Examining Students’ Mathematical Understanding of Geometric Transformations Using the Pirie-Kieren Model

Abstract

The content of curriculum designs and innovations made in the field of mathematics education in the last decade have been shaped with emphasis on the need for to have understanding while learning mathematics. For example, the recently updated curriculum for secondary mathematics in Turkey indicates that learning environments which not emphasize meaning or do not provide with an opportunity or possibility to create meaning from the mathematics that is being learned cannot properly meet the expectations of teaching (Ministry of Education [MoNE], 2013, p. 1). Similarly, according to the National Council of Teachers of Mathematics ([NCTM], 2000), students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge, (p. 20). How do then learn mathematics with understanding?Hiebert and Carpenter (1992) remind us that the major goal of mathematics education is to ensure mathematical understanding in the learning environment, and they indicate that researchers should focus on finding the answer to the question of how and in what ways make sense of mathematics. As widespread rhetoric, mathematical understanding is vital for teachers, researchers, and curriculum developers (Mousley, 2005, p. 553). When the nature of mathematical understanding is characterized, researchers can focus on studies that analyze students' understandings in detail, curriculum developers can design the curriculum according to the growth of mathematical understanding, and teachers can organize their teaching goals based on this growth (MacCullough, 2007). Researchers have endeavored to identify the complex nature of mathematical understanding with respect to the learning theories which they support and the interpretations they have made of the term understanding (Meel, 2003). An important part of this effort includes the work done by researchers to describe the development of students' mathematical understanding via mental processes based on the constructivist approach.One should consider the individual experiences, perceptions, and interactions carry out in their environment when analyzing the of mathematical understanding from the constructivist perspective (Pirie & Kieren, 1992). Many researchers have looked at learning from this perspective and provided distinct theories to characterize mathematical understanding (i.e., Davis, 1984; Sfard, 1991; Sierpinska, 1994; Skemp, 1978). One of these theories is the representation theory. This theory promotes the idea that understand mathematics as much as they can make sense of the different representations of mathematical concepts and build connections among these representations (Goldin, 2003; Hiebert & Carpenter, 1992; Janvier, 1987). Goldin (2003) defines the term representation as configuration of signs, characters, icons, or objects that can somehow stand for, or 'represent' something else (p. 276). Students work with representations of mathematical objects while dealing with mathematics across variety of situations (Duval, 2006). They can learn mathematics with understanding if they can accurately apply different representations of mathematical concept and construct the relationships between these representations (Lesh, Post, & Behr, 1987). Another theory that analyzes mathematical understanding from the constructivist perspective is the Pirie-Kieren theory, which was presented by Susan E. Pirie and Thomas Kieren in 1989. Pirie and Kieren (1994) define mathematical understanding as a whole, dynamic, leveled but non-linear, transcendently recursive process (p. 166). This theory emphasizes that the growth of mathematical understanding is not linear; by contrast, it improves dynamically with back and forth movements between mathematical ideas.Mathematical Understanding LevelsPirie and Kieren (1994) developed model in order to represent ideas while they were building the theory of the growth of mathematical understanding (see Figure 1). …