Mediterranean Journal for Research in Mathematics Education Vol. 10, 1-2, 187-213, 2011
An International Perspective between Problem Types in Textbooks and Students’ Understanding of Relational Equality
ROBERT M. CAPRARO: Texas A & M University & Aggie STEM Center MARY MARGARET CAPRARO: Texas A & M University & Aggie STEM Center EBRAR Z. YETKINER: Texas A & M University SENCER M. CORLU: Texas A & M University SERKAN OZEL: Bogazici University, Istanbul, Turkey SUN YE: West Virginia University HAE GYU KIM: Jeju National University, Jeju-do, Republic of Korea
ABSTRACT: This study broadens the international knowledge base about second- and sixth-grade students' understanding of the equal sign and possible explanatory power of their textbooks affording insights about educationally relevant factors related to the presentation of the equal sign. Participants from China(Beijing, Xia Peng), S. Korea(Jeju), Turkey(Istanbul), and the U.S.(central Texas) (N = 1823) were administered a language-free instrument to determine their conceptualization of the equal sign. Textbooks used by each sample were coded for presentation of the equal sign. Results showed “operation equals answer” in S. Korean second- and sixth-grade textbooks (18%, 16%) was substantially lower than in Chinese (41%, 28%), Turkish (40%, 27%), and central Texas textbooks (54%, 18%), respectively. The achievement of students from Beijing and Jeju was substantially higher than the students from Istanbul and central Texas. The results from the equal sign instrument are discussed in relation to the presentation of equal sign in the textbooks. Key words: Equal sign, International comparisons, Elementary mathematics, Early algebra. THEORETICAL BACKGROUND In this study, the connection between textbook use and conception of the equal sign is examined on a basis firmly rooted in international mathematics achievement and curricula comparisons. While international comparisons are useful as benchmarks to measure progress and change as new innovations are implemented, their most important application is to explore what mathematics is taught and how it is taught across countries. This study contributes uniquely to the literature by examining student
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performance in four countries on similar tasks identified from previous studies and then carefully examining a textbook from each country for how students encounter the equal sign. The U.S. has been commonly criticized for developing standards and curricula that are a “mile wide but an inch deep” (Schmidt, Houang, & Cogan, 2002, p. 3), and the U.S. has not been able to secure top rankings in international studies. In fact, U.S. students scored below most Asian countries such as China and S. Korea but higher than some other Organisation for Economic Co-operation and Development (OECD) (2006) countries such as Turkey on the Trends in International Mathematics and Science Study (TIMSS) in 2007 (Mullis, Martin, & Foy, 2008). For more than 30 years, researchers have examined students‟ understanding of the equal sign in the U.S., however, it has not been widely explored as a factor that may possibly influence large-scale international comparisons. This study built on previous findings by incorporating items used in previous research on second and sixth grade children‟s conception of the equal sign. The purpose of the current study was to a) broaden the international knowledge base about second and sixth graders‟ understanding of the equal sign within various uses and sentence structures that allows comparisons to earlier findings, b) provide international comparisons of second and sixth graders from specific regions in China (Beijing, Xia Peng), S. Korea (Jeju), Turkey (Istanbul), and the U.S. (central Texas) referred to in the rest of the paper as Beijing, Jeju, Istanbul, and central Texas, and c) explore student textbooks (curricula) underlying students‟ understandings of the equal sign in each region of each country to add to the existing framework for how the equal sign is presented to students. Two reasons for selecting second and sixth grades was first, relational symbols have been initially taught at the second grade across the proposed samples and examination of the second-grade textbook would indicate how the equal sign was presented and to facilitate comparisons to earlier findings (cf. Li, Ding, Capraro, & Capraro, 2008). Further, research indicated a U-shaped development of equivalence knowledge (e.g., 3rd graders do worse than 2nd graders) (McNeil, 2007). Given we have an indication that 2nd grades perform well we decided it would likely yield powerful insights. The second reason was several studies have examined aspects of the equal sign at the second- and sixth-grade levels, and it was these studies that provided the framework for the instrument and the study design. The items presented in previously published studies formed the basis and theoretical framework for designing this study‟s instrument. More specifically, the instrument contains items that examine the two broad categories of how the equal sign is traditionally presented in the U.S. (standard) and non-standard presentations (hypothesized by some to yield a relational understanding of the equal sign). Using second- and sixth-grade samples allowed us to build on the work of others and for our findings to build the next iteration of the theoretical framework. This study was informed by regional performance within a country based on its national performance on international comparisons. Because international comparisons use complex sampling techniques to ensure adequate and proportional sampling (PISA, 2002, 2006) across each nation, we expect our sampling within specific regions to result in rankings from this study to mirror those from international studies but we did not attempt to induct effects from this study to major international comparisons nor vice-
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versa. It was important to understand how textbooks might have influenced the development of students‟ understanding of the equal sign concept; therefore, an analysis of the textbooks used by the student participants was conducted. Equal Sign The concept of equality and the equal sign including students‟ misconceptions about the equal sign has been studied for over thirty years (Behr et al., 1980; National Council of Teachers of Mathematics [NCTM], 2000; Sáenz-Ludlow & Walgamuth, 1998; Thompson & Babcock, 1978). The 2006 Programme for International Student Assessment (PISA) (Baldi, Ying, Skemer, Green, Herget, & Xie, 2007) and 2007 TIMSS (Mullis et al., 2008) revealed that students from China, Hong Kong, Japan, and Singapore remained among the top achievers in algebra, which required understanding of the equal sign. Studies have shown that students have difficulty in understanding the meaning of the equal sign symbol (Bernstein, 1974; Ginsburg, 1977; Hiebert, 1984; Kieran, 1981; Li et al., 2008). These investigations were paramount because understanding of the equal sign has been linked to future algebraic success (Knuth, Stephens, McNeil, & Alibali, 2006) and continued success in higher mathematics (Usiskin, 1995). From earlier studies it has been articulated that some experiences convey to students that the equal sign functions as an operator. That is, a signal to do something, similar