International Journal of Educational Research 64 (2014) 92–106

Contents lists available at SciVerse ScienceDirect

International Journal of Educational Research

jo ur n al ho mep ag e: www .e lsev ier . c om / lo cate / i jed u res

Reasoning-and-proving in algebra: The case of two

reform-oriented U.S. textbooks

Jon D. Davis a,*, Dustin O. Smith a, Abhik R. Roy a, Yusuf K. Bilgic b

a Western Michigan University, United Statesb State University of New York-Geneseo, United States

A R T I C L E I N F O

Article history:

Received 1 December 2012

Received in revised form 21 May 2013

Accepted 7 June 2013

Available online 1 August 2013

Keywords:

Reasoning-and-proving

Algebra

Textbook

A B S T R A C T

This research study examined students’ opportunities to engage in reasoning-and-proving

(RP) within exposition and task components of two U.S. reform-oriented secondary

algebra textbooks. There were statistically significant differences between the two

textbooks in terms of the percentage of tasks coded as RP and statistically significant

differences in the percentages of tasks devoted to RP across different algebra topic areas

within each textbook. Differences also appeared in the role of technology in RP within both

textbooks. While this study is focused on two U.S. algebra textbooks, broader

recommendations will be made on textbook design with regard to RP. Moreover, the

framework presented in this study provides researchers and teachers with tools to

examine RP in textbooks and enacted classroom lessons.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction and background

Researchers from around the world have recently been turning their focus on what we refer to as proof-related constructs

(Davis, 2012; Hanna & de Bruyn, 1999; Stacey & Vincent, 2009; G. J. Stylianides, 2009; Thompson, Senk, & Johnson, 2012).This terminology will be used to refer to proof as well as its related constructs such as reasoning. The confluence of researchin proof-related constructs in mathematics textbooks is important for two reasons. Proof is a principal component of thepractice of mathematics (Hanna, 2007; Schoenfeld, 2009; Weber, 2008) while textbooks are a principal component of thepractice of teaching mathematics (Grouws & Smith, 2000; Tarr et al., 2008; Valverde, Bianchi, Wolfe, Schmidt, & Houang,2002). The study described here expands a framework developed by G. J. Stylianides, thus providing researchers and teacherswith additional tools with which analyses of reasoning-and-proving within the written curriculum can be conducted.Moreover, the results have implications for the design of textbooks that seek to incorporate RP.

This review is organized around six different themes. This review begins with a description of how secondarymathematics textbooks in the U.S. have been categorized in order to orient readers to issues with regard to curriculum.As this study involved analyses of the narrative portions of the textbook as well as student tasks the review describesprevious research involving proof-related constructs within these two areas. The review continues with a description ofresearch involving students’ proof engagement within and across textbooks. Next, the review discusses theconnectedness among different proof-related constructs. Last, the review includes a section on technological tools inproof-related constructs.

* Corresponding author. Tel.: +1 269 387 4591; fax: +1 269 387 4530.

E-mail address: jon.davis@wmich.edu (J.D. Davis).

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

http://dx.doi.org/10.1016/j.ijer.2013.06.012

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 93

1.1. Categorizing mathematics curricula in the United States

Mathematics curricula in the United States have been categorized as either reform-oriented or conventional. The firstcategory refers to curricula that relied on national reform documents such as the Curriculum and Evaluation Standards for

School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) as design templates. The second categoryrefers to textbooks that existed at the time of the development of reform-oriented curricula or have been created bypublishers through the assistance of editors and an invited set of authors. Conventional curricula used a back mappingprocess to match their curricular content and processes to national reform documents.

1.2. Opportunities for students to engage in proof-related constructs in textbook tasks

Studies in the United States (Davis, 2012; G. J. Stylianides, 2009; Thompson et al., 2012) and other countries (Nordstrom &Lofwall, 2005) highlight the limited number of student tasks related to proof in school mathematics textbooks. In the U.S.,Thompson et al. found that 5.4% of 9742 student tasks across 22 textbooks in the areas of exponent properties, logarithmproperties, and polynomials engage students in proof-related reasoning.1 Nordstrom and Lofwall examined the prevalence ofproof in five different courses in upper secondary school mathematics in Sweden covering the following content areas:algebra; geometry; statistics and probability; functions and calculus; exponents and logarithms; trigonometry; and complexnumbers. They found that about 2% of the tasks in these content areas asked students to construct proofs. G. J. Stylianides’examination of a middle school (ages 11–14) reform-oriented textbook series revealed that 5% of 4578 tasks asked studentsto construct what he described as demonstrations.2

An exception to these studies is work conducted by Hanna and de Bruyn (1999). They used a framework consisting ofthree categories: proof, discussion of proof, and non-proof. The second category involves discussions about the creation of aproof or directions about how to develop a proof. Proof items consisted of full or partial proofs. They found that 21% of 1086problems in one textbook involved the construction or discussion of proof.

1.3. Opportunities for students to read about proof-related constructs in student textbook narratives

Hanna and de Bruyn (1999) found that 22% of 465 narrative items in one popular 12th grade mathematics textbook and17% of 621 narrative items in another popular 12th grade mathematics textbook consisted of proof or the discussion of proof.Stacey and Vincent (2009) examined the modes of reasoning used in the narrative sections of seven different topicsappearing in nine textbooks for eighth grade Australian students. Modes of reasoning are related to Harel and Sowder’s(1998) proof schemes yet they differ from this construct in the following way. A proof scheme consists of the method bywhich a person or community removes doubt about the validity of an assertion. Modes of reasoning are presented in thetextbook and are not necessarily connected to a particular human agent as the textbook authors may have chosen thatexplanation for purely pedagogical reasons. Stacey and Vincent found a total of seven modes of reasoning within sevendifferent content areas in the different textbooks: deduction using a general case; deduction using a specific case; deductionusing a model; concordance of a rule with a model; experimental demonstration; appeal to authority; and qualitativeanalogy. In their analysis of exponent, logarithm, and polynomial properties in 22 secondary mathematics textbooks,Thompson et al. (2012) found that 69% of properties appearing in the narrative sections were justified in a variety of differentways, but 60% of this number contained valid proofs. Consequently, less than half of the properties appearing in thetextbooks were justified with a valid mathematical proof. They also found that 60% of the statements with no justificationwere accompanied by specific examples showing how to apply the property suggesting that examples might constitute asuitable justification for the validity of a property.

