Reasoning-and-proving in mathematics textbooks for prospective elementary teachers

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  • 1. Introduction

    ving in studentsat set curriculumational Governorsr reasoning-and-

    International Journal of Educational Research 64 (2014) 119131

    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

    In the United States, elementary teachers (grades 15 or 6, ages 611 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 ndings 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

    ndings for future research and teacher education.

    2013 Elsevier Ltd. All rights reserved.

    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 esOne 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 inOver the past decade there has been increased recognition of the importance of reasoning-and-promathematical education.2 For example, in the United States, the setting of this paper, policy documents thstandards for school mathematics (e.g., National Council of Teachers of Mathematics [NCTM], 2000; NAssociation Center for Best Practices & Council of Chief State School Ofcers, 2010) have made calls foproving to become central to all students mathematical experiences and across all school years.Reasoning-and-proving in mathematics textbooks forprospective elementary teachers

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

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

    A R T I C L E I N F O A B S T R A C T* 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

  • the te

    R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119131120indexes.This approach supports a consistent analysis of textbooks. In addition, it allows us to take the perspective of the

    student 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 theauthors 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: does it mean to do an indirect proof?), or an instructor who is basing the sequencing and ow of the course onxtbook. This turned our attention to the published maps of each textbook; specically, tables of contents andthis 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 nds 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 rst 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 difculties that many elementaryteachers face with reasoning-and-proving (e.g., Martin & Harel, 1989; Simon & Blume, 1996); (2) the research ndingthat students opportunities to learn are dependent on the quality of their teachers knowledge (e.g., Hill, Rowan, & Ball,2005); and (3) the recent research ndings 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 dening 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 ndings of our examination can offer abasis for reection 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 15 or 6, ages 611 years) teachers in the U.S. (a list of these textbooks is in Appendix A). The rststep 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 specic idea mentioned in class or study for a test (asking a question such as,what

  • R. McCrory, A.J. Stylianides / International Journal of Educational Research 64 (2014) 119131 121Although 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 prole of the corpus of textbooks, aiming to connect our ndings to current thinking about reasoning-and-proving.Tprohis sentence does not provide the student, or the instructor, with content from which they could learn reasoning-and-ving. In other cases, the term proof in the index could point to a denition of proof or an actual proof of a theorem.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.s the referenced page indicate how the term is connected to, or used in, other parts of the textbook?assumption, compound or conditional statements (including specically contrapositive and converse), deductive and indirectreasoning (or simply reasoning), denition, 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, falsication (or refutation),justication (or verication); and (3) terms that relate to different proof methods: contradiction, counterexample, indirectproof, 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 specic 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 nd 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 nding 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 nd relevant content, and we suggest that without references toat least some of these terms, students (and their instructors) may have difculty nding content ab...

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