to how it works on a calculator. Considering the equal sign as an operator places it in the same class of symbols as the addition, subtraction, multiplication and division signs instead of with other relational symbols such as the greater than (>) and less than (<) signs. This operational interpretation has been considered responsible for functional misconceptions among them, the one we term “running equal sign” e.g., 2+3=5*2=10-2=8. While the actual answer may in fact be correct, the representational form is inaccurate and laden with incorrect statements. This phenomenon was also termed “equality strings” (Knuth et al., p. 310) and combined into a bigger categorization referred to as process difficulties where students did not consider opposite sides of the equal sign to represent the same quantity (Capraro, Ding, Matteson, Li, & Capraro, 2007). The running equal sign sets up a contrasting condition (2+3=5+2=7) to the relational purpose of the equal sign where its role is to establish a “relationship” between what is contained on either side. Weaver (1971, 1973) investigated several factors that influenced success on various open-ended number sentences and found that student response was lower for items in which the operation was on the right of the equal sign. Ginsburg (1977) noted, “Most often, sentences do ask children to perform a calculation; if so, why should they interpret them otherwise?” (p. 85). In a study by Rittle-Johnson and Alibali (1999) only 31% of fourth- and fifth-graders correctly solved problems such as 3 + 4 + 5 = 3 + ___. Additionally, other researchers have found low percentages of students who developed accurate understanding of the equal sign: 20% (Baroody & Ginsburg, 1983); less than 10% (Falkner, Levi, & Carpenter, 1999); 28% (Li et al., 2008); and 18% (McNeil,
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2005). Thus, students possessed a limited understanding of the equal sign as an operator, that is, a signal for “doing something” rather than developing a relational understanding. In a slightly different study, Knuth et al. (2006) found that 32% of sixth graders offered a correct definition of the equal sign. Current researchers (Li et al., 2008; McNeil, 2008; McNeil et al., 2006; Seo & Ginsburg, 2003) specifically pointed out students‟ understandings of the equal sign depended on instructional conditions which have come to be used regularly in describing how the equal sign is used during instruction with mathematical sentences. The phrase “standard context” is used to describe the most typical instructional condition in presenting the equal sign. A+B=C is the standard context where the operation is on the left side of the equal sign and the answer immediately follows the equal sign. This format was also referred to as “operation equals answer”. Another 13 form of standard context is the use of the equal bar, is in the same form of 4 operations equal answer, however, here the line underneath takes the place of the equal sign. The second phrase is “non-standard context” which places the equal sign in some other orientation than operation equals answer. The non-standard context encompasses a broad range of possibilities, however, historically it was only disaggregated into two sub-categories: operations on the right (opposite of the standard context) and operations on both sides. However, today researchers recognize many variants and have refined the categorization to accommodate many forms. Seo and Ginsburg examined the use of two U.S. textbooks in which second graders typically saw the equal sign and related students‟ understandings of the equal sign to how it was presented. McNeil et al. examined four U.S. middle school textbook series and suggested that students‟ interpretations of the equal sign were likely to be shaped by how it was presented. Ironically, even teacher preparation books in the U.S. lacked a clear description and strategies for preservice teachers to use in teaching students the relational meaning of equality (Li et al.). Textbooks “Many teachers rely heavily on the material contained in adopted textbooks and supporting instructional materials when determining content coverage in their classrooms” (Phillips, 2008, p. 3). In fact, in elementary school mathematics classrooms, 78% of the participating teachers had their students complete textbook/worksheet problems (Malzahn, 2002). On a broader scale, Tornroos (2005) used an item-based analysis of textbooks and found moderately high correlations between textbook presentations of the concepts and student performance at the item level on TIMSS implying that teachers taught the content of their textbooks. As Reys, Reys, and Chavez (2004) stated, “The choice of textbook often determines what teachers will teach, how they will teach it, and how their students will learn” (p. 61). Further, this implied that an analysis of textbooks could produce useful data when looking for explanations concerning student achievement in mathematics.
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Equal Sign Research on Students from China, S. Korea, Turkey, and the U.S. The literature from each country identified that conceptual issues related to students‟ learning about the equal sign generally related to failing to understand that one side of an equation related to the other side (Choi, 2001 [Korea]; Ersoy & Erbas, 2002 [Turkey]; Fu & Wang, 2004 [China]; Kieran, 1981 [U.S.]; Research and Development Group of Mathematics Curriculum Standards, 2007 [China]). In fact, in three of the countries Korea, Turkey, and the U.S., students held an operational view of the equal sign at the elementary (Choi, 2001; Choi & Pang, 2008; Ersoy & Erbas, 2002), middle, and high school levels (Capraro et al., 2007; Do & Choi, 2003; Lee & Kim, 2003). Kieran (1981) pointed out that U.S. students believed that the equal sign was a “do something” signal, while Falkner et al. (1999) showed that students solved 8 + 4 = __ + 5 with 12 or 17. In Turkey and the U.S., conceptual misunderstandings with equalities were found to be related to students‟ thinking of equalities in standard context (Ersoy & Erbas, 2002; Li et al., 2006; Yaman, Toluk, & Olkun, 2003). It was reported in Turkey and the U.S. that students who possessed an operational view of the equal sign also believed that when an equation was written with the operation on the right side that it was reversed or “written wrong” (Baroody & Ginsburg, 1983; Dede, Yalin, & Argun, 2002; Diyifanwen, 2008; Nanchang Education Bureau, 2005). It was also reported in these two countries that students had difficulty when presented with equations with operations on both sides. In Turkey, students with an operational view of the equal sign considered the other side of the equal sign to be zero when nothing was provided on that side (i.e., 2 + 5x = ?; x = -2/5) (Dede, Yalin, & Argun, 2002). Moreover, students might have been predisposed to think of equality as calculating answers, and this kind of misconception could persist until they received direct instruction (Baroody & Ginsburg; Capraro et al., 2007). Across all countries when students had an operational view of the equal sign, they possessed one or more of the following equal sign misconceptions: the answer immediately follows the equal sign, an operation cannot be on the right side of the equal sign, the expression on one side of