1.4. Proof-related constructs varies within and across textbooks

Researchers have also sought to compare and contrast the presence of proof across textbooks for school mathematicsstudents within a certain age group. Among 22 different secondary school (ages 14–18) mathematics textbooks in the U.S.,Thompson et al. (2012) found that 14.7% of 532 student tasks involved proof-related reasoning in a reform-orientedmathematics program. On the other hand, 3.7% of 2042 tasks involved proof-related reasoning in a conventional textbookseries. In a similar vein, Davis (2012) found that 22% of 1158 tasks in a reform-oriented textbook unit were coded asreasoning-and-proving3 (RP), but comparatively fewer student tasks in the conventional textbook unit were considered to beRP (5% of 1129). Hanna and de Bruyn (1999) found that 21% of 1086 problems in the Foundations of Mathematics 12 textbookand 16% of 1491 problems in Mathematics 12 involved proof or the discussion of proof. Stacey and Vincent (2009) found that

1 Thompson et al. (2012) refer to proof-related reasoning as consisting of one or more of the following activities: finding a counterexample, making a

conjecture, investigating a conjecture, developing an argument, evaluating an argument, and correcting a mistake in an argument.2 G. J. Stylianides (2008) defined a demonstration as a proof or ‘‘valid argument based on accepted truths for or against a mathematical claim’’ (p. 11) that

do not depend ‘‘on the ‘representativeness’ of a particular case’’ (pp. 11-12).3 Davis (in press) used a framework constructed from the work of G. J. Stylianides (2009) who emphasized the integrated nature of reasoning-and-

proving as consisting of identifying patterns, constructing conjectures, develop non-proof arguments, and creating proofs.

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–10694

the proportion of topics with a deductive mode of reasoning across the nine eighth-grade textbooks varied from 0.5 to 1.0.Moreover, less than half of the textbooks contained either deductive or empirical reasoning for all of the seven mathematicaltopics investigated. Johnson, Thompson, and Senk (2010) reported that as students progressed in secondary school theyencountered more opportunities to engage in proof-related reasoning in the areas of exponents, logarithms, andpolynomials. They documented that 3.4% of 2838 tasks in beginning algebra, 5.4% of 3937 tasks in advanced algebra, and 7.7%of 2967 tasks in precalculus contained proof-related reasoning.

1.5. Connectedness among different proof-related constructs

In the studies reported earlier, researchers such as Hanna and de Bruyn (1999) or Stacey and Vincent (2008) haveexamined a proof or the development of arguments in Canadian and Australian textbooks, respectively. However, they didnot examine other processes that are important precursors of proof. Thompson et al. (2012) use the terminology proof-related reasoning to encompass a variety of different processes related to proof such as the development and investigation ofconjectures or the evaluation of arguments, but they did not examine the connections between conjecturing and thedevelopment of arguments. That is, their research did not illuminate how many conjectures were followed up by requests forstudents to develop arguments. G. J. Stylianides’ (2009) framework consists of several different components related to proofsuch as the identification of patterns and making conjectures, but, in addition, his framework also contains a seconddimensions involving the purposes of these components. If students’ pattern identification opportunities led to conjecturingopportunities these were denoted as conjecture precursors. The terminology conjecture non-precursor was used for patternsthat did not lead to conjectures. He found that in a middle school mathematics curriculum that 97% of plausible patterns and88% of definite patterns were conjecture non-precursors. The majority of conjectures (70%) were identified as proof non-precursors.

1.6. Technological tools in proof-related constructs

Technology in a variety of different forms can be used as a tool to help students identify patterns, construct conjectures,and develop valid arguments. Consider the task appearing in Fig. 1.

In this task, students use a computer algebra system (CAS) to graph a variety of different cubic polynomial functions andexamine these graphs for patterns in the number of maxima and minima that they contain. Later, students make a conjectureabout the number of extrema a cubic function may have. Last, students are asked to use the CAS to test the conjecture thatthey made earlier. A computer algebra system can also be used to complete the symbolic manipulation steps necessary for aproof involving this form of argument (Garry, 2003). In addition, graphing calculators with matrix operation capabilitiescould be used to show that matrix multiplication is in general, non-commutative through the calculation of a carefully

Fig. 1. A task involving a computer algebra system (Fey et al., 2009, p. 325).

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 95

chosen counterexample. Davis (2012) examined the use of technology in reasoning-and-proving in three U.S. secondarymathematics textbook units. On the one hand he found that approximately half of the pattern identification tasks and half ofthe conjecturing tasks used technology. On the other hand, none of the argument development tasks asked students to useany form of electronic technology. Additionally, none of the exposition elements that involved pattern identification,conjecture development, or argument construction in the three textbook units utilized technology. As secondarymathematics textbooks in the U.S. incorporate computer algebra systems into tasks it is important to understand if thispowerful technology is being used in proof-related constructs appearing in student tasks or in narrative sections of theseresources.

1.7. Summary and research questions

Several trends are notable in the studies reviewed above. First, students’ opportunities to engage in proof-related

constructs vary by textbook. Second, students’ opportunities to engage in proof-related reasoning appear to increase as theyprogress through secondary school (Thompson et al., 2012). Third, no studies have focused on more recently created U.S.reform-oriented secondary texts, nor have those studies that have been completed involved the entire set of topics appearingwithin a secondary advanced algebra course. Fourth, research on proof in student textbooks reviewed above often relies oncompletely different frameworks. This makes comparison across studies problematic. Consequently, this study focused onstudents’ opportunities to engage in reasoning-and-proving in two U.S. reform-oriented secondary (ages 14–18)mathematics textbooks designed for students learning from an advanced algebra course designed for students in grades 9–11 (ages 14–17) using a framework based upon the work of G. J. Stylianides (2009).

The following four research questions with regard to RP in two U.S. reform-oriented secondary mathematics advancedalgebra textbooks guided this study. First, what is the dispersion across RP categories within the exposition and taskcomponents of the two textbooks? Second, are there statistically significant differences in the percentage of tasks andpercentage of exposition sentences devoted to RP by unit and overall between the two textbooks? Third, what purposes dopatterns serve within the two textbooks? Fourth, how frequently is technology used by the textbook developers in theidentification of patterns, construction and testing of conjectures, and development of specific and general arguments withinthe two textbooks?

