REASONING-AND-PROVING IN GEOMETRY TEXTS Running Head: REASONING-AND-PROVING IN GEOMETRY TEXTS Reasoning-and-Proving Opportunities for Teachers in Secondary Geometry Textbooks Nicholas J. Gilbertson Michigan State University Lorraine M. Males University of Nebraska-Lincoln Kimberly C. Rogers Bowling Green State University – Ohio Samuel Otten University of Missouri 1 REASONING-AND-PROVING IN GEOMETRY TEXTS 2 Abstract Written curriculum materials provide an important window into the types of opportunities students have to learn how to reason-and-prove. This study extends previous work that focused solely on the introduction to proof chapter in geometry textbook student editions. Previous results indicate that relatively few opportunities exist for students to engage in general situations and reflecting on the nature of reasoning-and-proving. The present study suggests that while additional opportunities exist to engage students in reasoning-and-proving about general statements, relatively few opportunities exist for students to engage in reflecting on the nature of reasoning-and-proving within the introduction to proof chapter. Keywords: reasoning, proving, proof, geometry, curriculum, textbook REASONING-AND-PROVING IN GEOMETRY TEXTS 3 Reasoning and Proving Opportunities for Teachers in Secondary Geometry Textbooks Proof plays a fundamental role in establishing mathematical truths and communicating relationships. Yet, research has indicated that many K-12 students have difficulties with proving processes. For example, Chazan (1993) found that many students failed to see how a proof of a particular situation established the general truth of a set of mathematical objects. Others (e.g., Herbst & Brach, 2006; Schoenfeld, 1988; Soucy McCrone & Martin, 2009) have reported similar results of students’ limited conceptions of proving. To better understand how students learn to prove, recent studies (e.g., Stylianides 2007) have broadened the scope of proving to include reasoning processes that often serve as a pre-cursor to formal proof. Focusing on students’ conceptions or dispositions towards reasoning-and-proving, other works (e.g., Chazan, 1993; Harel & Sowder, 2007) have provided the field with case studies depicting students’ reasoningand-proving processes. Many factors, however, contribute to how students engage in such processes, including policies, teacher knowledge and practice, standards, assessments, and written curricula. This study focuses on written curricula because textbooks often play an important role in shaping students’ opportunities to learn (Stein, Remillard & Smith, 2007) through both teachers’ planning of lessons and the types of support provided for students. Multiple studies have characterized the nature of reasoning-and-proving opportunities in textbooks in the primary grades (Bieda, Drwencke, & Picard, 2014), middle grades (Stylianides, 2009), and secondary algebra (Davis et al., 2014; Thompson, Senk, & Johnson, 2012). Collectively, these studies provide evidence that in curricula, there are scant reasoning-andproving opportunities for K-12 students. Because these studies focused on content or grade levels outside of the high school geometry setting, some might argue that these results, while REASONING-AND-PROVING IN GEOMETRY TEXTS 4 compelling, may not be surprising considering proof’s traditional residence in high school geometry in the U.S. (Herbst, 2002). A previous analysis (Otten, Gilbertson, Males, and Clark, 2014) of the student editions of stand-alone high school geometry textbooks by the authors of this paper focused on investigating the nature and extent of reasoning-and-proving opportunities in student exercises and the introductory exposition sections across the textbooks. This study and the subsequent follow-up studies made use of the necessity principle (Harel & Tall, 1991), which maintains that students should see an intellectual need to engage in deductive reasoning. In its application to the study of written curriculum materials focused on reasoning-and-proving, we argue that deductive reasoning is best motivated by general situations (e.g., all triangles or all integers) because it requires arguing about all objects in a certain class. One key finding from the student-edition study was that the exposition of the textbooks often presented reasoning-and-proving opportunities with general mathematical situations (e.g., examples and explanations attending to feature of all triangles), while students’ in-class and homework exercises often provided opportunities for students to explore particular situations (e.g., a specific triangle with given measures). This distinction is important because the logical necessity for deductive reasoning is greater when students encounter general situations, such as those about infinite classes of objects. There is simply no other way to establish general truth except through deduction in general situations. As a result of this first study, we were prompted to further investigate the extent and nature of reasoning-and-proving opportunities in the textbook chapters that focused on introducing students to the foundations of proof, which ended up being the second chapter of