Do textbooks address known learning challenges in

Math Ed Res J https://doi.org/10.1007/s13394-018-0238-6 O R I G I N A L A RT I C L E

Do textbooks address known learning challenges in area measurement? A comparative analysis Dae S. Hong 1 & Kyong Mi Choi 2 & Cristina Runnalls 3 & Jihyun Hwang 3

Received: 10 November 2017 / Revised: 30 January 2018 / Accepted: 2 February 2018 # Mathematics Education Research Group of Australasia, Inc. 2018

Abstract This study compared area lessons from Korean textbooks and US standardbased textbooks to understand differences and similarities among these textbooks, as well as how these textbooks address known learning challenges in area measurement. Several well-known challenges have been identified in previous studies, such as covering, array structure, and linking array structure to area formula. We were interested in knowing if textbooks addressed these issues in their treatments of area measurement and, in doing so, provided students with opportunities to overcome or become familiar with known challenges. The results show that both countries’ textbooks demonstrated similar limitations; only few area and area-related lessons are covered and three important learning challenges in area measurement are not covered well, which need to be informed to practicing teachers. Keywords Area measurement . Textbooks . Curriculum

* Dae S. Hong [email protected] Kyong Mi Choi [email protected] Cristina Runnalls [email protected] Jihyun Hwang [email protected]

1

College of Education, University of Iowa, Iowa City, IA, USA

2

University of Virginia, Charlottesville, VA, USA

3

University of Iowa, Iowa City, IA, USA

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The results of international comparative studies such as the Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA) have given us insights about how students in other countries learn mathematics. The results consistently showed that East Asian students including China, Japan, and Korea, have done well compared to their American counterparts. In order to understand mathematics education in those countries, researchers have examined class practices (Hiebert et al. 2003), teachers’ pedagogy (Huang and Cai 2011), and students’ problem-solving skills (Cai 2004). Among many different ways to understand students’ opportunities to learn (OTL) mathematics, one important area we can examine is what textbooks offer to students for their learning, as textbooks play an important role in lesson enactment process— teachers use textbooks and other resources to plan, select, and possibly modify tasks for their mathematics lessons (Remillard et al. 2014; Remillard and Heck 2014). Although what textbooks offer to students is not the only way to understand students’ OTL, if textbooks demonstrated limitations in covering various mathematical topics, limited coverage can decrease students’ OTL (Smith et al. 2013) and can be a possible reason for students’ difficulties in learning certain mathematical topics. Because of the important role that textbooks play in lesson enactment process, researchers have often interpreted the results from textbook analysis, such as including more challenging tasks, tasks that require explanations, and tasks that address students’ challenges, as possible reasons for East Asian students’ consistently high performances in international assessments (Son and Hu 2016). Among many mathematical topics, measurement has widespread applications to real-life scenarios and problems (Lehrer 2003). However, findings from national assessments indicate that American students show poor performance in this domain compared to other domains. US students’ performance in measurement is weaker than any other content area on the National Assessment of Educational Progress (NAEP) and TIMSS (Mullis et al. 2012, 2016; Vasilyeva et al. 2013). While researchers conducting international comparative studies of textbooks have examined fraction lessons (Son and Senk 2010), arithmetic average (Cai et al. 2002), and linear functions (Son and Hu 2016), area lessons have not been examined and compared often. A recent textbook analysis study found conceptual limitations in popular American textbooks’ lessons on area measurement (Smith et al. 2016). Smith and his colleagues recommended cross-national curricular analyses so that curricular presentations of topics could be compared to national profiles of student performance in those countries. We chose the topics of area measurement in geometry because it is one of the most commonly used domains of measurement in everyday life (Outhred and Mitchelmore 2000) and it plays a foundational role in more advanced mathematics, from fractions to calculus (Smith et al. 2016). Also, many textbook analysis studies examined American textbooks that were developed prior to the development of common core state standards (Hong and Choi 2014; Son and Hu 2016; Son and Senk 2010). Thus, the purpose of this study was to examine area lessons in American common core-aligned textbooks and Korean textbooks to compare their differences and similarities, and explore how textbooks from these two countries cover and address area measurement and well-known learning challenges in area measurement. Since what textbooks offer to teachers and students play an important role in the process of teaching and learning of

Do textbooks address known learning challenges in area measurement?...

mathematics, we may be able to gain insights into reasons for differences in student achievement in TIMSS and PISA. The following are our research questions that we attempt to answer. 1. How do American and Korean textbooks distribute attention to area and arearelated lessons? 2. In what order do the curricula present concepts related to measuring area, and do the sequences differ significantly between textbooks? 3. How well do the curricula address well-known students’ challenges in learning area measurement?

Related literature How students learn area measurement and common challenges Understanding area requires learning and coordinating many ideas, which are challenging to young students (Battista 2004, 2007; Clements and Stephan 2004). Foundational concepts for area include covering a region without gaps or overlaps with equal-sized units, equal partitioning of a region, counting unit measures, iterating combined units, understanding row and column structure, and linking the number of squares to length and width (Battista 2007; Sarama and Clements 2009). Studies have shown that it is challenging for students to develop a good conceptual understanding of area. Students are not able to cover a two-dimensional region with equal-sized units. Instead, they often use unequal unit or leave gaps or overlaps (Battista 2004; Outhred and Mitchelmore 2000). According to Sarama and Clements (2009), students develop understanding of area concepts gradually, from not being able to partition a region without gaps or overlaps with equal-sized units, to understanding how and why multiplying the number of rows and columns produce area for a rectangular figure. Several studies demonstrate similar results (Battista and Clements 1996; Battista et al. 1998). These studies suggest that it takes time for students to be able to cover or partition a region without gaps or overlaps, and to eventually link row and column structures to area measurement. One important idea in the development of area is understanding row and column structure (array structure) (Battista 1999; Battista et al. 1998). Although it is challenging for students to see array structure, it is a critical idea in learning area measurement because once they are able to understand and visualize the array structure, they can iterate, cover, and link the structure to the area formula (Sarama and Clements 2009). Without experiencing array structure, students may use the area formula incorrectly, may not consider a unit square to be the unit of area, and may not be able to connect the number of squares in a covering and area (Kamii and Kysh 2006; Zacharos 2006). This eventually leads them to using the area formula length × width without understanding why it is valid and, possibly, for the wrong figures (Zacharos 2006). However, when students learn using the conceptual approaches of partitioning and covering (partitioning, filling a given space with equal-sized units, and seeing array structure), they are more likely to develop