2. Analytic framework

The analytic framework used in this study was derived from the work of G. J. Stylianides (2009) and appears in Fig. 2. Eachcomponent of the framework and their relationship to one another are described in the sections that follow. The studyreported on here uses the same framework to examine exposition and task components within two different reform-oriented curricula. We defined exposition components of the student textbook to consist of text that we interpreted aspresenting mathematical ideas to students. We defined task components of the student textbook to include written textcontaining imperatives or questions directed toward the reader. Such instances were considered tasks even if their answersappeared immediately after the problem. The dashed lines illustrate that patterns can lead to conjecturing, but may not.Although the same framework was used to examine student tasks as well as the expository sections of the two studenttextbook units, the unit of analysis for each of these student textbook components was different, as described within the nextsection.

2.1. Unit of analysis

The unit of analysis for task components of the student textbook consisted of a separate answer appearing in the teacherresource materials. A separate answer could consist of a number, phrase, or sentence appearing in the answers of theteacher’s edition of the student textbook. The unit of analysis for narrative components of the student textbook was one ormore sentences. On the one hand, a sentence in which a mathematical term was defined would be considered an instance ofreasoning-and-proving as described in more detail below. On the other hand, when an RP element consisted of a fairlylengthy proof, the RP element could have consisted of several sentences.

2.2. Identifying patterns

We agree with the integrated nature of RP as described by G. J. Stylianides (2009). The analytic framework used in thisstudy was constructed upon three sub-components comprising the mathematical component of G. J. Stylianides’ analyticframework for RP: identifying definite and plausible patterns; constructing a conjecture; and crafting arguments. The datafrom which patterns could potentially be identified are not tied to a particular representation and may exist in a variety ofdifferent representational forms (e.g., graphical). It is possible for a knowledgeable individual to provide evidence for adefinite pattern within a set of data. On the other hand, it is not possible to provide evidence for a plausible pattern over otherpatterns existing in the data. For instance, suppose that students are asked to find the next number in the sequence: 1, 3, 5,___. Assuming a linear relationship, 2n � 1 the next value is 7. However, the next number could be 13 if one assumes a cubicrelationship, n3� 6n2 + 13n � 7. Both of these denote plausible patterns as the data fit both interpretations. However,

Fig. 2. A framework for examining reasoning-and-proving in written curriculum resources.

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–10696

consider the following situation. Ten people are currently on a bus. If two people get on the bus at each stop, only one patternexists, n = 10 + 2s between the number of people on the bus n after s stops, thus illustrating a definite pattern.

The connection between pattern identification and conjecturing via solid (precursor) and dashed (non-precursor) lineswill be discussed in a later section. Pattern identification may lead directly to the development of arguments as noted by thedashed line in Fig. 2.

2.3. Crafting and testing conjectures

G. J. Stylianides (2009) describes two criteria in the definition of a conjecture. First, it ‘‘extends beyond the domain of casesthat gave rise to it’’ (p. 264). Second, a conjecture reflects uncertainty with regard to its validity. In order for a textbookexposition unit to be considered an instance of the development of a conjecture, both of these criteria needed to be present.That is, the sentence or set of sentences needed to go beyond the set of cases on which it was based and the statement neededto be accompanied by a word or series of words suggesting uncertainty. We amended the G. J. Stylianides framework withregard to conjecturing by adding the testing of conjectures to conjecturing or the development of conjectures. An instance oftesting a conjecture was present if the word test appeared with conjecture in the same sentence in either the expositioncomponents or in the task components of the student textbook. Conjectures may lead to the development of arguments asnoted by the dashed lines in Fig. 2.

2.4. Valid and invalid arguments

We considered a valid argument to be akin to a proof as defined by A. J. Stylianides (2007). We did not use G. J. Stylianides’(2009) invalid argument (non-proof) categories in this study. In order to discern valid from invalid arguments in thenarrative components we engaged in the process of proof validation (Selden & Selden, 2003). In task components of thestudent textbook, we used keywords such as, but not limited to, explain, justify, verify, and show to indicate that an argumentwas being requested from students. We examined the answers provided in the teacher resource materials to these tasks todetermine if students were capable of providing valid arguments. If we determined the answer provided in the teacher’sedition of the student textbook was a valid argument using the process of proof validation we considered the task an instance

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 97

of a valid argument. If the answer provided in the teacher’s edition of the student textbook was not a valid argument we usedthe previously presented textbook ideas to determine if a valid argument was within reach of the students. If so, the task wascoded as an RP instance in this category, otherwise it was not coded. As noted by the dashed line in Fig. 2, valid arguments canlead to the construction of proof building blocks such as theorems.

2.4.1. Categorizing valid arguments

G. J. Stylianides (2009) used generic examples and demonstrations to categorize valid proofs in his study. Due to a dearthof generic examples in the two textbooks analyzed, we did not include this category in our framework. Instead, previousanalyses (Davis, 2012) as well as our initial analyses of RP in the two textbook units suggested six valid argumentsubcategories: argument subcomponents-specific, argument subcomponents-general, argument-specific, argument-general,counterexample-specific, and counterexample-general. The fact that all of these different categories can be general or specific innature is shown in Fig. 2. The word specific refers to claims that relate to a specific set of numbers or quantities while the wordgeneral refers to claims involving an infinite set of elements. Argument subcomponents consisted of instances where studentswere asked to fill in either the statements or the explanations for a valid argument, complete a valid argument givensuggestions provided by the textbook authors about how to proceed, or examine and correct an invalid argument thatcontains some error. An example of an argument subcomponents-general category appears in Fig. 3.

The claim that students are asked to prove is that cos2 u + sin2 u = 1. Since this claim works for any angle u, it is a generalclaim. Students are given some ideas about how to proceed with the construction of the valid argument as seen in thepresentation of the three different methods, any one of which can be used to construct the argument. We assumed thatstudents would use one of these three methods to develop the valid argument placing it in the category of argumentsubcomponents as opposed to developing a new method, which would place this in the category of argument – general.

2.4.2. Practicing the writing of a proof

The last addition to the analytic framework within the valid argument category was the subcategory, proof-writing

exercises. As Schoenfeld (1992) has pointed out, problems in mathematics instruction typically consist of ‘‘routine exercises

Fig. 3. An example of the argument subcomponents-general category (EDC, 2009, p. 704).

