Investigation into How 8th Grade Students Define Fractals

Article excerpt

Recently, fractals began to emerge in mathematics and mathematics education. Studies of the learning and teaching of fractals (e.g., Fraboni & Moller, 2008; Goldenberg, 1991; Kern & Mauk, 1990; Naylor, 1999) often included activities that can be used in the classroom. However, there are few studies (e.g., Bowers, 1991; Bremer, 1997; Hughes, 2003; Karakus, 2011; Komorek, Duit, Bücker, & Naujack, 2001; Langille, 1996; Murratti & Frame, 2002) relating to how students understand fractals and the kinds of difficulties they face when learning them. One way to determine the students' understanding about fractals is to examine how students define them. Determining the students' concept definitions and concept images can provide information about their mental schema regarding fractals. In this context, students' definitions were focused to fractals.

Definitions in Mathematics Education

Definitions are considered fundamental in mathematics and mathematics education. National Council of Teachers of Mathematics (2000) emphasizes the importance of the students' perception of the roles definitions play, and the usage of conceptual definitions in mathematical studies starting in middle grades. Mathematical definitions have an important role in the concretization of a defined and exact concept, along with an understanding of the concept as powerful (Edwards & Ward, 2008). Tall and Vinner (1981) have dealt with the process of defining concept in students' learning of mathematics. Their model of concept image and concept definition provides the basis for analyzing students' representations of mathematical concept. Concept image is defined as, "to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and process." (Tall & Vinner, 1981, p. 152). Students' experiences are essential in the formation of a concept image. For example, if a student observes the perimeter of a rectangle increasing, he can surmise that if the perimeter of a rectangle increases then the area always increases also. For such a student this observation is part of his concept image and may cause problems when he encounters a situation where, as the perimeter increases, the area can reduce or remain fixed. Concept definition is defined as, "to be a form of words used to specify that concept." (Tall & Vinner, 1981, p. 152). Concept definition can be separated into two parts; formal concept definition and personal concept definition. Formal concept definition, which is an accurate explanation of the concept, is accepted by the mathematical community at large (Tall & Vinner, 1981). However, personal concept definition is the students' personal reconstruction of the definition (Tall & Vinner, 1981). Personal concept definition is part of concept image and, unlike formal definition; it is the student's alternative definition about a concept. Vinner (2002) suggests four situations for the relationship between concept definition and concept image. The focal point of the first three of these (see Figure 1-3) is the concept definition. In these situations, a mathematical task, such as proving a theorem, is completed in a mathematically acceptable way. On the other hand, in the last case (see Figure 4), the concept definition is not consulted during the problem-solving process and is not seen as mathematically acceptable.

For a student, a mathematical definition of the concept depends on what he/she accepts as a definition. In examining the students' understanding of integral, Rösken and Rolka (2007) indicated that students were quite unsuccessful in determining the formal definition of the integral, but their concept image was more dominant in their conceptual learning. This study shows that students' concept images are more effective than their formal concept definitions in conceptual learning. Examples, counterexamples and experiences are very important in the formation of students' personal concept definitions Wilson, (1990). …

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