D. S. Hong et al.

a better understanding of area (Huang 2017; Outhred and Mitchelmore 2000; Zacharos 2006). Korean students demonstrate similar struggles. Some elementary students were not able to explain why the area formula length × width worked (Na 2012), while others were not able to cover a region completely with equal-sized units (Lee 2010). These studies show that some conceptual challenges, covering two-dimensional space with equal-sized units and understanding and linking array structure to the area formula, are common among elementary students. These studies also indicated that it is important to provide curriculum materials (e.g., textbooks and other resources) and learning opportunities to students to learn area measurement with covering and array structure. Although many previous studies have shown common challenges in learning area measurement, those challenges have not been well addressed in some popular American textbooks (Smith et al. 2016). To explore the issue further, our analyses focused on textbook series not used in Smith et al. (2016) as well as textbooks from one of the topperforming countries in international assessments. In analyzing these textbooks, we can see if these textbooks provide learning opportunities for students to experience covering and array structure. We can then make curricular recommendations to address the issue. Textbooks in the curriculum enactment process Curriculum, in general, has several stages: written or formal, intended, and implemented or enacted (Remillard and Heck 2014). In most countries, Bformal^ or Bwritten^ curriculum is the curriculum recommended by officials, expected to be taught by teachers, and outlined as goals by school policies (Stein et al. 2007). BIntended^ curriculum represents teachers’ plans for instruction, while Benacted^ or Bimplemented^ curriculum is what teachers actually teach in their classes (Remillard 2005; Remillard and Heck 2014; Stein et al. 2007). US state standards are a component of the written curriculum (Stein et al. 2007), and what teachers and students experience in classes constitute enacted curriculum (Remillard 2005). Textbooks may be considered Bwritten^ or Bformal^ curriculum (Stein et al. 2007), Bintended^ curriculum (Cai and Howson 2013) or Bpotentially implemented^ curriculum (Valverde et al. 2002). Whether researchers refer to textbooks as Bwritten,^ Bformal,^ Bintended,^ or Bpotentially implemented^ curriculum, textbooks play an important role in the curriculum enactment process as teachers often select and possibly modify activities and tasks from textbooks as they plan their lessons (Polikoff 2015; Remillard et al. 2014; Stein et al. 2007). Not all contents in textbooks will be transformed to mathematics lessons directly (teachers will likely modify textbook contents). Because textbooks are important resources when teachers prepare lessons, analyzing how textbooks treat mathematical topics is important (Smith et al. 2016). Examining the treatment of mathematical topics in textbooks can tell us how much attention is given to that specific topic. In the case of measurement topics, textbooks often present them at the end, decreasing the likelihood of these topics being taught (Smith et al. 2016). Other researchers have stated that the focus of area lessons is often procedural (Lehrer 2003; Murphy 2012). In the curriculum enactment process, teachers select and possibly modify mathematical tasks and activities from textbooks and other curriculum materials (Remillard and Heck 2014). It

Do textbooks address known learning challenges in area measurement?...

is highly likely that textbook content will be modified when teachers enact their lessons, but since teachers use textbooks and other curriculum materials to prepare their lessons, textbook coverage will influence students’ opportunities to learn mathematics. In other words, when particular ideas and concepts are mentioned more often, OTL for students increases (Smith et al. 2013). Thus, textbook content and how teachers enact their lessons jointly influence what students experience in their classrooms (Smith et al. 2013). For example, if textbooks place more attention on procedures rather than concepts, it is possible that when teachers use those textbooks to prepare their lessons, their lesson plans may also place more focus on procedures. As a result, students may not experience important conceptual ideas in or outside of the classroom and this may be a possible reason for students’ frequent dependency on the area formula without knowing why it works (Smith et al. 2016). Also, for the same reason, if textbooks do not address well-known challenges, we can interpret that as a possible reason for students’ struggle because it is possible that those challenges are not well reflected in the teacher’s lesson plans (Smith et al. 2016). Lesson plans that focus on procedures and lack learning challenges may limit students’ opportunities to learn mathematics (Smith et al. 2013). Since the mathematics lesson planning process goes through several stages before implementation, from written curriculum to enacted curriculum, there are other influential factors, such as teachers’ knowledge, beliefs, and the use of other instructional materials, to prepare their lessons. Thus, we cannot say textbooks are the only reason for students’ struggle in learning area measurement. However, limited coverage of topics will limit OTL for students and is a possible explanation for their performances.

Methods Data sources Due to the number of elementary mathematics textbooks available for use in American classrooms, we were forced to choose some series over others for analysis. Three textbooks series—enVisionMath, Go Math, and MyMath—are common core-aligned textbooks from three major American publishers (Pearson, Houghton Mifflin, and McGraw-Hill, respectively). The first two series were examined in a previous study (Polikoff 2015). In Polikoff’s study, the Math Connects textbook series from McGraw-Hill was analyzed, but according to McGraw-Hill, the MyMath textbooks series are the new version of common corealigned textbooks from McGraw-Hill. enVisionMath and Go Math are two other popular elementary textbook series in the USA (Dossey et al. 2016; Sahm 2015), and according to McGraw-Hill, there are 3 million MyMath users in America. A recent study examined length and area lessons in other non-American Common Core textbooks (Smith et al. 2013, 2016). Also, several previous research studies on textbook content were conducted with textbooks that were developed before the introduction of the Common Core State Standards (CCSS) (Cai et al. 2002; Hong and Choi 2014; Son and Senk 2010; Valverde and Schmidt 2000). Therefore, our textbook analyses of three common core-aligned textbooks series may expand our understanding of how more current American textbooks treat area measurement.

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A total of nine American textbooks were examined, including three textbooks from grade 1 through grade 3 from each publisher. To find area and area-related lessons, we examined the table of contents of each textbook. We first identified those lessons that included the term Barea^ in the title. We then conducted a manual search to identify lessons that did not have the term Barea^ in the title, but were related to area measurement. Keywords for this search included the terms partitioning, covering, tiling, and array structure. Some of these topics are not found from the lessons with the term Barea^ in the title but they include some ideas of area measurement so we called them area-related lessons. For example, we found covering and partitioning are often introduced in fraction lessons, but they include ideas of area measurement so those lessons were also examined in this study. We examined all geometry lessons found in these searches, up to the initial introduction of the area formula. All textbooks were the latest editions available, published in 2014 or 2015. In all, 431 (Envision Math), 441 (My Math), and 550 (Go Math) items were analyzed. Items included exposition, worked examples, and exercise problems. Korea has a centralized education system, including a national curriculum. The government develops and provides guidelines for the school curriculum. There are two kinds of textbooks in Korea: government-published and government-authorized textbooks. The textbooks examined in this study, Elementary School Mathematics, were published by the government and the only mathematics textbooks used in elementary schools in Korea because all elementary textbooks in Korea are published by the government. We examined all Korean textbooks from grade 1 to 6 and found area and area-related topics (we did same search as we did for the American textbooks) in the grades 1, 3, and 5 textbooks. For Korean textbooks, we examined 129 items from grades 1, 3, and 5. Table 1 describes the number of pages and lessons examined in this study. Framework to analyze opportunities to learn in textbooks In this study, we investigated the OTL students have when teachers use textbooks as a major resource for mathematics lesson planning. There are several different ways to view students’ opportunities to learn mathematics when textbooks are used in the lesson enactment process (or when students use textbooks on their own to study). Smith et al. (2016) used the term the knowledge-in pieces, meaning that students’ learning is the collection and coordination of many different elements. We also think that learning is a collection of several different types of knowledge so we tried to understand different types of knowledge that textbooks can offer to teachers and students. Table 1 Textbooks used in the study Textbook series

Publisher

Publication date

Pages

Items

Lessons

envisionMath series

Pearson

2015

86

431

18

Go Math series

Houghton Mifflin

2015

133

550

15

MyMath series

McGraw-Hill

2014

69

441

12

Korean textbooks

The Ministry of Education in Korea

2015

31

129

6

Do textbooks address known learning challenges in area measurement?...