the equal sign is not related to the expression on the other side, or patterning strategies always provide suitable solutions. A patterning strategy is one where students can have success by finding a pattern and simply replicating the pattern, for example, 4+5+6= 7+8. Research in each country showed that textbooks may account, at least in part, for the equal sign misconception. Textbooks have been reported as generally presenting problems that required an answer immediately following the equal sign, easily leading to over generalization (Choi, 2004; Ersoy & Erbas, 2002; Li et al., 2006; McNeil et al., 2006). Consequently, research in each country suggested various instructional strategies to address this curricular deficiency. The presentation of various problem types was suggested, for example, 25 = ___ + 8, 50 = ___ + ___, and 48 = 40 + 8 = 35 + 13 (Choi; Development Group of Mathematics Textbooks, 2001; Kim, 2002; Knuth et al., 2006; Mc Neil & Alibali, 2005; Xie, 2002). In Turkey and Korea research supported the use of manipulatives to build a more robust understanding of the equal sign. Students who used manipulatives to solve equations
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overcame their limited understanding of the equal sign as an operational symbol (Yaman, Toluk, & Olkun, 2003). In addition to the importance of symbolic representations of the equal sign, research in S. Korea showed that students tended to better understand the concept of the equal sign through concrete manipulative activities (Choi, 2004; Choi & Pang, 2008; Do & Choi, 2003). The synthesis of the independent literature from each country showed that students exhibited similar manifestations of the equal sign misconceptions in each country. While the literature from China lacked any direct connection to the misconception, the literature was specific about processes and procedures for ensuring students‟ accurate internal representations for the concept of equality. The common threads across countries were that students (a) viewed the equal sign as an operator, (b) interpreted equations with operations on the right side only as being structurally incorrect, (c) performed the operation on the left side and inserted the answer into the blank on the right, and (d) believed the equal sign did not relate one side to the other. Thus we identified differences in textbook treatments of the equal sign in these regions that would allow us to comment on pressing conceptual issues that have emerged from prior research and form the basis for the following questions: (a) How do central Texas second- and sixth-grade students interpret the equal sign as compared to their peers in Beijing, Jeju, and Istanbul? and (b) How do country specific textbooks treatments potentially influence student understanding of the equal sign? METHODOLOGY Instrumentation Two different equal sign tests (ESTs) were developed to measure second and sixth graders‟ understanding of the equal sign respectively. Both tests included two classification matching items with no equal sign intended to evaluate students‟ basic arithmetic knowledge in order to eliminate students who did possess basic proficiency with arithmetic operations, thereby preventing the confounding effect of arithmetic deficiencies in the assessment of understanding of the equal sign. The classification items were not used in the analysis. To prevent redundancy, the test items and not the matching items were presented along with the results in Tables 3 and 4 for second and sixth graders respectively. For analytic purposes item 8 on the second grade test and item 9 on the sixth grade were disaggregated into component parts, which yielded 11 and 13 items, respectively. It was important to remove language as a possible mediator of the obtained results; therefore, the tests were designed to be language independent, thus no words were used in any of the mathematical tasks or directions. For the two classification matching items, students were instructed to match the quantity on the left with the same quantity on the right. The spoken directions for the main portion of the test were to complete each item by putting the appropriate number or numbers in the blanks provided. The items represented presentations of the equal sign that have been used in previous
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research on the equal sign such as operations on both sides (i.e., 4 + 2 + 3 = 4 +__), operation on the left or right side only (i.e., 4 +__= 5 or 7 =__ + __), or reflexive, an equality without any operations (6 =__). These three forms are those that were used by other researchers who have examined student understanding of the equal sign and can readily be seen on standardized tests. The items on the test could have been solved either using quantitative computations (using the numbers and operation to obtain equal quantities on both sides of the equal sign; 4+___ = ___ - 8) or qualitatively through reasoning (i.e., without needing to perform an operation in order to solve the problem). So when students solve a problem qualitatively we cannot make assumptions about their ideas of quantitative sameness. An example of qualitative solutions in the reflexive case, students could obtain the correct answer of 6 without performing a quantitative process by simply putting a six on the other side of the equal sign. Similarly, problems such as 3+5=5+___ could be solved by making both sides look alike (also referred to as a carry strategy, cf. Garber, Alibali, & Goldin-Meadow, 1998) without any quantitative processes or understanding of the equal sign. Qualitative items were added to the test for comparative purposes to previously published tests. Cronbach‟s alpha internal consistency reliability estimates for the second- and sixth-grade students in this study were .87 and .90, respectively. Internal consistency reliability estimates are constrained by score variance, therefore, homogeneity whether scoring all correct or any other possibility markedly diminishes the obtained estimates (Capraro & Capraro, 2003; Capraro, Capraro, & Henson, 2001). Participants The ESTs were administered to N = 699 second-grade students (Beijing = 169, Jeju = 152, Istanbul = 270, and central Texas = 108) and N = 1124 sixth-grade students (Beijing = 200, Jeju = 193, Istanbul = 334, and central Texas = 397). This sampling was not intended to characterize the performance of the country as a whole but the provinces or regions within a country. Participants were purposefully chosen to represent different SES groups, ethnicities, and genders, and to represent the composition of students at the second and sixth grades within the regions of interest (Gay, Mills, & Airasian, 2009). Therefore, the sampling technique was designed to allow generalization to students within a specific region of each country. Major demographics for each region were identified, and the samples were matched to the larger population. In central Texas, ethnicity, SES, and gender were critical. In Beijing and Jeju, SES was often controlled by urbanicity and there was only one ethnicity, therefore, only gender was important across these two countries. In Istanbul, there was one major ethnicity so SES and gender matched students to the population. Students were sampled from schools within the region; however, some schools within regions chose not to participate. Choosing not to participate did not seem related to any factor important to this study. Because it was important to explicitly compare the sample to the population under consideration (American Educational Research Association, 2006; National Research Council, 2005a; Zientek, Capraro, & Capraro,
