High school precalculus students' understanding of slope as measure

School Science and Mathematics, Feb 2001 by Stump, Sheryl L

Twenty-two high school precalculus students were interviewed to examine their understanding ofs[ope as measure. The students examined and discussed real-world situations involving slope: physical situations involving slope as a measure of steepness and functional situations involving slope as a measure of rate ofchange. For the various physical situations, students measured steepness with angles instead of ratios. Overall, they demonstrated a better understanding of slope in functional situations, but many students had trouble interpreting slope as a measure of rate of change. Instruction should focus on helping students form connections and providing opportunities for students to communicate their understanding.

The concept of slope has life in both school mathematics and life outside the classroom. Throughout the secondary mathematics curriculum, the concept of slope emerges in various forms. It appears algebraically in formulas and equations, geometrically in graphs, and trigonometrically as the tangent of an angle. In the study of calculus, slope appears as a limit. Real-world representations of slope exist in two different forms: physical situations, such as mountain roads, ski slopes, and wheelchair ramps; and functional situations, such as distance versus time or quantity versus cost. This investigation focused on these real-world situations and examined students' understanding of slope as a measure of steepness and slope as a measure of rate of change.

Research has documented various difficulties students have with the concept of slope. There are misconceptions associated with the calculation of slope, and when it is represented in decimal form, students have trouble considering slope as a ratio (Barr, 1980, 1981). They also have difficulty interpreting linear functions and their graphs (Barr, 1980, 1981; Moschkovich, 1990; Schoenfeld, Smith, & Arcavi, 1993), connecting graphs to linear equations (Kerslake, 1981), and connecting graphs to the notion of rate of change (Bell & Janvier, 1981; Janvier, 1981; McDermott, Rosenquist, & van Zee, 1987; Orton, 1984).

In order to help students make sense of the concept of slope, teachers often use real-world examples. It is believed that the use of these representations helps students develop understanding of abstract mathematics (Fennema & Franke, 1992). A previous investigation showed that teachers consider physical situations to be the most useful for developing understanding of the concept of slope (Stump, 1999). However, with recent recommendations emphasizing the study of functions in high school (National Council of Teachers of Mathematics, 1989, 2000), it is also important for students to understand the meaning of slope in the context of functional situations.

These two types of situations, physical and functional, involve two different notions: slope as a measure of steepness and slope as a measure of rate of change. Slope can be thought of as a ratio or a rate, and both of these measures require proportional reasoning (Lamon, 1995). Evidence suggests that students initially see ratios as additive structures (Hart, 1981; Singer & Resnick, 1992). The strategies students use to solve problems involving ratios may be affected by the context or by the numerical content or the problem (Karplus, Pulos, & Stage, 1983). The presence of context may evoke different reasoning processes (Heller, Post, Behr, & Lesh, 1990). Problems are more likely to elicit multiplicative responses from students if they involve time or money rates, "for every" statements, or a situation of containment (Kaput & West, 1994).

The literature presents differing views on the distinction between the concepts of ratio and rate. According to Vergnaud (1983, 1988), a ratio is a comparison between quantities of like nature, and a rate is a comparison of quantities of unlike nature. Thus, physical situations involve slope as a ratio, a measure of steepness; whereas functional situations involve slope as a rate, a measure of the relationship between two different variables. Thompson (1994) focused on the mental operations of the learner and defined ratio as "the result of comparing two quantities multiplicatively" (p. 191) and a rate as " a reflectively abstracted constant ratio" (p. 192). According to this scheme, in both physical situations and functional situations, slope can be thought of as either a ratio or a rate, depending on the level of reflective abstraction.

Simon and Blume (1994) analyzed preservice teachers' understanding of "ratio-as-measure" by studying their interaction with a physical situation involving slope. They found that preservice teachers did not see ratio as a measure of steepness in the same way they saw rate as a measure of velocity. Yet both of these very different measures are embodied in the concept of slope, and a robust understanding of slope includes the ability to employ both types of measures.

According to Noddings (1990), the perspective of constructivism assumes that the development ofa mathematical idea is contingent on (a) the networks of ideas presently existing in the mind of the learner and (b) purposive activity that induces transformation of these networks. Thus, Kloosterman and Gainey (1993) provided this advice for teachers: "Listen to students continually and base instruction on what they already know... Help students make connections among mathematical ideas" (p. 9).