To understand learning opportunities in textbooks and frame our analysis, we first examined previous textbook analysis studies (Charalambous et al. 2010; Hong and Choi 2014; Otten et al. 2014; Smith et al. 2016; Son and Senk 2010) to understand how they framed their analysis. First, when textbooks are analyzed, exposition (e.g., introductory paragraphs, text boxes with definitions, formulas, or theorems), worked examples (problems presented together with an explained solution), and exercise problems (mathematical items students are expected to solve) should be examined because they can provide potentially different learning opportunities for students. Teachers can use exposition to introduce mathematics content including definitions and formulas (students can also read exposition on their own to study), teachers and students can see how certain problems are solved with worked examples, and students have opportunities to engage in mathematical tasks with exercise problems. Finally, teachers can use all three areas to plan their lessons and implementing and enacting those lessons may lead to students’ opportunities to learn those topics in their classrooms. Second, in addition to searching for studies about textbook analysis in general, we also searched for studies that examined measurement lessons in textbooks (Lee and Smith 2011; Smith et al. 2013, 2016). Their framework specifically targets how textbooks cover topics of length and area, including the number of area lessons in textbooks, procedural and conceptual knowledge, and learning challenges. We adopted and modified (e.g., we added response type which was not considered in Smith and his colleagues’ study and used different terms to identify students’ learning challenges in area). In addition to the framework by Smith and his colleagues, we also paid attention to results from previous studies to understand students’ challenges in learning area measurement (Battista 2004; Sarama and Clements 2009; Zacharos 2006). Including tasks and items related to students’ learning challenges in area can increase students’ OTL to become familiar with those challenges. Taking into consideration previous textbook analysis and area measurement studies allowed us to develop our framework and explore different types of knowledge (learning opportunities) that textbooks offer. We examined three main areas, each focused on different learning opportunities: how much attention is given to area and area-related lessons, how area topics are treated and distributed, and how well-known challenges in area measurement are addressed. Table 2 describes our framework for this study. We will describe each focus in detail.

Table 2 Analysis framework of content and problems

Area of focus • Number of area and area-related lessons • Timing and topic sequence • Procedural and conceptual knowledge • Known challenges in learning area measurement - Covering with equal-sized units - Row and column array structure - Area formula and definition • Response type

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The number of area and area-related lessons One simple way to measure students’ opportunity to learn any specific content is the number of lessons that focus on that content (Smith et al. 2013). Knowing the number of area and area-related lessons in textbooks can provide information about how much attention is given to area measurement. To identify area lessons, we examined the table of contents of each textbook. We first identified those lessons that included the term Barea^ in the title. We then conducted a manual search to identify lessons that did not have the term Barea^ in the title, but were related to area measurement. We included all pages with at least one instance of such content to make sure to include any lessons that have to do with area measurement. Keywords for this search included the terms partitioning, covering, and tiling. The total number of area and area-related lessons was compared to the total number of lessons in each textbook to measure how much attention was given to area measurement. In our search of area and area-related topics, we were able to determine if textbooks provide any prior and necessary knowledge for topics related to area measurement. Timing and sequence When learning about measurement, there are tasks that students are likely able to do at a certain age (Sarama and Clements 2009). This implies that when and how mathematical topics are presented in textbooks can potentially influence how students learn those topics. Particularly for area measurement, many researchers have pointed out that students gradually learn related topics (Battista 2004; Outhred and Mitchelmore 2000; Sarama and Clements 2009), and there are different skills, such as covering and array structure, that students will often struggle with before a certain age. These studies recommend including area-related topics, such as partitioning, covering, and row and column structure, before introducing the area formula length × width. Lacking these early topics in textbooks may lead to missing those topics in teachers’ lesson plans (or students may not experience those topics if they studied on their own), which can keep students from experiencing and learning those topics, limiting their OTL to gradually experience those topics. We examined how each textbook series presented area measurement in regard to research recommendations for timing and sequence of area topics. Procedural, conceptual, and conventional knowledge Researchers often categorize the procedural and conceptual knowledge required when examining textbooks items (Smith et al. 2016; Son and Senk 2010). Hiebert and Lefvre (1986) defined conceptual knowledge as Bknowledge that is rich in relationships^ that cannot exist as Ban isolated piece of information.^ On the other hand, procedural knowledge is defined primarily as knowledge of sequential quality (Hiebert and Lefvre 1986). In the context of area measurement, procedural knowledge describes sequences of steps that are sufficient to solve area problems, such as using the area formula, counting unit squares, or covering a region with the same shape. In contrast to procedural knowledge, conceptual knowledge may be seen as recognizing that the area

Do textbooks address known learning challenges in area measurement?...

of a region is defined as the number of unit squares that completely cover that region. Classifying these two types of knowledge can be helpful in understanding how textbooks treat well-known conceptual challenges and difficulties, as well as their focus. Smith and his colleagues found 85 knowledge types, procedural and conceptual, when they examined area lessons. We adopted those categories to examine textbooks items. Although it is challenging to define an appropriate distribution of procedural and conceptual items when curriculum materials are analyzed, mathematics education researchers recommend building procedural fluency from conceptual understanding (National Council of Teachers of Mathematics 2014). In addition to these two knowledge types, Smith et al. (2016) also included a type of knowledge called Bconventional.^ This knowledge type represents choices about how to write certain terms in area measurement. For example, defining a 1 by 1 square as the standard unit of area measurement qualifies as conventional knowledge. Categorizing textbook items into knowledge types can indicate the curricular focus of the textbooks we compare. Placing more attention on procedures can limit students’ opportunities to learn area concepts and lead students to depend more on procedure. Known challenges in understanding area measurement As previously mentioned, researchers have examined well-known students’ difficulties and challenges when analyzing textbooks (Charalambous et al. 2010; Smith et al. 2016; Son and Senk 2010). In this study, we also examined whether the selected textbooks provided students opportunities to be exposed to difficulties and challenges in learning area measurement. By examining previous studies on students’ challenges in learning area, we identified several learning challenges in area measurement: covering with equal-sized units (without gaps or overlaps), array structure, and area definition and formula. Not including or limited coverage of these challenges can lead to lesson plans that do not reflect these challenges (or students may not experience these topics if they studied on their own) and, in turn, can limit students’ OTL. Response type When students work on mathematical tasks, there are various ways to respond to those tasks. Textbook analysis often includes those response types (Charalambous et al. 2010; Son and Senk 2010). Researchers believe that when mathematical tasks require explanation, students experience more opportunities to reason and think about those tasks. Previous studies have included different response types such as Bnumber only,^ Bexpression only,^ and Bexplanation.^ After initial examination, we found that all area items could be answered with either a short answer (number or expression) or explanation. Thus, the two categories of short response and explanation were used in our analysis. For response types, only exercise problems were examined. Unit of analysis Textbooks are composed of exposition, worked examples, and exercise problems. Previous textbook studies examined all three areas to examine potential learning

