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Assessment results are used to investigate relations between performance on a fraction number line estimation task and a circular area model estimation task for students with LD in Grades 6–8. Results indicate that students’ abilities to represent fractions on number lines and on circular area models are distinct skills. In addition, accurate fract...
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Despite the significance of visual fraction models for teaching and learning fraction multiplication, how students are supported to learn fraction multiplication through representations remains unknown. By examining the presentation of visual representations in the contexts of word problems and computational processes in representative Korean and U...
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... In school, teachers frequently use various shapes, such as circles or rectangles, to illustrate fractional concepts. However, the ability to use various representations (e.g., number lines or circular area models) to illustrate fractions requires distinct skills (Morano et al., 2019). Therefore, teachers may carefully select instructional materials and maintain consistency in the representations used to teach fractions. ...
This study reviews the literature on error patterns in mathematics among students with mathematics difficulty. We analyzed and synthesized the findings from 17 studies, focusing on the characteristics of error analysis studies, the mathematics topics examined, and the specific error patterns identified. The results revealed the following: (a) the criteria used to identify mathematics difficulties and the coding processes varied; (b) the mathematics topics investigated encompassed fractions (including fraction computation and representation), problem-solving, and general computation; and (c) a variety of common error types were identified across these mathematical domains. Implications for practitioners and researchers were discussed.
... Thus, the intervention in this study effectively increased students' ability to identify base-10 fractions as decimals. Prior studies have demonstrated that students with a strong understanding of rational number magnitude also develop a robust understanding of complex rational number concepts and skills (Morano et al., 2019;Van Hoof et al., 2018). This study supports that notion, as all students achieved proficiency on the magnitude probes (i.e., > 90%) before learning about fraction-to-decimal translation, a more complex rational number topic for fourth grade (NGAC & CCSO, 2010). ...
Competence with rational numbers is essential for mathematics proficiency in secondary mathematics. However, many students struggle with rational number concepts, and students with mathematics difficulties struggle even more. The purpose of this study was to examine the effects of an intervention that incorporated the use of explicit instruction and length models to teach the relationship between fractions and decimals to fourth-grade students with mathematics difficulties. To conduct this study, the researcher employed a multiple probe design across three intervention groups. The results of a visual analysis determined that a functional relation existed between a fraction and decimal intervention and increased performance on two dependent variables: base-10 fraction magnitude and fraction-to-decimal translation. The magnitude of change was calculated using a between-case standardized mean difference, and the results of a social validity questionnaire were presented.
... MD appear to struggle with many aspects of fraction skills, including fraction concepts, arithmetic, estimation of fraction magnitudes, and word problems (Bailey et al., 2015;Hansen et al., 2017;Hecht & Vagi, 2010;Hunt, 2015;Ikhwanudin et al., 2019;Mazzocco & Devlin, 2008;Mazzocco et al., 2013;Morano et al., 2019;Newton et al., 2014). These studies provide important insights into the general relationship between fraction knowledge and low mathematics achievement. ...
... For example, students with MD (who score below 35% on state assessment tests) were less accurate than TA students on fraction number line estimation tasks (0-1 and 0-5; Siegler & Pyke, 2013). In a recent study, students with diagnosed learning disabilities also showed poor performance on number line estimation tasks as compared with circular models (measure part-whole understanding) (Morano et al., 2019). The number line estimation task also showed stronger relation with mathematics achievement and fraction magnitude comparison tasks than circular models, indicating that the part-whole knowledge usually acquired through circle models might be easier to develop but magnitude knowledge (number line tasks) is more important for learning of fraction concepts (Morano et al., 2019). ...