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–10698

organized to provide practice on a particular mathematical technique that, typically, has just been demonstrated by thestudent’’ (p. 337). It is in a similar vein that we use the word exercises here to describe this subcategory. On their own,instances within this subcategory constituted valid proofs, yet the claims they sought to validate were sufficiently similar topreviously proven claims that arguments used to validate those claims could be used again to justify the new claim. Otherexamples that were considered proof-writing exercises were presentations in the exposition or problems within the taskcomponent of the student textbook that required students to apply a theorem or definition and justify their answer byappealing to these theorems or definitions.

2.5. Technology and proof building blocks

Technology was added to the G. J. Stylianides (2009) framework because it is an important tool for students toengage in RP. It can be used to construct sets of data from which plausible and definite patterns can be identified andconjectures constructed. In addition, it can be used in the development of proofs (Garry, 2003) and the two textbooksat the center of this study have both incorporated computer algebra systems (CAS) into their design (Cuoco, 2007;Usiskin, 2007). A range of electronic tools met the definitions of technology for this study. On one end of thiscontinuum of tools lies scientific calculators while at the other end lays representational toolkits that containspreadsheets, dynamic geometry systems, graphing calculators, and computer algebra systems. The dashed lines fromtechnology to patterns, conjectures, and arguments in Fig. 2 indicate that technology can play a role in each of theseactivities.

The proof building blocks category consists of the presentation within exposition components or the opportunity forstudents to construct within task components the wording of definitions, corollaries, theorems, etc. These were consideredproof building blocks since they could be used in the construction of valid arguments or proofs.

2.6. Purposes of patterns and conjectures

Similar to G. J. Stylianides (2009) we examined the purposes of definite patterns, plausible patterns,and conjectures. A problem that asked students to identify a definite or plausible pattern was considered a conjecture

precursor if students were later asked to make a conjecture or the textbook exposition presented a conjectureabout the mathematical idea related to that definite/plausible pattern. A problem that asked students to identify adefinite or plausible pattern was considered an argument precursor if students were later asked to develop an argumentor if the exposition presented an argument about the mathematical idea related to that definite/plausible pattern.While G. J. Stylianides examined the purpose of arguments (e.g., discovery), this was not examined as part of thisstudy.

G. J. Stylianides (2009) also included a component that he referred to as non-proof arguments, which consisted ofempirical arguments and rationales. The last category consisted of arguments that were missing one or more componentsthat would have resulted in a valid proof. As this study did not involve non-proof arguments neither empirical arguments norrationales were included in this analysis.

2.7. Curricula

In the United States, each year of school is designated by a grade. Elementary school typically encompasses gradesKindergarten through 5. Secondary school consists of two different components: middle school and high school. Middleschool spans grades 6–8 (ages 11–14) while high school spans grades 9–12 (ages 14–18). Traditionally in the United States,students would begin grade 9 (age 14–15) of secondary school taking a beginning algebra course, progress to geometry ingrade 10 the following year, take an advanced algebra course in grade 11, and enroll in a precalculus course in grade 12.More recently, however, students have been enrolling in beginning algebra in grade 7 or 8 of middle school (Dossey,Halvorsen, & McCrone, 2008). These students may then enroll in advanced algebra in grades 9–10 of high school. Thus thehigh school advanced algebra course is populated by students in a variety of different grades in high school (ages 14–18).The two textbooks analyzed as part of this study, Center for Mathematics Education’s (CME) Algebra 2 (EducationDevelopment Center [EDC], 2009) and the University of Chicago School Mathematics Project (UCSMP) Advanced Algebra

(Flanders et al., 2010), are designed for high school students enrolled in an advanced course in algebra. In addition, both ofthese textbooks are considered reform-oriented curricula as CME and UCSMP both have been created with the use of theNational Council of Teachers of Mathematics (NCTM) standards documents (1989, 2000) as design templates (Cuoco, 2007;Usiskin, 2007).

Similar content consisting of polynomials, exponential functions, logarithmic functions, trigonometry, matrices,sequences, and series appears in both textbooks. There was also material in one textbook that was not in the other. Forinstance, conic sections appeared in UCSMP, but not in CME. Similarly, Lagrange Interpolation appeared in CME, but not inUCSMP. Thus while there were similarities across the two units in terms of polynomials, there were also a number ofdifferent mathematics content areas contained in each unit. Although both textbooks were designed for students in anadvanced algebra course the UCSMP textbook contained more tasks overall (3973 vs. 2823) and more pages dedicated toexposition components (238 vs. 172) in the sampled sections (described below).

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 99

2.8. Sampling

In order to make comparisons about RP opportunities between the two texts, yet streamline the coding process astratified random sampling procedure was used where each chapter was considered to be a stratum (Schaeffer, Mendenhall,Ott, & Gerow, 2012). It is not possible to calculate variances for these data since they are qualitative in nature. Also, there areno significant differences in the cost of sampling one section vs. another. Consequently, we used proportional allocation(Schaeffer et al., p. 140). We sampled roughly half of the sections from each textbook. The UCSMP textbook contained threestrata: 8 sections, 9 sections, or 10 sections. The calculations resulted in sampling 4 sections from those chapters with 8sections, 5 sections from those chapters with 9 sections, and 5 sections from each chapter with 10 sections. This resulted in61 sections from the UCSMP textbook or 53% of the available sections of the textbook. The CME textbook contained six strata:10 sections, 12 sections, 13 sections, 14 sections, 15 sections, and 16 sections. Sampling half of the sections from each of thechapters resulted in a total of 56 sections or 52% of the text. The specific sections to be analyzed in each chapter of the UCSMPand CME textbooks were determined using a random number procedure.

2.9. Coding

Coding of each textbook began with a common chapter in polynomial functions. This material appeared as Chapter 11 inUCSMP and Chapter 2 in CME. The coding process began with the first three authors using the analytic framework to code thesame lessons from CME and discussing the results of that coding. After coding the same sections from the CME textbook tomake sure that we had a common understanding of the framework and that there were few differences in coding the first andthird authors independently coded randomly selected sections from an electronic version of the teacher’s edition of theUCSMP textbook while the second author coded the randomly selected sections from a hard copy of the teacher’s edition ofthe CME textbook.

