Analysis of proportional reasoning and misconceptions among students with mathematical learning disabilities

Journal of Mathematical Behavior 57 (2020) 100753

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Analysis of proportional reasoning and misconceptions among students with mathematical learning disabilities

T

S- h. Ima,*, Asha K. Jitendrab a

Teachers College, Columbia University, Department of Human Development, 525W 120TH ST, 465 Grace Dodge Hall, New York, NY 10027, United States b University of California-Riverside, Graduate School of Education, 900 University Avenue, 2127 Sproul Hall, Riverside, CA 92521, United States

ARTICLE INFO

ABSTRACT

Keywords: Proportional reasoning Students with mathematical learning disabilities Schema-based instruction Error analysis

We investigated not only the effects of schema-based instruction (SBI) on the mathematical outcomes of seventh-grade students with mathematical learning disabilities (MLD), but also extended prior work to analyze students’ written explanations on open-ended items involving ratio and proportion situations—ratio, proportion, and percent of change problems— to understand the ability to reason about proportions and identify misconceptions. The sample of 338 students with MLD [scored below the 25th percentile on a proportional problem solving (PPS) pretest] was taken from Jitendra, Harwell, Im, et al. (2019), which randomly assigned classrooms to either the SBI or control condition. Students with MLD in SBI classrooms outperformed their counterparts in control classrooms on proportional problem solving and general mathematics problem solving. Similar results, favoring the SBI condition, were found on the open-ended items; however, overall mean scores across pretest, posttest, and delayed posttest were low. Findings provide evidence for the limited understanding of fractional representations of ratios and highlight students’ persistent use of numerical and additive reasoning in explaining their low performance on the open-ended items.

1. Introduction Proportional reasoning, which is a critical bridge between the numerical, concrete mathematics of arithmetic and the abstraction that follows in many higher-level mathematic topics, develops over time through reasoning (e.g., Beckmann & Izsák, 2015; Fuson & Abrahamson, 2005; Lamon, 2007; Post, Behr, & Lesh, 1988). The development of proportional reasoning is crucial for understanding many mathematical ideas (e.g., fractions, functions, probability, statistics, algebra) needed to solve problems in mathematics, science, economics, and geography (Akatugba & Wallace, 2009; Boyer & Levine, 2015; Howe, Nunes, & Bryant, 2011; National Council of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 2000). Proportional reasoning is also essential to daily life situations such as grocery shopping by calculating best buys based on the unit price of a food item or performing measurement or currency conversions such as converting inches to centimeters or £ to $. Proportional reasoning, a major goal of instruction in upper elementary and middle school mathematics education in the United States, refers to the ability to reason about quantities in relative terms. Understanding the covariance of quantities and invariance of ratios is essential for a critical understanding of proportional reasoning (Fielding-Wells, Dole, & Makar, 2014; Lamon, 2007). The ability to reason with proportions requires discerning a multiplicative relationship between two quantities (a ratio) and extending the

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Corresponding author. E-mail addresses: [email protected] (S.-h. Im), [email protected] (A.K. Jitendra).

https://doi.org/10.1016/j.jmathb.2019.100753 Received 23 March 2019; Received in revised form 9 August 2019; Accepted 26 November 2019 0732-3123/ © 2019 Elsevier Inc. All rights reserved.

Journal of Mathematical Behavior 57 (2020) 100753

S.-h. Im and A.K. Jitendra

multiplicative relationship to other pairs of quantities. Consider, for instance, the following proportional reasoning problem that involves proportional relations: Suppose Jay and Kelsey start at the same time and are driving at a constant speed. Jay drove 60 miles in his car, and Kelsey drove 100 miles in her car. (1) How many miles has Kelsey driven in her car if Jay drove 90 miles in his car? (2) How many miles has Kelsey driven in her car, if Jay drove 180 miles in his car? Although both situations require understanding the multiplicative nature of proportions or the composite nature of ratios, the first question could be challenging to students because the multiplicative relation is not easily distinguishable given a non-integer ratio, 1.5 (Jay: 60 × 1.5 = 90; so, Kelsey: 100 × 1.5 = 150). Instead, they may use the additive strategy of looking at ratios of differences between the same variables (Jay: 60 miles + 30 = 90 miles; Kelsey: 100 miles + 30 = 130 miles). On the other hand, the second question is relatively easier because students are more likely to apply proportional reasoning and answer correctly because the multiplicative relation is more discerning from an integer ratio, 3 (Jay: 60 × 3 = 180; so, Kelsey: 100 × 3 = 300). Research on proportional reasoning suggests that children struggle to transition from additive reasoning to multiplicative and proportional reasoning (e.g., Nunes & Bryant, 2010; Van Dooren, De Bock, Evers, & Verschaffel, 2009). The shift from additive reasoning—“based on quantities connected by part-whole relations” (Nunes, Dorneles, Lin, & Rathgeb-Schnierer, 2016, p. 18)—to multiplicative reasoning—“based on quantities connected by one-to-one correspondences or ratios” (Nunes et al., 2016, p. 18)—involves understanding the multiplicative conceptual field (MCF) and the connected web of multiplicative ideas. This entails numerous understandings, including a grasp of the fundamental structure of multiplication and division as it relates to whole number multiplication, understanding ratio as a multiplicative comparison and a composed unit, making connections among ratios and rates and fractions, and understanding the ideas in situations involving different levels of cognitive complexity. In short, proportional reasoning is a complex way of thinking and its development includes a web of interconnected ideas—multiplication and division with whole numbers, fractions, ratios and rates, measurement, and percent (Hino & Kato, 2019). Solving proportional problems can be even more challenging for students with mathematical learning disabilities (MLD), who evidence persistent impairments in processing numerical information and learning arithmetic facts (American Psychiatric Association, 2013). Students with MLD often lack basic arithmetic skills, including the ability to compare and represent numerical magnitude information (Butterworth, Varma, & Laurillard, 2011), as well as evidence difficulties in domain-general abilities such as working memory, language, and attentive behavior (Cirino, Tolar, Fuchs, & Huston-Warren, 2016; Fuchs et al., 2010; Geary, 2004; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Hansen et al., 2015; Jordan et al., 2013; Raghubar, Barnes, & Hecht, 2010). In this paper, we examined written explanations and types of errors students with MLD made in proportional word problems in order to gain insights about factors that influence the difficulty of proportional problems. 2. Literature review In the following, we examine literature on proportional reasoning to understand why it is difficult for students, especially students with MLD. In so doing, we examine students’ common misconceptions that pose obstacles to the development of proportional understanding. It is important to note that there is a paucity of research focused on understanding developments in proportional reasoning among students with MLD (e.g., Grobecker, 1999; Hunt, 2015). In such instances, our review of the literature on proportional reasoning is informed by studies of typically achieving students or students who struggle to learn mathematics but do not have disabilities. We also review the literature concerning instructional approaches for improving proportional problem solving performance of students with MLD. 2.1. Why is proportional reasoning difficult? According to the Common Core State Standards for mathematics, instructional time in Grades 6 and 7 focuses on “connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems” … “developing understanding of and applying proportional relationships” (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 39; p. 46). However, students' informal introduction to ratios and rates starts with multiplication and division of whole numbers in Grade 3, which provides the basis for developing an understanding of fractions, ratios and rates, and proportional reasoning. Reasoning with whole number multiplication and division is connected to understanding of ratios and rates, which involves understanding multiplicative relationships between two same units (ratios) and two different units (rates). This learning sequence hints at some foreseeable challenges for many students with MLD who have not mastered many of the number concepts and number relationships, which in middle grades also requires a significant change in the way number is conceived. Much of early instruction focuses on multiplication as simply repeated addition. The conceptual change from making additive comparisons to forming ratios between two quantities as well as making and using multiple comparisons poses difficulties. In particular, many students with MLD and students with mathematics difficulties (MD) struggle with problems involving multiplicative reasoning (Tzur, Xin, Si, Kenney, & Guebert, 2010), which is the foundation of proportional reasoning. The limited research available suggests that students with MLD are less likely to find the multiplicative structure between two relations (i.e., second-order structure) compared to their counterparts without disabilities (Grobecker, 1999; Hunt, 2015). Furthermore, these students’ difficulties to compute ratio and proportion may be due to difficulties with receptive and expressive language and abstract reasoning (Grobecker, 1999). In the following, we discuss possible factors that influence students’ ability to reason about proportions. 2.1.1. Fractions Developing ratio sense, which is critical to proportional reasoning, involves building on fractional representation as a way to 2