... In a recent study, students with diagnosed learning disabilities also showed poor performance on number line estimation tasks as compared with circular models (measure part-whole understanding) (Morano et al., 2019). The number line estimation task also showed stronger relation with mathematics achievement and fraction magnitude comparison tasks than circular models, indicating that the part-whole knowledge usually acquired through circle models might be easier to develop but magnitude knowledge (number line tasks) is more important for learning of fraction concepts (Morano et al., 2019). Intervention studies also show that practicing number line tasks may lead to improved fraction knowledge in individuals with MD (Barbieri et al., 2020;Fuchs et al., 2013Fuchs et al., , 2016Saxe et al., 2013). ...
Fractions are challenging for both typically achieving children and adults. Although some prior research has focused on fraction difficulties of children with mathematics difficulties (MD), persistent difficulties encountered by adults with MD remain unknown. It is possible that these adults may be able to compensate for some deficits. Here, we administered an un-timed, paper-based fraction achievement test to adults with and without MD to compare their knowledge of fractions. Compared with controls, adults with MD performed worse in fraction number lines, fraction concepts, fraction arithmetic, and word problems. However, no difference in performance between the two groups was observed on symbolic representations. This suggests that adults with MD might be able to perform rote procedures such as transcoding from a verbal to a symbolic representation but are severely impaired for fraction number line, fraction concept, and fraction arithmetic. Exploratory error pattern analyses for fraction number line and fraction arithmetic further revealed mistakes similar to those observed in prior studies on children with MD, indicating core deficits in fraction understanding in individuals with MD.
... Fractions are among the most difficult concepts learners of mathematics encounter in their school life and children's errors in fraction learning have been investigated for many years (Charalambous and Pitta-Pantazi 2007;Hansen et al. 2017). One issue contributing to students' difficulty with learning fractions is confusion regarding the various conceptual interpretations including fractions as a part of a whole, part of a set of objects, a measure on a number line, a ratio, an operator and a quotient (Morano, Riccomini, and Lee 2019;Behr et al. 1992). Different visual representations are often used in teaching the different conceptual interpretations of fractions: For example, area models and set models are typically used to teach the part-whole interpretation of fractions, and number lines are recommended for teaching the measurement interpretation (Siegler, Thompson, and Schneider 2011). ...
... The integration of these multiple representations of fractions is regarded as particularly significant for students' conceptual understanding of fractions (e.g., Charalambous and Pitta-Pantazi 2007). However, research in the field shows that students often struggle in changing among the different forms of fraction representations (Morano, Riccomini, and Lee 2019). ...
Although raised in the early days of research on teacher noticing, the question of context specificity has remained largely unanswered to this day. In this study, we build on our prior research on a specific aspect of noticing, namely teachers’ analysis of how representations are dealt with in mathematics classroom situations. For the purpose of such analysis, we examined the role of context on the levels of mathematical content area and classroom situation . Using a vignette-based test instrument with 12 classroom situations from the content areas of fractions and functions, we investigated how teachers’ analyses regarding the use of representations are related concerning these two mathematical content areas. Beyond content areas, we were interested in the question of whether an overarching unidimensional competence construct can be inferred from the participants’ analyses of the different individual classroom situations. The 12 vignettes were analysed by N = 175 secondary mathematics teachers with different degrees of teaching experience and their written answers provided the data for this study. Our findings show that the data fit the Rasch model and that all classroom situations contributed in a meaningful way to the competence under investigation. There was no significant effect of the mathematical content area on the participants’ analyses regarding the use of multiple representations. The results of the study indicate that explicitly considering questions of context can strengthen research into teacher noticing.
... These results suggest that PSTs were more familiar and more comfortable with circle models of fractions than they were with other types of fraction VRs (e.g., length or set models). Previous research has shown that schoolage students also use circle model VRs most frequently to model fractions problems, even though they use bar models more effectively (Morano, Riccomini, & Lee, 2019). As with younger students, PSTs' preference for circle models of fractions is likely a direct result of their school experiences. ...