If students were asked to construct a conjecture about some mathematical idea, but they had not been given a chance toexplore data related to it for patterns, we assumed that students would engage in this activity first and applied this codebefore coding for a conjecture. Recall that according to our definition a conjecture moves beyond the data in some way andreflects uncertainty as to its validity. We used words such as ‘‘suggests’’, ‘‘I think that’’, or similar words in expositioncomponents along with some set of data to indicate the presence of a conjecture. If the word ‘‘conjecture’’ or ‘‘make a guess’’was used along with a statement that asked students to extrapolate from some set of data in the textbook exposition this wasalso considered an instance of the development of a conjecture.

An exception to the use of two criteria to determine a conjecture appeared in the task components of the studenttextbook. If student tasks did not use the word conjecture, they were still considered instances of the development of aconjecture if they asked students to move beyond the set of data in some way. In the UCSMP text, students were occasionallyasked to generalize from a set of data in some way as seen in the following excerpt.

‘‘25. Use a graphing utility to graph y = 10/x3, y = 10/x4, y = 10/x5. What pattern do you notice? What generalization canyou make about the value of n and the graph y = 10/xn?’’ (Flanders et al., 2010, p. 113)

In this example, students are asked to graph several rational functions and identify a pattern. The word generalization

used in the second component of this question along with the use of the variable n suggests that students need to go beyondthe data in some way. Indeed, the teacher resource materials implies that a conjecture is being sought after instead of anargument as the answer in the teacher’s edition to this task states: ‘‘If n is even, the graph will be its own reflection across they-axis; whereas if n is odd, it will not’’ (Flanders et al., 2010, p. 113). Note that this coding differs from G. J. Stylianides (2009)as he coded such instances as either plausible or definite patterns.

Occasionally, in the task components of the two textbooks, students were either asked to construct a set of data or weregiven a set of data, asked to located patterns in the data, and justify these patterns. Such instances were coded as thedevelopment of definite patterns and the construction of arguments, but not the construction or testing of conjectures. Thatis, we did not assume that students would construct or test conjectures as they moved from the identification of patterns tothe development of valid arguments.

To enable analyses to be conducted within Hyperresearch (Researchware, 2009) .txt files were prepared as follows. If weencountered an instance in the exposition component or task component of the student textbook that matched the analyticframework in the sections sampled from UCSMP, the verbatim text was copied from the electronic version and pasted into a.txt file. In the case of the CME text the words at the beginning and the words at the end of the instance were placed into a .txtfile. If the instance occurred within the exposition section, the code consisting of the word present was placed in the .txt file. Ifthe RP instance occurred in the task component of the student textbook the code consisting of the word develop was placed inthe .txt file. If the RP instance involved a pattern, it was further identified as a definite or plausible pattern using theframework. If the RP instance involved a pattern, the exposition and task components after that instance were examined forthe presence of a conjecture related to the pattern. If such a conjecture appeared the code was appended with the wordsconjecture precursor, if not, the code was appended with the words conjecture non-precursor. Next, exposition and taskcomponents of the student textbook were examined for valid arguments related to the pattern with the use of theframework. Such searches for conjectures and arguments related to the pattern occurred within the same section of thestudent textbook where they initially occurred and expanded out to sections appearing in chapters after the initial pattern

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106100

identification. If the textbook exposition presented or the task components provided students with opportunities to identifypatterns or construct conjectures from data that was generated with handheld or computer technology the word technology

was attached to the code. Likewise, if the symbolic manipulation capabilities of a CAS were used in one or more steps of avalid argument, the word technology was appended to the code. Instances that were considered proof building blocks ineither the exposition or task components of the student textbook were given the code develop or present followed by thewords proof building block and the type of proof building block (e.g., definition).

2.10. Analysis

Each text file was imported into Hyperresearch (Researchware, 2009) where frequencies for each RP subcomponentwithin the exposition and student task sections were calculated across each chapter and for each textbook. Hyperresearchsoftware was used to examine the data for definite pattern purposes and technology uses. Total tasks and tasks dedicated toRP were counted within in each sampled section and accumulated by chapter within each textbook. These values were usedto calculate a percent of RP tasks for each chapter in each textbook and for each textbook overall. The consistency of the RPpercentage by chapter within each textbook as well as the differences between the two textbooks in terms of percentage ofRP tasks were analyzed using Chi Square tests. A level of significance of 0.05 was used for all statistical tests.

2.11. Inter-rater reliability

We coded a total of three sections from CME until calculations involving the coding for one section by the first two authorsresulted in a high level of inter-rater reliability of Kappa = 0.8704. The first two authors next coded the first section in UCSMPindependently to insure that our coding was consistent within a new curriculum. The inter-rater reliability for coding ofUCSMP had a high level of agreement with Kappa = 0.8205.

3. Findings

3.1. RP components across textbooks by task and exposition component

Descriptive information across the RP components within the task components for each textbook is shown in Table 1. Asthe total RP tasks were slightly different across the two textbook units, we resorted to using percentages for comparisonpurposes. CME contained more opportunities for students to develop proof building blocks, identify patterns, and createspecific arguments. UCSMP contained more opportunities for students to develop conjectures, engage in proof-writingexercises, develop general arguments, and construct proof subcomponents. The largest difference between the texts was inthe area of proof-writing exercises and specific arguments.

Descriptive information across the RP components within the exposition components for each textbook appears inTable 2. On the one hand, CME presented more patterns, conjectures, and proof writing exercises in the student textbookexposition components than UCSMP. An example of a conjecture presented in the textbook exposition for CME appears inFig. 4. The CME textbook presents dialog between hypothetical students within the exposition sections of the textbook. Weconsidered Derman’s response to be a conjecture as the words ‘‘I wonder’’ indicate uncertainty as to its validity and he is

Table 2

Distribution of RP categories within exposition components of each textbook.

Textbook Proof building blocks Pattern Conjectures Argument Totals

Proof-writing exercises Specific General Subcomponents

UCSMP 238 (62.5%) 13 (3.4%) 3 (0.8%) 58 (15.2%) 22 (5.8%) 37 (9.7%) 10 (2.6%) 381 (100.0%)

CME 137 (48.1%) 40 (14.0%) 11 (3.9%) 67 (23.5%) 9 (3.2%) 16 (5.6%) 5 (1.8%) 285 (100.0%)

Totals 375 (56.3%) 53 (8.0%) 14 (2.1%) 125 (18.8%) 31 (4.1%) 53 (8.0%) 15 (2.3%) 666 (100.0%)

Table 1

Distribution of RP categories within task components of each textbook.

