Conceptual explanations and understanding fraction comparisons (original) (raw)
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We present the results of an in-depth qualitative study that examined ninth graders' conceptual and procedural knowledge of fractions as well as their approach to mathematics learning, in particular fraction learning. We traced individual differences, even extreme, in the way that students combine the two kinds of knowledge. We also provide preliminary evidence indicating that students with strong conceptual fraction knowledge adopt a deep approach to mathematics learning (associated with the intention to understand), whereas students with poor conceptual fraction knowledge adopt a superficial approach (associated with the intention to reproduce). These findings suggest that students differ in the way they reason and learn about fractions in systematic ways and could be used to inform future quantitative studies.
Drawing on a Theoretical Model to Study Students’ Understandings of Fractions
Educational Studies in Mathematics, 2007
Teaching and learning fractions has traditionally been one of the most problematic areas in primary school mathematics. Several studies have suggested that one of the main factors contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (i.e., part-whole, ratio, operator, quotient, and measure). Kieren was the first to establish that the concept of fractions is not a single construct, but consists of several interrelated subconstructs. Later on, in the early 1980s, Behr et al. built on Kieren’s conceptualization and suggested a theoretical model linking the five subconstructs of fractions to the operations of fractions, fraction equivalence, and problem solving. In the present study we used this theoretical model as a reference point to investigate students’ constructions of the different subconstructs of fractions. In particular, using structural equation modeling techniques to analyze data of 646 fifth and sixth graders’ performance on fractions, we examined the associations among the different subconstructs of fractions as well as the extent to which these subconstructs explain students’ performance on fraction operations and fraction equivalence. To a great extent, the data provided support to the associations included in the model, although, they also suggested some additional associations between the notions of the model. We discuss these findings taking into consideration the context in which the study was conducted and we provide implications for the teaching of fractions and suggestions for further research.
Individual Differences in Conceptual and Procedural Knowledge When Learning Fractions
Previous research on children's conceptual and procedural understanding of fractions, and other arithmetic skills, has led to contradictory conclusions. Some research suggests that children learn conceptual knowledge before procedural knowledge, some suggests that they learn procedural knowledge before conceptual knowledge, and other research suggests that they learn conceptual knowledge and procedural knowledge in tandem. We propose that these contradictory findings may be explained by considering individual differences in the way that children combine conceptual and procedural knowledge. A total of 318 Grade 4 and 5 students in the United Kingdom (mean age ϭ 9.0 years) completed a measure of fractions understanding, which included subscales of conceptual and procedural knowledge. A cluster analysis identified 5 distinct clusters that differed from each other in terms of their relative success with conceptual and procedural problems. The existence of these clusters suggests that there may be more than one way in which children draw on conceptual and procedural knowledge. Some children rely more on procedural knowledge and others more on conceptual knowledge, but these differences may not be related to developmental processes. The children in the 5 clusters differed in their total fractions performance and in their understanding of intensive quantities. These differences suggest that children who rely on conceptual knowledge may have an advantage compared to those who rely more exclusively on procedural knowledge.
Reciprocal relations of relative size in the instructional context of fractions as measures
Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education, 2016
The presented study is part of a bigger design and research enterprise in the teaching of fractions as measures. We analyze extracts of a teaching session with a single fifth grade student, in which he flexibly compared the relative sizes of the lengths of three drinking straws, skillfully using unitary, proper, and improper fractions. We identify aspects of his prior instructional experiences that supported the emergence of his relatively sophisticated ways of reasoning. Findings suggest that supporting students' reasoning about reciprocal relations of relative size can be a viable goal in an instructional agenda on fractions as measures. Cortina, J. L. & Visnovska, J. (in press, 2016). Reciprocal relations of relative size in the instructional context of fractions as measures. To be published in Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education, Szeged, Hungary: IGPME.
International Electronic Journal of Mathematics Education
There is a general consensus that both conceptual and procedural knowledge are essential for students' mathematical development. A common argument is that differences in mathematical performance are caused by differences in conceptual and procedural knowledge. Therefore, it is important to investigate to what extent such differences in conceptual and procedural knowledge are empirically evident at the level of individual students. Accordingly, the aim of the present study is to describe individual differences in conceptual and procedural knowledge using the example of fractions and to analyze their relationship to the covariates grade level, school type, school, class, gender, and general cognitive abilities. Data from 377 students in grades 8 and 9 from 18 classes at four schools in Germany was examined. A hierarchical cluster analysis showed five clusters which reflected individual differences in conceptual and procedural knowledge. The clusters were characterized by (a) equal strengths in conceptual and procedural knowledge, (b) relative strengths in procedural knowledge compared to conceptual knowledge, (c) relative weaknesses in procedural knowledge compared to conceptual knowledge. Cluster membership was not related to gender or grade level, whereas the school type, school, and grade level were relevant for cluster membership. A stronger correlation between conceptual knowledge and general cognitive abilities could only be confirmed to a limited extent.
Supporting students to reason about the relative size of proper and improper fractions
In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins, (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia), pp. 181-188. Sunshine Coast: MERGA., 2015
Fractions are a well-researched area, yet, student learning of fractions remains problematic. We outline a novel path to initial fraction learning and document its promise. Building on Freudenthal’s analysis of the fraction concept, we regard comparing, rather than fracturing, as the primary activity from which students are expected to make sense of fractions. Analysing a classroom design experiment conducted with a class of 14 fourth grade pupils, we identify two successive mathematical practices that emerged in the course of the experiment and indicate how their emergence was supported.
The impact of fraction magnitude knowledge on algebra performance and learning
Journal of Experimental Child Psychology, 2014
Knowledge of fractions is thought to be crucial for success with algebra, but empirical evidence supporting this conjecture is just beginning to emerge. In the current study, Algebra 1 students completed magnitude estimation tasks on three scales (0-1 [fractions], 0-1,000,000, and 0-62,571) just before beginning their unit on equation solving. Results indicated that fraction magnitude knowledge, and not whole number knowledge, was especially related to students' pretest knowledge of equation solving and encoding of equation features. Pretest fraction knowledge was also predictive of students' improvement in equation solving and equation encoding skills. Students' placement of unit fractions (e.g., those with a numerator of 1) was not especially useful for predicting algebra performance and learning in this population. Placement of nonunit fractions was more predictive, suggesting that proportional reasoning skills might be an important link between fraction knowledge and learning algebra.
Levels of students’ “conception” of fractions
Educational Studies in Mathematics
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.
Are conceptual knowledge and procedural knowledge empirically separable? The case of fractions
British Journal of Educational Psychology
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. Developing fraction knowledge is crucial for students' success in more advanced