaw of contraposition (p. 112, found in the chapter on reasoning and proof) but does not link it to the terms proof by

contradiction or indirect proof.

3.2.4. Counterexample (or proof by)

Ten textbooks include the term counterexample in their index. Several authors emphasize that one counterexampleis enough to refute a general mathematical claim whereas finding numerous examples does not prove a claim.For example, Billstein writes: ‘‘When we find an example that contradicts the conjecture, we provide a counterexampleand prove the conjecture false in general.. . . It is enough to exhibit only one counterexample’’ (p. 23). Billstein addsthat high school students have considerable difficulty understanding the idea of a counterexample, and confusedeductive and inductive reasoning. Parker (2004) connects further the notion of counterexample with the work ofteaching:

A counterexample to a mathematical statement is a [sic] example where the statement is false. To prove that astatement is true, one must prove it true for all numbers while to show that it is false requires only one

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counterexample. Counterexamples are incredibly useful when teaching mathematics. Use this idea of instructional

counterexamples to answer the questions below. (p. 123, emphasis in original)

The authors then describe three different situations with children claiming something that is false, and ask theprospective teachers to generate counterexamples in order to convince the children that their claims are false.

3.2.5. Deductive and inductive reasoning

Eleven textbooks include in their index the concepts deductive or inductive reasoning. All of them include both termsexcept Sowder, which includes inductive but not deductive reasoning. The textbooks identify these two types ofreasoning as the ones most often used in mathematics. The following quote from Masingila (2006) is illustrative of thisapproach:

Logical reasoning plays a role in many conversations that we have. When we try to convince someone to support acertain stance, we have to provide compelling and valid arguments. In mathematics, we use the same tools ofdeductive and inductive reasoning that are used in the courtroom in settling cases to solve mathematical problemsand prove mathematical theorems. In this section, we examine the rubrics of deductive and inductive reasoning andthe types of statements involved in logical arguments. (p. 11)

Most of the textbooks include these references to reasoning in the sections designated in the tables of contents; theindexes generally provide no additional pages where the terms are used or explained.

3.2.6. Definition

Only two of the textbooks (Bennett; Parker, 2004) index definition. Bennett talks about definitions in their discussion ofmathematical systems in their chapter titled Geometric figures:

A mathematical system consists of undefined terms, definitions, axioms, and theorems. There must always be somewords that are undefined. Line is an example of an undefined term in geometry. We all have an intuitive idea of what aline is, but trying to define it involves more words, such as straight, extends indefinitely, and has no thickness. Thesewords would also have to be defined. To avoid this problem of circularity, certain basic words such as point and line areundefined terms. These words are then used in definitions to define other words. Similarly, there must always besome statements, called axioms, that we assume to be true and do not try to prove. Finally, axioms, definitions, andundefined terms are used together with deductive reasoning to prove statements called theorems.’’ (p. 564,formatting in original)

Parker (2004) devotes a section to Definitions, Explanations, and Proofs (Section 5.1), emphasizing the importance ofdefinitions in mathematical reasoning. They give a list of four different definitions of even number that third-grade studentsmay know, and note:

The classroom repercussions are obvious. Suppose that a third grade teacher begins a discussion of even numberswithout mentioning any definition. As she talks, the students will each be trying to relate her words to the definitionthey have chosen. Different students will understand the discussion differently, and many will be confused. A betterteaching strategy proceeds as a mathematician would. The teacher chooses and clearly states one of the abovedefinitions and then links it to the others. In the end, students understand all four definitions. (p. 110)

3.2.7. Proof, reasoning

Five textbooks have the term proof in their index. The indexed pages sometimes include an actual proof or a generaldefinition or explanation of proof. Nine textbooks index reasoning. Some authors focus mostly on reasoning and mentionproof through a reference to the NCTM (2000) Reasoning and Proof Standard. For example, Bassarear has a section onreasoning and proof that begins with an example from a national standardized test of students giving answers that do notmake sense. He then asks the prospective teachers to read the NCTM Reasoning and Proof Standard (reproduced in thetextbook) to see ‘‘what NCTM has to say about reasoning’’ (p. 30). Finally, in the summary of the chapter the author notesabout reasoning and proof:

7. R

easoning and proof are not just things to be learned by older students; they are also an essential part of elementaryschool mathematics.

8. T

here are three kinds of reasoning used in understanding mathematical ideas and solving mathematical problems:inductive, deductive, and intuitive reasoning. (p. 58)

Long has 18 pages indexed by the word ‘‘proof’’, the most of any of the textbooks. Eleven of these pages are proofs oftheorems including the Pythagorean Theorem, the irrationality of H2, and Thales’ theorem. Several of the proofs are marked‘‘optional’’. None of the pages indexed give an explanation, or definition, of what constitutes a proof.

Masingila, who has reasoning but not proof in the index focuses mostly on reasoning rather than proof throughout thetextbook. Indicative of this emphasis is the introductory page of Chapter 1. Here, and at the start of each chapter, the authorsprovide a list of the NCTM standards covered in the chapter, naming each with a single word. Standard 2, Reasoning and Proof,

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131 129

is referred to simply as ‘‘Reasoning’’. Masingila offers a definition of proof in the Glossary (although proof does not appear inthe index): ‘‘Proof: A logical, irrefutable argument to demonstrate the truth of a mathematical result’’ (p. 417,).

Definitions of proof can be found in the other two textbooks; thus a total of three of these textbooks provide a definition ofproof that we were able to find through the index or table of contents. Immediately before presenting a proof of thePythagorean Theorem, Beckmann defines proof as follows:

A proof is a thorough, precise, logical explanation why something is true, based on assumptions or facts that arealready known or assumed to be true. So a proof is what establishes that a theorem is true. (p. 578, formatting inoriginal)

Parker (2004) defines proof as follows: ‘‘A proof is a clear, detailed explanation why a mathematical fact is true’’ (p. 110).The authors also discuss prospective teachers’ experiences with proofs and what proofs in elementary school should serve:

[A] proof is not an explanation of a mathematical fact, it is an explanation of why the fact is true. In that sense proofs aredetailed versions of classroom explanations. . . . You may recall proofs from your high school classes with someanxiety. This is most likely the unfortunate byproduct of encountering proofs without the necessary prerequisiteskills. To prepare students for the rigorous proofs of high school mathematics, students need exposure to proof-likethinking much earlier; they must develop what are often called ‘‘critical thinking skills.’’ Proofs in elementary schoolserve that purpose; they are informal and the focus is on getting students to reason out why an idea is true and toexplain it using models, pictures, or arithmetic (and later with letters and algebra). (p. 110, emphasis in original)

4. Discussion

In this discussion, it is important to keep in mind that our goal is not to evaluate individual textbooks, but to look across thecorpus of textbooks for similarities and differences, and to explore how this analysis informs our understanding of the role ofthese textbooks for teaching and learning reasoning-and-proving. In this analysis, we provided a review of the textbooks fromtwo perspectives, that of the students and that of their instructors. In addition, the analysis gives us a basis for discussingperspectives of the textbook authors, focusing on opportunities to learn reasoning-and-proving designed into the textbook.

First, from the viewpoint of the students (prospective elementary teachers): What opportunities can the textbook offer tostudents to study and learn about reasoning-and-proving? In most textbooks, we found that a student would have difficultyfinding explicit content about reasoning-and-proving when using the textbook independently, outside of class. Most of thecontent about reasoning-and-proving in these textbooks is embedded in examples and problems, not in explicit discussionabout reasoning-and-proving. In the majority of textbooks, there are few links across sections that create a flow of contentabout reasoning-and-proving, leaving explicit instruction almost entirely in the section or chapter with reasoning-and-proving as a main topic.

Second, from the point of view of instructors: In most cases, instructors would need to supplement the textbook withdiscussion of, and materials relating to, reasoning and proving if their goal was to help students develop a coherentunderstanding of the nature of reasoning-and-proving and to connect ideas about reasoning-and-proving to content outsideof logic and Euclidean geometry. Although there are problems in many of the textbooks that would support such instruction,the instructor would need to be quite deliberate and focused on this issue to take advantage of what is implicit, or in somecases buried, in the textbooks.

