|Sal Khan (CC BY-NC-ND Elvin Wong)|
Working on her dissertation in the mid-1990s, Sheryl Stump (now the Department Chairperson and a Professor of Mathematical Sciences at Ball State University) did some of the best work to date about how we define and conceive of slope. Stump (1999) found seven ways to interpret slope, including: (1) Geometric ratio, such as "rise over run" on a graph; (2) Algebraic ratio, such as "change in y over change in x"; (3) Physical property, referring to steepness; (4) Functional property, referring to the rate of change between two variables; (5) Parametric coefficient, referring to the "m" in the common equation for a line y=mx+b; (6) Trigonometric, as in the tangent of the angle of inclination; and finally (7) a Calculus conception, as in a derivative.
|(CC BY-NC-SA Raymond Johnson)|
All seven slope interpretations are closely related and together create a cohesive whole. The problem is, it's not immediately obvious why this should be so, especially to a student who is learning about slope. For example, if slope is steepness, then why would we multiply it by x and add the y-intercept to find a y-value (i.e., as in the equation y=mx+b)? And why does "rise over run" give us steepness anyway? Indeed, is "rise over run" even a number? Students with a robust understanding of slope can answer these questions. However, Stump and others have shown that many students -- even those who have memorized definitions and algorithms -- cannot.
|(CC BY Amber Rae)|
Despite all we know from research, we should not be surprised that there's still no clear "right way" to teach slope. Mathematics is complicated. Teaching and learning is complicated. We should never think there will ever be a "one-size-fits-all" approach. Instead, educators should learn from research and adapt it to fit their own unique situations. When Karim described teachers on Twitter debating "whether slope should always have units," we see the kind of incremental learning and adapting that moves math education forward. These conversations become difficult when Sal declares in his rebuttal video that "it's actually ridiculous to say that slope always requires units*" and Karim's math to be "very, very, very wrong." We absolutely believe that being correct (when possible) is important, but we need to focus less on trying to win a mathematical debate and focus more on the kinds of thoughtful, challenging, and nuanced conversations that help educators understand a concept well enough to develop better curriculum and pedagogy for their students.
|Khan Academy (CC BY-NC-ND Juan Tan Kwon)|
*Sal's point is that if two quantities are both measured in the same units, then the units "cancel" when the quantities are divided to find slope. As an example, he uses the case of vertical and horizontal distance, both measured in meters. The slope then has units of meters/meters, which "cancel". However, the situation is not so cut and dry, and indeed, has been considered by math educators before. For example, Judith Schwartz (1988) describes how units of lb/lb might still be a meaningful unit. Our point is not to say that one side is correct. Rather, we believe that the act of engaging in and understanding the debate is what is important, and that such a debate is cut short by declarative statements of "the right answer."
Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Heibert & M. J. Behr (Eds.), Number Concepts and Operations in the Middle Grades (Vol. 2, pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.
Stump, S. L. (1999). Secondary mathematics teachers' knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf