Accuracy of Area Model and Number Line Representations of Fractions for Students with Learning Disabilities

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 fraction magnitude estimation using number lines was more strongly related to other fractions skills (e.g., fraction magnitude comparison) than accurate fraction magnitude estimation using circular area models. Our findings call for greater integration of the number line into early fractions instruction, and highlight the importance of explicitly teaching students to make connections between different visual representations of fractions. Implications for research and practice are presented.     

Which Type of Rational Numbers Should Students Learn First?

Many children and adults have difficulty gaining a comprehensive understanding of rational numbers. Although fractions are taught before decimals and percentages in many countries, including the USA, a number of researchers have argued that decimals are easier to learn than fractions and therefore teaching them first might mitigate children’s difficulty with rational numbers in general. We evaluate this proposal by discussing evidence regarding whether decimals are in fact easier to understand than fractions and whether teaching decimals before fractions leads to superior learning. Our review indicates that decimals are not generally easier to understand than fractions, though they are easier on some tasks. Learners have similar difficulty in understanding fraction and decimal magnitudes, arithmetic, and density, as well as with converting from either notation to the other. There was too little research on knowledge of percentages to include them in the comparisons or to establish the ideal order of instruction of the three types of rational numbers. Although existing research is insufficient to determine the best sequence for teaching the three rational number formats, we recommend several types of research that could help in addressing the issue in the future.     

Levels of students’ “conception” of fractions

In this paper, we examine sixth grade students’ degree of conceptualization of fractions. A specially developed test aimed to measure students’ understanding of fractions along the three stages proposed by Sfard (1991) was administered to 321 sixth grade students. The Rasch model was applied to specify the reliability of the test across the sample and cluster analysis to locate groups by facility level. The analysis revealed six such levels. The characteristics of each level were specified according to Sfard’s framework and the results of the fraction test. Based on our findings, we draw implications for the learning and teaching of fractions and provide suggestions for future research.     

A model on the cognitive and affective factors for the use of representations at the learning of decimals

In a previous article of the same journal, we have discussed the interrelations of students’ beliefs and self‐efficacy beliefs for the use of representations and their respective cognitive performance on the learning of fraction addition. In the present paper, we confirm a similar structure of cognitive and affective factors on using representations for the concept of decimals and mainly we discuss the various interrelations among those factors. Data were collected from 1701 students in Grades 5–8 (11–14‐years‐old). Results revealed that multiple‐representation flexibility, ability on solving problems with various modes of representation, beliefs about the use of representations and self‐efficacy beliefs about using them constructed an integrated model with strong interrelations that has differences and similarities with the respective model concerning the concept of fractions.     

基于理解性任务的小数理解调查研究

为了解师生对小数的理解情况,研究开发了小数理解性任务,并对52名五年级学生和15名教师（6名职前,9名在职）进行了测试。结果表明:师生共运用了9种思维策略,其中“组成部分思维”与“等值思维”为两种主要策略;教师和学生的主要思维策略是一致的,教师的思维策略分布更复杂,但思维深度并不高于学生。9种思维策略可概念化为基于符号形式和基于符号意义两种思考路径。基于所建构的思维策略与思维路径,研究认为教学需对小数概念进行整合,建构多元理解下的小数概念整体:在宏观筹划上贯彻概念联系,在概念理解上聚焦概念结构,在概念发展中注重概念的双重性质。     

Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.     

Pizzas or no pizzas: An advantage of word problems in fraction arithmetic?

Fractions are an important but notoriously difficult domain in mathematics education. Situating fraction arithmetic problems in a realistic setting might help students overcome their difficulties by making fraction arithmetic less abstract. The current study therefore investigated to what extent students (106 sixth graders, 187 seventh graders, and 192 eighth graders) perform better on fraction arithmetic problems presented as word problems compared to these problems presented symbolically. Results showed that in multiplication of a fraction with a whole number and in all types of fraction division, word problems were easier than their symbolic counterparts. However, in addition, subtraction, and multiplication of two fractions, symbolic problems were easier. There were no performance differences by students’ grade, but higher conceptual fraction knowledge was associated with higher fraction arithmetic performance. Taken together this study showed that situating fraction arithmetic in a realistic setting may support or hinder performance, dependent on the problem demands.     

Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.

     

A comparison of the development of spatial conceptual abilities of students from two cultures

Efforts are underway to determine if there are any ways unique to Navajo thinking and thus to the way that they might learn. Studies have shown a consistent lag in achievement levels for Native Americans. The purpose of this investigation was to examine spatial thinking abilities of sixth and tenth grade students from 2 locales—a school on the Navajo Reservation and schools in Mesa, Arizona. A battery of 10 Piagetian-type tasks were administered individually to the subjects. A chi-square one sample procedure was used to test for significant differences between subsamples at each grade level. Significant differences were detected for two of the tasks at the sixth grade level, and one task at the tenth grade level. The overall findings of this study support the contention that there were no substantial time delays or advances in the development of selected spatial abilities of Navajo sixth and tenth grade students compared to those of parallel non-American Indian students. The concern of modifying instruction in science courses in order to adapt them to supposed 'different' spatial structures possessed by Navajo students appears to be unfounded.     

Counterintuitive effects of online feedback in middle school math: results from a randomized controlled trial in ASSISTments

This study compared the effects of three different feedback formats provided to sixth grade mathematics students within a web-based online learning platform, ASSISTments. A sample of 196 students were randomly assigned to one of three conditions: (1) text-based feedback; (2) image-based feedback; and (3) correctness only feedback. Regardless of condition, students solved a set of problems pertaining to the division of fractions by fractions. This mathematics content was representative of challenging sixth grade mathematics Common Core State Standard (6.NS.A.1). Students randomly assigned to receive text-based feedback (Condition A) or image-based feedback (Condition B) outperformed those randomly assigned to the correctness only group (Condition C). However, these differences were not statistically significant (F(2,108) = 1.394, p = .25). Results of this study also demonstrated a completion-bias. Students randomly assigned to Condition B were less likely to complete the problem set than those assigned to Conditions A and C. To conclude, we discuss the counterintuitive findings observed in this study and implications related to developing and implementing feedback in online learning environments for middle school mathematics.     

Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents

Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The integrated theory of numerical development posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of the magnitudes to which they refer, and this magnitude understanding is central to general mathematical competence. We investigated relations among fraction magnitude understanding, arithmetic and general mathematical abilities in countries differing in educational practices: U.S., China and Belgium. Despite country-specific differences in absolute level of fraction knowledge, 6th and 8th graders' fraction magnitude understanding was positively related to their general mathematical achievement in all countries, and this relation remained significant after controlling for fraction arithmetic knowledge in almost all combinations of country and age group. These findings suggest that instructional interventions should target learners' interpretation of fractions as magnitudes, e.g., by practicing translating fractions into positions on number lines.     

Algorithmic skill vs. conceptual understanding

With the help of Pascual-Leone's Theory of Constructive Operators, this study investigated the hypothesis that understanding of the long division algorithm requires a high cognitive level, or greater m-capacity, than does understanding of the fundamental concepts of division. Formal and preformal sixth grade students were tested on performance and understanding of a given division algorithm and division concepts.     

What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density

It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number based prior knowledge, certain intermediate states of understanding emerge. This paper presents a study of Greek and Flemish 9th grade students who solved a test about the infinity of numbers in an interval. The Flemish students outperformed the Greek ones. More importantly, the intermediate levels of understanding—where the type of the interval endpoints (i.e., natural numbers, decimals, or fractions) affects students’ judgments—were very similar in both groups. These results point to specific conceptual difficulties involved in the shift from natural to rational numbers and raise some questions regarding instruction in both countries.     

Does an integrated focus on fractions and decimals improve at-risk students’ rational number magnitude performance?

The purpose of this study was to assess whether intervention with an integrated focus on fraction and decimal magnitude provides added value in improving rational number performance over intervention focused exclusively on fractions. We randomly assigned 4th graders with poor whole-number performance to 3 conditions: a business-as-usual control group and 2 variants of a validated fraction magnitude (FM) intervention. One variant of FM intervention included an integrated component on fraction-decimal magnitude (FM + DM); the other included a fraction applications component (FM + FAPP) to more closely mirror the validated FM intervention and to control for intervention time. Cross-classified partially-nested analyses (N = 225) provided the basis for 3 conclusions. First, FM intervention improves 4th-graders’ fraction understanding and applications. Second, effects of FM intervention, even without a focus on decimals, transfer to decimal number line performance. Third, an intervention component integrating fraction-decimal magnitude does not provide added value over FM intervention on fraction or decimal performance, except on decimal tasks paralleling intervention tasks.     