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2008), we provided the comparable information by region composite. Sample and population demographical information for each region and ethnicity for the central Texas sample is displayed in Table 1. Participants took the EST without prompting or assistance, and due to the algorithmic nature of the test there were no translation concerns for answering the items. All participants in the current study used the Arabic numeral system. Students were given unlimited time to complete the tasks. Students (n = 33; Beijing = 7, Jeju = 2, Istanbul = 12, central Texas = 12) were excluded based on their low scores on the two classification matching items. Textbook Analysis Mathematics textbooks, issued to students, from each country in both second and sixth grades were coded page by page to determine the different ways for how equal sign tasks were presented. The textbooks coded for this study were those adopted for use by the student participants. In Jeju and Istanbul, the public school textbooks selected for coding were the only ones approved by the Ministries of Education aligned with each national curriculum and used by the students in the sample. The S. Korean textbooks coded were Mathematics 2-ga (Ministry of Education and Human Resources Development [MOE&HRD] 2000a), Mathematics 2-na (MOE&HRD, 2000b), Mathematics 6-ga (MOE&HRD, 2002a), and Mathematics 6-na (MOE&HRD, 2002b). The Turkish textbooks were Ilkogretim Matematik Ders Kitabi 6 [primary mathematics textbook 6] (Eden, 2006) and Ilkogretim Matematik Ders Kitabi 2 [primary mathematics textbook 2] (Ozgun, Pektas, & Serficeli, 2007). The textbooks that were used by the second- and sixth-grade Beijing students and analyzed for this study were Peoples Education Press (PEP) Elementary Mathematics Book (Lu & Wang, 2005) and Jiang Su Education Press (JSEP) Elementary Mathematics Book (Su & Wang, 2005). All of the central Texas second-grade students in this study used the Scott ForesmanAddison Wesley (Charles, Crown, & Fennell, 2007) text reported as being the most popular second-grade textbook (Malzahn, 2002). All of the central Texas sixth-grade students in this study used the Holt, Rinehart and Winston (Bennet et al., 2007) textbook and was reported as being the most widely adopted textbook in major markets. The coding was divided into two main categories – standard and non-standard contexts. These categories were explicated in previous research by Knuth et al., 2006 where problems in standard contexts led students to view the equal sign as an operator (place the answer in the blank or box) in contrast to those in non-standard context which conveyed a relational meaning of the equal sign that encouraged students to try and balance both sides of the equal sign. Traditional problems, operation on left side and the 11 3 missing number on the right of the equal sign only (2 + 3 = ___) and equal bar, i.e., , 14 were considered as standard contexts for the equal sign. All other presentations of
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International Perspective Table 1 Percentages for Samples by SES, Gender and Ethnicity and Comparison to the Population Females Sample Beijing Grade 2 Jeju Istanbul central Texas Beijing Grade 6 Jeju Istanbul 48 50 47 53 47 45 51 Pop 47 51 50 51 46 48 51 Males Sample 52 50 53 47 53 55 49 Pop 53 49 50 49 54 52 49 SES Sample Low 42 27 68 56 52 23 56 Pop Low 40 29 66 71 45 26 58 Asian Sample * * * 7 * * * Pop * * * 6 * * * Black Sample * * * 32 * * * Pop * * * 34 * * * Hispanic Sample * * * 38 * * * Pop * * * 37 * * * White Sample * * * 23 * * * Pop * * * 24 * * * 22
central 46 50 55 50 70 68 2 2 37 38 40 39 22 Texas Note. A = Asian, B = Black, H = Hispanic, and W = White. The population represents 2nd and 6th grade students who attended public schools (2007-2008) in the same regions as the sample. * not represented in the sample or the population
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the equal sign were considered non-standard. Those nine non-standard contexts included: name part of the operation (e.g. 4__4 = 8; place a + sign on the line), fill in missing numbers on the left-hand side operation (e.g. 5 + ____ = 9), no explicit operation on either side (1 foot = 12 inches; 8 = __ ), operation on the right side only (__ = 7 + 9), operations on both sides (6 + __ = 7 + __), use/insert relational symbols (< [is less than], > [is greater than], = [equals], [is not equal to], i.e., 6_<_ 9, and verbal (with words, i.e. are equal to, is the same as). Three categories emerged from the pageby-page textbook coding, without an equal sign (e.g. 3 + 2; match to an equivalent quantity or statement), using an arrow to connect two quantities (e.g. 7 -- 3 + 4), and the equals bar. Without an equal sign instances were presented in one of two ways, either matching with a group of statements on one side and some comparable grouping on the other side, or using arrows to connect two equal sets. Previous research only considered uses of the equal sign and how they led to student misconceptions. The understanding of the equal sign as a balance, the same as, or as a relational symbol can be shown in ways other than through the explicit use of an equal sign (Li et al., 2006). Therefore, as other researchers have identified, the implicit equal sign may also tacitly convey understanding of the equal sign. The non-traditional presentation was more likely to help students build a relational understanding of the equal sign (Baroody & Ginsburg, 1983; McNeil et al., 2006; Seo & Ginsburg, 2003). However, in the contrary case, the equals bar was an example of students performing an operation and then using
the answer to perform another operation. For example, in long division
students
perform a subtraction and then perform another subtraction – this is similar to presenting a running equal sign or operational interpretation of the equal sign. This is expanded in some textbooks as 12-7 = 5; then bring down the 3, = 53, then 53-49=4; so the remainder is 4. It is possible that this presentation may actually confound the equal sign misconception. The textbooks presented the equal sign in many more ways than we evaluated on the EST, however, our objective was not to measure presentation of the equal sign but rather to determine whether the various presentations of the equal sign in textbooks influenced students‟ understanding of equality and how the textbook presentations might enhance or expand the current working theoretical model. It is also important to note that mobility within and across regions in China, Korea, and Turkey was not problematic for the textbook comparison because each of these countries has a national curriculum and prescribed textbooks. In the U.S., curriculum and textbooks can change from school to school within a given district. However, in this study the participating schools were using the same textbook and scope and sequence for the last five years. This minimized intra-district mobility on the textbook analysis and subsequent connections to the equal sign items.