Textbook Proof building blocks Pattern Conjectures Argument Totals

Proof-writing exercises Specific General Subcomponents

UCSMP 4 (1.0%) 155 (38.2%) 34 (8.4%) 132 (32.5%) 29 (7.1%) 42 (10.3%) 10 (2.5%) 406 (100.0%)

CME 6 (1.3%) 183 (41.1%) 37 (8.3%) 106 (23.8%) 61 (13.7%) 45 (10.1%) 5 (1.1%) 445 (100.0%)

Totals 10 (1.2%) 338 (39.7%) 71 (8.3%) 238 (28.0%) 90 (10.6%) 87 (10.2%) 15 (1.8%) 851

Fig. 4. An example of a conjecture appearing within the exposition of the CME textbook (EDC, 2009, p. 611).

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 101

moving beyond the set of data, which gave rise to it. UCSMP presented more proof building blocks, argument-specific,argument-general, and subcomponents. In the next section, the percentages of RP tasks by each chapter within eachtextbook and across textbooks were analyzed using Chi Square tests.

3.2. Examining percentages of RP tasks by chapter and textbook

The percentages by chapter and overall of tasks that were categorized as RP within UCSMP and CME are shown in Tables 3and 4, respectively. The apportionment of RP tasks by chapter within the UCSMP textbook were statistically different fromone another x2(12, N = 3705) = 44.299, p < 0.001. For instance, almost 16% of the tasks appearing in the sections sampledfrom the chapter on polynomials in UCSMP involved RP whereas less than 6% of the tasks in the linear functions/sequenceschapter dealt with RP. The apportionment of RP tasks by chapter within the CME textbook were statistically different fromone another x2(7, N = 2823) = 60.351, p < 0.001. Overall, there were statistically significant differences between thepercentage of RP tasks between the two textbooks x2(1, N = 6528) = 36.310, p < 0.001. That is, the CME textbook providedstudents with a significantly higher number of RP tasks when these were considered as a percentage of total tasks.

The differences between the two textbooks in terms of their RP task percentages is seen in each textbook’s handling of thefactor theorem: x � r is a factor of P(x) if and only if P(r) = 0, that is, r is a zero of P. In the UCSMP textbook, the proof is

Table 3

Topics, tasks, RP tasks, and percentage of RP tasks in UCSMP.

Topic Tasks RP tasks RP tasks percent

Functions 284 39 13.73

Direct and Inverse Variation 290 38 13.10

Linear functions/sequences 260 15 5.77

Matrices 287 39 13.59

Linear systems 286 23 8.04

Quadratic functions 325 41 12.62

Powers 274 25 9.12

Inverses and radicals 249 29 11.65

Exponential/logarithmic 289 10 3.46

Trigonometry 313 33 10.54

Polynomials 536 85 15.86

Quadratic relations 281 25 8.90

Series and combinations 299 27 9.03

Overall 3973 429 10.80

Table 4

Topics, tasks, RP tasks, and percentage of RP tasks in CME.

Topic Tasks RP tasks RP tasks percent

Fitting functions to tables 229 60 26.20

Functions and polynomials 436 85 19.50

Complex numbers 447 25 5.59

Linear algebra 352 55 15.63

Exponential and logarithmic functions 472 73 15.47

Graphs and transformations 277 43 15.52

Sequences and series 353 55 15.58

Introduction to trigonometry 257 49 19.07

Overall 2823 445 15.76

Fig. 5. Information from the CME student textbook with regard to the factor theorem (EDC, 2009, p. 155).

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106102

presented in detail for students within the exposition section of the textbook. In CME, however, the burden of provingthis important theorem is placed on students as seen in Fig. 5. In both textbooks, the actual construction of the theoremitself, a proof building block, was completed by the textbook authors and appears in the exposition sections of bothtextbooks.

3.3. Examining percentages of RP sentences by chapter and textbook

The percentages by chapter and overall of exposition sentences that were categorized as RP within UCSMP and CME areshown in Tables 5 and 6, respectively. The apportionment of RP sentences by chapter within the UCSMP textbook werestatistically different from one another x2(12, N = 4085) = 275.989, p < 0.001. The apportionment of RP sentences by chapterwithin the CME textbook were statistically different from one another x2(7, N = 2987) = 56.269, p < 0.001. Overall, therewere statistically significant differences between the percentage of RP tasks between the two textbooks x2(1,N = 7072) = 26.267, p < 0.001. That is, the CME textbook provided students with a significantly higher percentage of RPsentences within exposition sections than the UCSMP textbook (30.23% vs. 24.75%).

Table 5

Topics, exposition sentences, RP sentences, and percentage of RP sentences in UCSMP.

Topic Sentences RP sentences RP sentences percent

Functions 251 26 9.39

Direct and inverse variation 324 49 13.14

Linear functions/sequences 294 32 9.82

Matrices 191 81 29.78

Linear systems 333 59 15.05

Quadratic functions 282 118 29.50

Powers 136 71 34.30

Inverses and radicals 196 48 19.67

Exponential/logarithmic 269 60 18.24

Trigonometry 183 141 43.52

Polynomials 195 78 28.57

Quadratic relations 197 140 41.54

Series and combinations 223 108 32.63

Overall 3074 1011 24.75

Table 6

Topics, sentences, RP sentences, and percentage of RP sentences in CME.

Topic Sentences RP sentences RP sentences percent

Fitting functions to tables 253 67 20.94

Functions and polynomials 295 105 26.25

Complex numbers 264 131 33.16

Linear algebra 397 126 24.09

Exponential and logarithmic functions 267 125 31.89

Graphs and transformations 202 118 36.88

Sequences and series 227 99 30.37

Introduction to trigonometry 179 132 42.44

Overall 2084 903 30.23

Fig. 6. Pattern and conjecture purposes within the UCSMP sections sampled.

Fig. 7. Pattern and conjecture purposes within the CME sections sampled.