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S.-h. Im and A.K. Jitendra

order and judge equivalence of ratios or rates. Fractions and ratios are complex ideas that are interconnected (Lamon, 2005). Several mathematical connections linking ratios and fractions indicate that ratios are expressed in fraction notation even though they do not have the same meaning, ratios are used to express part-part comparisons unlike fractions, and ratios can be meaningfully reinterpreted as fractions or quotients (Lamon, 2005; Lobato, Ellis, Charles, & Zbiek, 2010). As such, understanding of fractions is a prerequisite for proportional reasoning (e.g. Behr, Lesh, Post, & Silver, 1983; Charalambous & Pitta-Pantazi, 2007). There is also evidence that successful proportional reasoning supports the development of fractions (Howe et al., 2011; Lamon, 2007). Many students with MLD and MD have limited understanding of fractions as quantities in that they have “difficulties conceptualizing fractions centered on coordinating parts with respect to a referent whole and using notions of multiplicative structures or numerical composites as templates for partitioning fractions” (Hunt, Welch-Ptak, & Silva, 2016, p. 224). Incomplete understanding of fractional quantities could also affect operational aspects of fractions that are critical in higher-level mathematical contexts (National Mathematics Advisory Panel, 2008). Results of studies of fraction learning indicate that a range of general (attentive behavior, working memory) and mathematics-specific competencies (e.g., whole number estimation, multiplication fluency, nonsymbolic proportional reasoning, long division) predicted fraction outcomes for students with MLD (e.g., Cirino et al., 2016; Hansen et al., 2015; Jordan et al., 2013; Ye et al., 2016). Fraction expertise is quite low in students with MLD (e.g., Hansen, Jordan, & Rodrigues, 2017; Hunt, Tzur, & Westenskow, 2016; Tian & Siegler, 2017). Evidence suggests that these students “were 2.5 times more likely to experience low growth in fraction concepts than their peers who were not receiving special education and 11.5 times more likely to experience low growth in fraction procedures” (Hansen et al., 2017, p. 55). With regards to generally low student achievement in fractions, three differences between whole numbers and fractions emerged from the literature as potential reasons for difficulties with fractions even after several years of instruction (see Fuchs et al., 2016; Jordan, Resnick, Rodrigues, Hansen, & Dyson, 2017). First, fractions have multiple meanings. One perspective is that fractions express a part-to-whole relationship. Other meanings include the division operation and its result, which is a quotient. The second difference between whole numbers and fractions is magnitude. Unlike whole numbers, fraction magnitudes are determined by two fraction components holistically rather than separately. It requires conceptual change to understand that two numbers can be used to represent a single numerical magnitude. Third, the same fraction can be expressed with infinite different fractions by scaling each numerator and denominator up or down. For instance, 1 = 2 = 3 = … Understanding that there are 6 9 3 infinitely many ways to express a fraction is important when comparing two fractions by making two equivalent fractions. In sum, smooth transition between whole numbers and fractions is critical to developing proportional reasoning ability. 2.1.2. Complexity of required mathematical knowledge, multiple steps, and diverse strategies The multifaceted nature of proportional reasoning makes it difficult for students to solve proportion problems. One, proportional reasoning requires an understanding of the MCF. Two, the difficulty of a problem is affected by the number of steps involved. Three, the structure of proportion problems is such that multiple strategies can be used to arrive at the same answer and influences whether students are able to apply their reasoning to select a strategy based on problem characteristics (kinds of numbers involved). Vergnaud (1988) coined the term MCF to capture the notion that multiple mathematical concepts like multiplication and division operations, fraction, ratio, proportion, and linear functions in algebra contribute to the development of proportional reasoning. This aspect of understanding proportionality as the core structure of interconnected topics is challenging for students with MLD who have deficits in several areas. Developing understanding of proportional relationships in students with MLD is hindered by their limited or lack of understanding of the fundamental structure of whole number multiplication and division (e.g., Tzur et al., 2010). Furthermore, there is some evidence that the multiplication skills of middle school students with MLD is similar to typically achieving thirdgraders (Mabbott & Bisanz, 2008). As described earlier, these students have difficulties understanding fractions. In addition, students with MLD do not have a grasp of the meaning of decimals and percent (i.e., a special kind of ratio) and their connections to fractions (Hunt, 2015; Jitendra & Star, 2012; Jitendra, Woodward, & Star, 2011; Lewis, Xin, & Tzur, 2016; Mazzocco & Devlin, 2008; Mazzocco, Myers, Lewis, Hanich, & Murphy, 2013) that together constitute essential understanding of ratios and proportions. The ability to reason about proportions often involves multiple steps. Proportion problems require first identifying the underlying part-to-part or part-to-whole relation. For example, the problem discussed in the introduction involves a part-to-part relation. Based on the ratio relation, the next step would involve constructing two ratios to write a proportional equation. Last, it is important to evaluate the equivalence of two ratios in the problem. Students with MLD may have difficulty reasoning with four quantities simultaneously and understanding the scalar and functional perspectives in proportional reasoning (Gurganus, 2017). Multistep adjustment proportion problems may pose additional challenges compared to single-step proportion problems. Research on arithmetic word problem solving provide indirect evidence for the difficulty of multistep problems; children with MLD have more difficulty solving arithmetic word problems with two or more steps than single-step problems (Cawley, Parmar, Foley, Salmon, & Roy, 2001; Fuchs & Fuchs, 2002; Jitendra, Hoff, & Beck, 1999). Procedural flexibility in selecting among different strategies is another important consideration when solving proportion problems. Three different strategies—within (unit-rate), between (equivalent fractions), and cross-multiplication—are frequently observed for solving proportion problems (Ben-Chaim, Fey, Fitzgerald, Benedetto, & Miller, 1998; Lamon, 1993; Vig, Star, Dupuis, Lein, & a c Jitendra, 2015). In a proportional equation, a:b = c:d or b = d , the within strategy focuses on ratios (i.e., a to b or c to d) within each side of the equation. It compares two within ratios (e.g., a ÷ b = c ÷ d) to determine the unit-rate. In contrast, the between strategy focuses on ratios (i.e., a to c and b to d) across each side of the equation. The between strategy involves finding equivalent fractions. For the cross multiplication strategy, students cross multiply denominators and numerators, ad = bc. When within or between two ratios are integer ratios, both between and within strategies are easily applicable to evaluate the equivalence of two ratios in 4 6 proportional problems (e.g., 4 = 8 ). However, the two strategies may not be easily applicable with non-integer ratios (e.g., 6 = 9 ). 12

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Journal of Mathematical Behavior 57 (2020) 100753