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Student Data Analysis Percentage scores were used to compare second- and sixth-grade tests because they were comprised of different numbers of items. To provide more detailed information about achievement across regions, the 95% confidence intervals (CIs) for the percentage scores were computed and graphed. Effect size estimates were provided along with requisite 95% CIs in order to contextualize the magnitude of differences (cf. Capraro & Capraro, 2003). Istanbul was used as the baseline for computing the Cohen‟s d effect sizes because international test results indicated that Turkey should be the lowest performer. RESULTS Textbook Analysis Table 2 presents the percentages of how the equal sign was presented in textbooks as compared to the total number of equal sign instances. The use of standard context in S. Korean second- and sixth-grade textbooks (18% and 16%) was substantially lower than the use of the same context in Chinese (40.55% and 28.20%), Turkish (40% and 27%), and central Texas textbooks (54% and 18%), respectively. Reported examinations of textbooks with regards to equal-sign presentation suggested that textbooks in the U.S. differed broadly in their content coverage of the equal sign. McNeil et al. (2006) found the occurrences of the standard context were 70% in Saxon, 65% in Mathematics in Context, 49% in Prentice Hall, and 24% in the Connected Mathematics Program. It is important to note that there is a difference between the percentages of standard context reported in this study (54% and 18% by grade respectively, in the U.S.) as compared to McNeil et al., 2006. These differences could be attributed to the year of adoption or the sampling technique in that study. The Holt textbook (Bennet et al., 2007) used in the current study was adopted in 2007, while most of the texts in the previous study were adopted in 2004. Another factor accounting for differences from McNeil et al. could have been some textbooks in their study were National Science Foundation funded (cf. Tarr et al., 2008). Additionally, every page was examined in the current study as compared to a 50% random sample of pages in the previous study, possibly accounting for increased variance in the results. Without equal sign was one of the least commonly used forms in the Turkish and central Texas second-grade textbooks. On the other hand, without equal sign (or matching), by itself, comprised 65% of the S. Korean second-grade textbook. Although the presentation without equal sign was still one of the most commonly used in the sixth-grade S. Korean textbook, the percentage dropped to 16% resulting in a more even distribution of presentations. However, one of the most striking results was the high percentage of filling in missing numbers in the S. Korean sixth-grade textbook (19%), whereas, it was as low as 2%, 0.73% and 0.96% in Beijing, Turkish, and central Texas sixth-grade textbooks, respectively. The comparison of both grade levels showed that
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there was a consistent decrease in the percentage of use of the equals bar in S. Korean, Turkish, and central Texas textbooks and a noteworthy increase in no explicit operations on either side, operations on the right side only, and operations on both sides presentations. We also observed that the S. Korean textbooks did not use the name part of the operation context at either grade level. Filling in missing numbers was the only context that had a recognizable increase in S. Korean textbooks from grade 2 to grade 6, whereas, there was a substantive decrease in the frequencies of the same context from grade 2 to grade 6 in Beijing, Turkish, and central Texas textbooks. It is important to note that S. Korean textbooks had the largest increase in the operations on both sides from second to sixth grade (17 percentage point increase) while the central Texas textbook had a .19 percentage point increase. The instances of operations on both sides in Turkish textbooks increased 11 percentage points. Equal Sign Test Analysis Each task on the language independent EST was scored dichotomously, either correct or incorrect. One item (__+3 = 5 + 7 = __) on both the second- and sixth-grade tests, numbers 8 and 9, respectively, had two parts because each blank required a unique answer. Thus in total there were 11 possible correct answers on the second-grade test and 13 on the sixth-grade test. The first two items on each test were not part of the analysis. Because on international comparisons the U.S. outperformed Turkey, one might expect central Texas to outperform Istanbul on the EST, but the opposite was true. The means for the second (n = 270) and sixth (n = 334) grade Istanbul samples were 6.22 (SD = 2.88) and 9.36 (SD = 3.75), respectively. The means for central Texas second (n = 108) and sixth (n = 397) graders were 4.94 (SD = 2.90) and 8.31 (SD = 4.05), respectively. The results were more interpretable in pictorial format and easier for readers so graphic displays of confidence intervals (CIs) were used (e.g., Capraro, 2005; Sullivan, 2001). Figure 1 shows there were statistically significant differences between samples for the second-grade EST scores for Beijing and all others, Jeju and both Istanbul and central Texas, and between Istanbul and central Texas (see Cumming & Finch, 2001 for interpretation of CIs). The variance or measurement error for the Beijing sample was much smaller than for the other three regions. The variance was similar for both the central Texas and Jeju samples. The effect size estimates were calculated using Istanbul as the control group (baseline) because international tests consistently placed Turkey below the other three countries in this study. The Cohen‟s ds for the second-grade samples were 1.89 (Beijing), 0.27 (Jeju), and - 0.44 (central Texas). The 95% CIs for Cohen‟s ds for the second-grade samples were [1.60 – 2.18], [-0.03 – 0.57], and [-0.78 – -0.11] for Beijing, Jeju, and central Texas, respectively.
