Procedural Knowledge Of Fractions Research Articles

Exploring the Relationship Between Initial Mathematics Skill and a Sixth-Grade Fractions Intervention

This study explored whether initial skill moderated outcomes of Promoting Algebra Readiness, a Tier 2 sixth-grade mathematics intervention targeting conceptual and procedural knowledge of fractions. The study analyzed data from a quasi-experimental pilot study in which at-risk students ( N = 198) from Oregon middle schools were assigned to the treatment or control condition at the school level. Proximal and distal measures of math proficiency were collected in the fall (pretest) and spring (post-test). Analyses examined initial student achievement as a moderator of mathematics outcomes. Results indicated that intervention outcomes were not moderated by initial skill. Implications for tiered mathematics instruction and future mathematics intervention research are discussed.

Topic specificity of students’ conceptual and procedural fraction knowledge and its impact on errors

ABSTRACT When learning about fractions, teaching usually follows certain topics – e.g. part of the whole, fraction equivalence, and size comparison. Not yet answered is whether students’ conceptual and procedural knowledge of fractions is coherent between those topics. To answer this question, we performed a latent profile analysis of data from N = 1005 six-grade students regarding their conceptual and procedural knowledge in the three above-mentioned topics. Our results confirmed that the level of conceptual and procedural fraction knowledge can vary within students between topics. Furthermore, the analysis of students’ errors showed that conceptual knowledge of fractions within a specific topic is related to topic-specific error types – with conceptual errors (compared to non-conceptual errors) being demonstrated relatively more often by students with low conceptual knowledge. This indicates that to derive adequate support strategies for individual students, differentiated consideration of their topic-specific conceptual and procedural knowledge should be considered.

Individual Differences in Conceptual and Procedural Fraction Knowledge: What Makes the Difference and What Does it Look Like?

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.

A Study on Congruency Effects and Numerical Distance in Fraction Comparison by Expert Undergraduate Students

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants’ performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants’ accuracy rates or response times (although individual-level data suggest that there is an effect for response times). When fractions shared a common component, instead, our data display a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts’ reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.

SRSD Fractions: Using Self-Regulated Strategy Development to Support Students’ Conceptual and Procedural Fraction Knowledge

Fractions are a cornerstone skill for student success in higher-level mathematics and many practical skills. However, most students do not have proficient fraction conceptual or computational knowledge. Effective and efficient interventions are needed to remediate these skills, especially for students with or at-risk for disabilities. We present a curricular overview of a self-regulated strategy development (SRSD) framework, SRSD Fractions. SRSD Fractions addresses adding and subtracting fractions with unlike denominators, simplifying fractions, and converting fractions to mixed numbers. We summarize research on SRSD Fractions and lessons learned, including implications for future research and practice, especially for students with comorbid behavior and academic challenges.

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

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

Theoretical and methodological issues in the study of conceptual and procedural knowledge: Reflections on a series of studies on Greek secondary students’ knowledge of fractions

In this article we present an overview of four studies investigating Greek secondary students’ conceptual and procedural knowledge of fractions. We discuss the problem of defining conceptual and procedural knowledge, and the implications of adopting one particular definition over others. We draw on the studies and their results to discuss the problem of measuring conceptual and procedural knowledge; the issue of inter-individual differences in conceptual and procedural fraction knowledge; and related educational implications.

Blogging mathematics: Using technology to support mathematical explanations for learning fractions

Explanations with others help students learn yet little is known about how technology can support and augment these benefits. This paper describes an experiment that compared the effects of mathematical discourse (i.e., explaining, justifying, and arguing) with peers either face-to-face or using technology (a blog) on fraction learning. We hypothesized that blogs may provide benefits beyond face-to-face collaborations because a record of explanations is accessible for subsequent reflection, which allows students to revisit and revise their explanations. A quasi-experimental design with 134 fifth grade students (ages 9–11) was used to investigate the change in conceptual and procedural knowledge of fractions measured as change from pretest to posttest and delayed posttest. The results indicated that students in the blog condition showed the largest gains in conceptual knowledge from pretest to posttest and at delayed posttest. There was no significant difference between groups on procedural knowledge. The results suggest that the use of blogs may provide unique supports for learning because students are provided opportunities to explain, justify, and argue their thinking, as well as critique the reasoning of others through an interactive learning environment that affords the opportunity to clarify understandings and misconceptions that may not otherwise exist in a traditional face-to-face learning environment.

From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions

To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers—relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)—can be linked to early success with algebraic expressions.

Development of fraction concepts and procedures in U.S. and Chinese children

We compared knowledge of fraction concepts and procedures among sixth and eighth graders in China and the United States. As anticipated, Chinese middle school children had higher knowledge of fraction concepts and procedures than U.S. children in the same grades, and the difference in procedural knowledge was much larger than the difference in conceptual knowledge. Of particular interest, national differences in knowledge of fraction concepts were fully mediated by differences in knowledge of fraction procedures, and differences between the knowledge of Chinese and U.S. children were most pronounced among the lowest achieving children within each country. Based on these and previous results, a theoretical model of the mutually facilitative interaction between conceptual and procedural knowledge of fractions is proposed and discussed.

Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge?

This research analyses preservice teachers’ knowledge of fractions. Fractions are notoriously difficult for students to learn and for teachers to teach. Previous studies suggest that student learning of fractions may be limited by teacher understanding of fractions. If so, teacher education has a key role in solving the problem. We first reviewed literature regarding students’ knowledge of fractions. We did so because assessments of required content knowledge for teaching require review of the students’ understanding to determine the mathematics difficulties encountered by students. The preservice teachers were tested on their conceptual and procedural knowledge of fractions, and on their ability in explaining the rationale for a procedure or the conceptual meaning. The results revealed that preservice teachers’ knowledge of fractions indeed is limited and that last-year preservice teachers did not perform better than first-year preservice teachers. This research is situated within the broader domain of mathematical knowledge for teaching and suggests ways to improve instruction and student learning.

The contribution of domain-specific knowledge in predicting students' proportional word problem-solving performance

This study explored the extent to which domain-specific knowledge predicted proportional word problem-solving performance. We tested 411 seventh-grade students on conceptual and procedural fraction knowledge, conceptual and procedural proportion knowledge, and proportional word problem solving. Multiple regression analyses indicated that all four domain-specific knowledge variables (i.e., conceptual and procedural fraction knowledge, conceptual and procedural proportion knowledge) significantly predicted proportional word problem-solving performance. Conceptual fraction and procedural proportion knowledge contributed the most unique variance (10.0 and 6.7%, respectively, of the total variance) to proportional word problem solving. Procedural fraction and conceptual proportion knowledge each also contributed significant unique variance to proportional word problem solving explaining 5.6 and 2.8%, respectively. The results support the notion that both conceptual fraction and proportion knowledge and procedural fraction and proportion knowledge play a major role in understanding individual differences in proportional word problem-solving performance to inform interventions.

Preservice Teachers’ Conceptual and Procedural Knowledge of Fraction Operations: A Comparative Study of the United States and Taiwan

This study examined (a) the differences in preservice teachers’ procedural knowledge in four areas of fraction operations in Taiwan and the United States, (b) the differences in preservice teachers’ conceptual knowledge in four areas of fraction operations in Taiwan and the United States, and (c) correlation in preservice teachers’ conceptual knowledge and procedural knowledge of fractions in Taiwan and the United States. Participants were preservice teachers (N = 49) in a teacher education program in the United States and comparable Chinese preservice teachers (N = 47). Results indicated that Chinese preservice teachers performed better in procedural knowledge on fraction operations than American preservice teachers. No significant differences were found for conceptual knowledge on fraction division. Further, the correlation in this study showed that for Chinese and American preservice teachers, the relationship between conceptual and procedural knowledge of fraction operations was weak.

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

Recent research on children’s conceptual and procedural knowledge has suggested that there are individual differences in the ways that children combine these two types of knowledge across a number of mathematical topics. Cluster analyses have demonstrated that some children have more conceptual knowledge, some children have more procedural knowledge, and some children have an equal level of both. The current study investigated whether similar individual differences exist in children’s understanding of fractions and searches for explanations for these differences. Grade 6 students (n=119) and Grade 8 students (n=114) were given measures of conceptual and procedural knowledge of fractions as well as measures of general fraction knowledge, general conceptual ability, and general procedural ability. Grade 6 children demonstrated a four-cluster solution reflecting those who do poorly on procedural and conceptual fraction knowledge, those who do well on both, those whose strength is procedural knowledge, and those whose strength is conceptual knowledge. Grade 8 children demonstrated a two-cluster solution reflecting those whose strength is procedural knowledge and those whose strength is conceptual knowledge. Cluster in either grade, however, did not vary in distribution across schools and was not related to general conceptual ability or general procedural ability. Overall, these results provide a more detailed picture of individual differences in conceptual and procedural knowledge in mathematical cognition.

Mathematics anxiety in preservice teachers: Its relationship to their conceptual and procedural knowledge of fractions

The objective of the present study was to examine the relationship between mathematics anxiety and the procedural and conceptual knowledge of fractions in prospective teachers. Thirty-two preservice teachers enrolled in an elementary mathematics methods course were administered the Revised Mathematics Anxiety Rating Scale (RMARS, Baloglu, 2002). Procedural and conceptual knowledge of fractions was assessed with a validated paper and pencil test (Saxe, Gearhart, & Nasir, 2001) and with four additional problems created by the researchers. Results indicated that as mathematics anxiety scores increased, scores on the validated measure of procedural knowledge of fractions decreased. The same relationship was found between the RMARS and the validated measure of conceptual knowledge. The findings provide some insight on the cognitive and pedagogical factors associated with mathematics anxiety in the preservice teacher population, underscoring the importance of facilitating their proficiency in both mathematical procedures and concepts.

Introduction: Measuring Instruction

Introduction: Measuring Instruction