Finally, from the viewpoint of the textbook authors, we can hypothesize about what these textbook authors considerimportant for prospective elementary teachers to know about reasoning-and-proving, as reflected in what rose to the level ofthe tables of contents or indexes in their textbooks. Although we have no data about why an author included or omitted ideasabout reasoning-and-proving in these locations, the concepts or ideas that appear in those places may be an indication of theimportance assigned to them by the author. From this perspective, we see that authors varied in how they presentedreasoning-and-proving. Three divergent examples are Jensen with numerous proofs and no table of contents or indexreferences; Parker with reasoning-and-proving interwoven through the textbook and many references in both places; andBillstein with numerous index references mostly found in the chapter on problem solving in which reasoning-and-proving isexplained. Other textbooks fall nearer or further from each of these, particularly with respect to whether the issues ofreasoning-and-proving are primarily in a single chapter or section, and whether they are addressed explicitly. Many of thetextbooks approach reasoning-and-proving as equivalent to logic, concentrating on logical statements and forms ofreasoning. Our claims about what authors consider to be important are not strong: features of a textbook may reflectconstraints of the writing and publishing process such as the depth and breadth of the indexing process provided by thepublisher, the textbook’s history over several editions, or space limitations.

Specific elements of reasoning-and-proving that are scarce in these textbooks include explanations of assumptions,definitions, axioms, and theorems, and how they relate to each other. Even in chapters devoted to problem solving, logic, orreasoning-and-proving where most of the content was found, ideas about these key concepts were absent. For the textbooksthat do not index a term like assumption, it is possible that the term is never used in the text, and that, if used, it is notexplained. Although students now have many convenient ways to find the meaning and even illustrations and explanationsof mathematical terms (e.g., reliable sites on the Internet), our analysis provides an indication of the extent to which thesetextbooks can be used as reference for content about reasoning-and-proving. We consider the lack of explicit treatment of

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assumption and definition in most of the textbooks to be particularly problematic due to the fundamental role that they canplay in school mathematics not only in the context of reasoning-and-proving, but also in mathematical activity more broadly(Stylianides, 2007a).

This study raises the question of how best to present reasoning-and-proving in a textbook for future teachers. In most ofthese textbooks, the authors approach reasoning-and-proving implicitly, showing examples of sound mathematicalreasoning-and-proving but not explaining how those examples were created or how they exemplify issues of reasoning-and-proving. It is possible that these authors purposefully leave explicit work on reasoning-and-proving during interactions inthe teacher education classroom at the discretion of the instructor, but without delving into the authors’ intentions, this isonly a conjecture. We have no evidence about how these textbooks are used in practice, and it is clear that an instructor canprovide explicit instruction about reasoning-and-proving even in the absence of such content in the textbook. We argue,however, that most of these textbooks do not have a level of focus on reasoning-and-proving that would provide support forstudents’ individual efforts at understanding.

We recognize the problematic aspects of our method. Using the tables of contents and indexes almost certainly leavesout instances where reasoning-and-proving is addressed. Alternative methods, however, could be problematic in differentways. For example, a sampling method could work only with the assumption that the instances of reasoning-and-provingare somewhat evenly distributed. Reading and coding every page is generally prohibitive because of the size of most of thetextbooks, but even if this problem could be overcome, the method would not be necessarily informative. A possibilitywould be to combine our method with a sampling method to investigate whether the treatment of reasoning-and-provingvaried in different parts of a textbook (as those parts would be determined by the choice of sample), and to providequantitative data for analysis. The quantities we reported in the paper are only those that are comparable across textbooks,given our method.