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International Perspective Table 2 Percentage of Equal Sign by Presentation in Beijing, S. Korean, Turkish and Central Texas Textbooks: Grades 2 & 6
Beijing 40.55 40.55 NAa 59.45 31.85 5.20 7.50 5.90 0.70 0.15 1.80 3.50 S. Korea 18.29 5.19 13.10 81.71 65.08 0.00 7.92 3.69 0.88 0.00 1.06 0.88 Grade 2 Turkey 39.60 29.37 10.23 60.40 2.20 0.88 15.61 24.60 8.20 0.18 0.97 2.20 central Texas 54.29 27.60 26.69 45.71 2.93 1.25 2.00 17.91 2.64 2.89 1.86 7.63 Beijing 28.20 28.20 NAa 71.80 15.65 0.00 1.65 7.25 12.45 3.90 22.75 8.15 NAa S. Korea 16.43 7.96 8.47 83.56 16.22 0.00 6.73 18.77 16.94 4.49 18.37 1.33 0.71 Grade 6 Turkey 26.86 21.84 5.02 73.14 14.72 0.40 2.75 0.73 27.59 9.14 12.14 4.21 1.46 central Texas 17.84 14.92 2.92 82.17 34.80 0.00 4.84 0.96 22.09 7.63 2.05 4.70 5.10
Standard Contextual Presentation Operation on Left Side Only (e.g. 3 + 15 = ___ )
11 Equivalency Bar (e. g. 3 ) 14
Nonstandard Contextual Presentation Without Equal Sign (e.g. 7 + 3; match to an equivalent quantity) Name Part of the Operation (e.g. 4__4 = 8; place a + sign on the line) Using Arrow to Connect (e.g. 7 -- 3 + 4) Filling in Missing Numbers (e.g. 5 + ____ = 9) No Explicit Operations on Either Side (Reflexive) (e.g. 12 inches = 1 foot; 150=__) Operation on Right Side Only (e.g. ___ = 7 + 9) Operations on Both Sides (e.g. 6 + __ = 7 + __) Use/Insert Relational Symbols (e.g. 6 ___9; insert <, >, or = )
Verbal Representation (e.g. 7 + 3 is the same NAa 2.20 5.56 6.60 as___; 2 x 5 –possible solution) Note. Chinese book coding results are from Li et al. (2008), a These categories were not represented in Li et al.
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Figure 1. 95% CI for second- and sixth-grade students‟ percentage scores on EST by region. Figure 1 also shows there were statistically significant differences between the sixthgrade samples from Beijing and both Istanbul and central Texas, and Jeju and both Istanbul and central Texas, but not between Beijing and Jeju on the EST. There was a statistically significant difference between the Istanbul and central Texas samples. The variances were similar for the samples from Beijing, Istanbul, and central Texas. For sixth grade, the Cohen‟s ds were 0.89 (Beijing), 0.85 (Jeju), and -0.27 (central Texas). The 95% CIs for Cohen‟s ds for the sixth-grade samples were [1.10 – 0.57], [1.12 – 0.56], and [-0.05 – -0.49] for Beijing, Jeju, and central Texas, respectively. Item-based results for the EST are reported below in the following categories: standard, operations only on the right, operations on both sides, and carry/pattern strategy. The results of grade 6 and grade 2 are summarized in Tables 3 and 4 for all presentations of equality. Second grade, item 6 (4+ __ = 5) had the highest success rate across all samples (see Table 3). Item 9 on the sixth-grade test and item 8 on the second-grade test (__+3 = 5+7 = __) consisted of two equal signs connecting two equations, containing two presentations in one question: (a) operations on both sides and (b) standard context. Only 26% of the second-grade central Texas sample answered the item correctly, with 17% of those who answered it incorrectly correctly putting a 9 in the first blank, but 70% of those who got it wrong correctly placed a 12 in the second blank. When considering only the standard context in item 9 on the sixth-grade test, it had high
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correct response rates from every sample (see 9b in Table 4), which was aligned with results from our textbook analysis. The correct response rate for the standard context part of item 8 on the second-grade test was also high across all countries. Similarly, the majority of sixth-grade students in this sample across the four countries answered the question in standard context (i.e., 47 + __= 63) correctly (i.e., the percentage of correct answers was above 89% in each country) as displayed in Table 4. Table 3 Grade 2 Percent Correct Item Analyses by Region Context Items Beijing 1) 3+5 = 5+ __ 2) 4+2+3 = 4+ __ 3) 3+ ___ = ___ +1 4) 6 = __ 5) 8+3 = __ + 5 6) 4+ __ = 5 7) 8+3 = 5 +__ 8) __+3 = 5+7 = __
a
Sample Jeju 75 61 56 67 49 93 54 41 43 78 55 62 22% Istanbul 78 40 56 77 36 80 49 17 30 70 49 49 8% central Texas 66 27 36 65 18 86 26 12 17 70 43 33 8%
NS NS NS NS NS NS NS NS NS S NS NS
97 95 97 95 92 99 95 88 92 94 97 97 74%
_*_ +3 = 5 + 7 =__ _ +3 = 5 + 7 =_*_
b
9) 7 = ___ + ___ 10) 3+5 = 4+ __ Overall Total Correct
Note. a considers only the first portion of item 8, b considers only the second portion of item 8 NS = nonstandard presentation; S= standard presentation.
