6.1 Introduction
In the last chapter I analyzed students‘ ideas about the mathematical concepts of differential and integral from the perspectives of resources and conceptual metaphors. I categorized students‘ use of mathematical resources (i.e., symbolic forms) and metaphors in their mathematical discourse in physics contexts. In this chapter I address the second research question: How do students combine their knowledge in calculus and physics to set up integrals in physics? The purpose of this question is to explore students‘ overall strategies of applying mathematical integrals in physics. I use the language of conceptual blending to describe how students combine their knowledge in mathematics and physics.
Previous studies in physics education research have provided evidence that students‘ difficulties with using mathematics in physics is often caused by physics contexts rather than mathematical aspects alone. Analyzing students‘ blending of knowledge of mathematics and physics would help us generate deeper insights into understanding how students transfer their learning from mathematics to physics, or how the physics context affects their problem solving strategies. I conducted a qualitative analysis of student work and in this chapter we will talk about our data analysis and the several different strategies that students invented towards setting up integrals. The main findings are from a recently submitted article (Hu & Rebello, 2013c).
6.2 Data Analysis: Identifying the Blends
I present a detailed analysis that focuses on one physics problem in the context of resistance (Figure 5.3). The problem involves finding the total resistance of a cylindrical resistor with non-constant resistivity. When the resistivity is not a constant, one must first chop the cylinder into infinitesimally thin disks and find the resistance dR of each disk (Figure 6.1). We find the total resistance by summing up i.e., integrating over the resistance, dR of each disk. Finally, we substitute the resistivity function ρ(x) provided in this equation and evaluate the integral.
57
Figure 6.1 Partial solution to the interview task in Figure 5.3.
The problem was presented to students during the fifth interview session. There were two main reasons for choosing this problem for a detailed analysis. First, all of the students were familiar with the resistance concept but they had not seen any specific example similar to the problem nor had they received any instruction on how to find the resistance of a resistor with non-uniform resistivity. Students had to invent their own integral equation as there was no specific integral equation on resistance that they could draw upon from their memory. Second, students used diverse strategies and reasoning when solving this problem. This problem seemed neither too easy nor too difficult for most students. Most students were able to invent strategies to solve this problem and explain their thinking.
Our unit of analysis was a student discussion group. We transcribed the video files for all five groups working on the task. We conducted a qualitative analysis and our unit of analysis was a student discussion group. I transcribed the video files for all five groups working on the task. Next, I conducted a qualitative analysis using a phenomenographic approach. An important aspect of our approach was that I did not categorize students‘ descriptions based on pre-decided categories. Rather, the categories emerged from the data. Then, I identified the important themes that emerged from students‘ conversations or written work on the board. Next, I examined these emergent themes through the lens of the conceptual blending framework. For each theme, I determined the input spaces and blended spaces by analyzing the ways in which students connected their ideas from different domains. Finally, I generated a holistic description of how the students blended their ideas. This holistic picture describes how the knowledge
d x L L 0 0 x
The basic physics equation for resistance with constant resistivity is given by . The resistance of one infinite thin disk dR can be expressed as with dx the thickness of the disk. The total resistance R is the integral of dR, that is ∫ ∫ ( )
58
elements from each input space are projected to the blended space and how a new structure emerges from the blend.
6.3 Description of Blends
In this section, I present four distinct conceptual blends identified from five groups of students working on the resistor problem (Figure 5.3). I noticed that students within the same group did not necessarily follow the same approach, thus, there was often more than one blend created in a group. I claim that blending is a dynamic process. That is, students do not have explicit ideas in their mind and construct the solutions spontaneously. I found that students constructed one blend at the beginning and later changed the blend as they proceeded through the problem. Some blends eventually led students to solve the problem correctly while others hampered students‘ efforts to setting up the correct integral. When constructing a blend, students often recruited an organizing frame, which determines how students put together the knowledge elements projected from input mental spaces. Hence, the organizing frame describes the structure of student reasoning. I will mainly focus on the organizing frame adopted by the students. This analysis allows us to classify student work into four frame clusters: Integral-Sum Across frame, Differential Algorithm frame, Equation Mapping frame, and Chopping &Adding frame. I will illustrate the four different frames using examples of student work.