Individual differences in conceptual and procedural knowledge when learning fractions.

Abstract

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.

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

“…The 'part-whole' sub-construct refers to a continuous quantity, a set or an object divided into parts of equal size Lamon, 97 1999). The 'ratio' sub-construct concerns the notion of a comparison between two quantities and as such, it is considered to be a comparative index rather than a number (Hallett, Nunes, & Bryant, 2010;. The 'operator' sub-construct comprises the application of a function to a number, an object or a set.…”

Teaching Fractions in Elementary School

“…The 'part-whole' sub-construct refers to a continuous quantity, a set or an object divided into parts of equal size . The 'ratio' sub-construct concerns the notion of a comparison between two quantities and as such, it is considered to be a comparative index rather than a number (Hallett, Nunes, & Bryant, 2010;. The 'operator' sub-construct comprises the application of a function to a number, an object or a set.…”

“…(2) The 'ratio' sub-construct concerns the notion of a comparison between two quantities and as such, it is considered to be a comparative index rather than a number (Carraher, 1996;Hallett, Nunes, & Bryant, 2010;. The orange juice experiment by has been widely used to measure students' understanding of this sub-construct (e.g., John and Mary are making lemonade.…”

Cognitive Factors' Influence on Fraction Development in Students of Varying Ability Levels

“…These clusters differed in the magnitude of differences between expected procedural and conceptual understanding but again replicated four of the five clusters described by Hallet et al (2010). For fifth-grade students the one cluster that was not replicated was the lower conceptual (average procedural) cluster.…”

“…Alternatively the student could have difficulty figuring out that she should actually be looking for 75% of the $25 shirt. Students with working memory issues have difficulty managing the competing facts and procedures that need to be done to As working memory has the potential to disrupt student acquisition of fraction knowledge, perhaps differences in working memory or other similar cognitive factors can explain the different learning patterns found by Hallet et al (2010) and Hecht an Vagi (2012). …”