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opportunities (Charalambous et al. 2010; Hong and Choi 2014; Otten et al. 2014), often using frequencies or the number of times a particular topic appeared on textbook pages. When the frequencies of textbook items are considered, they are usually weighted equally (Ding 2016; Polikoff 2015; Smith et al. 2016). While it would be possible to develop an alternative weighting scheme, such as weighting by importance, equal weighting is the most logical and defensible approach because it is challenging to determine which items are more important than others in terms of providing learning opportunities to students. Since we used frequencies, we needed to discuss and decide on the unit of analysis. Figure 1 shows one example of unit of analysis we used in this study. Each exposition, worked example, and exercise problem in textbooks has its own instructional purpose (potentially different OTL). Each exposition introduces mathematics content including definitions, formulas, and procedures; each worked example demonstrates how certain problems are solved; and each exercise problem gives students opportunities to engage in problem-solving. Thus, when we

Fig. 1 An Example showing unit of analysis (Go Math, 2015b, p. 779)

Do textbooks address known learning challenges in area measurement?...

discussed our unit of analysis, we first considered each worked example, exercise problem, and exposition as one unit of analysis as each item provides OTL to teachers and students. Figure 1 is a sample page from one of the textbook series. This page includes one worked example and two exercise problems—there are three items (three units) to analyze. All three items in Fig. 1 are about partitioning (or covering) a rectangle with tiles to array structure. The worked example demonstrates how this problem can be solved. With the worked example, students can solve the next two problems in the same way. In all, we have three items, where each worked example and problem is one unit of analysis, on this page to code. Also, each item was given two or three codes—this will be explained further with coding examples. For all other textbook pages, we used the same method to identify expositions, worked examples, and exercise problems to count the number of units to be coded from each page. We will provide more coding examples to describe in more details how we identified and coded each item. Coding procedures and examples The authors met several times to understand the analytic framework, including developing an understanding of procedural and conceptual knowledge, response types, and students’ challenges in area measurement. Then, we assigned codes to each identified textbook item. Each exposition, worked example, and exercise problem was counted as one item. Figures 2, 3, and 4 show examples of how we coded each item. First, we decided there are three items (three exercise problems or three units) to be coded. Students are asked to use the area formula to compute the area of each room. In terms of topic, these items are coded as area formula because students just need to multiply numbers to get the correct answers. In terms of procedural and conceptual knowledge types, these were coded as procedural (only multiplying two numbers is required). Finally for response type, these were coded as short response (only numbers are required). We only had one item (one exercise problem or one unit) to code in Fig. 3. This item was coded as counting squares without array structure because it was about counting

Translation: This is the floor plan for Chul Soo’s home. Find out the area of each room. Parents’ Room Chul Soo’s Room Brother’s Room Fig. 2 Coding examples from a Korean textbook (The Ministry of Education in Korea, 2015, p. 139)

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Fig. 3 Coding example from one American series (Go Math, 2015 c, p. 651)

the number of squares in the picture. It was also coded as a procedural knowledge type requiring a short response (just a number) because it was largely a procedural counting task without linking the number of squares to an area formula. There was one item (one exposition or one unit) to code in Fig. 4, as it is on several related components about one topic, namely covering a region with equal unit. It was also coded as conceptual, as the process of using iteration to cover a region was laid out in a sequential manner. Response type was not considered for this item as it was from an exposition section. Reliability Each textbook included exposition, worked examples, and exercise problems. After discussing the established codes, two authors coded approximately 20% of the textbook items to check inter-rater reliability. The coders gave, at most, three codes to each item, because there are three possible codes (i.e., procedural vs. conceptual, knowledge needed, and response type) for each item (except for exposition). After comparing codes for sample items and finding an acceptable high inter-rater reliability, the authors coded all textbook items jointly to produce a final set of tables for analysis, resolving coding differences of individual items when they arose. To determine reliability, we applied a generalizability theory D study (Alkhrausi 2012). This technique produced a reliable coefficient of 0.964.

Fig. 4 Coding examples from one American series (enVisionMath, 2015 b, p. 718)

Do textbooks address known learning challenges in area measurement?...

Results The results of our analyses of number of area and area-related measurement lessons, timing and sequence, and procedure and conceptual knowledge types address research questions 1 and 2. Attention provided by each textbook series to the different knowledge types needed to understand area measurement addresses research question 3. Area measurement lessons in textbooks Table 3 shows the number and percentage of area and area-related lessons in each textbook. These lessons include topics such as area definition, area formula (area lessons), partitioning, and covering a region (area related). Our results indicated that little curricula attention is given to area and area-related topics in these textbook series, supporting earlier findings by Smith et al. (2016). Each textbook series contained between 1.4 and 7.6% of lessons focused on area measurement. These percentages are less than the 1 to 12% range found by Smith et al. (2016) and demonstrate a limited to modest coverage of area lessons across every textbook series. Time and sequence Table 4 describes the timing and sequence of area and area-related lessons for each textbook series. We examined the table of contents from each series to create this table. With regard to the sequencing of topics, Korean and American textbooks differed. The Korean textbook series first introduces area in grade 1, with a lesson titled BComparing Area.^ This is a short lesson that asks students to compare and make visual judgments between two objects (see Fig. 5). These items introduce comparison of the two-dimensional space that each object covers and encourage students to begin developing a notion of area (These items are all coded as Bother topics). In the three American textbook series examined, we did not find any similar Table 3 Number and percentage of area and area-related lessons to the total lessons

enVision Math

Go Math

MyMath

Korean textbooks

Grade

Area and area-related lessons

Total

1

5 (4.5%)

110

2

8 (6.9%)

116

3

5 (4.2%)

119

1

3 (3.0%)

101

2

4 (3.6%)

110

3

8 (7.6%)

105

1

3 (3.2%)

95

2

2 (2.2%)

92

3

7 (6.2%)

113

1

1 (1.4%)

70

3

1 (1.5%)

68

5

4 (5.7%)

70

D. S. Hong et al. Table 4 Timing and sequence of area lessons envision Math

MyMath

Go Math

Grade Lesson and topics

Korean textbooks

Lesson and topics

Lesson and topics

Lesson and topics

1

- Comparing areas (making visual comparative judgement)

- Making equal parts - Equal parts halves, - Describing equal parts quarters and fourths of whole objects - Making halves and fourths of rectangles and circles - Identifying halves and fourths - Draw pictures

- Equal or unequal parts - Halves - Fourths

2

None

- Identifying halves, - Halves, thirds and fourths fourths and eighths - Area: partition - Unit fractions and rectangles regions - Non-unit fractions and regions - Equal parts of a whole - Comparing fractional parts - Counting fractional parts - Draw a picture - Measuring area

- Partition rectangles - Equal parts - Show equal parts of a whole - Describe equal parts

3

- Partitioning shapes to equal parts.

