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 s 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.     

Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction

As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.     

对称、倒数与取实数值的复有理分式--对数学问题1216的探研

从对称与倒数的角度出发 ,对数学问题 12 16作了较深入的探研和推广 .     

土壤无机磷形态生物有效性研究

盆栽试验和幼苗试验表明 ,土壤中无机磷形态的生物有效性有明显差异 ,其顺序为Ca2 -P >Al-P >Fe -P、Ca8-P >O -P、Ca10 -P ,但在不同的土壤条件下又有差异     

Gestures and conceptual integration in mathematical talk

Spontaneous gesture produced in conjunction with speech is considered as both a source of data about mathematical thinking, and as an integral modality in communication and cognition. The analysis draws on a corpus of more than 200 gestures collected during 3 h of interviews with prospective elementary school teachers on the topic of fractions. The analysis examines how gestures express meaning, utilizing the framework of cognitive linguistics to argue that gestures are both composed of, and provide inputs to, conceptual blends for mathematical ideas, and a standard typology drawn from gesture studies is extended to address the function of gestures within mathematics more appropriately. Electronic supplementary material  The online version of the original article (doi:) contains supplementary material, which is available to authorized users.     

Instructional practices related to prospective elementary school teachers’ motivation for fractions

This study was undertaken in order to better understand prospective elementary school teachers’ motivations for working with fractions before and after taking a course designed to deepen their understanding of mathematics, as well as what instructional practices might be related to any changes detected in their motivations. Eighty-five education students were given a motivation questionnaire at the beginning and end of the semester, and observations were made of the 9 days when fractions were taught. Three levels of teacher data were collected to understand instructional practices. Students’ ratings of the importance and usefulness of fractions (value), self-concept of ability, and anxiety were near the center of the scale at pre-test, with only value in the desired direction. At posttest, value and self-concept of ability increased while anxiety decreased, but these changes differed somewhat by instructor. In particular, reform-oriented practices, such as engaging students in high-level discourse, seemed to be associated with lowered anxiety.     

Conceptual explanations and understanding fraction comparisons

Explanations are used as indicators of understanding in mathematics, and conceptual explanations are often taken to signal deeper understanding of a domain than more superficial explanations. However, students who are able to produce a conceptual explanation in one problem or context may not be able to extend that understanding more generally. In this study we challenge the notion that conceptual explanations indicate general understanding by showing that – although conceptual explanations are strongly associated with correct answers – they are not employed equally across different contexts, and the highest performing students tend to use more general explanations, which may or may not be conceptual. Overall, our results suggest that explanations of fraction magnitudes follow a learning trajectory reflected in students’ accuracy and explanations: weak students focus on concrete, non-conceptual features, stronger students use concepts to explain their answers, and the highest performers tend to use general (but not necessarily conceptual) rules.     

Impacts of a university–school partnership on middle school students' fractional knowledge: A quasiexperimental study

In this quasiexperimental study, the authors investigated the effects of university within school partnership model, within which faculty members acted as teacher-researchers to improve fractional knowledge among middle school (Grades 5–8) students. Students in nine Grade 6 mathematics classes from two public middle schools in Turkey were assigned to two conditions: University within school model instruction and traditional instruction. Pre- and posttest data showed that the students exposed to instruction through the university within school partnership model significantly outperformed their traditional instruction peers on the fractions test. Results indicated that students made significant gains in fractional knowledge in the experimental classrooms and in different subgroup populations. It was suggested that a substantial amount of mathematical infusion through partnership could have a positive impact on middle school students' fractional knowledge. The educational implications of the study were also discussed.