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 103

3.4. Purposes of patterns and conjectures

We also examined the differences between the two textbook sections sampled in terms of the purposes for theidentification of patterns and construction of conjectures. That is, how frequently was the identification of patterns followedby the construction of conjectures or the development of valid arguments? Fig. 6 illustrates the breakdown in the UCSMPsections sampled in the purposes of patterns and conjectures. A total of 16% of students’ opportunities to identify definitepatterns resulted in opportunities for students to construct conjectures. Overall, 19% of students’ opportunities to identifydefinite patterns resulted in opportunities for students to develop arguments. Fig. 7 shows the breakdown in the CMEsections sampled in the purposes of patterns and conjectures. Similar to UCSMP 10% of the definite pattern identificationtasks were connected to conjecture development. A total of 12% of the pattern identification tasks resulted in thedevelopment of arguments. These findings suggest that patterns in both textbooks were rarely used to develop conjectures,test conjectures, or develop valid arguments. On many occasions the identification of a definite pattern in both the UCSMPtextbook and CME textbook appeared to be an end in itself.

3.5. Examinations of technology use in RP across textbooks

Within UCSMP, 51% of students’ pattern detection opportunities were conducted on data that were generated with theuse of technology. In the case of conjectures, this dropped down to 26%. In the area of arguments, 5% of general argumentsinvolved technology and 20% of opportunities within subcomponents involved technology. Thus the most frequent use oftechnology in UCSMP was in the area of pattern detection with a gradual decrease in technology use from patterns throughconjectures and into general arguments. In the sampled sections from the CME textbook, 21% of patterns involved data thatwere created with the use of technology. A higher percentage of conjectures involved technology (35%) while 5% of generalarguments involved technology. No argument subcomponents instances in CME involved technology. Altogether there werea total of 48 instances of technology use in CME compared with 95 in UCSMP. While technology was touted as a designprinciple in both textbooks (Cuoco, 2007; Usiskin, 2007) it did not appear as frequently in CME as it did in UCSMP.

4. Discussion and conclusion

This study used a framework adapted from the work of G. J. Stylianides (2009) to examine RP opportunities withinexposition and task components appearing in two reform-oriented secondary advanced algebra textbooks. Quantitative and

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106104

qualitative analyses were conducted on the prevalence of RP within the task and exposition components of each of thetextbooks, the purposes of patterns, and the use of technology in RP. Similarities and differences with regard to RP werefound between the two textbooks. These findings will be summarized and their implications discussed in the sections thatfollow.

4.1. Similarities across texts

In both texts, the most prevalent RP category was the identification of definite patterns. Neither text contained manyopportunities for students to engage in the identification of plausible patterns. G. J. Stylianides (2009) found evidence ofthese at the middle school level reform-oriented textbook series while Davis’ (2012) examination of RP in three differenthigh school mathematics textbook units found very few instances of plausible patterns. These previous research studies aswell as the analyses in this study suggest that textbook developers may see little use for this RP category at the high schoollevel.

In both textbooks very few definite pattern opportunities were tied to the development of arguments. The disconnectbetween definite pattern identification and the construction of arguments is troubling as such a finding discounts theimportance of the experimental side of mathematics (Lakatos, 1976; NCTM, 1989, 2000; Steen, 1990). It may also be the casethat students who are not given opportunities to identify patterns may not be as successful in generating valid arguments asthe examination of patterns may help students understand why patterns exist which may reveal how an argument can beconstructed. Given the important role of the textbook in helping to shape classroom lessons (Tarr et al., 2008), it is likely thatif the textbook does not provide students with opportunities to engage in the identification of patterns and the constructionof conjectures on the road toward developing proofs, that students and their teachers are not likely to engage in theseactivities of their own accord. In addition, the prevalence of definite patterns that are not connected to arguments mayreinforce a belief among students that empirical evidence is sufficient to determine the validity of a mathematical idea (Harel& Sowder, 1998).

Conjecturing did not appear as frequently as other RP components such as pattern identification in either textbook. Inaddition, a total of 16% and 10% of definite patterns were connected to the development of conjectures in UCSMP and CME,respectively. Consequently, students may not undervalue the role of conjectures in the development of mathematicalknowledge. Providing students with fewer opportunities to engage in the development of conjectures when compared withother RP components within the written curriculum may reduce the importance of this act in the eyes of students, which canin turn affect the quantity of mathematical discourse during the enacted curriculum. That is, conjectures presented bystudents during classroom conversations may cause other students to engage in discussions whereby they developcounterexamples or other types of valid arguments as they seek to refute or validate those conjectures.

4.2. Differences between texts

While Davis (2012) found a higher prevalence of RP opportunities in a reform-oriented textbook unit than in aconventional U.S. textbook unit, this study suggests that there are notable differences between U.S. reform-orientedtextbooks. Chi Square analyses involving the percentage of tasks and percentage of exposition sentences dedicated to RPshowed that CME provided students with more opportunities to engage in reasoning-and-proving than did UCSMP.Specifically, the CME textbook also contained more instances of pattern identification and conjecture development in thenarrative component through student dialog than did UCSMP. Consequently, there seemed to be a greater alignmentbetween the presentation and development of RP in CME.

National reform initiatives in the United States (Common Core State Standards Initiative [CCSSI], 2010; NCTM, 1989,2000) as well as other countries such as Ireland (National Council for Curriculum and Assessment, n.d.) have called for theinstantiation of reasoning and proof across different mathematics content strands. Recall that Thompson et al. (2012) founddifferential opportunities for students across content areas in algebra. This study also found differences in RP tasks amongthe different chapters in each textbook, suggesting that different content areas within the algebra content strand containdifferential RP task opportunities. These findings suggest that textbook authors need to more carefully attend to RP whenconstructing classroom activities for students. The framework used in this study provides one such way that textbookdesigners can more thoughtfully plan for RP in written curriculum resources.

4.3. Situating RP in these texts with regard to previous research

In detailing the thought process used to choose exponentials, logarithms, and polynomials as the site for an examinationof RP, Thompson et al. (2012) noted that there were few instances of proof-related reasoning in other algebra content areas.G. J. Stylianides (2009) also found variability among different mathematics content areas such as number theory, geometry,and algebra. The findings of this study align with the results of these previous studies in that students experience differentopportunities to engage in RP among a variety of mathematics content areas. Moreover, these different opportunities extendto smaller subareas within algebra. We considered the low frequency of RP in the UCSMP chapter on exponential andlogarithms to be surprising given that Thompson et al. (2012) found this category to contain the most RP opportunities across22 United States high school mathematics textbooks. In addition, it was unexpected that the CME chapter on complex

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106 105

numbers would contain so many fewer opportunities for students to engage in RP tasks given the breakdown of RP tasks inthe other CME chapters.