It would be instructive to compare how textbooks used in teacher education programs in different countries designopportunities for prospective teachers to learn about reasoning-and-proving, especially in a sample of contrasting countriesbased on prospective teachers’ mathematical knowledge recently assessed in an international study (Schmidt, Blomeke, &Tatto, 2011; Tatto & Senk, 2011). Our analysis shows that reasoning-and-proving is not designed as an interwoven practice inU.S. textbooks for teachers, but it may be the case that it is treated quite differently in other countries. Documenting thedifferent ways of designing reasoning-and-proving learning opportunities for prospective teachers in a larger collection ofinternational textbooks, and linking those ways with teachers’ actual learning of reasoning-and-proving in courses that usethe textbooks, could help test our conjecture that explicit attention to reasoning-and-proving is necessary for effectivelearning of reasoning-and-proving. The methodology we developed and followed in this study could find application inanalyses of textbooks for teachers in other countries.

Acknowledgements

The authors wish to thank Helen Siedel who was an integral partner in our early work on analyzing textbooks. Thisresearch was supported by the Center for Proficiency in Teaching Mathematics at the University of Michigan, a projectfunded by the National Science Foundation (NSF Grant # 0227586), by Michigan State University, and by a CAREER grant tothe first author (NSF Grant # 0447611).

Appendix A. List of Textbooks

Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed.). Brooks/Cole. ISBN: 978-084005463-0.Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed.). Boston, MA: Pearson Education. ISBN: 978-

032164694-1.Bennett, A. B., Burton, L. J., & Nelson, L. T. (2012). Mathematics for elementary teachers: A conceptual approach (9th ed.). McGraw-

Hill. ISBN: 978-007351957-9.Billstein, R., Libeskind, S., & Lott, J. W. (2013). A problem solving approach to mathematics for elementary school teachers (9th ed.).

Boston, MA: Pearson. ISBN: 978-032175666-4.Brumbaugh, D. K., Moch, P. L., & Wilkinson, M. (2005). Mathematics content for elementary teachers. Routledge. ISBN-13: 978-

0805842470.Darken, B. (2007). Fundamental mathematics for elementary and middle school teachers (3rd ed). Kendall/Hunt. ISBN: 978-

075756914-2.Jensen, G. R. (2003). Arithmetic for teachers: With applications and topics from geometry. American Mathematical Society. ISBN-

13: 978-0821834183.Jones, P., Lopez, K. D., & Price, L. E. (2000). A mathematical foundation for elementary teachers (Revised ed). New York: Addison

Wesley Higher Education. ISBN-13: 978-0201347166.Long, C. T., DeTemple, D. W., & Millman, R. S. (2012). Mathematical reasoning for elementary teachers (6th ed.). Pearson. ISBN:

978-032169312-9.Masingila, J. O., Lester, F. K., & Raymond, A. M. (2006). Mathematics for elementary teachers via problem solving: Student resource

manual (2nd ed.). Tichenor Publishing. ISBN: 978-1427502926.

R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119–131 131

Musser, G. L., Burger, W. F., & Peterson, B. E. (2010). Mathematics for elementary school teachers: A contemporary approach (9thed.). New York: John Wiley & Sons. ISBN: 978-0470531341.

O‘Daffer, P., Charles, R., Cooney, T., Dossey, J., & Schielack, J. (2008). Mathematics for elementary school teachers (4th ed.). Boston:Pearson Education. ISBN-13: 978-0321448040.

Parker, T. H., & Baldridge, S. J. (2004). Elementary mathematics for teachers. Okemos, MI: Sefton-Ash Publishing. ISBN:0974814008.

Parker, T. H., & Baldridge, S. J. (2006). Elementary geometry for teachers. MI: Quebecor World. ISBN: 0974814024.Sonnabend, T. (2010). Mathematics for elementary teachers: An interactive approach for grades k-8 (4th ed.). Brooks/Cole.

ISBN-13: 978-0495561668Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing mathematics for elementary school teachers. W. H. Freeman.

ISBN-13: 978-0716771968.Wheeler, R. E., & Wheeler, E. R. (2009). Modern mathematics for Elementary Educators (13th ed.). Kendall/Hunt Publishing

Company. ISBN-13: 978-0757562044.

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