An Exploratory Study of the Impact of a Manipulatives-Intensive Fractions Unit during a Middle Grades Methods Course on Prospective Teachers’ Relational Understanding of Fractions

Abstract

“A critical pillar of a strong PreK–12 education is a well-qualified teacher in every classroom” (Conference Board of Mathematical Sciences, 2012, p. 1). In order for teachers to help students build procedural fluency from conceptual understanding and meet the standards set forth by governing agencies, prospective teachers will need deep conceptual understanding of the mathematics they teach and experiences learning mathematics in ways that they will be expected to teach (Kilpatrick, Swafford, & Findell, 2001). However, at least one study has suggested that there exists a preparation gap that may contribute to a lack of student achievement (Schmidt et al., 2007). Furthermore, prospective teachers who will likely be expected to use manipulatives in their future teaching practice may have limited experience using those materials to demonstrate their knowledge. Data from the National Assessment of Educational Progress over the last decade indicates that student achievement in Grades 4 and 8 in the Number & Operations domain has only increased by seven points and two points, respectively (https://nces.ed.gov/). Since fraction proficiency is thought to be a predictor of later success in algebra (Booth, Newton, & Twiss-Garrity, 2014; National Mathematics Advisory Panel, 2008; Usiskin, 2007), further study on rational numbers may be beneficial. This study, which was conducted at a mid-sized, four-year university in the southeastern United States, used a case-study design to examine the impact of a manipulatives-intensive middle grades mathematics methods course on prospective teachers’ procedural and conceptual knowledge of fraction multiplication and division and the connections between the two types of knowledge, i.e. relational understanding (Skemp, 1987). Data for four participants were collected through tests of knowledge; observations; various assessments given by the instructor of the course; and one-on-one, task-based and semi-structured interviews.