# The Examination of Representations Used by Classroom Teacher Candidates in Solving Mathematical Problems

## Article excerpt

The concept of using multiple representations has an important place in the teaching of mathematics (Cai, 2005; Cobb, Yackel, & Wood, 1992; Goldin, 1998; Janvier, 1987), especially in the understanding of mathematical concepts and interpreting them from different points of view through the use of multiple representations (Cathcart, Pothier, Vance, & Bezuk, 2006; Hjalmarson, 2007; Pape & Tchoshanov, 2001). According to Tripathi (2008), the use of multiple representations in teaching mathematics is a strong instrument that eases the understanding of mathematical subjects for students. Also, the use of multiple representations strengthens the understanding of students for learning how to form and solve problems in a mathematics course.

The new standards of the American National Council of Teachers of Mathematics (NCTM) published in 2000 especially emphasize the importance of the concept of "representation." Within this context, according to the NCTM, the use of diagrams, graphics, tables and symbols, as well as transitioning between them, is of capital importance in expressing mathematical thoughts and relations. Stemming from the NCTM (2000) standards, representations are also a part of the required abilities in the mathematics curriculum of Turkey for use in problem solving, communication (verbal lectures, written statements, images, graphics, concrete concepts), and associations (Milli Egitim Bakanligi [MEB], 2005, 2009). Stylianou (2010) also states that students should be effective in using representations and their transitions for solving mathematics and understanding mathematical concepts.

As a general term, a representation is a way to show an actual situation from a different point of view (Even, 1998; Goldin & Kaput, 1996). In mathematics, however, teaching representation is a part of forming or shaping a mathematical concept. The different representations that teachers use during in-class activities affect the knowledge, and accordingly, the success of students (Cai, 2005; Neria & Amit, 2004; Stylianou & Silver, 2004). That is why the understanding of representations and how to use different types of representations should be an active part of the teaching process (Hjalmarson, 2007; Pape & Tchoshanov, 2001). Researchers mostly focus on two different representation types regarding the classification of representations. These are internal and external representations (Cai, 2005; Goldin, 1998; Goldin & Shteingold, 2001). Within this scope, internal representations are defined as those that express a reality using mental models (Cai, 2005; Hiebert & Carpenter, 1992), cognitive diagrams developed via experiences, or abstractions of mathematical thoughts (Pape & Tchoshanov, 2001). External representations are the expression of a person's thoughts regarding a certain reality by use of visual objects (Cai, 2005) or the use of written or verbal words (Goldin & Shteingold, 2001) involving numbers, algebraic equations, graphics, tables, diagrams or charts (Pape & Tchoshanov, 2001). Also, concrete structures such as tables, graphics, images, diagrams that are used in problem solving, or the defining of mathematical concepts are considered to be external representations (Goldin & Janvier, 1998). In order to deepen the understanding of students, teachers must define a concept by using the multiple representation method, or more plainly, using different types of representations. Instead of using just one model, teachers must present a concept by using different representations and then make suitable transitions between them (Ball, 1990).

Representations, which are effective in forming or shaping a mathematical concept (Goldin & Shteingold, 2001), are part of the abilities required in mathematics curriculum, including problem solving, communication and association. Accordingly, in the problem solving process, the path of solving the problem must be given importance. …

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