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Table 4 Grade 6 Percent Correct Item Analyses by Region Context Items Beijing 1) 13 + 51 = 51 + __ 2) 6 + 3 + 7 = 5 + __ 3) 8 + ___ = ___ + 7 4) 160 = ___ 5) 15 - 7 = __ + 5 6) 6 x __ = 40 - __ 7) 47 + __= 63 8) 15 - 7 = 5 + ___ 9) __ + 3 = 5 + 7 = __
a
Sample Jeju 93 89 97 100 92 92 96 91 86 87 93 95 94 86 60% Istanbul 75 67 77 82 61 59 91 69 55 60 86 79 71 60 28% central Texas 69 63 70 70 49 51 89 60 43 48 83 70 58 52 16%
NS NS NS NS NS NS NS NS NS NS S NS NS NS
95 93 96 93 96 88 95 95 84 89 91 93 93 94 49%
__ + 3 = 5 + 7 = __
__ + 3 = 5 + 7 = b __ 10) __ + 5 = 2 x 8 11) 13 + 51 = 24 + __ 12) 8 = __ - 8
Overall Total Correct
Note. a considers only the first portion of item 9, b considers only the second portion of item 9 NS = nonstandard presentation; S= standard presentation. Operation on the right side only. Students from Beijing did not have apparent difficulties with this presentation at the second or sixth grade level. Istanbul, and central Texas students‟ performances were low on this item at both grade levels. Although the
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percentage improved from second to sixth grade, this item remained as one of the most difficult at the sixth-grade level for these two countries. Operations on both sides. Operations on both sides were found to be an effective presentation for promoting relational understanding of the equal sign (Carpenter, Franke, & Levi, 2003; Kieran, 1981). At the second-grade level, the correct response rate for the operations on both sides was low within each country respective to other item types for that country. Although the second-grade Beijing sample achieved better than the other samples, their achievement for operations on both sides when not including carry pattern items was lower than that for other problem types. The Istanbul and central Texas samples had apparent difficulties with the operations on both sides presentation. At the sixth-grade level, the Beijing and Jeju samples had the highest correct answer rates. The average performance for the Istanbul and central Texas samples on operations on both sides was low although better than their average performance on the operation on the right side only. The difference between items 5 (8+3 = __ + 5) and 7 (8+3 = 5 +__ ) for second grade was the blank was moved from immediately following the equal sign to following the operator. Performance on item 7 was better than on item 5, either because of positioning of the equal sign or because of item ordering providing an important experience. While doing a more fine-grained analysis of specific incorrect solutions within the secondgrade sample, it was discovered that (Beijing, 33%; Jeju, 44%; Istanbul, 52%; and central Texas, 47%) showed similar rates of the equal sign misconception by placing an 11 in the blank for item 5. Carry/pattern presentation. To determine if students were more likely to answer items correctly that did not require them to compute quantitative sameness, the items that comprised this group were analyzed again in addition to their respective group analysis. The second-grade test items 1, 2, 3, and 4 and items 1, 3, and 4 on the sixth-grade level test could be answered correctly using a carry/pattern strategy. The carry/pattern items 1, 3, and 4 at both grades had some of the highest correct answer rates within each country. On item 2 of the second-grade test (4+2+3 = 4+ __), none of the Beijing sample, 12% of Jeju, 22% of Istanbul, and 36% of the central Texas samples placed either 2 or 3 in the blank possibly applying a carry/pattern strategy simply placing a 2 or 3 in the blank because there was one on the other side. However, across all samples, only one student answered with 2+3 in the blank (carry). On the third item (3+ ___ = ___ +1), students could have simply used a carry/pattern strategy simply placing a 1 on the left and 3 on the right because of the pattern or because there was one on the other side. Because no students chose to solve the problem using some other solution strategy, it is not clear if a carry/pattern was being used or if students would have been able to use some other numbers to make the sentence true such as, 2 and 4. The fourth item on both the second- and sixth-grade tests could have been solved using either a qualitative or quantitative approach. At the sixth-grade level, the Beijing sample (26%) had the highest percentage of quantitative answers for 160 = __ (e.g., 80x2, 40x4, 160/1), followed by Jeju (18%), central Texas (14%), and Istanbul (10%). Of all the
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second-grade participants only one student from Beijing provided a quantitative answer for 6 = __ (i.e., 7-1). To determine if there was a difference in the items hypothesized to be able to be answered qualitatively - that is without the need for computation, and quantitative items - ones requiring computation, the correct response rates were compared. Only 10% of sixth-graders in the Istanbul sample and 14% in the central Texas sample who answered all qualitative items correctly scored at or above the 25th percentile for the quantitative items. However, all of the Beijing and all but two of the Jeju students who answered all the qualitative items correctly were at or above the 25th percentile on the quantitative items. This indicates that student performance was different when students could use some form of patterning to answer the item (qualitatively) as compared to when they had to use some form of computation to solve the item. The second-grade results in Beijing and Jeju were similar to their sixth-grade results. In comparison the central Texas percentage dropped to 5%, whereas, in Istanbul it increased to 12%. CONCLUSIONS AND DISCUSSION In thinking meta-analytically about how the results from this study fit with those of other studies, one must consider the outcome effect (NRC, 2005b). Similarly, when researchers examined studies about the effect of cigarette smoking and longivity the studies differed in quantity, duration, and mediating and moderating variables – however, the point was clear a 2% effect across studies. So it is essential to examine the effects of studies purporting to report the effects related to the study at hand. The Texas sample did not perform as well as the international comparison groups, it is also evident that the magnitude of understanding the equal sign is not as high (second grade: 8%, sixth grade: 16% answering all items correctly) as other reported estimates (cf. Baroody & Ginsburg, 1983 [20%]; Falkner et al., 1999 [<10%]; McNeil, 2005 [18%]; RittleJohnson & Alibali, 1999 [31%]). These results are more modest than contemporary research studies showing improved understanding (cf. Knuth et al., 2006 [32%]; Li et al., 2008 [28%]). Important differences emerge related to what and how textbooks might present the equal sign. The Beijing textbook uses operations on both sides nearly twice a much at six grade than the Istanbul or central Texas textbooks. Commensurately, in the Beijing text both without an equal sign and insert a relational symbol are decreased by six grade, both of which have been linked to equal sign misconceptions. Perhaps the relationship between decreasing the use of implied operations or implied equivalence and bolstering the use of operations on both sides of the equal sign have bolstered performance in both of the two higher achieving samples. Then it would be wise for textbooks publishers in the U.S. to decrease instances where students are expected to compute without the use of the equal sign, and insert the correct relational symbol (implied operations) in favor of operations on both sides of the equal sign. Regardless of the potential outcome for dealing with the equal sign misconception, removing most if not all of the instances of asking students to compute and answer without the use of an equal sign would be a