- Covering region - Hands on: - Area and units understanding - Standard units area - Area of squares and - Measure area rectangles - Hands on: tile - Area and the distributive rectangles property to find area - Area of rectangles - Hands on: area model of distributive property - Area of composite figures - Area and perimeter

- Understanding area - Measure area - Use area models - Area of rectangles - Area of combined rectangles - Same perimeter, different areas - Same area, different perimeters - Relate shapes, fractions and area

5

- Unit squares - More unit squares - Finding area of rectangles - Finding area of combined rectangles

lessons at any grade level. Although each American textbook series devote several lessons to comparing lengths of different objects, comparison items related to areas were not found.

Do textbooks address known learning challenges in area measurement?...

Translation: Compare the area of two mats. Fig. 5 One task from 1st grade Korean textbook (The Ministry of Education in Korea, 2014a, p. 130)

After a brief introduction to area in grade 1, Korean textbooks have one lesson about partitioning shapes into equal parts in grade 3 (no area items in grade 2). Lessons on area then begin in grade 5, where the textbook introduces unit squares, area of rectangles, and the area formula length × width. In contrast, all three American textbook series include several lessons in grades 1 and 2 about partitioning regions, such as rectangles and circles. These lessons include creating, identifying, and counting equal parts, although these lessons are related more to fractions than to area concepts. For example, the lesson titles in envision Math grade 2 focus primarily on unit fractions and regions, although many tasks ask students to partition shapes into equal pieces. It is only in grade 3 that topics such as covering, array structure, area formula, and area of rectangles are introduced. In terms of timing, Korean textbooks introduce area first in grade 1, but more involved ideas of partitioning, understanding, and finding area of rectangles are found earlier in the American textbook series. According to learning trajectory research, there are certain topics that students are able to do at a given age (Sarama and Clements 2009). For example, around the age of kindergarten, students can cover a given space; around the age of grade 1, they can see local structure of some (but not all) rows and columns; and at the age of grade 2, they can use partial structure and iteration to understand global row and column structure of the array. However, within each of the examined textbook series, these topics are introduced later or not at all. Since studies have shown that elementary students (2nd and 3rd graders) struggle with area-related topics (Battista 2004; Outhred and Mitchelmore 2000), this could be problematic. Until grade 2, students using any of the American textbook series are only expected to divide rectangles and circles into equal parts, with limited experience in covering or seeing the structure of rows and columns. Similarly, until grade 5, students using the Korean textbook series are only expected to compare area and partition into equal shapes. Introduction of the area formula in grade 3 (or grade 5) with limited experience in array structuring or covering may lead to students’ inability to link the array structure of rectangles to the area formula. This will eventually lead students to develop a more procedural understanding of area measurement. We were unable to find any studies that investigated learning trajectory of Korean students, but studies have shown that Korean students experience similar difficulties in the early ages (Lee 2010; Na 2012). Thus, Korean students may experience issues similar to those of American students when they learn about area measurement. In all, there were issues with the timing of area topics in every textbook series, where all three series do not provide enough necessary prior topics in area measurement. Important

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topics, such as covering and filling two-dimensional space, are missing in the early treatment of area lessons from both countries’ textbooks. Procedural, conceptual, and conventional knowledge As seen in Table 5, the majority of items in both countries’ textbooks are procedural, with very few items including conceptual or conventional knowledge. This finding supports previous work examining American textbooks, where procedural items accounted for more than 87% of items (Smith et al. 2016). Such findings imply that the focus of area lessons in both American and Korean textbooks is more about procedures than concepts. Again, this can be one way to lead both countries’ students to a more procedural understanding of area. Lastly, conventional knowledge appear in grade 3 textbooks only (grade 5 for Korean). This is because of the introduction of the definition of unit square. Students’ learning challenges in understanding area measurement Table 6 shows the distribution of each area—related knowledge type presented in American and Korean textbooks. We describe each type in more detail below. Covering Being able to cover a given region without gaps or overlaps is an important precursor to understanding the area definition and area formula. Students who are able to cover a region with a unit without gaps or overlaps will more easily progress to understanding an array structure on that region (e.g., rows and columns) (Sarama and Clements 2009). Covering may be introduced with drawing, using tiles or iterating. Once students are able to cover a region completely with equal-sized units without gaps or overlaps, they will gradually learn to see partial array structure. Eventually, they can link the number Table 5 Percent distribution of conceptual, procedural, and conventional knowledge in textbooks

enVision Math

Go Math

My Math

Korean textbooks

Grade

Procedural

Conceptual

Conventional

1

99.5

0.5

0

2

98.5

1.5

0

3

82.9

11.7

5.4

1

100

0

0

2

93.5

6.5

0

3

89.4

10.1

0.5

1

100

0

0

2

100

0

0

3

86.2

10.9

2.9

1

100

0

0

3

100

0

0

5

93.2

1.1

5.7

Do textbooks address known learning challenges in area measurement?... Table 6 Percent distribution of knowledge type in textbooks Grade Covering Partition (without Counting squares Array Area Other array structure) (without array structure definition topics structure) and formula enVision Math