S.-h. Im and A.K. Jitendra

The cross multiplication strategy would be a more suitable alternative to evaluate non-integer ratios (4 × 9 = 6 × 6). Thus, to succeed in both non-integer and integer proportional problems, students need to know about these diverse strategies (e.g., BenChaim, Keret, & Ilany, 2012). The difficulty to reason about proportions among students with MLD may be due to the focus on procedural fluency in special education and teaching them algorithmic methods (cross multiplication) rather than developing reasoning abilities by having them use multiple methods to solve and then comparing to determine which method is more efficient (Jitendra & Star, 2012). At the same time, students with MLD may not have a repertoire of strategies because the premise is that teaching a variety of methods may be inappropriate given their limited working memory capacity (Geary, 2004). 2.2. Common misconceptions during proportional problem solving Mathematical misconceptions are inaccurate ideas based on faulty thinking and understanding about a mathematical idea or concept. Such misconceptions may occur when learners inappropriately apply or generalize their previous knowledge and reasoning to new contexts. For example, students’ faulty understanding and inaccurate application of whole number knowledge and additive reasoning when solving proportion problems induce misconceptions that interfere with the ability to understand ratios and proportional reasoning (Dougherty, Bryant, Bryant, & Shin, 2016). Prior studies have found the commonality of an additive approach in young children and adolescents’ responses on proportional word problem solving (e.g., Fernández, Llinares, Van Dooren, De Bock, & Verschaffel, 2012; Lo & Watanabe, 1997). This primitive idea of reasoning additively is exacerbated when the type of ratio is noninteger (Fernández et al., 2012; Riehl & Steinthorsdottir, 2017; Van Dooren et al., 2009). Misailidou and Williams (2003a, 2003b) provide robust evidence for common errors of proportional word problem solving using the results of item response analysis and interview data from a relatively large sample of late elementary and early middle school students. Results showed that the most prevalent erroneous strategy across 13 proportional problems was the additive strategy. Other observed erroneous strategies were incorrect build-up (i.e., mixing multiplicative and additive approaches for handling the remainder), doubling/halving, and constant sum strategies. Interestingly, the incorrect build-up strategy reflects the mixed use of additive and multiplicative reasoning to address the remainder in non-integer ratio problems. In line with the easiness of doubling and half factors, students also showed their overuse of doubling/halving explanation in proportional problem solving. Difficulties with reasoning involving ratios and proportions are typically reflected in student explanations and error patterns in their answers (see Jitendra, Woodward et al., 2011). Jitendra, Woodward et al. (2011) evaluated seventh-grade students’ proportional thinking during individual interviews to gain insight into their understanding of ratios and proportions using three tasks. The first task required identifying and explaining which of two plants grew more when shown two plants that were of different heights at the start and their new heights after two months of growth – both plants grew 3 inches. The second task was a picture matching task that required identifying cards that represented the same ratios. The third task required calculating how many cookies could be made with 1 egg and with 3 eggs when a recipe for making 40 cookies used 2 eggs. Results showed that students with mathematical difficulties (i.e., low average performance on a standardized math achievement test) found the percent change problem to be the most difficult. Even though many students recognized the plant growth problem, none reconsidered their “absolute” answers on this task (e.g., “they grew the same”) in spite of the interviewer probing by asking if there was another way to think about the plant growth. On the other two tasks, few students demonstrated multiplicative thinking. For example, students were unfamiliar with the picture matching task and tended to be distracted by the superficial features of the problem; none used formal ratio notation to represent and compare the different ratios. Although students were most familiar with the recipe task, they tended to rely on the “count by 20” strategy (i.e., “1 egg makes 20, so 2 eggs make 40, so 3 eggs make 60.”) to solve the problem. 2.3. Improving proportional problem solving performance of students with MLD Research on the development of proportional reasoning was prominent in the late 1980s and early 1990s (e.g., Behr, Wachsmuth, Post, & Lesh, 1984; Carpenter, Fennema, & Romberg, 1993; Harel, Behr, Post, & Lesh, 1992). Scholars in mathematics education have been interested in children’s learning to gain insights about strategies to inform teaching. Prior intervention studies have examined different approaches (e.g., inquiry methods that encourage students to construct knowledge of proportionality through collaborative problem solving activities, use of pictorial representations or manipulative models) to developing proportional reasoning (e.g., Adjiage & Pluvinage, 2007; Ben-Chaim et al., 1998; Fujimura, 2001; Miyakawa & Winslow, 2009; Özgün-Koca & Altay, 2009). Intervention research on ratios and proportional problem solving with students with MLD and MD are few and limited in several ways. First, most interventions did not focus on the broad domain of ratios and proportional relationships (e.g., Hunt, 2014; Jitendra, DiPipi, & Perron-Jones, 2002; Moore & Carnine, 1989; Xin, Jitendra, & Deatline-Buchman, 2005). Second, the majority of studies involved a limited number of students and used a teaching experiment or single-case research designs to evaluate the effectiveness of intervention approaches to the teaching and learning of ratios and proportions, limiting causal inferences or external generalization (Hunt, 2014, 2015; Hunt & Vasquez, 2014; Jitendra et al., 2002). Third, researchers rather than teachers delivered all instruction (e.g., Hunt, 2014, 2015; Hunt & Vasquez, 2014). The focus of this study is on schema-based instruction (SBI), a research-based instructional program designed to help students make sense of their reasoning about proportions in word problem contexts. Recent studies of SBI with middle school students have addressed the above limitations by using a randomized controlled design, comprised large samples of teachers and students, and the SBI program included a comprehensive coverage of state content standards related to ratios and proportional relationships (e.g., 4

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Jitendra et al., 2015; Jitendra, Harwell, Im, Karl, & Slater, 2019; Jitendra, Harwell, Dupuis, & Karl, 2017; Jitendra, Star, Dupuis, & Rodriguez, 2013; Jitendra, Star, Rodriguez, Lindell, & Someki, 2011). SBI’s theoretical framework, which is an elaboration of schema theory (see Carpenter & Moser, 1984; Marshall, 1990) is guided by research on expert problem solvers as well as cognitive models of mathematical problem solving (Mayer, 1999). The SBI program provides instruction not only on the concepts of ratios and proportional relationships, but also focuses on solving proportion word problems. Problem solving instruction in SBI emphasizes knowledge of procedures (e.g., problem representation, planning) for a given class of problems such as ratio, proportion, and percent of change (see Marshall, 1990; Mayer, 1999). Teachers use instructional practices (e.g., guided questions to engage students in conversations about their thinking and problem solving) to help students recognize common underlying problem structures, represent problems using visual-schematic diagrams, plan how to solve problems, and solve and check the reasonableness of answers. Furthermore, SBI emphasizes metacognition skills and encourages students to “think about what they are doing and why they are doing it, evaluate the steps they are taking to solve the problem, and connect new concepts to what they already know” (Woodward et al., 2012, p. 17). Results of several studies support the effectiveness of SBI in improving students’ proportional reasoning for a range of classroom/ teacher and student characteristics (Jitendra, Harwell, Karl, Im, & Slater, 2019). Furthermore, findings of prior SBI studies on teaching ratio and proportion problem solving indicate positive outcomes for students with MLD (e.g., Jitendra, Dupuis, Star, & Rodriguez, 2016; Jitendra, Harwell, Karl, Simonson, & Slater, 2017; Jitendra, Harwell, Im, Karl, & Slater, 2018; Jitendra, Harwell, Karl, et al., 2017). 3. Aims and research questions Given the complexity of proportional problem solving and associated misconceptions, we analyzed the reasoning skills (i.e., identifying the underlying ratios and evaluating the equivalence of ratios) of students with MLD and their erroneous explanations in solving proportional word problems rather than assessing their correct solutions only as in previous SBI studies. One means of capturing evidence of reasoning and associated errors is from students’ written work that can identify the presence of the concepts required for successful proportional problem solving. Although the primary purpose of this paper was to qualitatively describe the rubric scores of reasoning and error coding schemes for short-response tasks involving proportional reasoning for students with MLD, we also evaluated the effect of the SBI program by comparing treatment and control students’ proportional problem solving and overall mathematical problem solving performance to provide a context for characterizing students’ proportional reasoning ability and specific error types when interpreting and solving comparison proportion problems. In this research, we investigated the following questions: (1) (a) Is SBI effective in enhancing the proportional problem-solving skills of seventh graders’ with MLD (scored below the 25th percentile on a measure of proportional problem solving, PPS) compared to a business-as-usual instruction control group and whether students’ proportional problem solving skills would be maintained 9 weeks after the termination of the intervention? (b) Does SBI result in increased achievement compared to a control group on overall mathematical problem solving performance after a focused period of time spent on ratios and proportional relationships? (2) Does SBI enhance the proportional reasoning ability of students with MLD as evidenced by their written explanations on openended items compared to a business-as-usual instruction group? (3) (a) What are most common errors at pretreatment when interpreting and solving comparison proportion problems? (b) Do these errors reduce or persist after receiving the assigned intervention? (c) Do the pattern of errors differ between the two conditions? 4. Methods 4.1. Participants and setting The sample in the current study was taken from a larger study (Jitendra, Harwell, Im, et al., 2019), that included a heterogeneous pool of seventh-grade students. The target population was middle school students and teachers in the western and southeastern region of the U.S. Fifty-nine teachers from 36 middle schools in 5 public school districts participated in the Jitendra, Harwell, Im, et al. (2019),study. Once a teacher had been selected, one of their seventh-grade mathematics classes was selected at random to participate. Jitendra, Harwell, Im et al. (2019) used a randomized cluster design with longitudinal data (pretest, posttest, delayed posttest) in which 59 classrooms were initially assigned at random to SBI or control (“business-as-usual”) conditions. Teachers/ classrooms served as clusters. The current study used a subset of data from the Jitendra, Harwell, Im et al. (2019),study that consisted of students with MLD. We operationalized MLD as scores below the 25th percentile on a measure of proportional problem solving (PPS, see Section 4.3.1. Measure of proportional problem solving). Although there is considerable variability in defining and operationalizing a mathematical learning disability, the 25th percentile cutoff is commonly used to identify students as having mathematics disabilities in the literature (see Fuchs, Fuchs, & Prentice, 2004; Zheng, Flynn, & Swanson, 2012). Based on scores on the PPS measure administered at pretreatment, 338 students from among the 1492 seventh graders who participated in the larger study were identified as having MLD. As expected, the mean PPS pretest scores of students with MLD (M = 5.24, SD = 1.74) was much lower than scores of students without MLD (M = 12.68, SD = 4.01). A similar pattern was found on the measure of general problem solving (see Section 4.3.2. Measure of mathematical general problem solving), in that students with MLD scored lower (M = 9.24, SD = 3.49) than students without MLD on 5

