Reasoning and proof in eighth-grade mathematics textbooks in China

International Journal of Educational Research 98 (2019) 77–90

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Reasoning and proof in eighth-grade mathematics textbooks in China Di Zhang, Chunxia Qi

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Faculty of Education, Beijing Normal University, No. 19, XinJieKouWai St., Haidian District, Beijing 100875, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Mathematics textbook Reasoning and proof China Curriculum

Despite the importance of reasoning and proof (RP) opportunities in mathematics textbooks has been focused on by many researches, the methodologies were lack of consistency. To make it more accessible to obtain specific details about RP in the mathematics textbooks and compare RP across Oriental and Western mathematics textbooks, this study provides a unified analytical framework to present RP within the exercises and narratives. Meanwhile, this paper utilized this new framework to examine two volumes of Chinese reform-oriented eighth-grade mathematics textbooks. We connected the results with prior researches, and discussed the features of Chinese textbook design regarding to RP compared with U.S. textbooks. Moreover, this framework could also be applicable to examine RP in the enacted lessons in future studies.

1. Introduction Similar to Lithner (2003), who mentioned that one student was puzzled about the rules of powers when performing operations such as a5·a3, we find that a Chinese student also has this problem even though he is already be the eleventh grade. He is unsure about whether the result of the operation mode 54 ·55 is 59 or 520. At first glance, it may seem that the Chinese student has learned the operation rules of powers without fully understanding the basic definition of power (e.g., Schoenfeld, 1992; Stein, Grover, & Henningsen, 1996): generally, multiplication a·a·a···a with n factors is recorded as an. If this Chinese student has already understood the definition, he would know that 54 ·55 means 5·5·5·5·5·5·5·5·5, and he would obtain the right answer. However, when we prompt him about the definition, he is suddenly enlightened. It seems that some students can remember and understand mathematical concepts, but in situations such as the operating mode 54 ·55, they are not able to reason mathematically based on what they have learned. Lithner (2004) noted that these students did not consider mathematical reasoning as a major strategy, even if they had mastered basic knowledge of mathematics. Currently, an agreement has been made that mathematical reasoning plays a key role in students’ mathematical learning (e.g., Schoenfeld, 1994; Ross, 1998; NCTM (National Council of Teachers of Mathematics), 2000; Ball, Hoyles, Jahnke, & MovshovitzHadar, 2002; Yackel & Hanna, 2003; Mariotti, 2006). Though we will not deny that mathematics reasoning and proof can be taught to middle school students, and even to children in younger grades (e.g., Hawkins, Pea, Glick, & Scribner, 1984; Light, Blaye, Gilly, & Girotto, 1989; Galotti, Komatsu, & Voelz, 1997; Reid, 2002; Ball & Bass, 2003; Stylianides, 2005, 2007; Stylianides, Stylianides, & Shilling-Traina, 2013), it is nonetheless difficult to teach, especially geometric reasoning and proof (Fan, Qi, Liu, Wang, & Lin, 2017; Fan, Mailizar, Alafaleq, & Wang, 2018; Weber, 2001). Textbooks not only connect the intentions of curriculum standards with the mathematical achievement of students (McKnight

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Corresponding author. E-mail addresses: [email protected] (D. Zhang), [email protected] (C. Qi).

https://doi.org/10.1016/j.ijer.2019.08.015 Received 1 January 2019; Received in revised form 23 July 2019; Accepted 29 August 2019 0883-0355/ © 2019 Elsevier Ltd. All rights reserved.

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et al., 1987; Schmidt et al., 2001) but also act as the instantiation of the implementation of curriculum standards (Valverde, Bianchi, Wolfe, Schmidt, & Houang, 2002). In general, textbooks usually shape teachers’ instructional actions in the classroom, thus affecting students’ learning directly (Cohen, Raudenbush, & Ball, 2003; Leinhardt, Zigmond, & Cooley, 1981; Ni & Cai, 2011). That is, mathematics textbooks have a critical impact on classroom teaching and students’ opportunity to learn (Thompson, 2014). In China, mathematics textbooks are teachers’ main references for their daily teaching activities, as well as the main resources from which students learn mathematics (Fan, Chen, Zhu, Qiu, & Hu, 2004). As mentioned previously, a senior high school student did not get used to reasoning mathematically even if he had been taught the relevant mathematical knowledge at his junior high stage. It seems that students in their junior secondary years do not form a habit of engaging in the reasoning and proof, which is quite related to the reasoning and proof opportunities provided by their mathematics textbooks. In a sense, the first step to improving students’ reasoning and proof abilities is to clarify how reasoning and proof presents in junior high school mathematics textbooks. In this paper, we focus on the opportunities that eighth-grade mathematics textbooks provide for students to learn reasoning and proof (RP). More specifically, we will try to answer the following research questions: 1 What types of RP are introduced in both the narratives and exercises of mathematics textbooks? 2 How are types of RP distributed across different topics in mathematics textbooks? 2. Literature review Researchers are prone to using the two concepts “reasoning” and “proof” without clearly distinguishing them. Recently, instead of remaining distinct terms, the two elements have become a united term in the field of mathematics education (e.g., Cai & Cirillo, 2014; Stylianides, 2008, 2009; Bieda, Ji, Drwencke, & Picard, 2014; Thompson, Senk, & Johnson, 2012; Thompson, 2014; Wong & Sutherland, 2018). However, not all researchers choose the exact same definition of RP in their analyses. For example, Stylianides (2008) defined reasoning-and- proving as “the major activities that are frequently involved in the process of making sense of and establishing mathematical knowledge: identifying patterns, making conjectures, providing non-proof arguments, and providing proofs”. In somewhat of a contrast, Thompson et al. (2012) defined the term proof-related reasoning in the following way: “includes making and investigating conjectures, developing and evaluating deductive arguments, and other experiences, such as finding counterexamples or correcting mistakes in logical arguments, that are foundational elements of mathematical reasoning”. Although some definitions of RP are incongruent, under which scholars developed their own frameworks to conduct the analyses of mathematics textbooks. Among them, content and exercises pertained to RP in mathematics textbooks were the main objects of investigation. Some researchers tend to choose the geometry content, which seems to be much more related to RP (e.g., Hanna & de Bruyn, 1999; Hanna, 2000; Herbst, 2002; Herbst et al., 2009; Sears, 2012). For example, Hanna and de Bruyn (1999) found that only geometry in Canadian textbooks provided opportunities for students to learn proof in an appropriate way. However, RP could distribute across different content areas and topics (Stylianides, 2014). Recently, with the depth of study, scholars no longer restricted their RP research to geometry; they extended their efforts into other contents in mathematics textbooks, such as numbers and algebra. For instance, Stacey and Vincent (2009) examined how modes of reasoning were presented across 7 topics in numbers, space, algebra and measurement in 9 8th-grade mathematics textbooks in Australia. They reported that most mathematical rules across different contents were presented with explanations, but the explanatory texts were not used for thinking but rather for deriving rules or justifying the prepared textbook exercises. Similarly, Thompson et al. (2012) focused on RP in 3 topics (exponents, logarithms, and polynomials) in algebra across 20 U.S. high school mathematics textbooks. They found that approximately half of the properties of the selected 3 topics were justified with arguments, and the left half consisted of unjustified properties; in other words, “without additional instruction, students may have limited experiences with proof-related reasoning” (Thompson et al., 2012, pp. 282). In general, from these studies, we can observe that the research in the field has given more attention to Western countries (Fan et al., 2018); only a few studies focus on Eastern countries (e.g., Fujita & Jones, 2014; Miyakawa, 2012). Among them, very few studies are from China. Thus, research from China could provide a wider international perspective in this research area. 3. Analytical framework 3.1. Guiding sources for developing our analytical framework The analytical framework of our study mainly originates from existing frameworks (Stylianides, 2009; Thompson et al., 2012). Stylianides (2009) categorized RP into identifying a pattern, making a conjecture, providing a proof, and providing a non-proof argument. Identifying a pattern and making a conjecture are processes in which students try to make generalizations based on the given data. Providing a proof is giving valid arguments (they are built on accepted facts such as definitions, axioms, and theorems that are shared by a community for or against claims) to mathematical claims (Stylianides, 2007, 2009). Stylianides (2009) further distinguished proof into generic example and demonstration. The former can be seen as using a representative case to make the truth of mathematical claims explicit (Balacheff, 1988; Stylianides, 2009). Stylianides (2009) defined the latter as proof that use valid arguments (e.g., counterexample, mathematical induction) except those that depend on representative cases. Balacheff (1988) referred to it as a “thought experiment” and Harel and Sowder (1998) referenced it as an “axiomatic proof”. In addition, a non-proof argument is “an argument for or against a mathematical claim that does not qualify as a proof” (Stylianides, 2009, pp.266). According to Stylianides (2009), it can be classified into two kinds: an empirical argument and a rationale. In his framework, the former has a similar meaning to 78