Go Math

My Math

1

0

2

10.7

86.3

2.6

0

0

0.4

3

0.9

0

41.4

0.9

17.1

39.7

1

0

81.3

0

0

0

18.7

2

6.7

83

0

7.3

0

3

3

6.2

0

23.2

3.1

51

16.5

1

0

100

0

0

0

0

2

1.9

81.9

15

1.2

0

0

3

16.2

0

31.8

3.5

15.6

32.9

0

0

0

0

0

100

Korean Textbooks 1

99.2

0

0

0

0.8

3

0

100

0

0

0

0

5

17.2

0

26.4

11.5

17.4

27.5

of units covering the region to length and width, a pivotal component of developing a conceptual understanding of area measurement (Sarama and Clements 2009). In all three American textbook series, items related to covering are included in grades 2 and 3. When covering was included, however, the three textbook series include other topics such as partition without array structure, counting unit squares, and area formula. Such a lack of inclusion is problematic, as previous studies have shown that it is challenging to second graders to cover a region completely without gaps or overlaps. They were either unable to draw a complete covering or they covered the region with inconsistently shaped or sized units (Battista et al. 1998; Outhred and Mitchelmore 2000). Even if students are not able to cover a region without gaps or overlaps in earlier ages, it is important to introduce those tasks early so that students can experience opportunities to gradually understand the concept of covering a region without gaps or overlaps (Outhred and Mitchelmore 2000). A sudden introduction of the topic in grade 2 or grade 3 may be insufficient to develop the skill without significant difficulty. Compared to the American textbooks, Korean textbooks include covering items much later, only introducing them in the fifth grade. There are no items related to covering in any earlier grades. Korean textbooks also include more additional topics. All of the covering items included are about using tiles to cover the given region. Items that allow students to draw in their own units are not found. Similar to students in other countries, sparse coverage and late timing may explain first and second grade Korean students’ challenges in covering activities (Lee 2010). We also noticed that when textbooks include covering items, the terms Bgaps^ and Boverlaps^ are used only few times: less than 10 times in each American textbook series and never in the Korean textbooks. Since students often struggle with covering a region without gaps and overlaps (Outhred and Mitchelmore 2000; Sarama and Clements 2009), careful attention to covering and explicit remarks about why gaps and overlaps are important will lead students to a more conceptual understanding of what it means to measure area.

D. S. Hong et al.

Array structure Being able to divide or partition rectangles into rows and columns is an important conceptual idea in area measurement. Once students are able to construct a correct partition of a rectangle and cover the region without any gaps or overlaps, they can proceed to either skip counting or iterating the structure to find the area. This can lead to understanding and linking why and how the area formula works. Items in this group include drawing, tiling, or partitioning a region into rows or columns (the terms Brows^ or Bcolumns^ need to be included or students need to have opportunities to show array structure) and then counting them or using partial array structure (e.g., rows or columns) to compute the area (Fig. 6). In the American textbook series, partitioning a rectangle into rows and columns first appear in grade 2 and then grade 3. All three series introduce the concept in grade 2, but the idea of iteration is only included in Go Math. All three series make note of the array structure when they introduce tiling or the area formula, but only a limited number of items are included (ranging from 0.9 to 7.3%). In all three American textbook series, students have limited opportunities to develop either a partial or complete understanding of row and column structure. In Korean textbooks, the topic of array structures appears in grade 5. Unlike the American textbooks, the Korean textbook provides opportunities for students to visualize an array structure and link the structure to area measurement. From there, they have opportunities to see how area formula and array structure are related. For example, the items in Fig. 7 ask students to count unit squares in partial rows and columns and then count the total to get the area. The fifth grade Korean textbook devotes approximately 11% of its area-related items to array structure. However, after a few items such as those in Fig. 7, the Korean textbook quickly moves to decomposing figures into rectangles and applying the formula (e.g., Fig. 8). This reflects the more procedurefocused approach of the Korean textbook series. Also, since items related to array structure are included late in the grade progression, there remains the question of whether Korean students are able to fully understand array structure given their limited experience with the concept previously. Area definition and formula The definition of area is introduced in the exposition section of all three American textbook series. Items were coded as area formula if they showed that multiplying two numbers gives the area, or if they used the length × width formula to compute the area, as in Fig. 9. In all three American textbook series, area is defined in grade 3 as Bthe

Fig. 6 Using array structure to count the total number of squares (Go Math, 2015b, p. 742)

Do textbooks address known learning challenges in area measurement?...

Translation: 1. Count the number of squares to find out the area of rectangles. 2. Complete the table to find out the area. Figure Row Column Ga Na Da

Area

Write the formula for the area of a rectangle.

Fig. 7 Using array structure to introduce area formula (The Ministry of Education in Korea, 2015, p. 138)

number of unit squares needed to cover a figure.^ On the other hand, area is not formally defined in Korean textbooks. There are several items that discuss covering a region with a unit (not necessarily unit squares), but a formal definition of area is not provided. Both countries’ textbooks use array structure to introduce the area formula. The Korean textbook lets students derive the formula after working on tasks counting unit squares (see Fig. 7). The American textbook series also use an array structure in grade 3 (see Fig. 9). Items in Fig. 9 demonstrate how the number of squares in each row and the area are related to each other. As previously mentioned, only a limited number of array structure items are included in both countries’ textbooks. Since both American and Korean students will have limited experiences with array structure, it would likely be challenging to link array structure to the area formula, even when it is presented carefully in exposition. With limited opportunities to explore how array structure and the area formula are related, students are likely to resort to more procedural approaches

Translation: Find the area of the figure. Fig. 8 Procedure-based item in Korean textbook (The Ministry of Education in Korea, 2015, p. 140)

D. S. Hong et al.

Fig. 9 Items using array structure to introduce area formula (Go Math, 2015c, p. 655)

to area. Such a tendency is reinforced by every textbook, as they all move quickly to items like those in Fig. 10, where students are only required to use a procedure, multiplying length and width. What is important conceptually in understanding the area formula is to recognize the correspondence between the number of unit squares, number of rows and columns, and area, so that students can see why multiplying length and width works. To enhance students’ experiences, it would be better to provide them with opportunities to draw partitions and coverings, identify rows and columns, and

Fig. 10 Procedure-based area items (MyMath, 2014c, p. 786)

Do textbooks address known learning challenges in area measurement?...

develop an awareness of the links between array structure and the area formula. With textbooks from both countries, students have only limited opportunities to link array structure to the area formula. Other area topics In grade 1, all items in the Korean textbook are related to comparing two-dimensional space (Fig. 5). The most frequent item (over 80% in grade 1 and grade 2) in the three American textbook series is about partitioning a shape into equal parts, with limited connections to array structure. Items such as the one in Fig. 11 are about partitioning different shapes into equal parts, following the common core state standard of Bpartition circles and rectangles into two, three, or four equal shares.^ Korean textbooks include this topic in grade three, but coverage of the topic includes only one lesson, embedded within the unit topic of Fractions and Decimals. Items in the lesson are once again more about fractions than area measurement (see Fig. 12). Another frequent topic in both countries’ textbooks is counting unit squares in a shape. Figure 13 shows one set of items that is about counting unit squares, included in both grade 2 and grade 3 American textbooks (ranging from 23 to 41% of items in grade 3 textbooks). Items such as that in Fig. 13 are found immediately after the terms Bgaps^ and Boverlaps^ are introduced. We found similar results with Korean textbooks as well, with approximately 26% of items in grade 5 Korean textbooks focusing on counting unit squares. Also, in grade 3 American textbooks, there are several items about using area model to represent distributive property. In grade 5 Korean textbooks, there are several items about decomposing a figure into smaller shapes to compute the area (Fig. 8). With such a heavy focus on counting squares, paired with limited experience with array structures in previous years, it will be challenging for students to connect array structure to the area formula. Prior studies have shown that even with drawn lines and squares, the connection between counting unit squares and area is not apparent to elementary students (Battista 2004; Battista et al. 1998). Given these issues, it will be difficult to use items in Fig. 13 to address the challenges of covering or array structure. Students may count unit squares procedurally without seeing array structure or understanding the purpose of not having gaps or overlaps. With limited opportunities to experience covering and array structure in grades 1

Fig. 11 Sample partitioning item (enVision Math, 2015, p. 597)

D. S. Hong et al.

Translation: Discuss different ways to divide squares into four equal parts. Fig. 12 Sample partition items from Korean textbook (The Ministry of Education in Korea, 2014b, p. 195)

and 2, it will be challenging for students to see array structure when they are trying work on items like the one in Fig. 14. Response types For response types, only exercise problems were examined. Previous studies have used response types to conclude that Asian countries’ textbooks provide more opportunities for students to explain their thinking (Son and Senk 2010). As we see from Table 7, our results do not agree. Most items in both countries’ textbooks require only short responses, with few explanations. Since students struggle with linking the number of squares and the area formula, it would be more meaningful to ask students to verbalize connections between the number of squares in each row or column to the area formula, but such opportunities are rarely presented in either country’s textbooks.