all six textbooks (Otten, Males, & Gilbertson, 2014). While some textbooks introduced proof REASONING-AND-PROVING IN GEOMETRY TEXTS 5 formally and others did not, each Chapter 2 had a focus on the language and logic of deductive reasoning which forms the foundation of proof. Not surprisingly, these chapters provided more opportunities for students to reflect on the process of reasoning-and-proving than the other chapters in the textbooks. A limitation of both of these studies, however, was that student experiences with curriculum go beyond their direct interactions with the student materials. Teachers’ engagement with curricular support materials can provide both additional examples (e.g., questions to ask students) and additional opportunities for students to engage in reasoning-and-proving. Indeed, other curriculum studies exist that have considered the use of teacher materials (e.g., Stylianides, 2009) in analyzing reasoning-and-proving opportunities for students. In order to address this concern, we undertook the current study with the goal of determining to what extent additional opportunities exist in written curriculum materials—specifically, teacher editions—that may support students’ opportunities to learn reasoning-and-proving. Research Questions The overarching goal for this study was to investigate the types of additional opportunities that exist in teacher support materials (specifically, teacher editions) of secondary geometry textbooks that support student development of reasoning-and-proving. For this paper we report on two research questions related to our previous work: 1) How do the opportunities for students to engage in reasoning-and-proving exercises about general and particular statements compare across student and teacher materials? 2) How do the opportunities for students to engage in exercises focused on reflecting on the process of reasoning-and-proving compare across student and teacher materials? Method REASONING-AND-PROVING IN GEOMETRY TEXTS 6 The textbooks used in this study were six stand-alone (i.e., non-integrated) secondary geometry textbooks. These textbooks (table 1) were selected because they are from textbook series that together reach almost 90% of the secondary population in the U.S. (Banilower, Smith, Weiss, Malzahn, Campbell, & Weis, 2013). In the “introduction to proof” chapter (Chapter 2) we coded all lessons, including the chapter introduction and the chapter review. This chapter included an explicit lesson about deductive reasoning—an important topic related to formal proof—in all textbooks with the exception of UCSMP which focused on geometric language and logical structures needed in deductive reasoning such as definitions, conditional statements and converses. Chapter 2 was chosen because we hypothesized based on our previous work that this chapter would have the strongest potential for investigating how students engage in questions about the process of reasoning-and-proving and reflecting on this process. Table 1 Geometry textbooks analyzed in this study Titlea Geometry (CME) Geometry (Glencoe) Geometry (Holt) Discovering Geometry (Key) Geometry (Prentice) Publisher Authors Year Pearson Glencoe McGraw Hill Holt McDougal CME Project Carter, Cuevas, Day, Malloy, & Cummins Burger, Chard, Kennedy, Leinwand, Renfro, Roby, Seymour, & Waits Serra 2009 2010 No. of Sectionsb 20 10 2011 9 2008 8 Key Curriculum Press Pearson Prentice Hall Wright Group McGraw Hill Bass, Charles, Hall, 2009 7 Johnson, & Kennedy Geometry (UCSMP) Benson, Klein, Miller, 2009 8 Capuzzi-Feuerstein, Fletcher, Marino, Powell, Jakucyn, & Usiskin a The term given parenthetically is how we will refer to each textbook. b This column reports the summation of one chapter introduction, all lessons, and one review in each Chapter 2. REASONING-AND-PROVING IN GEOMETRY TEXTS 7 To draw comparisons, differences, and generalizations across texts, each Chapter 2 had three main sections that were coded for reasoning-and-proving opportunities: 1) the chapter introduction, 2) canonical lessons, and 3) chapter reviews. We analyzed the chapter introduction because it provided the opportunity for the authors to communicate important information regarding content, pedagogy, student learning, and the chapter’s role in the mathematical storyline of the textbook. All the canonical lessons were coded, that is, the numbered lessons that appear in both the student and teacher edition. Within each canonical lesson, solutions to problems, and any statements about reasoning and proving were coded. Additionally, if a solution was referenced in another part of the book because of space considerations these were coded as if they appeared in the canonical lesson. The only notable exception to not being coded were images of supplementary materials (e.g. pictures of worksheets) displayed to the side on the wrap-around section of the teacher text. Finally, the chapter review was coded, because the review represents what the author may view as important for students to learn in the chapter. In all three sections, we coded statements, exercises, and their provided answers about reasoning-and-proving. Each coder had prior experience with coding of written curriculum