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benefit to students when they transition to formal algebra where they learn that equations and expressions are two very different things and removing expressions from both the second and sixth grades where they learn to compute from them might lead to unexpected benefits later. Equal Sign Misconception Among various equal sign presentations, operations on both sides was the most effective for understanding equality (Carpenter et al., 2003; Kieran, 1981). This presentation transmits the relational meaning of the equal sign more effectively than the standard or non-standard presentations and allows students to equate two arithmetic identities (e.g., 6 + 3 + 5 = 7 + __). The equalities that have an operation on one side and a single number on the other may emphasize operational understanding of the equal sign (i.e., do the operation). Even though the operation on both sides presentation supports the relational understanding of the equal sign, the absence of emphasis on this presentation in textbooks may result in students' failing to develop the appropriate understandings (McNeil & Alibali, 2005). Recently, McNeil (2008) found that the arithmetic presentation of the equal sign hindered student performance; “Given the same verbal lessons, children solved fewer math equivalence problems correctly on average after receiving lessons in the arithmetic condition than after receiving lessons in the [non-arithmetic] condition” (p. 1530). This terminology refers to the ways students balance both sides of the equal sign without having to perform arithmetic computations to arrive at admissible solutions. When considering our international comparisons and the textbook analyses, the samples from Istanbul and central Texas exhibit the lowest performance on the EST even though their textbooks present this reflexive (non-arithmetic) use of equality more frequently than textbooks used by the two samples. It is also important to note that the Beijing sample has the highest performance on the EST, but the textbook analysis shows that in second grade there is a greater percentage of standard arithmetic problem types as compared to the other countries. It is possible that a greater frequency of textbook problems with operations on both sides of the equal sign may account for the stark differences between samples. Operations on both sides is one of the least frequent presentations in second grade textbooks in all participating countries. However, in Beijing, S. Korea, and Turkey, operations on both sides increases at the sixth grade level. Only in central Texas textbooks does this frequency remain consistent and lower than comparison textbooks. The misconception that the answer follows the equal sign is evident in all second-grade samples although it is limited in the samples from Beijing and Jeju. Among the students who missed the item 8+3 = __ + 5, 33% in Beijing, 44% in Jeju, 52% in Istanbul, and 47% in central Texas placed 11 in the blank, displaying an answer follows the equal sign misconception. However, in Beijing only 8% of the students answered this item incorrectly and 33% of that group placed an 11 in the blank, a marker for the
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misconception. This same misconception is evidenced in the Istanbul and central Texas samples to a greater degree and with a larger effect. Additionally, the coding scheme suggested from the extant literature and items on previous tests proved insufficient for a complete and satisfactory coding of the current textbooks. There are several reasons for why this might be. First, it is possible that previous items were focused narrowly on the position of the equal sign and not on the general idea of equality. Therefore, the need to consider items where students are expected to match equivalent representations without an equal sign were not considered. Secondly, most previous textbook analyses used a random selection of pages in textbooks for coding. While a random sampling remains a prudent way to examine large volumes, it is possible that items are missed where the idea of equality is being conveyed in the material. Finally, we believe that a major contribution to the literature is the expansion of the categories for coding and future instruments designed to examine the equal sign should include items representing each of the nine categories. Emergent Factors for Future Research Consideration There are three factors that arise as part of this study that we present here for thought in formulating future studies. The first factor, seemingly one of syntax, is the use of the blank. The blank is expected to hold the place for a single number and not to be a placeholder for an expression. However, one second grade student when answering the non-arithmetic reflexive problem, 6 = __, wrote 6 = 7 – 1, while a larger percentage of sixth grade students use an expanded notion syntax (160 = 100+60) in answering 160 =__. These relationships were scored correct as long as the equality was true. It was more likely for students to have answered most of the items correctly (88%) than not (11%) when they inserted an expression in the blank. For the second factor, the non-arithmetic items yield limited insights about students‟ solution patterns. One would expect students who incorrectly answer most arithmetic items with operations on both sides to still be able to answer all the non-arithmetic items. However, the non-arithmetic items are not consistently the easiest items. In central Texas and Istanbul item 10 (__ + 5 = 2 x 8), on the sixth-grade test, an arithmetic item, was answered correctly by a greater percentage of participants than the non-arithmetic items 1 (13 + 51 = 51+___) or 3 (8 + ___ = ___ + 7). Less than onequarter of these sixth graders who missed the non-arithmetic item 1 (13 + 51 = 51 + __), placed 64 in the blank. This suggests that the arithmetic solution was employed (i.e., 13+51 = 64; so 64 was placed in the blank) resulting in an incorrect answer of 64 as compared to the anticipated correct solution of 13 that would have been obtained through a non-arithmetic balancing solution. The over application of an arithmetic solution seems to be related to the equal sign misconception. Thirdly, the use of the equals bar in the textbook analysis appeared as a conundrum. In this study, we consider the equals bar as part of the standard context because its basic interpretation is the same as the horizontal presentation of problems. That is, the equals bar is read the same way as the operation equals answer. However, another presentation
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of the equals bar is revealed in long division when students encounter multiple equals
bars (e.g., ). In the long division process, the bar implies a running equal sign interpretation. However, long division is introduced after conceptualization of the equal sign should take place. One general understanding of the equals bar is a “command to compute” and is not intended to convey equivalence. However, in early grades, students encounter the vertical presentations of basic addition and subtraction problems that are read in the same way as horizontal problems reading the command to compute “bar” as equals. Might this nexus in mathematical development between simple vertical addition or subtraction problems and the “command to compute” in long division fuel students‟ equal sign misconception? Regardless, the implied misconception by the use of the equals bar in long division may be important and require further study to understand how students interpret the equals bar. This may explicitly link the equals bar to the emerging theoretical model. LIMITATIONS Our sampling techniques as compared to that of international samples differ in important ways. The “National Project Managers for international studies must identify appropriate stratification variables to reduce sampling variance when appropriate” (PISA, 2006, p. 6) and the operations manual describe the sampling technique as a standard PISA two-stage stratified cluster design (PISA, 2002). Therefore, the nature of the international design would provide a reasonably well-drawn sample that would represent the country. Thus, well-drawn samples from smaller areas of the country (regions) should mirror the national performance at the ranking level of analysis. We are not saying the smaller well drawn samples can be used as proxy measures for national sampling, what we believe is that at the ranking level –well drawn smaller samples should adequately mirror the national trend. Therefore, we only compare at the ranking level between our study and national samples. We report our estimates of effect based on the regions within country. It is important to note that small poorly drawn samples can be highly idiosyncratic and produce greatly fluctuating estimates of an effect. REFERENCES American Educational Research Association (2006). Standards for reporting on empirical social science research in AERA publications. Educational Researcher, 35(6), 33–40. Baldi, S., Ying, J., Skemer, M., Green, P. J., Herget, D., & Xie, H. (2007). Highlights from PISA 2006: Performance of U.S. 15-year-old students in science and
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