Summary and discussion This study compared area lessons from Korean textbooks and American standard-based textbooks to understand how textbooks treat area measurement lessons. We were interested in knowing how each textbook covers area lessons and if they addressed learning challenges in area measurement and, in doing so, if they provided students with opportunities to overcome or become familiar with known challenges and area

Fig. 13 Items that let students count unit squares without mentioning array structure (MyMath, 2014c, p.767)

Do textbooks address known learning challenges in area measurement?...

Fig. 14 One iteration problem in textbook (Go Math, 2015c, p. 680)

concepts. We analyzed textbook items, including exposition, worked examples, and exercise problems, to understand the learning opportunities available to teachers and students when using these textbooks. The overall timing and sequence and response types were examined as well. Our results indicated that textbooks from both countries paid modest or limited attention to area measurement lessons (less than 7.6% across all four series). In terms of timing, the American textbook series introduce area-related topics, partitioning, covering, array structure, and area formula much earlier than Korean textbooks. While American textbooks spread out many area-related lessons over grades 1, 2, and 3, Korean textbooks present the majority of area-related lessons in grade 5. In terms of sequence, the American textbooks progress through partitioning, tiling, and then presenting the area definition and formula. According to learning trajectory research, there are certain topics that students are able to do at a given age (Sarama and Clements 2009). For example, students are able to cover a region and see partial array structure in kindergarten and grade 1. However, the textbook series we examined introduces such topics either later or not at all. Korean textbooks introduce comparing areas in grade 1, followed by one lesson about partitioning shapes into equal parts in grade 3, and then lessons on area in grade 5. In grade 5, the textbook introduces the ideas of unit squares all the way up to finding the area of rectangles nearly simultaneously. Such findings indicate that both timing and sequencing are an issue for both countries’ textbooks. As a result, when the area formula is introduced to students (in grade 3 for American students and grade 5 for Korean students), opportunities to learn are limited due to limited coverage of covering and array structure items in previous grades. Table 7 Distribution of items in response type

enVision Math

Go Math

My Math

Korean Textbooks

Grade

Explain

Short response

1

32

68

2

8.9

91.1

3

8

92

1

6.1

93.9

2

14.7

85.3

3

18.3

81.7

1

5.7

94.3

2

4.3

95.7

3

13.8

86.2

1

18.2

81.8

3

21.5

78.5

5

12.5

87.5

D. S. Hong et al.

Both countries’ textbooks place strong focus on procedures rather than concepts, and on short responses (e.g., number, expression) rather than explanation. For example, many tasks involving area formula are procedural rather than conceptual because each textbook moves quickly to just multiplying two numbers to get area after the initial introduction of the area formula. Moreover, our results reveal that both countries’ textbooks provide only limited opportunities for students to become familiar with well-known challenges to area measurement, such as covering a region completely, seeing array structure, or linking array structure to the area formula (Battista 2004; Clements and Sarama 2004; Outhred and Mitchelmore 2000). For example, the most frequent items in American textbooks are procedurally focused tasks that focus on partitioning regions without array structure and counting unit squares without array structure. These two topics are also frequent topics in 3rd and 5th grade Korean textbooks. Such items are unlikely to provide opportunities to explore an array structure or develop students’ skills for drawing coverings of regions. Compounded with issues of sequencing and timing, a procedural focus, restricted response type, and limited coverage of important conceptual area, ideas are highly likely to lead to area lessons that do not reflect these challenges and concepts. In turn, such lessons can limit students’ learning opportunities to explore a more conceptual understanding of area measurement. BMile wide, inch deep^ is a phrase coined to describe the fragmented distribution of American curricular content at the topical level, indicating too many topics repeated with little depth in subsequent grades. Among the three American textbook series we examined, it appears that in grades 1 and 2, there are many items (over 80% of all area measurement items) about partitioning a region into equal parts without sufficient attention given to row or column structuring or covering space (e.g., Fig. 11). Such topics do not become more difficult from one grade level to another, instead, repeating themselves without adding substantial depth. On the other hand, there are no arearelated items in grades 2 and 4, only one lesson in grades 1 and 3, and all of the rest in grade 5 in the Korean textbooks. Although not a Bmile wide,^ such a distribution of items means that the entire depth of the subject of area measurement is placed all in one grade level. Lacking area-focused items in American textbooks and not including area items at all in Korean textbooks can lead to difficulties learning area measurement for elementary students. These issues are important to address at both the textbook and classroom level in order to provide more enhanced learning opportunities to students.

Conclusion What may we conclude from our findings? As we mentioned previously, although textbooks do not provide mathematics lessons directly (content will likely be modified by teachers), they are one of the main resources teachers use when planning lessons. With the issues identified in these textbook series thus far, it is possible that limitations in textbooks can lead to area lesson plans that do not reflect challenges and important concepts of area measurement. In turn, limited coverage may lead to limiting elementary students’ learning opportunities and they may be inclined to adopt a more procedural understanding of area, without attaining a conceptual understanding. Our findings echo similar issues found in other popular American textbooks (Smith et al.

Do textbooks address known learning challenges in area measurement?...