materials and each coder also coded at least one section within each textbook. A minimum of 30% of each textbook’s sections were doubled coded by a different coder to confirm reliability. The teacher editions included additional examples or prompts to ask students questions related to reasoning-and-proving. These exercises often came in the form similar to problems in the student edition, or by a prompt from the authors to “Ask students…”. Because the focus of this paper is on these additional exercises only, we provide an overview of the pertinent components of the analytic framework used. In this study, we made use of the analytic framework for coding exercises in the student edition (Otten, Gilbertson, Males & Clark, 2014) building off the work of REASONING-AND-PROVING IN GEOMETRY TEXTS 8 Thompson, Senk and Johnson (2012) and Stylianides (2009). This previous framework was used for this section of the codes because the dimensions we were analyzing in the teacher edition additional exercises were similar to those in the exercises from the student editions. Exercises were coded along three dimensions. The first dimension was the type of mathematical situation or statement being reasoned around, specifically, is the exercises about a general situation (e.g. all triangles), a particular situation (e.g. a 3-4-5 right triangle) or a general situation with particular instantiation, which is typically a general situation question followed by a specific diagram. The second dimension is the expected student activity, which included making a conjecture, filling in the blanks of a conjecture, investigating a statement, constructing a proof, developing a rationale or non-proof argument, outlining a proof, filling in the blanks of a proof, evaluating a proof, finding a counterexample. The third dimension was the environment for exploration; that is did the question explicitly call for the student to reason or prove using deductive reasoning, empirical reasoning, or was the reasoning left implicit? A final category of exercises focused on the process or nature of reasoning-and-proving. These exercises were only coded along the second dimension, expected student activity, because these exercises were qualitatively different than the other reasoning-and-proving exercises. Results The aim of this study is to characterize the types of opportunities present for teachers in written curriculum materials to support student development of reasoning-and-proving. The scope of this study is broader than the scope of this paper, so we present only two sets of results here which highlight how the teacher materials provide different opportunities for students to potentially engage in reasoning-and-proving as mediated by the teacher. Research Question 1 focused on the types of opportunities students had to encounter general and particular situations REASONING-AND-PROVING IN GEOMETRY TEXTS 9 in both student and teacher edition exercises. Table 2 compares the relative frequencies of general and particular exercises given in the student edition and additional prompts given in the teacher edition. Within the student editions, all texts have a higher frequency of particular situations than general ones, except UCSMP which is nearly equal. The additional exercises contained in the teacher editions, however, had only Glencoe and UCSMP with much higher relative frequencies of particular situations over general ones. Unlike their student editions, Holt, CME, and Key’s teacher edition had at least as many general reasoning-and-proving exercises as they did particular exercises. While CME (n=4) and Key (n=14) had relatively few additional exercises in the teacher editions, the other four textbooks ranged from 35 to 69 additional exercises in the teacher editions. Table 2 Percentages of Statement-types in Reasoning-and-Proving Exercises within Chapter 2 Student Edition Teacher Edition General Particular General Particular Textbook Statements Statements Statements Statements (%) (%) (%) (%) CME 45 53 25 25 Glencoe 38 52 33 59 Holt 33 49 39 39 Key 30 51 29 21 Prentice 24 71 40 49 UCSMP 47 45 33 67 Note. Percentages do not always sum to 100% because categories such as exercises about reflecting on the process of reasoning-and-proving, and general statements with particular instantiations have been omitted. Student Edition data previously published in Otten, Males, and Gilbertson (2014). Research Question 2 focused on the opportunities for students to engage in problems that prompt students to reflect on the nature of reasoning-and-proving. Whereas many other problems (e.g., those that were coded as “general” or “particular” statements) involved students actively reasoning about a particular mathematical object or class of objects, the reflection-type questions REASONING-AND-PROVING IN GEOMETRY TEXTS 10 asked students to consider aspects of the reasoning-and-proving process. For example, in the Glencoe student edition, students were given two conditionals and prompted in the exercise to, “Explain why the Law of Syllogism cannot be used to draw a conclusion from these conditionals” (p. 122). Similarly, an additional exercise in the teacher edition from Holt was, “When you write