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“naive empiricism” (Balacheff, 1988) and “empirical proof schemes” (Harel & Sowder, 1998), and it can be viewed as utilizing a particular case or a few cases that are inconclusive in their support of the truth of the claim. The latter is defined as the valid arguments that are used to support claims that are neither qualified as proof nor empirical (Stylianides, 2009, 2008). Similarly, Thompson et al. (2012) developed another framework of RP; they used proof-related reasoning to represent RP. This terminology has some similarities with the framework of Stylianides, it comprises “making and investigating conjectures, developing and evaluating deductive arguments, and having other experiences, such as finding counterexamples or correcting mistakes in logical arguments” (Thompson et al., 2012, pp. 258). In practice, two frameworks were proposed by Thompson et al. (2012) and then applied in analysing both the narratives and exercises within textbooks. For the exercise sets, the three dimensions in the framework mentioned above were further divided into seven subcategories: make a conjecture, investigate a conjecture (students are asked to decide whether an assertion or conjecture is true or not and to explain why), develop an argument (students are asked to prove a claim), evaluate an argument (students are asked to judge the validity of an argument), counterexample (students are asked to give a counterexample to prove a claim is false), correct or identify a mistake (students are asked to find or to correct a mistake in the reasoning), and principles of proof (students are asked to interpret the formation of a particular argument without providing the whole proof). In addition, the framework employed in the narratives was categorized into general argument (or proof), specific argument (it is using a specific case to justify and can be viewed as a particular example of a general argument), left to the student (meaning that students will be asked to justify by themselves) and no justification (no statement or any clue was provided to support the truth of a claim). 3.2. Pilot study We used the frameworks of Stylianides (2009) and Thompson et al. (2012) to conduct a pilot coding scheme for the exercises and narratives in eighth-grade mathematics textbooks. We found that either framework was slightly too narrow to cover all possible categories of RP in the textbooks. For example, “make a conjecture” in both of their frameworks mainly referred to formulating a generalized hypothesis according to the given cases, while some of the exercises and narratives provided students with other kinds of opportunities to conjecture (see Fig. 3), which has a meaning similar to that of the term “predict” used in TIMSS (2003). Polya (1968) viewed inductive reasoning (similar to “make a conjecture” in the frameworks of Stylianides and Thompson et al.) and analogy reasoning as two particular kinds of plausible reasoning that could be “more philosophical to consider the general idea” (Polya, 1968, pp.v). In addition, there are other isolated cases of plausible reasoning. Polya (1954) thought there were neither fixed standard nor clarity as good as demonstrative logic of plausible reasoning, but opportunities should be provided for students to engage in guessing or plausible reasoning. Therefore, to cover all types of exercise sets and narratives, we use the term “generalize” to represent making a mathematical generalization; and “predict” is used to represent conjectures that exclude actions performed by generalizing. We further broadened the connotation of “make a conjecture” and used it as plausible reasoning, which is similar to Harel and Sowder (2007) “conjecture” (an assertion made by an individual who is uncertain of its truth, pp.808). 3.3. The framework we used in our study Through the pilot study, we found that neither the framework of Stylianides nor that of Thompson et al. could fully encompass all types of RP that appeared in the narratives and exercises of Chinese mathematics textbooks. Thus, we also referenced Polya’s (1968, 1954) statement and reasoning framework of TIMSS (2003). In summary, we built the structure of our analysis framework based on their collective work. This framework will be applied to the examination of the role of RP in both exercises and narratives within eighth-grade mathematics textbooks in China. In our pilot study, we extended the notion of “making a conjecture” and modified its subcategories. To make our framework usable for all kinds of potential RP opportunities in the exercises and narratives of Chinese mathematics books, we add one type of proof that has not yet appeared during our work. Both Stylianides (2009) and Thompson et al. (2012) emphasized the importance of counterexamples in helping students learn formal proof. Furthermore, Fan et al. (2018) analysed geometric proof in secondary mathematics textbooks in China and chose the series of the Beijing Normal University Press (BNUP) Edition and the framework in their research already contained the concept of the counterexample. Therefore, we tend to examine the exercises and narratives that provided opportunities for students to use counterexamples in their proofs. In addition, some exercises and narratives in the textbooks require students to judge whether the argument is valid. Therefore, we added “evaluate an argument” to our framework. In addition, we provided some exercises and narratives that did not contain RP and coded them as “others”. Based on existing frameworks and our analysis mentioned above, we summarize the analytical framework in Table 1. Meanwhile, we illustrate the framework below by some examples. Fig. 1 shows an example of identifying a pattern. The students are asked to identify the relationship between the abscissa of several pairs of corresponding points in two graphics before and after translation, which could be seen as a pattern. Fig. 2 illustrates the example of generalizing. After filling the blanks of the formulas in item (1), students are required to make a conjecture and generalize the potential rules hidden behind the above specific cases in item (2). Unlike the example shown in Fig. 2, Fig. 3 shows another kind of conjecture. Some mathematical claims (e.g., predicting whether △DEF is an equilateral triangle or not) needed to be predicted at first. Then, students were asked to make deductive arguments for each claim, which could be considered to be demonstrations. Fig. 4 not only provides opportunities to generalize and demonstrate but also encourages students to use examples to support their 79

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Table 1 Analytical framework for coding RP. (IP)

Identifying a Pattern

(MC)

Making a Conjecture

(PP)

Providing a Proof

(PNP)

Providing a Non-proof Argument

(EVA)

Evaluating an Argument

(G) (P) (GE) (D) (C) (EA) (R)

Generalize Predict Generic Example Demonstration Counterexample Empirical Argument Rationale

Notes. (1) The categories of “identifying a pattern”, “providing a proof”, and “providing a non-proof argument” in second column were same with those in Stylianides’ framework. But the meaning of “making a conjecture” is extended compared to his framework. (2) The subcategories of “generic example”, “demonstration”, “empirical argument”, and “rationale” in fourth column were also same with those in his framework. In addition, the meaning of “generalize” was similar to “conjecture” in his framework.

Fig. 1. Identifying a pattern for corresponding points in two graphics before and after translation (Ma, 2014b, p. 73).

Fig. 2. Generalize for the rules of several formulas (Ma, 2014b, p. 63).

Fig. 3. Prediction and demonstration for equilateral triangles and midpoints (Ma, 2014b, p. 13).

Fig. 4. Generalization, demonstration and empirical argument for the properties of fractions (Ma, 2014b, p. 133).

statements, which could be classified as empirical arguments. The example illustrated in Fig. 5 provides an opportunity to evaluate the validity of the process of obtaining a given solution to inequality. This process is categorized as evaluating an argument.

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Fig. 5. Evaluate an Argument for the process of solving inequality (Ma, 2014b, p. 48).