G. J. Stylianides (2009) found that about 40% of the tasks appearing in a reform-oriented middle school mathematicsprogram contained RP instances. However, using a similar framework in this study students were presented with feweropportunities to engage in RP in the second algebra course at the high school level. This is somewhat troubling as nationalreform initiatives in the U.S. advocate for RP to be embedded throughout students’ learning experiences across the K-12school continuum. Moreover, as many students prepare to move to university education systems in which proof plays a moreprominent role, it is important that students’ experiences with these processes increase rather than remain at the same levelas middle school or decrease. Yet it is not clear what percentage of task or narrative components should involve RP in orderfor students to become proficient at engaging in this sophisticated behavior. This connection between the writtencurriculum and the learned curriculum has yet to be investigated with regard to RP. Bieda’s (2010) examination of RP withinan enacted middle school curriculum suggests that the percentage of RP tasks that students engage in during the classroomlesson is substantially less than the number appearing in the written textbook curriculum.

4.4. Textbook design

The act of identifying patterns was the most frequent RP category within the task components of both student textbooks.However, as mentioned earlier students’ pattern identification was often not tied to conjecture development or theconstruction of arguments. In order to provide students with more opportunities to engage in the construction of conjecturesand arguments perhaps textbook developers could provide students with fewer opportunities to identify patterns and morecoordinated sets of activities that show the importance of pattern identification and conjecture development as precursors tothe construction of arguments. That is, textbook developers could use research on students’ engagement with RP todetermine which mathematical ideas are within the conceptual reach of pupils and purposefully design tasks that providestudents with opportunities to identify patterns, construct conjectures, create general and specific arguments, and put intowords the theorems that result from this work (proof building blocks). In addition, while pattern identification is animportant subcomponent of RP textbook developers could present the following within the exposition component of thestudent textbook: sets of data, plausible and definite patterns embedded in the data, and several potential conjectures (validand invalid). One vehicle by which this could be accomplished was demonstrated in the CME textbook with the use of dialogbetween hypothetical students. Curriculum developers could then inject more specific and general argument opportunitiesinto the task components of student textbooks.

References

Bieda, K. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41,351–382.

Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Author.Cuoco, A. (2007). The case of the CME Project. In C. R. Hirsch (Ed.), Perspectives on the development of school mathematics curricula (pp. 119–127). Reston, VA:

National Council of Teachers of Mathematics.Davis, J. D. (2012). An examination of reasoning and proof opportunities in three differently organized secondary mathematics textbook units. Mathematics

Education Research Journal, 24, 467–491.Dossey, J., Halvorsen, K., & McCrone, S. (2008). Mathematics education in the United States 2008: A capsule summary fact book. Reston, VA: National Council of

Teachers of Mathematics.Education Development Center. (2009). Algebra 2. Boston: Pearson Education.Flanders, J., Lassak, M., Sech, J., Eggerding, M., Karafiol, P. J., McMullin, L., et al. (2010). The University of Chicago school mathematics project: Advanced algebra (3rd

ed.). Chicago: Wright Group.Garry, T. (2003). Computing, conjecturing, and confirming with a CAS tool. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer algebra

systems in secondary school mathematics education (pp. 137–150). Reston, VA: National Council of Teachers of Mathematics.Grouws, D. A., & Smith, M. S. (2000). NAEP findings on the preparation and practices of mathematics teachers. In E. A. Silver & P. A. Kennedy (Eds.), Results from the

seventh mathematics assessment of the National Assessment of Educational Progress (pp. 107–139). Reston, VA: National Council of Teachers of Mathematics.Hanna, G. (2007). The ongoing value of proof. In W. M. Roth, L. Verschaffel (series Eds.), P. Boero (vol. Ed.), New directions in mathematics and science education: Vol.

1. Theorems in school: From history, epistemology and cognition to classroom practice (pp. 3–18). Rotterdam, The Netherlands: Sense.Hanna, G., & de Bruyn, Y. (1999). Opportunity to learn proof in Ontario grade twelve mathematics texts. Ontario Mathematics Gazette, 37(4), 23–29.Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. CBMS Issues in Mathematics Education, 7, 234–283.Johnson, G. J., Thompson, D. R., & Senk, S. L. (2010). Proof-related reasoning in high school textbooks. Mathematics Teacher, 103(6), 410–418.Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.National Council for Curriculum and Assessment. (n.d.). Leaving certificate mathematics syllabus: Foundation, ordinary & higher Level: 2014 examination. Author.National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of

Mathematics.National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Nordstrom, K., & Lofwall, C. (2005). Proof in Swedish upper secondary school mathematics textbooks – the issue of transparency. In Proceedings of the 4th congress

of the European Society for Research in Mathematics Education (CERME4) (pp. 448–457). Spain: ERME.Researchware. (2009). Hyperresearch 2.8.3 [Computer software]. Randolph, MA: ResearchWare Inc.Schaeffer, R. L., Mendenhall, W., Ott, R. L., & Gerow, K. (2012). Elementary survey sampling (7th ed.). Boston: Brooks/Cole.Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of

research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.Schoenfeld, A. H. (2009). Series editor’s foreword. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16

perspective (pp. xii–xvi). New York: Routledge.Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in

Mathematics Education, 34, 4–36.

J.D. Davis et al. / International Journal of Educational Research 64 (2014) 92–106106

Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72,271–288.

Steen, L. A. (Ed.). (1990). On the shoulders of giants: New approaches to numeracy. Washington, DC: National Academy Press.Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321.Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28, 9–16.Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288.Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle-grades mathematics curricula and the classroom learning

environment on student achievement. Journal for Research in Mathematics Education, 39, 247–280.Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in

Mathematics Education, 43, 253–295.Usiskin, Z. (2007). The case of the University of Chicago School Mathematics Project – secondary component. In C. R. Hirsch (Ed.), Perspectives on the design and

development of school mathematics curricula (pp. 173–182). Reston, VA: National Council of Teachers of Mathematics.Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice

through the world of textbooks. Dordrecht, The Netherlands: Kluwer.Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39, 431–459.