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Table 1 Student and teacher demographic and background information. Treatment SBI

Student Information Age Sex Race

ELL SpEd Missing (age, sex, Race, ELL, & SpEd) FRL

Teacher Information Sex Location Math courses taken Education courses taken Years experience in math

Control

Total

n

%

n

%

n

%

M (SD) Female Male Asian Black Hispanic Multiracial White Yes No Yes No

12.65 95 95 3 33 61 4 89 32 158 39 151 3

0.5 49.2 49.2 1.6 17.1 31.6 2.1 46.1 16.6 81.9 20.2 78.2 1.6

12.56 69 72 8 13 63 4 53 25 116 15 126 4

0.5 47.6 49.7 5.5 9.0 43.4 2.8 36.6 17.2 80.0 10.3 86.9 2.8

12.61 164 167 11 46 124 8 142 57 274 54 277 7

0.5 48.5 49.4 3.3 13.6 36.7 2.4 42.0 16.9 81.1 16.0 82.0 2.1

Yes No Missing

88 22 83

45.6 11.4 43.0

38 8 99

26.2 5.5 68.3

126 30 182

37.3 8.9 53.8

Female Male Suburban Urban Rural M (SD) M (SD) M (SD)

28 3 26 3 2 7.55 3.52 11.00

90.3 9.7 83.9 9.7 6.5 4.9 3.5 6.8

18 6 22 0 2 11.33 5.08 10.21

75.0 25.0 91.7 0.0 8.3 10.4 11.0 10.5

46 9 48 3 4 9.20 4.23 10.65

83.6 16.4 87.3 5.5 7.3 8.0 7.8 8.5

Note. FRL = students eligible for free or reduced priced lunch; ELL = English language learner; SBI = schema-based instruction; SpEd = students qualified for special education services.

the pretest (M = 12.55, SD = 4.15). Thus, scores on the two assessments provide support for students meeting the MLD criteria. Of the 338 students with MLD, 193 were in SBI classrooms and 145 were in business-as-usual control classrooms. On the PPS pretest, the performance of SBI students was comparable to that of the control students. The mean PPS score was 5.17 (SD = 1.72) for SBI and 5.35 (SD = 1.76) for the control group. Table 1 presents student and teacher demographic data. In general, student demographic patterns were similar across the treatment and control groups. For example, the percentages of White, Black, Hispanic, and Asian students in the treatment group were 46.1 %, 17.1 %, 31.6 %, and 1.6 %, respectively; for the control group these percentages were 36.6 %, 9.0 %, 43.4 %, and 5.5 %, respectively. These findings support our assumption that students with MLD were equally distributed across the treatment and control classrooms. Similar results emerged for teacher characteristics. 4.2. Instructional procedures Teachers in the two conditions implemented two instructional units (Ratio and Proportion; Percent) using the assigned curricular program. Control teachers used their district-mandated textbooks and typical practices to teach the content in the two units, whereas treatment teachers used the SBI program to teach the same content over the same time period (45 min daily, five days a week, for approximately 6 weeks). The SBI program is comprised of 21 lessons, 10 lessons in each unit and an additional lesson reviewing the content from both units. All lessons could be completed in about 30 days, with some lessons taking more than a day to implement (see Appendix in online supplementary materials for scope and sequence of the SBI program). The first unit included four lessons to develop the meaning of ratios, equivalent ratios, rates, and scale factors (see Appendix in the online supplementary materials for sample excerpts of teaching notes from Lessons 1, 2, & 5) that are critical to understanding proportions. The majority of the remaining lessons provided practice solving problems involving ratios, proportions, and scale drawings in word problem contexts (see Appendix in the online supplementary materials for a sample excerpt of teaching notes from Lesson 7 for solving a proportion problem). The second unit extended that understanding of ratios/rates to understanding the meaning of percent (as well as fractions and decimals as alternative representations) and provided practice solving word problems involving part-whole comparisons and percent of change. Percent of change problems included solving simple and complex problems involving sales taxes, discounts, tips, and simple interest as well as multistep adjustment problems. One of the lessons in this unit used a Jeopardy game format to engage students as they categorized problems into the different problem types (ratio, proportion, percent). 6

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The culminating lesson in each unit used real-world scenarios (i.e., designing a recording studio and constructing a digital planetarium) and provided further practice with proportional reasoning as students worked in small groups to solve a variety of problems involving ratios and proportional relationships. Mathematical tasks in these lessons complemented the mathematical ideas in each unit by making direct connections to everyday life. For example, Designing a Recording Studio in Unit One illustrates how to represent and solve ratio and proportion problems by discussing how to redesign the school’s band room into a recording studio (see Appendix in the online supplementary materials for a sample tasks from Lesson 10). Instruction in the SBI program focuses on applying problem solving procedures for a given class of problems, including checks to monitor and reflect on the problem-solving process (e.g., when, how, and why to use multiple strategies—equivalent fractions, unit rate, cross multiplication—for a given class of problems). Key SBI instructional features for proportional problem-solving include: (a) identifying the type of problem (i.e., ratio, proportion, or percent) by reading, retelling, and examining information in the problem as well as thinking about how problems within and across types are similar or different, thus connecting the problem to already solved problems, (b) representing critical information in the problem using an appropriate representation that illustrates the relationships between relevant quantities in the problem, (c) estimating the answer and determining what strategy (equivalent fractions, unit rate, cross multiplication) to use (one that is most efficient based on the quantities in the problem), (d) solving the problem and using the estimated answer from the previous step to evaluate the work and determine whether the answer made sense (see Appendix in the online supplementary materials for a sample excerpt of script from Lesson 7 illustrating how the problem can be solved). 4.3. Assessments We administered the PPS and general mathematical problem solving assessments prior to and immediately following the intervention with retention of PPS data collected nine weeks following intervention. 4.3.1. Measure of proportional problem solving The PPS test consisted of 22 multiple-choice items (dichotomously scored) and three short-response tasks with four open-ended items (total score range 0–30). The test was developed using released items from NAEP, TIMSS, and past state mathematics assessments, with Jitendra, Harwell, Im, et al. (2019) providing psychometric evidence supporting the PPS. The content of this test, which covers topics of ratio, proportion, and percent, closely parallels the content taught in the two conditions. The same assessment was given three times (pretest, posttest, and delayed posttest given nine weeks after the intervention). The coefficient omega reliabilities (Dunn, Baguley, & Brunsden, 2014) for the pretest, posttest, and delayed posttest were 0.77, 0.84, and 0.84. The multiplechoice items allowed for an objective analysis of student performance and provided limited evidence of students’ proportional reasoning. For example, students might select the correct response from among several options but not have the opportunity to justify their solution. We developed or modified short-response tasks to address each topic—ratio, proportion, and percent of change—and ensured that the contexts were appropriate for the students in our study. These tasks offered opportunities for students to demonstrate their proportional reasoning by communicating mathematically through writing. The percent of change task (Books Read) included two open-ended items. For the Books Read task shown in Fig. 1, student explanations could address both additive (Item 18 part 1[P1]) and proportional (Item 18 part 2[P2]) reasoning on the two-open ended items. Given that the aim and research questions in the present study focused on proportional reasoning, we analyzed students’ written work on three open-ended items (Items 5, 10, and 18 P2) assessing proportional reasoning and excluded further analysis of Item 18 P1 assessing additive reasoning in the remainder of this paper. 4.3.2. Measure of mathematical general problem solving The Process and Applications subtest of the Group Mathematics Assessment and Diagnostic Evaluation (GMADE; Pearson, 2004) was used to assess students’ mathematics problem solving performance. This measure consists of 30 multiple-choice questions, including multiple-step problems and process problems. The test assessed students’ ability to comprehend mathematical language and concepts and apply relevant operations to solve word problems across multiple content areas (e.g., algebra, geometry, number and operations). All items were scored as correct or incorrect. The coefficient omega reliabilities for this measure were 0.69 for the pretest and 0.76 for the posttest. 4.4. Developing rubrics to capture evidence of proportional reasoning on the three open-ended items Students’ written explanations on the three open-ended items (Items 5, 10, and 18 P2)— ratio, proportion, and percent of change problems—were scored using the rubric described below. The two main objectives for scoring all three items are (1) whether students identify the underlying ratios with an appropriate expression including fractions, decimals, percent, “to,” or “:” and (2) whether they correctly evaluate the equivalence of ratios by drawing attention to multiplicative relations among quantities. Any explanations satisfying these two criteria were scored 2. Otherwise, they were scored 1 (only satisfied one criteria) or 0 (did not satisfy any criteria). Specifically, for the Skateboards to Soccer Balls problem, the two criteria include the identification of part-to-part ratios and an explanation as to why/how the two boxes are different. The sample correct explanation is that the ratio of skateboards to soccer balls in the problem is 1:2 whereas the ratio of skateboards to soccer balls in box (ii) is 2:3, which makes the two ratios nonequivalent 1 2 (2 ). For the Parking Lots problem, two criteria were the identification of part-to-whole ratios and explanations as to why/how the 3 7