4. Methodology 4.1. Textbook selection As there are diverse series of mathematics textbooks being used in Chinese junior secondary schools (Xu, 2013), we ultimately chose the latest series published in 2014 by Beijing Normal University Press, considering that this series reflects the advanced rationale of new national mathematics curriculum reform. As a reform-based series, it is in line with the recommendations put forward by the Mathematics Curriculum Standard for Compulsory Education (2011 version)[MCSCE (2011)] (Ministry of Education, People’s Republic of China, 2012), and Fan et al. (2018) mentioned that many key members of this series of textbooks also belonged to the team that was responsible for the national mathematics curriculum reform (Ma, 2014a). In our analysis, we examined the RP opportunities that were directly provided by the two eighth-grade textbooks rather than the implementation of textbooks in the classroom. 4.2. Unit of analysis To make our results comparable, we consider adopting similar unit of analysis to those used in the existing studies. The analysis unit of exercises was the task that had a separate answer (Stylianides, 2009, 2005), which could be found in teacher’s edition of textbooks. The number of each RP category in exercises were counted separately. We met the situation that one exercise contained several tasks, then we counted the number of its separate tasks (Thompson et al., 2012; Otten, Gilbertson, Males, & Clark, 2014). For example, if one exercise comprised task (1), (2), (3), (4), this exercise counted as 4. The unit of analysis for narratives was one sentence or more sentences that included at least one mathematical term (Davis, Smith, Roy, & Bilgic, 2014). For those narratives that included several sentences but only contained one purpose or single theme, we viewed this situation as an analysis unit and marked one code only. What we make some changes when analyzing the unit for narratives was that we identified the RP type of each unit of narrative also took teachers’ edition for reference, which the existing studies didn’t mention that. 4.3. Procedure To code the two selected mathematics textbooks, we asked two doctoral students, one of whom majored in mathematics education and the other of whom was familiar with RP research, to conduct the pilot study and official analysis, respectively. First, we requested that one of them perform the pilot work with me. We chose 5 units for the pilot study. Then, we discussed each category and subcategory of the frameworks of Stylianides and Thompson et al. to ensure that our understanding was coherent. After that, we decided to start our pilot study from the exercise sets in the student textbooks and used the teacher’s editions as references to check the authors’ intention of each exercise at the same time. We determined which RP type that each task belong to from the authors’ perspective of teacher’s editions towards each task. The teacher’s editions contained the suggested answers, which reflected the author’s expectations for the students (Stylianides, 2009, 2005). We coded the RP type of each task based on the suggested answers. We chose Stylianides’s framework as the basis to independently code those selected chapters section by section. After finishing each section, we would compare our codes and discuss disagreements until we reached a consensus. During the pilot process, we found limitations of the existing framework, as we mentioned before, so we revised our initial framework and added categories and subcategories into it. After several discussions, we reached a very high consistency between our codes. No other problems were found until we completed coding the entire sample chapters. In the next stage, we focused on the narrative parts of the mathematics textbooks. We used the framework developed within the pilot analysis of exercises to perform the initial coding. Similar to the pilot work of coding the exercises, we decided which RP type that each unit of narrative belong to according to the authors’ intention of teacher’s editions of textbooks towards the corresponding narratives. In teacher’s editions, each unit of narrative was companied with at least one paragraph of explanatory guidance, which contained the intention of textbook authors to the narratives. We coded the RP type of each unit of narrative based on the explanatory guidance. We gradually coded all the narratives of the 5 selected chapters independently and reached a very high agreement level between us. When finishing the pilot analysis, we were faithful to our analytical framework. Nevertheless, to have a strict and dependable coding system, we asked the other doctoral student who had not participated in the development of our framework to conduct the official analysis with me. First, we explained our analytical framework to him in detail and showed him the coded sample chapters at 81

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Table 2 Frequency of RP in Exercise Sets. Eighth-grade Textbook

No. of Lessons(1) Analysed

No. of Exercises(2)

No. of RP Exercises(3) (%)

RP Exercises per Lesson

Volume One Volume Two Grand Total

42 32 74

751 784 1535

225 (30.0) 270 (34.4) 495 (32.2)

5.4 8.4 6.7

Notes. (1) The number of lessons in the table included 13 unit revisions and 2 general revisions. Volume One contained 7 unit revisions and 1 general revision, and Volume Two contained 6 unit revisions and 1 general revision. (2) The number of exercises was the base for the percentages. It was the total number of exercises coded in the textbooks, including in-class exercises, after-class exercises, unit revision exercises and general revision exercises. (3) We counted the number of RP codes as the number of RP exercises because some exercises were coded as more than one type of RP.

the same time. Then, he took some time to study it and discussed his understanding of the framework with me. Next, we began to independently analyse the two mathematics textbooks in their entirety. After that, we compared all our codes and identified disagreements. Then, we had a discussion about all the disagreements and finally reached a consensus. To check the reliability, we compared the codes of the other rater with our codes to calculate the interrater agreement of our coding scheme (Stylianides, 2005). For the exercise sets, we calculated two reliability values. The first reliability value indicated whether each exercise offered an RP opportunity to students. The interrater agreement was 98.4% (k = .961). The second reliability value indicated the determinations of how to arrange the RP exercises to the nine subcategories. The interrater agreement was 99.8%. For the narratives, we also calculated two reliability values. The first reliability value indicated whether each narrative offered an RP opportunity to students. The interrater agreement was 98.4% (k = .962). The second reliability value indicated the determinations of how to arrange the RP narratives to the seven subcategories. The interrater agreement was 98.1%. 5. Results In this section, we presented our study results of the RP opportunities in eighth-grade mathematics textbooks with respect to exercises and narratives. 5.1. RP in exercise sets 5.1.1. RP in exercise sets by lesson Table 2 reports the frequencies and percentages of RP exercises in the lessons of eighth-grade mathematics textbooks. Overall, the RP opportunities provided by the textbooks increased as students deepened their mathematical learning. Although the number of lessons decreased from 42 in Volume One to 32 in Volume Two, the number of RP exercises increased from 225 in Volume One to 270 in Volume Two. In addition, Volume Two contained more exercises (784) than did Volume One (751), but its proportion of RP exercises (34.4%) was higher than that of Volume One (30.0%). Additionally, Volume Two, which had a lower frequency of lessons and a higher frequency of RP exercises, had a higher number of RP exercises per lesson (8.4) than did Volume One (5.4). In general, the RP exercises constituted slightly more than one-third of the total number of exercises, which was lower than the percentage found in the Connected Mathematics Project (CMP) in the U.S. (approximately 40%) (see Stylianides, 2009). The entire frequency of RP exercises per lesson was 6.7, which was lower than that of U.S. textbooks, the lowest of which were Key and CME (7.6 and 8.7, respectively) (see Otten, Gilbertson et al., 2014). 5.1.2. Type of RP in exercise sets by textbooks Table 3 summarizes the distribution by percentages of RP types in exercise sets by textbooks. Overall, the proportions of RP exercises in eighth-grade textbooks varied by type. The eighth-grade textbooks had the greatest percentage of exercises devoted to providing a proof and making a conjecture (42.4% and 36.0%, respectively), most of which offered students opportunities to demonstrate and predict (41.2% and 31.9%, respectively). The lowest percentage of exercises pertained to evaluating an argument (3.4%). In addition, the RP types between the most and the least were, respectively, identifying a pattern and providing a non-proof argument (11.1% versus 7.1%). Similarly, the most frequent RP type in Volume Two was providing a proof (54.4%), followed by making a conjecture, which comprised 30.4% of the exercises. In contrast, Volume One had the greatest percentage of exercises devoted to making a conjecture (42.7%), followed by providing a proof (28%). The lowest percentage of RP exercises in Volume Two required students to provide a non-proof argument (1.1%), and the lowest percentage of RP exercises in Volume One required students to evaluating an argument (2.2%). Across the two volumes, the overall percentage of RP exercises in Volume One (30.0%) was approximately equal to that in Volume Two (34.4%). However, the corresponding percentages for each RP type differed for each volume. The exercises in Volume Two were more likely to offer opportunities for students to engage in providing a proof (18.8%) than were exercises in Volume One (8.4%) (Pearson χ2(1) = 34.9, p < .001), but less likely to provide opportunities for providing a non-proof argument (0.4%) than were exercises in Volume One (4.3%) (Pearson χ2(1) = 25.9, p < .001). The corresponding proportions of the three RP types that remained in the two volumes were relatively the same: Volume One had only a slightly higher percentage of exercises devoted to identifying a pattern and making a conjecture than did Volume Two (3.9% versus 3.3%, 12.8% versus 10.5%, respectively) and a 82

