Academic journal article Kuram ve Uygulamada Egitim Bilimleri

Mathematics Teachers' Covariational Reasoning Levels and Predictions about Students' Covariational Reasoning Abilities*

Academic journal article Kuram ve Uygulamada Egitim Bilimleri

Mathematics Teachers' Covariational Reasoning Levels and Predictions about Students' Covariational Reasoning Abilities*

Article excerpt

Abstract

Various studies suggest that covariational reasoning plays an important role on understanding the fundamental ideas of calculus and modeling dynamic functional events. The purpose of this study was to investigate a group of mathematics teachers' covariational reasoning abilities and predictions about their students. Data were collected through interviews conducted with five secondary mathematics teachers to reveal about their covariational reasoning abilities as they worked through a model-eliciting activity, predictions about their students' possible approaches to solve the given problem, possible mistakes in solving the problem, and misconceptions they possibly held. The results showed that not only the teachers' covariational reasoning abilities were weak and lack depth but also their predictions about students' reasoning abilities bounded by their own thoughts related to the problem.

Key Words

Covariational Reasoning, Functions, Mathematical Modeling, Mathematics Education, Secondary Mathematics Teachers.

The notion of change is fundamental for the major concepts of calculus, a critical course for students majoring in mathematics, physics, engineering, and others (Carlson, Larsen & Lesh, 2003; Carlson & Oehrtman, 2005; Cottrill et al., 1996; Kaput, 1994; Saldanha & Thompson, 1998; Thompson, 1994a; Zandieh, 2000). Studies show that students' covariational reasoning abilities have a major role to construct and interpret the models of continuously changing events (Carlson, Jacobs, Coe, Larsen & Hsu, 2002; Kaput, 1992, 1994; Monk, 1992; Rasmussen, 2001). These studies also demonstrated that this type of reasoning (i.e., covariational reasoning) ability plays an important role on modeling dynamic functional situations. However, even high performing students have difficulties in modeling dynamic functional situations (Carlson et al., 2002; Carlson & Oehrtman, 2005; Köklü, 2007; Monk & Nemirovsky, 1994). In particular, students' difficulties are associated with lack of imaging and coordinating the simultaneous changes of variables, namely their covariational reasoning abilities (Carlson & Oehrtman, 2005). Although there are various definitions of covariational reasoning in the literature (e.g., Carlson et al., 2002; Confrey & Smith, 1995; Saldanha & Thompson, 1998), the common trait in these definitions is that covariational reasoning is imagining and coordinating the changes in two quantities simultaneously. In this study, we adopted the definition provided by Carlson et al. (2002) to describe covariational reasoning: "the cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other" (p. 354).

Researchers emphasize that the concept of function should be introduced as covariation, that is coordinating changes in one variable with the other variable, and student should be provided with more opportunity to explore the concept of rate of change in earlier grades (Confrey & Smith, 1995; Kaput, 1994; Köklü, 2007; Thompson, 1994b). In Principles and Standards for School Mathematics, NCTM (2000) emphasizes the need for more function related tasks in the classrooms and stresses the importance of the students' understanding of the concept of rate of change in real-world situations. In this context, modeling activities provide opportunities to achieve these aims (Carlson et al., 2003; Lesh & Doerr, 2003b; Lesh, Hole, Hoover, Kelly & Post, 2000). For effective instruction, on the other hand, it is important for teachers to know the concepts students have difficulty with and the ways to overcome the common misconceptions or misunderstandings. However, research studies suggest that although it is an important component of pedagogical content knowledge (Shulman, 1987) there are discrepancies between teachers' predictions about students' difficulties and students' actual difficulties (Even, 1993; Hadjidemetriou & Williams, 2002). …

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