Journal of Mathematical Behavior 57 (2020) 100753

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(caption on next page) 8

Journal of Mathematical Behavior 57 (2020) 100753

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Fig. 1. Three short response tasks: (1) Skateboards to Soccer Balls, part-to-part ratio problem (top), (2) Parking Lots, part-to-whole ratios –proportion problem (middle), and (3) Books Read, part-to-whole ratios – percent of change problem (bottom).

two parking lots are different by considering the total space. The sample correct explanation is that parking lot A is 70 % full (full to total; 14:20 or 14 ) whereas parking lot B is 60 % full (full to total; 6:10 or 6 ). Thus, parking lot B is emptier than parking lot A. For the 20 10 Books Read problem, the criteria include recognizing the percent of change from September to December and describing why/how these two percent changes are not equivalent. The sample correct explanation is that Jill had (8–5) ÷ 5 × 100 = 60 % increase in the number of books read, whereas Dushawn had (9–6) ÷ 6 × 100 = 50 % increase in the number of books. Thus, the percent of change in books read is more for Jill than Dushawn. All three open-ended items were constructed to elicit students’ explanation in solving comparison proportional problems. It is important to note that all problem situations were familiar, everyday life situations and were accompanied by visual diagrams. Thus, interpretation of problem contexts was not considered in scoring criteria. Interrater reliabilities using intraclass correlations were above 0.88 on the pretest, posttest, and delayed posttest (Jitendra, Harwell, Im et al., 2019), 4.5. Developing error coding schemes to identify misconceptions underlying incorrect answers on the three open-ended items We further analyzed students’ incorrect answers to identify their reasoning and misconceptions. We entered all students’ incorrect answers (scored 0) across pretest, posttest, and delayed posttest into an Excel spreadsheet and identified frequently used explanation words that appeared more than two times. We used these words to infer students’ error types. Table 2 summarizes types of errors/ misconceptions and their corresponding explanation words and examples of incorrect answers on the three open-ended items. For the Skateboards to Soccer Balls problem (Item 5, ratio problem), frequently used explanation words were more (or less), not same, different, even or odd numbers, simplify, go into, multiply, divide, and add 1. These words reflected students’ errors that were mostly linked to numerical, additive, and pre-multiplicative reasoning. For the Parking Lots problem (Item 10, proportion problem), the frequently used explanation words were more (or less) empty spaces, more (or less) parked cars, and greater (or smaller) parking lots. These words indicated students’ failure to recognize part-whole ratios needed to solve the problem. Most incorrect answers were derived from comparing part-to-part relations. For the Books Read problem (Item 18 P2, percent of change problem), frequently used explanation words for justifying Jill’s reasoning were read more (or less) books, read the same amount of books, and had 3 books increase between September and December. Again, these words reflected students’ errors that only focused on changes between two parts rather than emphasizing the part-whole relation between them. In addition, we separately coded the “no” response or “I don’t know” response error. The remaining incorrect answers were categorized as “Other.” These were mostly incomplete and undecipherable responses, computation errors, and problem-comprehension errors. The current study did not distinguish computation errors and problem-comprehension errors for the following reasons. All items primarily assessed students’ mathematical reasoning. They did not require complex calculation and problems involved real-world contexts. Furthermore, less than 5 % of the incorrect answers in each open-ended item entailed computation and/or language relevant errors. In short, we categorized all incorrect answers using the above mentioned error types and computed the percentage of each error type across PPS pretest, posttest, and delayed posttest for both SBI and control conditions. 4.6. Data analysis To address the first research question concerning the evaluation of the SBI program, we conducted independent t-tests on the PPS and GMADE outcome variables by condition. Hedges’ g, which is considered a less biased estimate of effect size than Cohen’s d, was used to compute treatment effect size. To address the second question about changes in the proportion of correct and incorrect answers on the open-ended items, we conducted a series of nonparametric Cochran’s Q tests (analogous to one-way repeated measures ANOVA) and McNemar post-hoc tests. Given multiple post-hoc pairwise comparisons, we used the Dunn-Bonferroni correction to control for compounding error rates in which an overall (i.e., experiment wise) Type I error rate (e.g., α = 0.05) was .05 divided by the number of tests, producing = 3 = .017 . To address our third research question with regard to types of errors, we computed the frequency of each error type and used the two proportion z-tests to compare between-condition differences in error types. We assessed reliabilities of the PPS and GMADE outcome measures using the coefficient omega value, which represents an estimated ratio of true score variance to observed scored variance (Dunn et al., 2014) 5. Results 5.1. SBI effect on proportional problem solving and mathematical problem solving First, we evaluated the effect of the SBI program by comparing the PPS and GMADE overall scores across the pretest, posttest, and delayed posttest by condition (Table 3). Results indicated no significant differences between the two conditions on the PPS and GMADE pretest scores (ps > .191). In sum, pretreatment scores of students with MLD in the SBI and control conditions were comparable in terms of proportional reasoning and general mathematical problem-solving skills. Results indicated significant differences between conditions on both PPS (t = 5.201, p < .001, g = 0.56) and GMADE (t = 2.325, 9

10

Numerical

Item 18 Part 2, Book Read [part-to-whole ratios; percent of change]

Additive

Whole (total number of spaces) not considered (a) Whole (total number of spaces) not considered (b) Part (number of empty spaces) not considered

Item 10, Parking Lots [part-to-whole ratios; proportion]

Pre-multiplicative

Additive

Numerical

Item 5, Skateboards to Soccer Balls [part-to-part ratio; ratio]

Perceptual comparison of numbers (based on physical similarity between numbers) Additive comparison (based on constant difference between ratios)

Compared only the parts (empty spaces) in the part-whole ratios (agree) Compared only the parts (parked cars) in the part-whole ratios (disagree) Compared only the whole in the part-whole ratios (disagree)

Perceptual comparison of numbers (based on physical similarity between numbers) Additive comparison (comparison between two quantities is absolute rather than relative) Pre-multiplicative (e.g., numerical pattern comparison)

Types of student errors/misconceptions

Items

read the same amount of books, had 3 books increase between September and December

more (or less) books

greater (or smaller) parking lots

more (or less) parked cars

more (or less) empty spaces

simplify, go into, multiply, divide, double

more (or less), not same, different, even or odd numbers added one more in each line or row

Explanation words

Table 2 Types of errors and misconceptions and their corresponding explanation words, and examples of incorrect answers.