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Table 3 Distribution by Percentages of RP Types in Exercise Sets by Textbooks. RP Category IP IP(1) Total(2) MC G(1) P(1) Total(2) PP GE(1) D(1) C(1) Total(2) PNP EA(1) R(1) Total(2) EVA EVA(1) Total(2)

Volume One

Volume Two

All Eighth-grade Textbooks

12.9 (3.9) 12.9 (3.9)

9.7 (3.3) 9.7 (3.3)

11.1 (3.6) 11.1 (3.6)

4.4 (1.3) 38.2 (11.5) 42.7 (12.8)

3.7 (1.3) 26.7(9.2) 30.4 (10.5)

4.0 (1.3) 31.9 (10.3) 36.0 (11.6)

0 (0.0) 26.7 (8.0) 1.3 (0.4) 28 (8.4)

0 (0.0) 53.3 (18.4) 1.1 (0.4) 54.4 (18.8)

0 (0.0) 41.2 (13.3) 1.2 (0.4) 42.4 (13.7)

2.7(0.8) 11.6 (3.5) 14.2 (4.3)

1.1 (0.4) 0 (0.0) 1.1 (0.4)

1.8 (0.6) 5.3 (1.7) 7.1 (2.3)

2.2 (0.7) 2.2 (0.7)

4.4 (1.5) 4.4 (1.5)

3.4 (1.1) 3.4 (1.1)

Notes. (1) The first value in each cell shows the percentage (rounded to 1 decimal place) of each type of RP exercise relative to the total number of RP exercises in each textbook. The second value in parentheses shows the proportion (rounded to 1 decimal place) of each type of RP exercise of all of the exercises in each textbook. (2) The values in this row represent the sum of the values in the cells above.

lower percentage devoted to evaluating an argument than did Volume Two (0.7% versus 1.5%).

5.1.3. Type of RP in exercise sets by content areas Table 4 summarizes the distribution by percentages of RP types in exercise sets by content areas. The numbers of exercises were Table 4 Distribution by Percentages of RP Exercises in Content Areas. RP Category

IP IP(1) Total(3) MC G(1) P(1) Total(3) PP GE(1) D(1) C(1) Total(3) PNP EA(1) R(1) Total(3) EVA EVA(1) Total(3) Grand Total(4)

Content Area Algebra(2) (7, 930)

Geometry(2) (7, 528)

Statistics(2) (2, 77)

16.1 (1.5) 16.1 (1.5)

11.2 (7.8) 11.2 (7.8)

0 (0.0) 0 (0.0)

11.5 (1.1) 25.3 (2.4) 36.8 (3.4)

2.7 (1.9) 29.0 (20.1) 31.7 (22.0)

0 (0.0) 71.4 (39.0) 71.4 (39.0)

0 (0.0) 14.9 (1.4) 0 (0.0) 14.9 (1.4)

0 (0.0) 52.2 (36.2) 1.6 (1.1) 53.8 (37.3)

0 0 0 0

8 (0.8) 11.5 (1.1) 19.5 (1.8)

0.5 (0.4) 1.1 (0.8) 1.6 (1.1)

0 (0.0) 28.6 (15.6) 28.6 (15.6)

12.6 (1.2) 12.6 (1.2) 87 (9.4)

1.6 (1.1) 1.6 (1.1) 366(69.3)

0 (0.0) 0 (0.0) 42 (54.5)

(0.0) (0.0) (0.0) (0.0)

Notes. (1) The first value in each cell shows what percentage (rounded to 1 decimal place) each type of RP exercise is of the total number of RP exercises in each content area. The second value in parenthesis shows what proportion (rounded to 1 decimal place) each type of RP exercises is of the entire number of exercises in each content area. (2) The first value in parentheses shows the number of units in each content. Here, we viewed the 3 different contents of the general exercises as 1 unit. Algebra contained 6 units and 1 unit of general exercises, for a total of 7 units. Geometry included 6 units and 1 unit of general exercises, for a total of 7 units. Statistics included 1 unit and 1 unit of general exercises, for a total of 2 units. The second value in parentheses represents the total number of exercises in the content area. (3) The values in this row represent the sum of the values in the cells above. (4) The first value in each cell represents the total number of RP exercises in each content area. The second value in parentheses represents the proportions (rounded to 1 decimal place) these RP exercises represent of the entire number of exercises in each content area. 83

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quite different among the content areas. Overall, the highest number of exercises belonged to algebra (930), and the lowest number of exercises pertained to statistics (77). However, the algebra content had the lowest percentage of RP exercises (9.4%), and the geometry content had the highest percentage (69.3%). Within the content areas, the distribution of RP exercises by percentages was uneven. For geometry content, the exercises that required students providing a proof comprised the largest proportion (53.8%), and the exercises that offered students opportunities to engage in providing a non-proof argument and evaluating an argument comprised the lowest proportion (1.6%). For algebra and statistics, the situation differed from that of geometry. Both algebra and statistics had the greatest proportions of exercises devoted to making a conjecture (36.8% and 71.4%, respectively). The lowest proportion of RP exercises for algebra was directed towards evaluating an argument (12.6%), which was similar to the corresponding situation for statistics (0%). Additionally, the opportunities for identifying a pattern and providing a proof were non-existent in statistics (0%). Across the content areas, the percentages of RP exercises in geometry (69.3%) and statistics (54.5%) were much larger than those in algebra (9.4%), which indicated that exercises in geometry and statistics were more likely to provide students opportunities to engage in RP than exercises in algebra (Pearson χ2(2) = 572.2, p < .001). Within the RP categories, the distribution of RP exercises varied by content area. Specifically, both the mathematical exercises in algebra and statistics were less likely to offer students opportunities to engage in identifying a pattern (Pearson χ2(2) = 41.2, p < .001) and providing a proof (Pearson χ2(2) = 380.7, p < .001) than exercises in geometry (1.4%–1.5% of algebra exercises and 0% of statistics exercises required students to identify a pattern and provide a proof; the corresponding values for geometry were 7.8% and 37.3%, respectively). The geometry exercises were more likely to provide students with opportunities to engage in making a conjecture than were algebra exercises (22.0% versus 3.4%) (Pearson χ2(1) = 126.8, p < .001), while they were less likely to provide students with opportunities to engage in making a conjecture than were exercises in statistics (22.0% versus 39.0%) (Pearson χ2(2) = 10.6, p < .01). The statistics exercises (15.6%) were more likely to offer opportunities for providing a non-proof argument than exercises in algebra (1.8%) and geometry (1.1%) (Pearson χ2(2) = 65.1, p < .001). The percentages of exercises that were designed to encourage students to evaluate an argument were almost equal in the three content areas. 5.2. RP in narratives 5.2.1. RP in narratives by lessons Table 5 shows the frequency and percentage of RP narratives in eighth-grade mathematics textbooks. Similar to the situation in student exercises, the presentation of RP in the narratives intensified from Volume One to Volume Two. As the number of lessons decreased from 34 in Volume One to 25 in Volume Two, the number of RP narratives also decreased from 571 in Volume One to 466 in Volume Two, but the proportion of RP narratives in Volume Two was approximately double that in Volume Two (22.6%). Moreover, Volume Two, which has fewer lessons and a higher frequency of RP narratives, has more than twice the number of RP narratives per lesson than Volume One (8.3 versus 3.8). Overall, the RP narratives constituted slightly more than one-third of the total narratives in eighth-grade mathematics textbooks, the number of which was almost equal to the total number in three U.S. secondary mathematics textbooks (PH, DA, CPM) (35%) (see Davis, 2012) but was lower than that in U.S. high school mathematics textbooks (50.7%) (see Thompson et al., 2012). The total number of RP narratives per lesson was 5.7, more than that of six U.S. geometry textbooks (CME, Glencoe, Holt, Key, Prentice, UCSMP) (see Otten, Gilbertson et al., 2014). 5.2.2. Type of RP in narratives by textbooks Table 6 summarizes the distribution by percentages of RP types in narratives by textbooks. Overall, the proportion differed within the RP types in the narrative portions. Among the RP categories, the narratives that required students to make a conjecture and to provide a proof comprised the largest proportion (36.9% and 35.4%, respectively), most of which asked students to make predictions and demonstrations (22.0% and 34.5%, respectively). The lowest percentage of narratives pertained to evaluating an argument (0.3%). In addition, the RP types between the most and the least were, respectively, identifying a pattern and providing a non-proof argument (16.1% versus 11.3%). Similarly, the most frequent RP type in Volume Two was providing a proof (42.5%), followed by making a conjecture, which comprised 38.6% of the narratives. In contrast, Volume One had the greatest percentage of narratives devoted to making a conjecture (34.1%), followed by providing a proof (24.0%). The lowest percentage of RP narratives in both Volume One and Volume Two required students to evaluate an argument, the percentages of which was 0.8% and 0%, respectively. Table 5 Frequency of RP in Narratives. Eighth-grade Textbook

