A Cross-age study of students' understanding of fractals

Abstract

The purpose of this study is to examine how students understand fractals depending on age. Students' understandings were examined in four dimensions: defining fractals, determining fractals, finding fractal patterns rules and mathematical operations with fractals. The study was conducted with 187 students (grades 8, 9, 10) by using a two-tier test consisting of nine questions prepared based on the literature and Turkish mathematics and geometry curriculums. The findings showed that in all grades, students may have misunderstandings and lack of knowledge about fractals. Moreover, students can identify and determine the fractals, but when the grade level increased, this success decreases. Although students were able to intuitively determine a shape as fractal or not, they had some problems in finding pattern rules and formulizing them.Keywords: Fractal Geometry. Students' Understandings. Cross-age Study. elaboradas por meio da literatura nos currículos turcos em matemática e geometria. Os resultados mostraram que em todos os graus, os alunos podem ter falta e conhecimentos equivocados sobre fractais. Além disso, os estudantes podem identificar e determinar os fractais, mas quando o nível de ensino aumenta, a identificação e a determinação dos fractais diminui. Embora, os alunos pudessem determinar intuitivamente uma forma como fractal ou não, eles tiveram alguns problemas em encontrar regras e padrões na formulação dos fractais. ResumoPalavras-chave: Geometria Fractal. Entendimentos dos Alunos. Faixa-etária do Estudo.

Investigation into how 8th Grade Students Define Fractals

“…In this context, it can be stated that students have difficulty in understanding the concept of a fractal. Similarly, in their studies, Bowers (1991), Langille (1996), Komorek et al (2001), Murratti and Frame (2002), Karakuş (2013), and Karakuş and Karataş (2014) stated that students experienced difficulties defining fractals and specifying their characteristics. The most common inaccurate fractal definition was: "fractal is an iterated/recursive shape. "…”

“…In the first subsection, (Fraboni & Moller, 2008;Goldenberg, 1991;Kern & Mauk, 1990;Naylor, 1999) activities were frequently developed for the teaching and learning of fractals for teachers to apply in the classroom. Other studies (e.g., Bowers, 1991;Bremer, 1997;Günay & Kabaca, 2013;Hughes, 2003;Karakuş, 2011Karakuş, , 2013Karakuş & Karataş, 2014;Komorek et al, 2001;Langille, 1996;Murratti & Frame, 2002) focused on how fractals could be integrated into the existing mathematics curriculum, the difficulties faced in the teaching and learning of fractals, and the effect of fractals on changing attitudes towards mathematics. For example, Bowers (1991) determined that students learning fractals have difficulties in three specific areas.…”

“…The definition used for deciding about selfsimilarity, by comparing any part of the object and entire object, was deemed inadequate. Karakuş (2013) examined elementary and secondary school students' understanding about fractals depending on age. He found that some lack of knowledge and misunderstanding about fractals existed in all (grades 8-10) grades.…”

A Learning Trajectory to the Understanding of the Curve Length Concept

“…What is observed nowadays is that, in relation to the Calculus basic concepts, there is still a valuation of technical procedures in detriment of the exploitation of intuitive capacity and graphic representation. Research results as in Vinner (1989), Tall (1994), Meyer and Igliori (2003), Bossé and Bahr (2008), Nasser (2009), Cury (2009), Karatas, Guven and Cekmez (2011), Silva (2011), Rasmussen, Marrongelle and Borba (2014), among others, point to the difficulties in the teaching process of Calculus and the obstacles for the comprehension of the concepts. Yet, they indicate that the students have a better performance when doing activities in which there is the prevalence of exercises that focus on operational and technical aspects.…”

How Does Pre-Service Mathematics Teacher Prove the Limit of a Function by Formal Definition?

“…Teaching and learning the concept of limit have been important and interesting research subjects in mathematics education. The results of the study on students' understanding on the concept of limit show that students have delicate conceptual representation on the limit (Davis & Vinner, 1986;Karatas et al, 2011;Beynon & Zollman, 2015). It is problematic for most students to apply the concept of limit intuitively into a formal concept.…”