a paragraph proof, how can you be sure that you haven’t left out any of the reasons?” (p. 121). Table 3 below, shows the number of exercises in the student edition and teacher edition of these types of problems. Table 3 Exercises that Focused on Reflecting on the Nature of Reasoning-and-Proving Textbook Student Edition Teacher Edition CME 2 2 Glencoe 25 5 Holt 35 15 Key 14 7 Prentice 3 4 UCSMP 7 0 Note. Student Edition data previously published in Otten, Males, and Gilbertson (2014). We know from our analyses of the student editions that such opportunities to reflect on reasoning-and-proving are rare in the other chapters of the geometry textbooks. In the introduction to proof chapters, there are some opportunities in the student editions to reflect on reasoning-and-proving (with the exception of UCSMP, which actually has more reflection opportunities later than the other five textbooks), across all textbooks there are few additional opportunities via additional exercises to help support students with reflecting on the nature of reasoning-and-proving. Discussion Previous findings from (Otten,Gilbertson, Males & Clark, 2014; Otten, Males, & Gilbertson, 2014) show that students have more opportunities to do exercises that are particular REASONING-AND-PROVING IN GEOMETRY TEXTS 11 in nature and have limited opportunities to engage in reflecting on the process of reasoning-andproving via exercises in the introduction to proof chapter. The data in table 2 does not categorically refute the result from our previous analysis of student editions (Otten, Gilbertson, Males & Clark 2014), namely that students are more likely to encounter particular statements than general ones in exercises. There are however three textbooks where there are as many general statements as particular ones in the additional exercises. For teachers, this may mean that they are better able to supplement exercises with general statements to the few opportunities of this statement type that currently exist only in the student editions. For curriculum authors, placing these general statements in the additional exercises may also increase the likelihood that a teacher will provide students with support for engaging in this general type of reasoning, either through a class discussion or collaborative work in class. While the data in table 2 focused on students reasoning-and-proving opportunities, the data in table 3 focuses on opportunities to reflect on the nature of reasoning and proving. In table 3 there is a general trend across textbooks of few additional opportunities via suggested prompts or exercises for students to engage in reflecting on the process of reasoning-and-proving. Holt was the only textbook in the sample to have more than ten such opportunities, but relatively speaking the additional exercises represented less than half of these types of problems in the student editions. This suggests that if teachers are going to help students reflect on the nature of reasoning-and-proving, that they may have to extend the given opportunities primarily from the student editions. This raises the question as to how teachers might supplement opportunities to enhance the existing set of opportunities for students to engage in reflecting on reasoning-andproving. REASONING-AND-PROVING IN GEOMETRY TEXTS 12 Conclusion This final set of data in table 3 is noteworthy because it points to two important limitations in this study. First, we have restricted our sample to only the introduction to proof chapter. Previous analysis (Otten, Males & Gilbertson, 2014) suggests that there may be some additional opportunities in the remaining chapters. Second, the notion of “teacher support” extends beyond simply additional exercises. Curriculum materials can support teachers in many ways, including describing the capabilities of certain exercises when enacted, such as providing anticipated student misconceptions, describing pedagogical strategies, or helping the teacher make sense of the mathematics content. As teachers support students engaging in reasoning-and-proving processes, the textbooks used afford different opportunities in communicating the nature of reasoning-and-proof. Since the teacher editions provide additional reasoning-and-proving exercises and opportunities for reflecting on proof, some related next questions include; what additional supports are available to teachers to help students with learning to reason-and-prove, to what extent are teachers using these curricular resources, how does teachers’ implementation of these resources affect students’ opportunities to learn about reasoning-and-proving in the classroom, and what additional resources do teachers need to support their instruction of reasoning-and-proving? REASONING-AND-PROVING IN GEOMETRY TEXTS 13 References Banilower, E. R., Smith, P. S., Weiss, I. R., Malzahn, K. M., Campbell, K. M., & Weis, A. M. (2013). Report of the 2012 National Survey of Science and Mathematics Education. Chapel Hill, NC: Horizon Research, Inc. Bass, L. E., Charles, R. I., Hall, B., Johnson, A., & Kennedy, D. (2009). Prentice Hall mathematics: Geometry. Boston, MA: Pearson Prentice Hall. 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