No. of Lessons(1) Analysed

No. of Narratives(2)

No. of RP Narratives(3) (%)

RP Narratives per Lesson

Volume One Volume Two Grand Total

34 25 59

571 466 1037

129 (22.6) 207(44.0) 336 (32.4)

3.8 8.3 5.7

Notes. (1) Here, we did not include unit revisions and general revisions. (2) The number of narratives is the base for the percentages. It was the total number of narratives coded in the textbooks. (3) We counted the number of RP codes as the number of RP narratives because some narrative portions were coded as more than one type of RP. 84

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Table 6 Distribution by Percentages of RP Types in Narratives by Textbooks. RP Category IP IP(1) Total(2) MC G(1) P(1) Total(2) PP GE(1) D(1) C(1) Total(2) PNP EA(1) R(1) Total(2) EVA EVA(1) Total(2)

Volume One

Volume Two

Entire Eighth-grade Textbooks

20.2 (4.6) 20.2 (4.6)

13.5 (6.0) 13.5 (6.0)

16.1 (5.2) 16.1 (5.2)

14.0 (3.2) 20.2 (4.6) 34.1 (7.7)

15.5 (6.9) 23.2 (10.3) 38.6 (17.2)

14.9 (4.8) 22.0 (7.1) 36.9 (12.0)

0.8 (0.2) 23.3 (5.3) 0 (0.0) 24.0 (5.4)

0 (0.0) 41.5 (18.5) 1.0 (0.4) 42.5 (18.9)

0.3 (0.1) 34.5 (11.2) 0.6 (0.2) 35.4 (11.5)

3.9(0.9) 17.1 (3.9) 20.9 (4.7)

1.9 (0.9) 3.4 (1.5) 5.3 (2.4)

2.7 (0.9) 8.6 (2.8) 11.3 (3.7)

0.8 (0.2) 0.8 (0.2)

0 (0.0) 0 (0.0)

0.3 (0.1) 0.3 (0.1)

Notes. (1) The first value in each cell shows the percentage (rounded to 1 decimal place) comprised by each type of RP narrative of the total number of RP narratives in each textbook. The second value in parentheses shows what proportion (rounded to 1 decimal place) comprised by each type of RP narrative of the entire number of narratives in each textbook. (2) The values in this row represent the sum of the values in the cells above.

Across the volumes, the overall proportion of RP narratives in Volume Two was almost twice the proportion in Volume One (44.0% versus 22.6%). However, the corresponding percentages for each RP type differed for each volume. Compared to Volume Two, Volume One provided less than half of the number of opportunities for making a conjecture (7.7% versus 17.2%, respectively), and less than one third the number of opportunities for providing a proof (5.4% versus 18.9%, respectively). Meanwhile, the proportion of identifying a pattern in Volume One was slightly less than the corresponding percentage in Volume Two (4.6% versus 6.0%, respectively). However, the percentages of the remaining RP types in Volume One were more than their corresponding proportions in Volume Two: Volume One had almost twice percentage of narratives devoted to providing a non-proof argument, and a slightly higher percentage devoted to evaluating an argument than did Volume Two (4.7% versus 2.4%, 0.2% and 0%, respectively).

5.2.3. Type of RP in narratives by content areas Table 7 summarizes the distribution by percentages of RP types in narratives by content areas. Among the total number of narratives (1037), half were about algebra (519), four-fifths of the remaining narratives were geometry content (408), and the lowest number belonged to statistics (104). However, the proportion of narratives with respect to RP was the lowest in algebra (17.0%), the percentages of RP narratives in statistics (20.2%) were slightly higher than in algebra, and over half of the narratives in geometry contained RP (55.6%). The distribution of RP types varied in each content narratives. For geometry content, the narratives that required students providing a proof comprised the largest proportion (46.3%), and the narratives that offered students opportunities to engage in providing a non-proof argument and evaluating an argument comprised the lowest proportion (6.2% and 0%, respectively). For algebra and statistics, the situation differed from that of geometry. Both algebra and statistics had the greatest proportions of narratives devoted to making a conjecture (45.5% and 57.1%, respectively). For algebra, the following types were identifying a pattern, providing a non-proof argument and providing a proof, the percentages of which were 20.5%, 17.0% and 15.9%, respectively, and the proportion of RP narratives was directed towards evaluating an argument (1.1%). In addition, the remaining RP type in statistics was only providing a non-proof argument (42.9%), and other RP types were non-existent (0%). Across the content areas, the percentages of RP narratives in geometry (55.6%) were much larger than those in algebra (17.0%) and statistics (20.2%), which suggested that geometry narratives were more likely to provide students with opportunities to engage in RP than were narratives in other two contents (Pearson χ2(2) = 156.3, p < .001). Within the RP categories, the distribution of RP narratives varied by content area. Specifically, the geometry narratives were more likely to provide students with opportunities to engage in identifying a pattern (8.8%, 3.5%, 0%) (Pearson χ2(2) = 18.8, p < .001), making a conjecture (17.6%, 7.7%, 11.5%) (Pearson χ2(2) = 20.2, p < .001), as well as providing a proof (25.7%, 2.7%, 0%) (Pearson χ2(2) = 130.2, p < .001) than narratives in algebra and statistics, while they were less likely to provide students with opportunities to engage in providing a non-proof argument than were narratives in statistics (3.4% versus 8.7%) (Pearson χ2(1) = 8.1, p < .01), but slightly more than that of algebra (2.9%). The percentages of narratives that were designed to encourage students to evaluate an argument were almost non-exist in all three content areas.

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Table 7 Distribution by Percentages of RP Narratives in Content Areas. RP Category

IP IP(1) Total(3) MC G(1) P(1) Total(3) PP GE(1) D(1) C(1) Total(3) PNP EA(1) R(1) Total(3) EVA EVA(1) Total(3) Grand Total(4)

Algebra(2) (6, 519)

Content Area Geometry(2) (6, 408)

Statistics(2) (1, 104)

20.5 (3.5) 20.5 (3.5)

15.9 (8.8) 15.9 (8.8)

0.0 (0.0) 0.0 (0.0)

28.4 (4.8) 17.0 (2.9) 45.5 (7.7)

11.0 (6.1) 20.7 (11.5) 31.7 (17.6)

0.0 (0.0) 57.1 (11.5) 57.1 (11.5)

0 (0.0) 15.9 (2.7) 0 (0.0) 15.9 (2.7)

0.4 (0.2) 44.9 (25.0) 0.9 (0.5) 46.3 (25.7)

0 0 0 0

5.7 (1.0) 11.4 (1.9) 17.0 (2.9)

1.8 (1.0) 4.4 (2.5) 6.2 (3.4)

0 (0.0) 42.9 (8.7) 42.9 (8.7)

1.1 (0.2) 1.1 (0.2) 88 (17.0)

0 (0.0) 0 (0.0) 227(55.6)

0 (0.0) 0 (0.0) 21 (20.2)

(0.0) (0.0) (0.0) (0.0)

Notes. (1) The first value in each cell shows the percentage (rounded to 1 decimal place) of each type of RP narrative of the total number of RP narratives in each content area. The second value in parentheses shows the proportion (rounded to 1 decimal place) of each type of RP narrative of the total number of narratives in each content area. (2) The first value in parentheses shows the number of units in each content area (here, we did not include the unit of general revisions). The second value in parentheses represents the total number of narratives in the content area. (3) The values in this row represent the sum of the values in the cells above. (4) The first value in each cell represents the total number of RP narratives in each content area. The second value in parentheses represents the proportions (rounded to 1 decimal place) these RP narratives represent of the total number of narratives in each content area.