On fractions and non-standard representations: Pre-service teachers' concepts

The present study examined the reasoning strategies and arguments given by pre-service school teachers as they solved two problems regarding fractions in different symbolic representations. In the first problem, the pre-service school teachers were asked to compare between two different fractions having the same numerical representation, and in the second problem, they were asked to compare between different notational representations of the same fraction. Numeration systems in bases other than ten were used to generate various representations of fractions. All students were asked to provide justifications to their responses. Strategies and arguments relative to pre-service teachers' concepts of fractions and place value were identified and analyzed based on results of 38 individual clinical interviews, and written responses of 124 students. It was found that the majority of students believe that fractions change their numerical value under different symbolic representations.     

Characteristics of 5th Graders' Logical Development Through Learning Division with Decimals

When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of “division with decimals” in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals, caused by certain conceptions which children have implicitly or explicitly. In this paper we discuss how children develop their logical reasoning beyond such difficulties/misconceptions in the process of making sense of division with decimals in the classroom setting. We then suggest that children's explanations based on two kinds of reversibility (inversion and reciprocity) are effective in overcoming the difficulties/misconceptions related to division with decimals, and that they enable children to conceive multiplication and division as a system of operations.     

Monitoring Student Achievement for Accountability: The Demonstration of a Model

Abstract

This study demonstrated a procedural model that can be applied by any school to assess, guide, and account for the progress of its students as well as to analyze its own effectiveness. The model uses equivalent achievement tests to monitor student achievement in subject areas at grade levels, between grade levels, and across subgroups of students. Multiple regression analyses of test scores between grades identify factors associated with achievement Using sixth and eighth grade Comprehensive Tests of Basic Skills scores in a matched longitudinal sample of 208 students, the study found small differences in average achievement between boys and girls. Differences between corresponding sixth and eighth grade test means were higher in mathematics than in language. From the sixth grade to the eighth, there was a widening gap in average achievement between high and low I.Q. groups. In multiple regressions of eighth grade test scores on sixth grade measures, I.Q., study skills, and reading were prevalent in the regression equations, but clusters of measures associated with achievement differed between high and low’ LQ. groups. The results of the study have implications for developing and evaluating the achievement of students with varying mental abilities.     

An Investigation of Lebanese G7-12 Students’ Misconceptions and Difficulties in Genetics and Their Genetics Literacy

Lebanese educators claim that middle and secondary school students exhibit poor understanding of genetics due to misconceptions and difficulties that hinder progression in conceptual understanding of major genetics concepts and phenomena across different grade levels. They attributed these problems to Lebanon’s ill-structured genetics curriculum which needs a thorough revision in light of curricular reform models that take into account student misconceptions, cognitive abilities, and past experiences. Despite these claims, no empirical tests were done. Consequently, this study aimed to investigate G7-12 Lebanese students’ misconceptions and difficulties in genetics in an attempt to design a curriculum that would enhance student understanding of genetics. Using quantitative and qualitative data collection methods, we obtained an in-depth understanding of the nature of the misconceptions and difficulties encountered by students in grades 7–12, determined the level of students’ genetics literacy, and explored the progression of their level of conceptual understanding of major genetics concepts across grade levels. A questionnaire was administered to 729 students (G7-12) in 6 schools and was followed by semi-structured interviews with 62 students to validate the questionnaire results, gain further understanding of students’ misconceptions, and assess their level of genetics literacy. Findings showed that patterns of inheritance, the deterministic nature of genes, and the nature of genetic information were found to be among the most difficult concepts learned. Students also showed inadequate understanding of many basic genetics concepts which persist across grade levels. Furthermore, results indicated that students across all grade levels exhibited a low level of genetics literacy. Implications for practice and research are discussed.     

Conceptual distinctions and preferential alignment across rational number representations

European Journal of Psychology of Education - Rational numbers can be represented in multiple formats (e.g., fractions, decimals, and percentages), and a rational number notation can be used to...