Jill is wrong because they both read 3 more books from September to December (5:6 = 5 + 3:6 + 3 = 8:9).

Jill had 13 and Dushawn had 15 so Jill did not read more

Parking lot A has 6 empty spaces and parking lot B only has 4, and 6 is bigger number than 4. Parking lot B is emptier than parking lot A because parking lot A has more cars than B. Parking lot A is a bigger parking lot than parking lot B. So lot B is emptier.

Its double the balls and skateboards in (i) its two skateboards and one soccer ball but in (ii) its two soccer balls and 3 skateboards.

There are more skateboards and soccer balls than the box above. They added one skateboard and soccer ball to each row.

Examples of incorrect answers

S.-h. Im and A.K. Jitendra

Journal of Mathematical Behavior 57 (2020) 100753

Journal of Mathematical Behavior 57 (2020) 100753

S.-h. Im and A.K. Jitendra

Table 3 Descriptive statistics and independent t-tests of PPS and GMADE overall scores and the PPS scores only on short-response tasks. SBI

PPS pretest PPS posttest PPS delayed posttest GMADE pretest GMADE posttest PPS pretest only SR PPS posttest only SR PPS delayed posttest only SR

Control

N

M

SD

N

M

SD

193 182 169 189 180 193 182 169

5.17 11.03 9.57 9.02 11.07 0.20 1.13 0.79

1.72 4.67 4.14 3.56 3.76 0.51 1.49 1.23

145 130 115 137 117 145 130 115

5.35 8.72 8.89 9.53 10.03 0.23 0.49 0.59

1.76 3.16 2.92 3.38 3.78 0.54 0.91 1.09

df

t

p

g

336 310 282 324 295 336 310 282

−0.972 5.201 1.626 −1.309 2.325 −0.534 4.354 1.415

.332 < .001 .105 .191 .021 .594 < .001 .158

−0.10 0.56 0.18 −0.15 0.28 −0.06 0.50 0.17

Note. For the PPS and GMADE overall scores, the total possible score is 30. For the PPS short-response (SR) score, the total possible score is 8. Hedges’ g effect size was calculated as the two conditions’ mean difference divided by the pooled standard deviation (Hedges & Olkin, 1985).

p = .021, g = 0.28) posttest overall scores, with students with MLD in the SBI condition (PPS: M = 11.03, SD = 4.67; GMADE: M = 11.07, SD = 3.76) outperforming their counterparts in the control condition (PPS: M = 8.72, SD = 3.16; GMADE: M = 10.03, SD = 3.78). However, there was no significant SBI effect at a 9-week follow-up on the PPS delayed posttest (t = 1.626, p = .105, g = 0.18). Second, we conducted the same analysis on scores from three short-response tasks on the PPS. Again, pretreatment scores were not significantly different (t = - 0.534, p = .594, g = − 0.06) between the two conditions. On the PPS posttest and delayed posttest, results were similar to the findings for overall scores in that there was a significant difference between conditions only on the posttest (t = 4.354, p < .001, g = 0.50). Regardless of the assigned condition, it is important to note that the obtained mean short-response scores across pretest, posttest, and delayed posttest were very low compared to the total possible score of 8. The mean scores across tests were less than 15 % of the total possible score. The findings suggest that the SBI program is more effective in improving proportional-problem solving and general mathematical problem-solving skills of students with MLD compared to their counterparts in the control condition. However, the effect of SBI is minimal for the short-response tasks. 5.2. SBI effect on proportional reasoning ability We examined the data in more detail by tallying the number of students who provided incorrect, partially correct, and correct answers on their written explanation for each item across pretest, posttest, and delayed posttest (see Table 4). Given the lack of or small number of students who provided partially correct or correct answers, we merged these two categories into one category as “correct” and then contrasted with the incorrect answer category. Results indicated significant changes on all four open-ended items (Qs > 16.059, ps < .001) for students in the SBI condition. Following SBI, the percentage of students whose responses changed from incorrect on the pretest to correct on the posttest were as follows: 26.6 % for Item 5; 14.4 % for Item 10; 26.3 % for Item 18 P1; and 9.4 % for Item 18 P2. Results of multiple post-hoc pairwise McNemar tests indicated that the changes were statistically significant (ps < .001). Figs. 2–4 present examples of students’ written work to illustrate positive changes from pretest to posttest on Items 5, 10, and 18 P2. In the control condition, the only statistically significant change was on Item 18 P1 (Q = 18.053, p < .001). About 20 % of students’ responses shifted from incorrect on the pretest to correct on the posttest (p < .001). There were no significant changes on Table 4 Percentages of incorrect, partially correct, and correct answers on the four open-ended items across tests by condition. Items

Item 5 Item 10 Item 18, P1 Item 18, P2

Answers

Incorrect Partially correct Correct Incorrect Partially correct Correct Incorrect Partially correct Correct Incorrect Partially correct Correct

Pre

Post

Delayed post

SBI (n = 193)

Control (n = 145)

SBI (n = 182)

Control (n = 130)

SBI (n = 169)

Control (n = 115)

91.7 7.3 1.0 95.9 4.1 0 95.9 2.6 1.6 99.5 0.5 0

91.7 6.2 2.1 98.6 1.4 0 94.5 2.1 3.4 97.9 2.1 0

75.3 12.6 12.1 84.6 9.9 5.5 74.2 7.1 18.7 90.7 7.7 1.6

93.1 3.8 3.1 96.2 3.8 0 79.2 8.5 12.3 97.7 2.3 0

83.4 6.5 10.1 90.5 5.9 3.6 78.1 8.3 13.6 95.9 4.1 0

89.6 5.2 5.2 97.4 2.6 0 77.4 7.8 14.8 96.5 3.5 0

11

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Fig. 2. SBI student’s written work on pretest (scored 0) and posttest (scored 2) for Item 5.

Fig. 3. SBI student’s written work on pretest (scored 0) and posttest (scored 1) for Item 10.

Fig. 4. SBI student’s written work on pretest (scored 0) and posttest (scored 2) for Item 18 P2.

the other three open-ended items (Qs < 1.750, ps > .417). In sum, these findings suggest that some students with MLD in both conditions demonstrated improved understanding of the additive relation on Item 18P1. Although students with MLD in the SBI condition also showed improvement in their proportional reasoning ability on the remaining three open-ended items, the improvements were minimal. Most students had difficulties explaining their proportional reasoning on the open-ended items. 5.3. Misconceptions based on error analysis of incorrect answers We analyzed the specific error types based on incorrect answers on Items 5, 10, and 18 P2 to examine misconceptions in proportional reasoning among students with MLD. The focus of this analysis was three-fold and examined the following questions: (1) What types of errors were most common at pretreatment for students with MLD? (2) Which types of errors reduced or were persistent at posttreatment? and (3) Do the pattern of error types differ by condition? As shown in Fig. 5, the most common error for the part-to-part ratio comparison problem at pretest involved numeral reasoning. Most students with MLD in both SBI (37.3 %) and control (54.9 %) conditions referred to the numerical equivalence of each part, or the total number of two parts to justify their answers. For example, a common incorrect answer was “It is not equivalent because it has 12

Journal of Mathematical Behavior 57 (2020) 100753

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Fig. 5. Student error types on Item 5 (Skateboards to Soccer Balls, part-to-part ratio problem).

more soccer balls and skateboards than box i and the box above.” The prevalence of this numerical reasoning did not reduce on the posttest and delayed posttest. Interestingly, on the posttest, the proportion of pre-multiplicative reasoning was higher in the SBI condition (15.3 %) than in the control condition (5.8 %). Results of the z-test confirmed the significant difference between the two conditions, z = 2.459, p = .014. Some students with MLD in the SBI condition attempted to figure out a multiplicative relation or ratio between two parts in the same box. For example, there are 9 skateboards and 6 soccer balls and neither of the numbers go into each other. Such explanations likely provide some evidence of the positive effect of the SBI program with regard to shifting students’ focus from perceptual comparison of numbers to the multiplicative relation between two parts.