6. Discussion In this research, we have investigated RP opportunities embedded in eighth-grade BNUP mathematics textbooks in China. There were 1535 exercises and 1037 narratives across three topics (algebra, geometry and statistics). 6.1. Summary and conclusions 6.1.1. RP exercises The examination of exercise sets indicated that more than one-third of the total exercises provided opportunities for students to engage in RP activities, the percentage of which was less than that found in the eighth-grade CMP (geometry and algebra) in the U.S. (43%) (Stylianides, 2005). Considering the content area, the number of RP opportunities in algebra (9.4%) was very low compared to that of algebra in the CMP (40%) (Stylianides, 2005) and more than that of algebra in 20 high school textbooks in the U.S. (approximately 6%) (Thompson et al., 2012), but less than the percentage in the reform-oriented high school textbooks (CPM) (Davis, 2012). For geometry content, Chinese textbooks (69.3%) were more likely to provide RP opportunities for students than U.S. geometry textbooks (approximately 25%) (Otten, Gilbertson et al., 2014) and geometry of CMP (39%) (Stylianides, 2005). It is worth noting that the textbooks of the BNUP and the CMP are standard-based textbook series in China and the U.S., respectively, and eighthgrade Chinese textbooks were less likely than American textbooks to offer RP opportunities to students. In China, it seemed that geometry takes the most responsibility to provide opportunities for students to engage in RP activities, which differed from the situation in the U.S. However, the results of the BNUP edition at the eighth-grade level could not be seen as a direct representation of all middle school textbooks, especially the high school mathematics curricula. Therefore, the results of its comparison with American high school textbooks should be treated with caution. Chinese textbooks did not provide opportunities for identifying a pattern as much as CMP did in U.S. (3.6% versus 11%), but the opportunities for generalizing a hypothesis according to given cases were similar to those offered by CMP textbooks (approximately 1%) (Stylianides, 2005). In contrast to CMP textbooks, Chinese textbooks emphasized predicting as another form of plausible reasoning, which became the main type of making a conjecture in China. Perhaps we can find some clues about this difference from newly revised Chinese MCSCE (2011) and CMP guidance in U.S. The Chinese MCSCE (2011) stated the following: “At the compulsory education stage, we should pay attention to the rationality of students’ thinking and not overemphasize the form of reasoning… teachers should design appropriate learning activities and guide students to discover some rules through observation, trial, estimation, induction, analogy, drawing and other activities, guess some conclusions, and develop plausible reasoning ability” (Ministry of Education, People’s Republic of China, 2012, p. 54). The authors of the CMP claimed the following: “Throughout the curriculum, students are encouraged to look for patterns, make conjectures, provide evidence for their conjectures, refine their conjectures and 86

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strategies, connect their knowledge, and extend their findings” (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2002, p. 8). It could be found that CMP textbooks highlighted the generation process of how a pattern became a conjecture, which mainly required inductive reasoning. While Chinese MCSCE (2011) emphasized drawing some conclusions based on plausible reasoning, which of course included inductive reasoning, it actually relied more on the comprehensive forms of reasoning to make conjectures. It may be appropriate to allocate a very low percentage of space to the presentation of empirical arguments in Chinese textbooks. As Stylianides (2009) argued, empirical arguments might make students mistake the examination of examples for the provision of a formal proof. However, he further found that there was little connection between students’ misconception of proof and the frequency of a particular RP type that appeared in the textbooks. Students tended to rely on examples to demonstrate the truth of mathematical statements (Knuth, Choppin, Slaughter, & Sutherland, 2002; Williams et al., 2011), although the opportunities for empirical arguments in the textbooks were rare (Stylianides, 2005). Since the percentage of empirical arguments in the written curriculum was not the cause of students’ misunderstanding of proof, does it have any function in helping students make more credible conjectures? If the answer is yes, perhaps we can enhance the proportion of empirical arguments in the design of future textbooks. The opportunities designed for rationales were rare in the exercises of Chinese textbooks, and statistics exercises were more likely to provide opportunities for rationales than the exercises in algebra and geometry (Pearson χ2(2) = 94.1, p < .001). However, more than one-quarter of U.S. eighth-grade CMP exercises designed opportunities for rationales, and the percentages in the three content areas (algebra, geometry and number theory) were almost equal (Stylianides, 2005). In recent years, the senior high school entrance examination in China included one item (either statistics or geometry) that required students to provide rationales. This situation to some extent encouraged students to give rationales “when the production of a proof is impractical, impossible, or undesirable” (Stylianides, 2009). Although Stylianides (2009) stated three desirable reasons for students to engage in rationales, whether the proportion of rationales in the textbooks should be high or low is still inconclusive. In addition to having the potential to promote students’ transformation from empirical arguments to proof, we question whether it has a potential influence on students’ development of their awareness of providing evidence when making a mathematical statement during their daily mathematics learning. The proportion of providing a proof in Chinese textbooks (13.7%) was much larger than that in eighth-grade CMP textbooks (5%) in the U.S. (Stylianides, 2005). Specifically, algebra exercises in Chinese mathematics textbooks (1.4%) were less likely to engage students in providing a proof than were algebra tasks in U.S. mathematics textbooks (e.g., Stylianides, 2005; Davis et al., 2014). Most algebra exercises required students to perform the operation based on operational rule that contained deductive reasoning (e.g., solving an inequality), and few algebra exercises were designed in a way that needed to be demonstrated. This situation differed from the analyses of geometry content in Chinese written curricula. The geometry exercises in Chinese mathematics textbooks (37.3%) were much more likely to offer students opportunities to engage in providing a proof than were exercises in American geometry textbooks (e.g., Stylianides, 2005; Thompson et al., 2012; Otten, Gilbertson et al., 2014; Otten, Males, & Gilbertson, 2014; Sears & Chávez, 2014). In fact, both in China and in the U.S., proof was traditionally tied to geometry in school mathematics (Bao, 2002; Harel & Sowder, 1998; Herbst, 2002. Consistent with the notion that “RP needed to be introduced to students before high school stage”, as many researchers had called for early on (see reviews in Stylianides, 2005), proof in China was introduced at the beginning of grade 8 (Fan et al., 2018). Perhaps the time at which proof was emphasized to students was not early enough, but the high frequency with which they appeared in exercises could have compensated for their delayed appearance in school mathematics textbooks. In China, however, the reasoning requirement decreased from Mathematics Syllabus (People’s Educational Press, 2000) to Mathematics Standards (Ministry of Education, 2001) during the curriculum reform in 2001, and reasoning complexity decreased greatly from old standard-based mathematics textbooks to the new edition, with most of the changes being related to geometric proof. Bao (2002, 2004) concluded this situation with the following statement: “New Chinese mathematics curriculum appeared to lean towards a ‘Definition-based curriculum’ from a ‘Theorem-based curriculum’” . Unfortunately, in recent years, some areas in China have attempted to reduce the complexity of geometric proof in daily mathematics teaching and examination because of students’ poor performance on geometric proof. From the perspective of the role that geometric proof played on the cultivation of students’ logical thinking, it was not a recommended step to blindly reduce the difficulty of proof. In contrast, we should try to improve students’ understanding or performance to proof through exceptional teaching or through developing teachers’ knowledge of mathematics, especially RP. Stylianides (2005, 2009) proposed a model to describe the distribution of RP exercises across grade levels according to the results of developmental psychology. He indicated that the Non-proof arguments should be reduced across the grades, whereas the proof arguments should be increased. The proof opportunities provided by textbook exercises were more than doubled from Volume One to Volume Two, and the opportunities of Non-proof arguments reduced largely from Volume One to Volume Two. One might argue that this organization of proof and non-proof arguments in eighth-grade Chinese textbooks was reasonable based on that model. This arrangement of proof and non-proof might be an advantage of eighth-grade Chinese textbooks; however, our article still could not tell us whether Chinese textbooks had an appropriate distribution of RP exercises across the grades. In addition, here raised another question: to what extent were the opportunities of proof and non-proof arguments arranged in textbooks across each volume or grade enough? This question needs to be further investigated. 6.1.2. RP narratives From our analysis of the narratives, it could be found that 32.4% of the narratives were related to RP. Among them, the narratives that required students to make a conjecture and to provide a proof comprised the largest proportion (12.0% and 11.5%, respectively), most of which asked students to make predictions and demonstrations (7.1% and 11.2%, respectively). Additionally, 5.2% were identifying patterns, and 3.7% were with respect to providing a non-proof argument, and the opportunities for students to evaluate an argument were only 0.1%. It could be found that the narratives provided similar percentages of opportunities for students to engage 87

