Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience

Teaching argument writing in math class: challenges and solutions to improve the performance of 4th and 5th graders with disabilities

Incorporating argument writing as a learning activity has been found to increase students' mathematics performance. However, teachers report receiving little to no preservice or inservice preparation to use writing to support students' learning. This is especially concerning for special education teachers who provide highly specialized mathematics instruction (i.e., Tier 3) to students with mathematics disabilities (MLD). The purpose of this study was to examine the effectiveness of teachers providing content-focused open-ended questioning strategies, which included both argument writing and foundational fraction content, using Practice-Based Professional Development (PBPD) and Self-Regulated Strategy Development (SRSD) for implementing a writing-to-learn strategy called FACT-R2C2. We report the relative number of higher-order mathematical content questions that teachers asked during instruction, from among three different-level question types: Level 1: yes/no questions focused on the mathematics content; Level 2: one-word responses focused on the mathematics content; and Level 3: higher-order open-ended responses centered around four mathematical practices from the Common Core State Standards for Mathematics. Within a well-controlled single-case multiple-baseline design, seven special education teachers were randomly assigned to each PBPD + FACT-R2C2 intervention tier. Results indicated that: (1) teachers' relative use of Level 3 questions increased following the introduction of the FACT intervention; (2) this increase was apart from the professional development training that the teachers had initially received; and (3) students' writing quality improved to some extent with the increase in teachers' relative use of Level 3 questions. Implications and future directions are discussed.

On the Role of English as a Foreign Language Learners' Individual Differences in Their Use of Grammar Learning Strategies

As the use of strategies facilitates the tedious process of language learning, a multitude of studies have been conducted on language learning strategies and their educational consequences. Nonetheless, grammar learning strategies (GLSs) have not been widely studied. Moreover, no review study has been carried out to illustrate the role of individual differences in the use of GLSs. To address the existing gaps, the present review study intends to explain the role of English as a foreign language (EFL) learners’ individual differences (i.e., desire to learn a second language, motivation, and willingness to communicate) in their employment of GLSs. The favorable impact of individual difference variables on grammar learning strategy use was proved using the theoretical and empirical evidence. Future research directions and pedagogical implications are also discussed.

Math anxiety differentially impairs symbolic, but not nonsymbolic, fraction skills across development

Although important for the acquisition of later math skills, fractions are notoriously difficult. Previous studies have shown that higher math anxiety (MA) is associated with lower performance in symbolic fraction tasks in adults and have suggested that MA may negatively impact the acquisition of fractions in children. However, the effects of MA on fraction skills in school-aged children remain underexplored. We, therefore, investigated the impact of MA on the performance of younger (second and third graders) and older (fifth and sixth graders) children in math fluency (MF), written calculation, fraction knowledge (FK), and symbolic fraction and nonsymbolic ratio processing. On the basis of our prior work suggesting a perceptual foundation for fraction processing, we predicted that symbolic, but not nonsymbolic, math skills (especially fractions) would be impaired by MA. As predicted, higher MA was associated with lower performance in general mathematics achievement and symbolic fraction tasks, but nonsymbolic ratio processing was not affected by MA in either age group. Furthermore, working memory capacity partially mediated the effects of MA on general mathematics achievement, FK, and symbolic fraction processing. These results suggest that understanding the bidirectional interactions between MA and fractions may be important for helping children acquire these critical skills.

Taking the Relational Structure of Fractions Seriously: Relational Reasoning Predicts Fraction Knowledge in Elementary School Children

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 194 second grade and 145 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall mathematics knowledge and skill for second graders but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children's fractions skills.

Symbolic fractions elicit an analog magnitude representation in school-age children

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

Are conceptual knowledge and procedural knowledge empirically separable? The case of fractions

BACKGROUND

Concerning students' difficulties with fractions, many explanatory approaches are based on the distinction between conceptual knowledge and procedural knowledge. For further research in this field, it is thus crucial to make these constructs accessible to valid measurement.

AIMS

In this study, we aim at developing a test instrument that affords valid measurement of students' conceptual and procedural fraction knowledge, including in particular empirical validation of this distinction.

SAMPLE

The data used in this study were from 8th- and 9th-grade students (N = 235) in Germany.

METHODS

Facilitated by expert discussions, items from previous studies were developed further and assigned to either a conceptual scale or procedural scale. Confirmatory factor analysis was used to investigate the underlying structure of the data including model comparisons (1-dimensional; conceptual-procedural, verbal-non-verbal). Further analyses in terms of validation focused on reliability and on correlations of the knowledge types with general cognitive abilities.

RESULTS

It was found that the theoretically assumed 2-dimensional model fitted the data best. Correlations of the two knowledge types with general cognitive abilities differed significantly. Furthermore, the latent constructs could be reliably estimated from its indicators.

CONCLUSIONS

Our findings indicate that the empirical separation of conceptual and procedural fraction knowledge is possible: A theoretically grounded test instrument was developed that allows measuring the knowledge types with a sufficient degree of validity. These findings address a research gap that was pointed out repeatedly and gives rise to further research into reasons and remedies for students' difficulties in dealing with fractions.

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

Adição de frações por estimativa a partir do referencial de metade e de inteiro

O presente estudo investigou o papel desempenhado pelos referenciais de inteiro e de metade na resolução de adição de frações por estimativa. Crianças (8-9 anos) de classe média alunas do ensino fundamental resolveram as mesmas adições de frações em diferentes situações: por estimativa, usando o referencial de metade (Tarefa 1), por estimativa, usando o referencial de inteiro (Tarefa 2) e utilizando o simbolismo matemático apenas (Tarefa 3). Observou-se que embora não resolvessem as adições por meio do simbolismo matemático, as crianças faziam estimativas apropriadas com base no referencial de inteiro e de metade, tendo o mesmo bom desempenho com ambos os referenciais. Concluiu-se que pontos de referência desempenham papel importante no raciocínio lógico-matemático.