2016), reinforcing the importance of informing textbook authors and developers of existing limitations in American textbooks. Thus, we can recommend the need to include some area topics (covering, partitioning, and array structure) early to provide and build necessary and prior knowledge. Also, to address more in-depth ideas of covering without gaps or overlap, develop an array structure and fostering connections between that array structure and the formula for area. Moreover, although we cannot say that textbooks are the only cause of students’ learning challenges, it is important to inform teachers of these results so they can attempt to address these issues when they plan their lessons. For example, professional development could focus on developing modified tasks from these textbooks so that teachers can incorporate opportunities to address challenges in area measurement into their own classrooms. Items such as those in Figs. 6 and 11 are found in grade 2 textbooks. Modifying the items in Fig. 11 so that students have opportunities to see partial array structure (e.g., What is a row or column in this picture? How many equal parts are in each row and column?) and introducing more items like that in Fig. 6 to grade 2 classrooms can provide meaningful early experiences with array structure. These modifications can be followed in grade 3 with modified tasks such as those in Figs. 10 and 13 (asking students to count unit squares in each row or column for Fig. 13 and asking them to draw array structure to link unit squares and area for Fig. 10). Items in Fig. 15 are also found in grade 3. Students are introduced to tiling and explicitly asked to identify whether there are any gaps or overlaps. However, students can easily answer these items with a simple Byes^ or Bno^ and have no opportunity to construct a covering or tiling themselves. Since past work has shown that students tend to struggle to cover a region when drawing coverings independently (Battista et al. 1998; Outhred and Mitchelmore 2000), presenting opportunities to do this without tiles or physical manipulatives is also critical. To expand upon these tasks, students should be asked to explain why there were no gaps or overlaps and what this means in the context of finding area. This involves asking questions such as: Why do we care when there are overlaps? Why do gaps matter for area? It would also be beneficial if students are asked to explain why 4 square inches or 8 squares inches are not correct. Students should also be asked to independently draw their own coverings with same-sized units, both with and without physical manipulatives. In this way, students have opportunities to cover a region without gaps or overlaps and link array structure to the area formula. While developing a conceptual understanding of area will remain challenging to 2nd and 3rd graders, early experiences with these modified tasks can provide the critical background needed to have gradual understanding of area in a conceptual way. Since both countries’ textbooks have similar issues, modifying these textbook tasks would be recommended to both Korean and American teachers. For Korean textbooks, it may be beneficial to include more area measurement items in grades 1 to 3 so that students can experience covering and array structure earlier. More items that are similar to Fig. 7 would also be useful in developing a meaningful link between array structure and area measurement. Similar to modifying items like that in Fig. 11, items such as those in Fig. 12 can also be modified so students can have opportunities to experience array structure. In terms of international comparative studies, we cannot say that the learning opportunities these textbooks offer are directly related to American students’

D. S. Hong et al.

Fig. 15 Sample tiling task (Go Math, 2015b, p. 649)

performances in TIMSS. However, our results show that how these textbooks treat area measurement can be one of the reasons for American students’ TIMSS results because limited coverage in textbooks may lead to lesson plans that do not reflect important area measurement topics and limited learning opportunities for students. For Korean educators, although we are not able to generalize our results, an important question remains: BWhat insights do we gain from our results to understand Korean students’ performances despite what textbooks offer to students and teachers?^ We did not find that Korean textbooks presented area lessons in a way that provided better learning opportunities for Korean students and, in particular, Korean textbooks shared many of the same conceptual limitations as the American

Do textbooks address known learning challenges in area measurement?...

textbooks. However, Korean students performed equally well in measurement compared to other content areas in TIMSS 2011 and 2015 unlike American students (Mullis et al. 2012, 2016) and a recent report found that Korean teachers use textbooks as their main resources (Pang 2012). With further examination of TIMSS results, we are able to find two possible ways to think about this. The TIMSS 2015 report shows that Korean students have more available educational resources at home and more Korean students enter primary schools with numeracy skills than students in the USA (Mullis et al. 2016). These results may partially explain our results that there may be other contributing factors for Korean students’ performances. Previous studies that compared East Asian students and American students have shown that East Asian students are superior in computational and procedural problems but their performances are relatively similar to American students on open-ended items (Cai 2005; Cai and Nie 2007). TIMSS and PISA item analyses reveal that many items are multiple-choice or shortresponse items (over 80% of TIMSS and 60% of PISA items, respectively) (Neidorf et al. 2006). Thus, it may be more complex to understand TIMSS and PISA results than what ranking and scores indicate (Cai 2005; Cai and Nie 2007). With procedural and short response items in Korean textbooks, Korean students’ high overall achievement may be more about doing well on multiple choice and short response items. Since Korean students have similar struggles as American students (Lee 2010; Na 2012), we may need to understand the results of TIMSS and PISA more carefully (Wang et al. 2017). To explore this issue further, as Smith et al. (2016) noted, further studies are required to examine the link between curriculum use and students’ performances in assessments in order to make more distinct claims about influence of textbooks on students’ performances. Also, it will be beneficial to examine and compare other mathematical topics to determine if Korean mathematics textbooks provide better learning opportunities to teachers and students when other topics are considered. For future research, it will be interesting to examine the results of TIMSS and PISA of students’ performances on different item types. Results from these further studies may enable us to have a better understanding about Korean students’ international assessment results. Smith and his colleagues also asked whether the conceptual limitations demonstrated in American textbooks were unique or not. As our results show, similar conceptual limitations are demonstrated in Korean textbooks as well. Teachers will likely be called to fill in some of the Bgaps^ between textbooks and known learning challenges. While textbook lessons are likely modified by teachers (Son and Kim 2016), what teachers do in the process of planning and implementing curriculum materials is important. Teachers will likely continue to select certain tasks from textbooks and implement them in class. However, teachers often have similar learning challenges as students (Baturo and Nason 1996; Murphy 2012). Thus, teachers need to be informed of limitations in textbooks and obtain the proper support to fill the gaps between textbooks and students’ learning challenges in area measurement. Finally, our study has limitations because we only examined area and area-related lessons. It will be interesting to examine other measurement topics (and other mathematical topics) to see if these textbook series address learning challenges in those topics or not. Such studies can expand our understanding about what textbooks bring to teachers and students in terms of providing OTL.

D. S. Hong et al.

Appendix Textbooks Analyzed enVision Math Common Core edition 2.0, Grade 1 (2015 a). New York, NY: Pearson/ Scott Foresman– Addison Wesley. enVision Math Common Core edition 2.0, Grade 2 (2015 b). New York, NY: Pearson/Scott Foresman– Addison Wesley. enVision Math Common Core edition 2.0, Grade 3 (2015 c). New York, NY: Pearson/Scott Foresman– Addison Wesley. Go math! Common Core edition, Grade 1 (2015 a) (student edition e-book). Orlando, FL: Houghton Mifflin Harcourt. Go math! Common Core edition, Grade 2 (2015 b) (student edition e-book). Orlando, FL: Houghton Mifflin Harcourt. Go math! Common Core edition, Grade 3 (2015 c) (student edition e-book). Orlando, FL: Houghton Mifflin Harcourt. MyMath, Grade 1. (2014 a). New York, NY: Macmillan/McGraw-Hill. MyMath, Grade 2. (2014 b). New York, NY: Macmillan/McGraw-Hill. MyMath, Grade 3. (2014 c). New York, NY: Macmillan/McGraw-Hill. The Ministry of Education in Korea (2014a) Mathematics 1. Seoul, Korea. The Ministry of Education in Korea (2014b) Mathematics 3. Seoul, Korea. The Ministry of Education in Korea (2015) Mathematics 5. Seoul, Korea.

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