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in RP activities as the exercise parts did, as well as the similar distribution of RP categories. The reform-oriented eighth-grade textbooks had a consistent consideration of the RP opportunities across the volumes, which was in line with the requirements of the Chinese MCSCE (2011), “the development of reasoning ability should be throughout students’ entire mathematics learning process” (Ministry of Education, People’s Republic of China, 2012, p. 9). Though it is unclear how many RP should be embedded in narratives, students’ experiences of engaging in RP process grow up from Volume One to Volume Two. As we mentioned before, Stylianides (2005, 2009) put forward a hierarchy model of different arguments based on mathematical sophistication across grade levels, which suggested no empirical arguments, decreasing rationales, and more proofs across the grades. These features can be found from Volume One to Volume Two, which seems that Chinese textbook authors did a good job on embedding RP in narratives across Volumes. Chinese students could then experience more RP processes with their deepening of learning. Some researchers were worried that teachers could skip some potential RP opportunities in the mathematics textbooks in that they did not get used to teaching each lesson in the written curriculum (Ball & Cohen, 1996; Tarr, Chávez, Reys, & Reys, 2006; Thompson et al., 2012). This phenomenon may lead to a disadvantageous setting for students to learn RP under the situation of limit RP opportunities, especially limited proof in the textbooks. Fortunately, textbooks in China were the main resources for teachers to prepare their daily lessons. For example, textbooks are usually the main concerns when teachers decide what to teach and how (Fan et al., 2004). As Fan et al. (2004) told us that textbooks were also the main sources for students to conduct their mathematics learning. Chinese students would like to read the textbooks directly by themselves both because of teachers’ requirements and driven by self-motivations. In addition, slightly different from what Begle (1973) said that “most student learning is directed by text rather than the teachers”, Chinese student’s mathematics learning was not only through reading the narratives by themselves but also through classroom teaching as Chinese teachers usually selected most of example problems into their teaching practice. To some extent, students would have high possibilities to learn RP provided in the narratives of Chinese mathematics textbooks. 6.2. Limitations In this study, we selected one contemporary reform-oriented middle school textbook series in China, and the eighth-grade textbooks were the only two volumes being analysed here. Our result could not represent all the textbooks in China but provided a small perspective to see RP in Chinese mathematics textbooks. To better understand the details of RP in textbooks in China, more comprehensive analyses of different textbook editions are needed. In addition, we chose the percentages rather than counts to compare across the content areas or textbooks in the Results part and Summary and Conclusions section. However, it might bring some misconception about the real RP opportunities in different contents or sometimes cover the actual RP opportunities that emerged in the classroom. For example, making a conjecture took up 3.4% and 39.0% in the exercises of algebra and statistics, respectively, which probably would make us feel that students have much more conjecturing opportunities in statistics than in algebra. However, the counts of the exercises in statistics were only 77, while the algebra had 930 exercises, which means that the real opportunities to make a conjecture were almost equal between algebra and statistics (31.6 versus 30). Considering their implementation in a teacher’s daily practice, a statistics lesson is more likely to provide students opportunities to make a conjecture than an algebra lesson. Another limitation was that our research did not contain the related observation of how the RP in this textbooks was implemented in the classroom. Therefore, even if we made an optimistic conclusion according to some previous studies that Chinese teachers have a very high possibility to implement most of the textbooks, the situation of their actual implementation is unknown yet. Teachers’ textbook use was influenced by many factors, such as their knowledge, beliefs, perceptions of text, views about the nature of learning and purpose of school education, or concepts of external pressures (Remillard & Heck, 2014; Remillard, 1999). The same textbook might be used differently in the classrooms by different teachers (e.g., Thompson & Senk, 2014; Chávez, 2003). Even for a single teacher whose textbook use of the same lesson might change in two classes (e.g., Eisenmann & Even, 2009), let alone from the novice stage to the expert stage. Mostly, novice teachers are more likely to teach based on textbooks, while expert teachers tend to use textbooks more creatively (e.g., Bush, 1986; Qi, Zhang, & Huang, 2014; Borko & Livingston, 1989). This is not to say that there are more RP opportunities in the classroom of teachers who are loyal to the textbooks because it is very likely that the RP opportunities contained in the textbooks content are insufficient. In contrast, the RP opportunities are not necessarily less in the classrooms of teachers who are not loyal to the textbooks. The real RP opportunities that Chinese students could have in the classroom are related to how Chinese teachers use the textbooks, which must be investigated further. Acknowledgement We want to thank Doctor Xiaofeng Du and Lianming Hao who generously offered their time, insights in our study. And we also wish to thank anonymous reviewers for their valuable comments on an earlier version of this paper. References Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.). Mathematics, teachers and children (pp. 216–235). London, England: Hodder & Stoughton. Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.). A research companion to principles and standards for school mathematics (pp. 27–44). Reston, VA: National Council of Teachers of Mathematics.

