Examining tenth grade students' ways of thinking and ways of understanding according to DNR system in context of circle

The DNR system, introduced by Harel (2008), is a conceptual framework that deals with teaching and learning of mathematics. In this framework, Harel (2008) introduces the triad of determinants, Mental Act-Ways of Understandings-Ways of Thinking, to analyze students' acts of a particular mental act. The notions of way of understanding and way of thinking simply can be thought as two different categories of knowledge. Ways of understanding refer to products, such as definitions, conjectures, theorems, proofs, problems, and solutions, whereas ways of thinking refer to the mathematical practices used to create such products. Examples of ways of thinking include empirical reasoning, deductive reasoning, structural reasoning, heuristics, and beliefs about the nature of mathematical knowledge and the process of its acquisition (Harel, 2013). The purpose of this study is to investigate tenth grade sutends' ways of thinking and ways of understanding within the framework of DNR in the context of circle. In this qualitatively designed study, the data was collected through a focus group study with four high school students. They answered four circle problems designed by Harel, Rabin, Stevens ve Fuller (2008). The data was analyzed qualitatively by using content analysis technique. Findings indicated that students used algebraic way of thinking as well as geometric way of thinking.