Fig. 6. Student error types on Item 10 (Parking Lots, part-to-whole ratios –proportion problem). 13

Journal of Mathematical Behavior 57 (2020) 100753

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Fig. 7. Student error types on Item 18 part 2 (Books Read, part-to-whole ratios – percent of change problem).

For the proportion problem (i.e., Item 10), the most common error on the pretest was students’ reference to quantities related to empty spaces in the parking lots (see Fig. 6). That is, students with MLD in both groups frequently discussed the discrepancy in only the parts of the part-to-whole ratios (SBI = 45.4 %; Control = 62.2 %). A second common error was also based on comparing only parts—examining the difference between the number of parked cars in the two parking lots. In both conditions, these two error types did not reduce over time. However, there was some increase with regard to students referring to parking lot sizes, reflecting their awareness and the need to consider the whole to determine a ratio. There were no significant differences between treatment and control conditions in terms of patterns of error types on this problem. The percent of change problem (i.e., Item 18 P2) was most challenging for students with MLD. As shown in Fig. 7, a common error on the pretest was “no response” or “I don’t know” (SBI = 41.1 %; Control = 33.1 %). Without understanding the concept of proportion, it would be hard to accept that Jill read more books than Dushawn in two months because the same change of three books conflicts with Jill’s reasoning. The following common error, referring to the number of books read by Jill is 13 (5 + 8) and the number of books read by Dushawn is 15 (6 + 9) also reflects students’ alternative thinking for making sense on the basis of their numerical reasoning. This ill-conceived reasoning did not decrease following instruction in ratio and proportional relationships. More surprisingly, the majority of students (SBI = 55.5 %; control = 42.3 %) explicitly disagreed with Jill’s reasoning by focusing on the number of books read or the same change of three books on the delayed posttest. Taken together, the results of error analysis suggest that the numerical comparison responses of students with MLD remained intractable despite learning about ratio and proportional relationships. They did not recognize the underlying multiplicative relations in proportional problems. Although the SBI intervention emphasizes visualizing ratio, proportion, and percent of change problems, its impact was not fully realized by students with MLD. Furthermore, the focus on fraction and ratio expressions in their answers was low in both conditions. 6. Discussion Prior research suggests that children and adolescents’ misconceptions related to ratios and proportional reasoning may be due to not fully understanding the relation between additive and multiplicative comparisons. However, less is known about misconceptions and errors among students with MLD. To the best of our knowledge, with the exception of the study by Jitendra, Woodward, et al. (2011), this study is one of the first attempts to examine misconceptions related to ratios and proportional reasoning of students with MLD within the context of a randomized controlled trial evaluating the effects of research-based intervention, SBI, on student learning. In the present study, we not only examined the effects of SBI on the mathematical outcomes of students with MLD, but also conducted an extensive analysis of students’ proportional reasoning ability and characterized student misconceptions to provide insights into how students in both conditions progress in their ability to reason about proportions in general and changes in student misconceptions, in particular. Although our preliminary results indicated that seventh-grade students with MLD in SBI classrooms outperformed their counterparts in control classrooms on proportional problem solving and general mathematics problem solving, an analysis of students’ 14

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written work on three open-ended proportion items showed a lack or limited understanding of fractional and ratio representations and persistent numerical and additive errors masking the otherwise positive effects of SBI. Specifically, errors and misconceptions were more obvious (a) when students with MLD did not know how to express ratios as fractional notations, (b) when students with MLD did not identify the type of ratio (part-to-part or part-to-whole ratio), and (c) when problems required converting ratios to percent or entailed multiple steps. In the following section, we discuss these findings by the research questions guiding the study. 6.1. SBI effect on proportional problem solving and general mathematics problem solving The first research question examined the effect of SBI on the proportional and general mathematical problem solving performance of seventh-grade students with MLD, and whether there was a retention of improved proportional problem solving performance at a 9-week follow-up. Compared to the control condition, SBI had a significant positive effect on the PPS and GMADE posttests. The results of our study are consistent with previous studies regarding the effects of SBI in improving mathematical outcomes of students with MLD (Jitendra, Dupuis et al., 2016, 2018; Jitendra, Harwell, Dupuis et al., 2017; Jitendra, Harwell, Karl et al., 2017). It is encouraging that the present study’s effects sizes of 0.56 and 0.28 SDs for the PPS and GMADE posttests are quite similar to the significant effect sizes of 0.50 and 0.26 SDs in Jitendra, Harwell, Im et al. (2019). Furthermore, the current study’s significant effect size of 0.50 SD for the four open-ended items on the PPS posttest is higher than the significant effect size of 0.37 SD found in Jitendra, Harwell, Im, et al. (2019). What are some possible reasons for the stronger results for students with MLD in the SBI condition compared to students in the control condition? First, SBI with its focus on the underlying problem structure required students to categorize problems into a few problems types by discerning the relevant quantities and their relations, which possibly reduced working memory load allowing for more efficient and effective learning (Kalyuga, 2009). Second, visual-schematic diagrams in SBI may have helped students organize information in the problem during the initial phase of the problem solving process (i.e., problem representation) to further reduce the cognitive memory demands and enable the learner to focus on problem solution (i.e., devise a plan to solve the problem). Evidence suggests that visual representational approaches improve mathematical problem solving of students with MLD (Gersten et al., 2009; Jitendra, Nelson, Pulles, Kiss, & Houseworth, 2016). Third, SBI promoted meaningful learning in that appropriate guidance (e.g., teacher think-aloud on how the problem can be solved or opportunities to explore different ways of solving the problem) provided throughout the learning process may have enabled the learner to understand and solve not only proportional problems but also mathematical problems involving other topics (e.g., algebra, geometry). In contrast, SBI did not have a significant effect on the PPS delayed posttest; the effect size of 0.18 SD, although positive, was much lower than the PPS posttest effect of 0.56 SD. Contrary to prior research (Jitendra, Harwell, Dupuis, et al., 2017; Jitendra, Harwell, Karl, et al., 2017). we were surprised to find no evidence of SBI students’ retention of improved problem solving performance. One potential explanation is that these students evidenced significantly more difficulties in mathematics (scored at or below the 25th percentile) compared to students identified as having MD (e.g., scored at or below the 35th percentile) in previous SBI studies. An alternative explanation is that these students’ difficulties may be specific to solving percent problems, within the larger category of proportion reasoning problems, which are more challenging than ratio and proportion problems (see Jitendra, Harwell, Dupuis, et al., 2017). Nevertheless, these students may need more intensive instruction (e.g., additional time and use of small groups, problem analysis and frequent progress monitoring) with increased opportunities to integrate and apply the critical components of SBI to solve increasingly complex problems to improve learning of ratios, proportions, and percent, thus, producing lasting effects (see Fuchs et al., 2008). 6.2. SBI effect on the ability to reason about proportions and changes in misconceptions Regarding the second research question, an analysis of students’ written work showed that only students in the SBI condition improved in their proportional reasoning ability in that there were statistically significant changes from incorrect answers on the pretest to partially correct or correct answers on the posttest and delayed posttest for all four open-ended items. Results of further analysis of incorrect answers provide preliminary evidence for the effect of SBI in terms of shifting students’ focus from perceptual comparison of numbers to multiplicative comparison of quantities (Research Question 3). For example, there was an increase in premultiplicative reasoning from pretest (8.5 %) to posttest (15.3 %) on the part-to-part comparison problem, Item 5. However, the ability to reason about proportions and the changes in misconceptions was somewhat limited. For example, only 5.5 % and 1.6 % of SBI students’ responses on the proportion and percent of change comparison problems, Items 10 and 18 P2, respectively, were correct. Interestingly, students in the control group did not demonstrate significant changes across the three waves of testing in terms of shifting from incorrect answers to partially correct or correct answers on the open-ended items (except on the additive reasoning problem, Item 18P1). Moreover, these students’ responses showed no changes in misconceptions—the overuse of numerical and additive reasoning was not reduced following instruction. Conceptual change, which is a gradual process that takes a long time probably stretches longer for students with MLD and may explain their overuse of additive reasoning in multiplicative situations (Durkin & Rittle-Johnson, 2015; Vamvakoussi & Vosniadou, 2010; Vosniadou & Verschaffel, 2004). Why is the improvement in proportional reasoning ability slow and somewhat limited for students with MLD? One explanation could be the lack of understanding of fractions. The first criterion on our open-ended items’ scoring rubric was the use of fractions or appropriate mathematics expression (i.e., “:” or “to”) to express ratios. The majority of students with MLD in both conditions inadequately used fractions, decimals, or percent in their written work to express underlying relations in the problem. Despite receiving instruction, most students with MLD continued to rely on whole numbers to compare quantities in the problems. For example, on 15