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Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–14. Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.). Proceedings of the International Congress of Mathematicians, III. Bao, J. (2002). Comparative study on composite difficulty of Chinese and British school mathematics curricula. Unpublished doctoral dissertation. Shanghai: East China Normal University. Bao, J. (2004). A comparative study on composite difficulty between new and old Chinese mathematics textbooks. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.). How Chinese learn mathematics: Perspectives from insiders (pp. 208–227). World Scientific. Begle, E. G. (1973). Some lessons learned by SMSG. Mathematics Teacher, 66(3), 207–214. Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71–80. Borko, H., & Livingston, C. (1989). Cognition and improvisation: Differences in mathematics instruction by expert and novice teachers. American Educational Research Journal, 26(4), 473–498. Bush, W. S. (1986). Preservice teachers’ sources of decisions in teaching secondary mathematics. Journal for Research in Mathematics Education, 17(1), 21–30. Cai, J., & Cirillo, M. (2014). What do we know about reasoning and proving? Opportunities and missing opportunities from curriculum analyses. International Journal of Educational Research, 64(5), 132–140. Chávez, Ó. (2003). From the textbook to the enacted curriculum: Textbook use in the middle school mathematics classroom. Unpublished doctoral dissertation. University of Missouri-Columbia. Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 119–142. Davis, J. D. (2012). An examination of reasoning and proof opportunities in three differently organized secondary mathematics textbook units. Mathematics Education Research Journal, 24(4), 467–491. Davis, J. D., Smith, D. O., Roy, A. R., & Bilgic, Y. K. (2014). Reasoning-and-proving in algebra: The case of two reform-oriented US textbooks. International Journal of Educational Research, 64, 92–106. Eisenmann, T., & Even, R. (2009). Similarities and differences in the types of algebraic activities in two classes taught by the same teacher. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.). Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 152–170). New York: Routledge. Fan, L., Chen, J., Zhu, Y., Qiu, X., & Hu, J. (2004). Textbook use within and beyond mathematics classrooms: A study of 12 secondary schools in Kunming and Fuzhou of China. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.). How Chinese learn mathematics: Perspectives from insiders (pp. 228–261). World Scientific. Fan, L., Mailizar, M., Alafaleq, M., & Wang, Y. (2018). A comparative study on the presentation of geometric proof in secondary mathematics textbooks in China, Indonesia, and Saudi Arabia. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.). Research on mathematics textbooks and teachers’ resources: Advances and issues (pp. 53–65). Springer. Fan, L., Qi, C., Liu, X., Wang, Y., & Lin, M. (2017). Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An intervention-based study with two Chinese classrooms. Educational Studies in Mathematics, 96(2), 229–248. Fujita, T., & Jones, K. (2014). Reasoning-and-proving in geometry in school mathematics textbooks in Japan. International Journal of Educational Research, 64, 81–91. Galotti, K. M., Komatsu, L. K., & Voelz, S. (1997). Children’s differential performance on deductive and inductive syllogisms. Developmental Psychology, 33(1), 70–78. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5–23. Hanna, G., & de Bruyn, Y. (1999). Opportunity to learn proof in Ontario grade twelve mathematics texts. Ontario Mathematics Gazette, 37, 23–29. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.). Research in collegiate mathematics education III (pp. 234–283). Providence, R.I: American Mathematical Society. Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.). Second handbook of research on mathematics teaching and learning (pp. 805–842). Greenwich, CT: Information Age Publishing. Hawkins, J., Pea, R. D., Glick, J., & Scribner, S. (1984). Merds that laugh don’t like mushrooms”: Evidence for deductive reasoning by preschoolers. Developmental Psychology, 20(4), 584. Herbst, P. G. (2002). Establishing a custom of proving in american school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283–312. Herbst, P., Chen, C., Weiss, M., González, G., Nachlieli, T., Hamlin, M., Brach, C., et al. (2009). Doing proofs” in geometry classrooms. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.). Teaching and learning proof across the grades: A K-16 perspective (pp. 250–268). New York: Routledge. IEA (2003). TIMSS assessment frameworks and specifications 2003. Chestnut Hill, M.A: TIMSS International Study Centre. Knuth, E. J., Choppin, J., Slaughter, M., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Weigel, R. L. Bryant, & K. Nooney (Vol. Eds.), Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education: Vol. 4Athens, GA: Clearinghouse for Science, Mathematics, and Environmental Education pp. 1693–1670. Lappan, G., Fey, J., Fitzgerald, W., Friel, S., & Phillips, E. (2002). Getting to know connected mathematics: An implementation guide. Glenview, IL: Prentice-Hall. Leinhardt, G., Zigmond, N., & Cooley, W. W. (1981). Reading instruction and its effects. American Educational Research Journal, 18(3), 343–361. Light, P., Blaye, A., Gilly, M., & Girotto, V. (1989). Pragmatic schemas and logical reasoning in 6- to 8-year-old children. Cognitive Development, 4(1), 49–64. Lithner, J. (2003). Students’ mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52, 29–55. Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. The Journal of Mathematical Behavior, 23(4), 405–427. Ma, F. (Ed.). (2014). Mathematics (grade 8, volume two)Beijing: Beijing Normal University Press. Ma, F. (Ed.). (2014). Mathematics (grade 7-9)Beijing: Beijing Normal University Press. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez, & P. Boero (Eds.). Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam, The Netherlands: Sense Publishers. McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. J., et al. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Champaign, IL: Stipes Publishing. Ministry of Education, People’s Republic of China (2012). Mathematics curriculum standard for compulsory education (2011 version). Beijing: Beijing Normal University Press. Ministry of Education (2001). National mathematics standards (experimental edition). Beijing: Beijing Normal University Press. Miyakawa, T. (2012). Proof in geometry: A comparative analysis of French and Japanese textbooks. In T. Y. Tso (Vol. Ed.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education: Vol. 3, (pp. 225–232). Ni, Y., & Cai, J. (2011). Searching for evidence of curricular effect on the teaching and learning of mathematics: Lessons learned from the two projects. International Journal of Educational Research, 50(2), 137–143. NCTM (National Council of Teachers of Mathematics) (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51–79. Otten, S., Males, L. M., & Gilbertson, N. J. (2014). The introduction of proof in secondary geometry textbooks. International Journal of Educational Research, 64, 107–118. People’s Educational Press (2000). National mathematics teaching syllabus for compulsory education. Beijing: Author. Polya, G. (1954). Mathematics and plausible reasoning. Volume I: Induction and analogy in mathematics. Princeton, NJ: Princeton University Press. Polya, G. (1968). Mathematics and plausible reasoning. Volume II: Patterns of plausible inference (second edition). Princeton, NJ: Princeton University Press. Qi, C., Zhang, X., & Huang, D. (2014). Research on textbooks used in china for teaching geometric transformations in secondary school: From the perspective of the

89

International Journal of Educational Research 98 (2019) 77–90

D. Zhang and C. Qi

teachers’ role. In K. Jones, C. Bokhove, G. Howson, & L. Fan (Eds.). Proceedings of the International Conference on Mathematics Textbook Research and Development (ICMT-2014) (pp. 91–104). Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33, 5–29. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Remillard, J. T., & Heck, D. J. (2014). Conceptualizing the curriculum enactment process in mathematics education. ZDM, 46(5), 705–718. Ross, K. A. (1998). Doing and proving: The place of algorithms and proofs in school mathematics. The American Mathematical Monthly, 105(3), 252–255. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., Cogan, L. S., et al. (2001). Why schools matter: A cross-national comparison of curriculum and learning. San Francisco, CA: Jossey-Bass. 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. 334–371). New York: Macmillan. Schoenfeld, A. H. (1994). What do we know about mathematics curricula? The Journal of Mathematical Behavior, 13(1), 55–80. Sears, R. (2012). An examination of how teachers use curriculum materials for the teaching of proof in high school geometry. Unpublished doctoral dissertation. Columbia: University of Missouri. Sears, R., & Chávez, Ó. (2014). Opportunities to engage with proof: The nature of proof tasks in two geometry textbooks and its influence on enacted lessons. ZDM, 46(5), 767–780. Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72(3), 271. Stein, M., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488. Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321. Stylianides, G. J. (2005). Investigating students’ opportunities to develop proficiency in reasoning and proving: A curricular perspective. Unpublished doctoral dissertation. Ann Arbor, USA: University of Michigan. Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16. Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11, 258–288. Stylianides, G. J. (2014). Textbook analyses on reasoning-and-proving: Significance and methodological challenges. International Journal of Educational Research, 64, 63–70. Stylianides, G. J., Stylianides, A. J., & Shilling-Traina, L. N. (2013). Prospective teachers’ challenges in teaching reasoning-and-proving. International Journal of Science and Mathematics Education, 11(6), 1463–1490. Tarr, J. E., Chávez, Ó., Reys, R. E., & Reys, B. J. (2006). From the written to the enacted curricula: The intermediary role of middle school mathematics teachers in shaping students’ opportunity to learn. School Science and Mathematics, 106(4), 191–201. Thompson, D. R. (2014). Reasoning-and-proving in the written curriculum: Lessons and implications for teachers, curriculum designers, and researchers. International Journal of Educational Research, 64, 141–148. Thompson, D. R., & Senk, S. L. (2014). The same geometry textbook does not mean the same classroom enactment. ZDM, 46(5), 781–795. 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(3), 253–295. 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. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119. Williams, C., Akinsiku, O., Walkington, C., Cooper, J., Ellis, A., Kalish, C., Knuth, E., et al. (2011). Understanding students’ similarity and typicality judgments in and out of mathematics. In L. R. Wiest, & T. Lamberg (Eds.). Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1180–1189). Wong, K. C., & Sutherland, R. (2018). Reasoning-and-proving in algebra in school mathematics textbooks in Hong Kong. In A. J. Stylianides, & G. Harel (Eds.). Advances in mathematics education research on proof and proving (pp. 185–198). Springer. Xu, B. (2013). The development of school mathematics textbooks in China since 1950. ZDM Mathematics Education, 45(5), 725–736. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.). A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.

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