Using Open-Response Fraction Items to Explore the Relationship Between Instructional Modalities and Students’ Solution Strategies

www.ijemst.com Using Open-Response Fraction Items to Explore the Relationship Between Instructional Modalities and Students’ Solution Strategies ISSN: 2147-611X Jessica F. Shumway1, Patricia S. Moyer-Packenham1, Joseph M. Baker2, Arla Westenskow1, Katie L. AndersonPence3, Stephen I. Tucker4, Jennifer Boyer-Thurgood1, Kerry E. Jordan1 1 Utah State University 2 Stanford University 3 University of Colorado, Colorado Springs 4 Virginia Commonwealth University To cite this article: Shumway, J.F., Moyer-Packenham, P.S., Baker, J.M., Westenskow, A., Anderson-Pence, K.L., Tucker, S.I., Boyer-Thurgood, J., & Jordan, K.E. (2016). Using open-response fraction items to explore the relationship between instructional modalities and students’ solution strategies. International Journal of Education in Mathematics, Science and Technology, 4(2), 112-132. DOI:10.18404/ijemst.20845 This article may be used for research, teaching, and private study purposes. 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Jordan Article Info Abstract Article History The purpose of this study was to explore the relationship between instructional modality used for teaching fractions and third- and fourth-grade students’ responses and strategies to open-response fraction items. The participants were 155 third-grade and 200 fourth-grade students from 17 public school classrooms. Students within each class were randomly assigned to two instructional treatment groups: a virtual manipulatives representations (VMR) instruction group and a physical manipulatives and textbook representations (PMTR) instruction group. A conversion mixed methods analysis was used to examine quantitative and qualitative data. The quantitative analysis showed achievement outcomes were the same for both groups. The qualitative analysis revealed shifts in learning that were otherwise hidden with solely quantitative achievement results. Specifically, the results indicated VMR group success in understanding fractions as relationships and PMTR group success in maintaining conceptualization of the whole. Overall, the results of this study corroborate previous research indicating the importance of both types of instructional modalities, showing that virtual manipulatives and physical manipulatives are effective instructional tools with positive effects on student learning. The study expands existing research by offering an opportunity to explore the nuances of students’ fractions understanding and provide a window into students’ shifts in fraction learning. Received: 17 February 2015 Accepted: 29 September 2015 Keywords Virtual manipulatives Physical manipulatives Fractions Instructional modalities Open-response items Introduction Elementary teachers use a variety of instructional modalities when teaching children early fraction concepts. Their instruction often includes physical, pictorial, and symbolic representations. Some teachers use virtual manipulatives (Moyer, Bolyard, & Spikell, 2002), which combine representations (e.g., pictorial and symbolic) and representational modalities (e.g., visual and haptic). Studies indicate that using multiple representations and modalities in fraction instruction develops and expands students’ understanding of fractions (Behr, Lesh, Post, & Silver, 1983; Moyer-Packenham & Westenskow, 2013; Sowell, 1989). The purpose of this study was to explore the relationship between instructional modalities used for learning fraction concepts—specifically using virtual manipulatives or physical manipulatives with textbooks—and students’ solution strategies on open-response fraction items. We employed a conversion mixed methods approach (Teddlie & Tashakkori, 2006) to analyze open-response items, which we coded and quantitized for quantitative and qualitative analysis. Open-response items provide windows into students’ thinking processes and strategies for solving mathematics tasks (Cai, 2000; Cai, Magone, Wang, & Lane, 1996; Lane, 1993). This study complements and extends previous studies by using open-response items to examine these phenomena in depth using qualitative analysis with a large sample of participants (n = 355). The study was framed as a comparison between the learning outcomes of two groups of students using different modalities for learning fraction concepts (i.e., virtual and physical manipulatives). As you will read, our MannWhitney U analysis corroborated prior research (e.g., Burns & Hamm, 2011; Manches et al., 2010; Melideo & Dodson, 2009; Mendiburo & Hasselbring, 2011; Moyer-Packenham et al., 2013) indicating no numerical achievement differences between the groups. Hence, in the paper, we aimed to explore the more nuanced patterns in students’ responses and strategies through a qualitative analysis. We selected specific student work Int J Educ Math Sci Technol 113 examples for the Results section to highlight patterns and interesting features of students’ responses and strategies on these open-response items. The examples we selected highlight key themes that emerged in our analyses, namely, shifts in learning from pretest to posttest and small differences between the groups’ responses and strategies. Furthermore, through the process of this rigorous qualitative analysis of 355 students’ strategies, we developed a classification scheme of the strategies that emerged (see Appendix A), which we anticipate will be helpful to the research community. Representations, Instructional Modalities, and Fraction Learning As children develop their understandings of number and quantities from whole numbers to rational numbers, they often struggle with understanding that a fraction represents a relationship. Children have difficulty understanding the meaning of the denominator, keeping track of the whole, and thinking multiplicatively (Behr & Post, 1992; Kamii & Clark, 1995; Smith, 2002). To help children overcome these challenges, representations are often at the heart of teaching and learning the persistently difficult concept of fractions . Research (e.g., Cramer, Post, & delMas, 2002; Sowell, 1989) and mathemat (...truncated)