To provide effective fractions instruction, teachers need deep understanding of fraction concepts (i.e., content knowledge) and skill in using visual representations of fractions to support student learning (i.e., pedagogical content knowledge); yet, research indicates that many pre- and in-service teachers lack knowledge in both areas. The present study examines the performance of 55 pre-service teachers (PSTs) on tasks assessing their procedural and conceptual knowledge of fractions, including a test of their ability to generate visual representations and story problems to model fraction multiplication and division. PSTs demonstrated high levels of procedural and conceptual knowledge of fraction addition and subtraction, moderate procedural knowledge for fraction multiplication and division, and relatively weak conceptual knowledge of fraction multiplication and division. Weak conceptual knowledge was related to inability to accurately model fraction multiplication and division using visual representations and story problems. Implications for research and practice are presented.
Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta‐analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number ( r = 0.33) and fractions number ( r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta‐analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced ( β = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students’ fractions learning.
Research Highlights
Two meta‐analyses examined the link between the number line and mathematics performance.
The first revealed significant relations across domains (e.g., algebra and computation).
The second controlled for general cognitive ability and resulted in reduced but still significant relations.
The relation between number line and fractions performance was stronger than relations to other domains.
Fractions are an integral part of the mathematics curriculum. Most students acquire proficiency with these concepts during the course of their elementary education and are usually able to perform basic fractions operations when reaching middle-school age. However, a considerable number of students require extra help to not fall further and further behind in the curriculum. In this study, we extended the use of a simple strategy (Look, Ask, Pick; Test & Ellis, 2005) that holds the potential to help students with problems understanding and working with fractions catch up with their classmates. We applied a multiple-baseline design across four struggling sixth graders. After receiving the instruction, all participants' performance on fractions improved significantly; moreover, they viewed the strategy as highly useful. Limitations of the study, future directions of research, and implications for teachers regarding the instructional utility of the intervention are discussed.
Fractions are an important concept used throughout advanced mathematical ideas and everyday life. Students need a solid understanding of fraction concepts, such as magnitude, equivalence, and operations. However, many students, including students with learning disabilities (LD) in mathematics, struggle with fractions. In this article, the use of multiple representations are discussed to support and teach fraction concepts to students with LD in mathematics. Multiple representations to include manipulatives, drawings, and numbers lines. Throughout the article, the use of explicit instruction to teach students with LD in mathematics how to use multiple representations to understand and solve fraction problems will be presented.
The purpose of this study was to investigate the effects of the concrete-representational-abstract–integrated (CRA-I) sequence on students with learning disabilities’ performance when learning fraction and decimal concepts. Three elementary students in Grades 4 and 6 participated in a single-case multiple probe across behaviors study. The intervention involved explicit use of fraction blocks, coins, base ten blocks, number lines, pictures, and abstract symbols to teach unit fractions, fraction and decimal equivalence, addition of fractions with unlike denominators, and writing fractions as decimals. The researchers demonstrated a functional relation between CRA-I and three different behaviors related to fraction concepts: decreased error in estimating fraction magnitude, addition of unlike fractions, and writing fractions as decimals. The findings show promise in the use of CRA-I for teaching fraction concepts to students with learning disabilities.
This study aimed to identify learners’ level of number sense and arithmetic ability in solving fractional problems using area and vertical line models, and to derive specific implications for teaching fraction. We analyzed responses from 227 sixth-graders attending 11 elementary schools in each of the 11 education support offices in Seoul to the number sense and arithmetic ability test paper developed based on the three-fraction representations of area models, structured vertical-line models, and empty vertical-line models. The results confirmed that the mean and incorrect answers by question, errors which habitually mistook the entire model for 1, without checking the entire given model in a range larger than 1, occurred frequently. Additionally, there were cases of problems being solved mechanically without using models. The meaning and use of the fractional model should be explicitly presented in teaching and learning situations, and opportunities to access fractional representation in various forms and ranges should be provided. It is also necessary to offer learners opportunities to implement problem-solving strategies on their own, based on representations, so as to promote their number sense and arithmetic abilities.
