Gilbertson, Males, Rogers, Otten - Reasoning-and-Proving in Secondary Geometry NCTM 2014

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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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?
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