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Item 18 P2, percent of change problem, few students with MLD in both conditions expressed or mentioned percent of change in their written work on the posttest when comparing the change of books read by Jill and Dushawn from September to December. Understanding use of fractional notations and the ability to operate with fractions is critical for successful proportional problem solving (Jitendra, Lein, Star, & Dupuis, 2013). Previous research suggests that foundational to proportional reasoning is fraction knowledge (Charalambous & Pitta-Pantazi, 2007) and that the ability to reason with proportions supports fraction understanding (Howe et al., 2011). A second explanation for students’ limited proportional reasoning ability may be due to the misapplication and overgeneralization of numerical and additive reasoning when solving problems involving ratio or proportion situations. Another criterion of our scoring rubric was identifying multiplicative relations among quantities to evaluate the equivalence of ratios. Again, the failure to use fractions to represent ratios disrupts students’ attentional shift toward multiplicative relations. Instead, students with MLD reverted to their familiar approaches—numerical and additive reasoning. In all three proportional reasoning open-ended items, most students with MLD did not satisfy the second criterion as they engaged in numerical reasoning or used an additive approach when comparing quantities. This finding confirmed scholars’ claims regarding students’ persistent use of additive reasoning in proportional situations (Fernández et al., 2012; Van Dooren, De Bock, & Verschaffel, 2010). It is important to note that we included specific information about types of students’ errors or misconceptions based on types of proportional problems (ratio, proportion, and percent of change). For the ratio problem, we found students had difficulty matching part-to-part and lacked understanding of how to compare non-integer and integer ratios and/or explain why they are different. In general, students with MLD were less likely to identify a multiplication relation in a non-integer ratio, 2:3 in box (ii) compared to their ability to identify double ratios, 4:8 and 3:6 in the other pictures (see Fig. 1). For the proportion problem involving part-to-whole ratio, we found that students often determined the equivalence of two ratios by only comparing part-to-part (i.e., only empty spaces in parking lots A and B) or whole-to-whole (only parking lot sizes of A and B – the total number of empty and parked spaces). Such comparisons reflect students’ negligence of higher-order relations (i.e., secondorder structure – the number of empty parking spaces compared to the corresponding total number of parking spaces in a specific parking lot) needed to evaluate the equivalence of two ratios (Clark & Kamii, 1996; Grobecker, 1999; Lamon, 1993). For the percent of change problem, students did not know how to use percent when making comparisons. Although they learned about the percent expression (part divided by 100), they did not recognize the necessity of converting ratios into percent, which is critical to describe the comparison in terms of the percent of change. Moreover, multiple problem-solving steps in the percent of change problem imposed additional cognitive load and resulted in the lowest performance compared to the other open-ended proportional problems. To alleviate the observed misconceptions in proportional problem solving, researchers have emphasized the use of appropriate instructional approaches to shift student reasoning from additive to multiplicative and proportional. One successful instructional practice could be a compare and contrast case approach (Alfieri, Nokes-Malach, & Schunn, 2013; Rittle-Johnson & Star, 2007). For example, Durkin and Rittle-Johnson (2012) reported that fourth and fifth graders who studied both correct and incorrect examples of placing decimal values on the 0–1 number line showed better conceptual and procedural knowledge of decimals and a reduction in whole number misconception errors (e.g., 0.65 is smaller than 0.365, because 65 < 365). Similarly, comparing and contrasting additive and proportional problems may have a positive effect in reducing the misuse of additive reasoning when solving proportional problems. Third, difficulty with proportional reasoning may be explained by characteristics of students with MLD, who are deficient in general cognitive skills (Geary, 2004). Evidence suggests that working memory capacity influences these students’ responses to interventions targeting word problem-solving skills (Rittle-Johnson & Jordan, 2016). For example, Swanson (2014) noted that among third graders with mathematics difficulties, students with higher working memory capacity were more likely to benefit from instructional strategies, such as representing problems using diagrams than their counterparts with lower working memory capacity. A study by Booth and Koedinger (2012) indicated that younger students and those with lower mathematical ability were less likely to benefit from visual representations than older students and those with higher mathematical ability when solving algebra problems. 6.3. Limitations and directions for future research We identified several limitations of our study. First, we focused on a limited number of comparison proportional problems with a discrete format. This poses restrictions to the generalizability of our findings to other problems within the domain of proportional reasoning. Future studies could find out whether students’ errors—particularly, erroneously solving proportion problems additively—would occur as often in missing-value proportion problems and in continuous format problems with another structure (e.g., comparing the concentration of two juice-water mixtures in Boyer & Levine, 2012). Future research is warranted to understand whether other problem types would still explain students’ proportional reasoning ability and misconceptions observed in this study. A second limitation relates to examining students’ thinking and misconceptions based on their written work on open-ended items. Future studies could use a combination of interview and think-aloud methods to explore the processes used by students to solve proportional problems. A third limitation is that we used a subset of data from Jitendra, Harwell, Im et al. (2019) to select students based on a criterion of scoring below the 25th percentile on the PPS test. One could wonder whether students’ proportional reasoning ability and errors would occur more often if we selected students using the strictest criterion related to their mathematics skills (scoring below the 10th percentile). Future studies could find out whether there are more variations in erroneous responses when considering a diverse sample of students.

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6.4. Educational implications Notwithstanding the above-mentioned limitations, we can derive some educational implications based on the findings of this study. Given that diminished understandings of fractional representation as well as misapplication and overgeneralization of numerical and additive reasoning when solving problems involving proportion situations explains the occurrence of errors or misconceptions, it is worth considering how such errors, especially ineffectual reasoning, can be remediated and prevented. Interventions may need to not only develop robust understanding of integers, fractions, decimals, and percent, but also target discriminating skills to distinguish comparison situations that require additive or multiplicative reasoning. First, interventions should use real-world models to develop understanding of fractions, decimals, unit rates, and proportions (Cramer et al., 2018; Fuchs et al., 2013, 2016; National Council of Teachers of Mathematics, 2000). Given that students with MLD in the study were able to identify double or half ratios rather than a multiplication relation in a non-integer ratio, activities could focus on double or half ratios to discuss multiplicative relations in introductory lessons. Second, educational activities should be targeted at overcoming students’ overreliance on numerical or additive reasoning when solving problems involving comparison situations. Problems involving both additive and multiplicative reasoning such as Items 18 P1 and 18 P2 used in this study could be appropriate for the purpose of developing discrimination skills and help overcome the overreliance on additive reasoning. Activities addressing students’ overreliance on additive reasoning could involve small group discussion in which students are prompted to compare and contrast different types of reasoning for a specific problem as a way to negate students’ misconceptions and elicit attentional shift from a focus on additive to multiplicative relations among quantities in proportion problems. Third, instruction should target the web of interconnected ideas that are at the core of proportionality. In addition to teaching the meaning of ratios and rates, scale factor, fractions, decimals, and percent, students with MLD should be primed about or explicitly taught the meaning of multiplication of whole numbers and make explicit the connection with ratios and rates, fractions, decimals, proportions, and percent. This meaning of multiplication should be independent of repeated addition, which does not work when multiplying fractions or decimals. Instead, the meaning of multiplication should focus on rate or “groups of” notion of multiplication or one-to-many correspondence, which is essential to promote multiplicative thinking and proportional reasoning (Hino & Kato, 2019). Instruction should be aimed at deepening students understanding of the relational concept (i.e., x times as many [much] as), which can then be used to extend the meaning of multiplication to fractions and decimals (Hino & Kato, 2019). 6.5. Conclusions In this article, we focused specifically on students' ability to reason about proportions. The findings of this study, highlighting students’ persistent use of numerical and additive reasoning, raise important questions about instruction. Early intervention for students with MLD to prevent misconceptions rather than remediating misconceptions can be efficient. 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