Incorporating Representation-Based Instruction Into Mathematics Teaching: Engaging Middle Schoolers With Multiple Representations of Adding Fractions

Introduction

At first glance, mathematics can be seen as an abstract world involving concepts, algorithms and symbols without any connection with real world (e.g. Cramer, 2003). For this reason, researchers emphasize the necessity of teaching mathematics as a unified system of concepts and operations based on certain patterns and relationships that exist in the real world (Nair & Pool, 1991; Resnick & Ford, 1984). From this necessity, mathematics educators have paid attention to how to teach mathematics to the students effectively for a long time. In this sense, many researchers argue for the importance of using appropriate representations in teaching and learning of mathematics in terms of having complete understandings of mathematical concepts, expressing mathematical ideas and relationship between concepts (Dreyfus & Eisenberg, 1996; Duval, 2006; Goldin & Shteingold, 2001). Furthermore, it is emphasized that representing a mathematical concept in multiple ways and making flexible transitions within and between different modes of representations have critical importance in terms of reaching a perfect and complete internalization of a mathematical concept (Dufour-Janvier, Bednarz & Belanger, 1987; Kaput, Blanton & Moreno, 2008; Lesh, 1999; National Council of Teachers of Mathematics [NCTM], 2000). Hence, the use of representations has been a crucial topic in learning of mathematics over the past three decades in standards of school mathematics for developing students’ abilities to use appropriate representations and to make correct and robust translations among them (Common Core State Standards Initiative [CCSSI], 2010; Ministry of National Education [MoNE, 2013]; National Council of Teachers of Mathematics [NCTM], 2000; Van de Walle, Karp & Bay-Williams, 2010). However, studies focusing on students’ abilities in use of representations indicate that middle school students have inadequate knowledge and ability to select appropriate representations and to transform from one representation to the others (Gagatsis & Elia, 2004; Neria & Amit, 2004).

Students’ poor representation transformation skills adversely affect their mathematical learning and problem-solving performances (Gagatsis & Shiakalli, 2004; Lesh, Post & Behr 1987). In this sense, researchers have begun to incorporate multiple representation-based learning environments into middle school mathematics. They reached substantial empirical evidence for the positive effects of multiple representation-based instructions on students’ conceptual understanding (Çıkla, 2004; Schnotz & Bannert, 2003; Tunç-Pekkan, 2015) especially in selection and use of appropriate type of representations. In this sense, we think that it is important to understand the effects of multiple representation-based environments in mathematics classrooms. Hence, researchers who seek and design alternative pedagogical instructions can be more aware of what kind of learning occurred in representation transformation processes. We also think that researchers may have chance to support students’ learning via representations by choosing and utilizing an appropriate type of methods, manipulatives or activities.