I think just about every Algebra 1 student I ever taught came to me from Prealgebra knowing what slope was. At least they thought they knew what slope was. They could usually echo the words "rise over run," and I admit that very early in my career I probably would have found that somewhat satisfactory. But with each new Algebra 1 class (I taught 14 sections in 6 years), my students' limited understandings of slope became more frustrating. Honestly, it wasn't until my last year of teaching that I really felt I had the kinds of problems, activities, and explanations to help students construct an understanding of slope that I was happy with.
In discussions with my graduate school colleagues Fred Peck and Michael Matassa, I found that my experience wasn't unique. We were interested in exploring slope further, and that led us to an article by Sheryl Stump. Her name seemed familiar, and as soon as I saw her picture I realized that I'd had lunch with her at a conference just a few weeks before. I suppose if I'd found the article earlier I could have talked to her about slope instead of swapping stories about our common Midwestern roots, but she's been kind enough to reply to my emails when we've wanted to know more.
I think one of the reasons I liked Stump's article was that it focused on teachers instead of students. After all, the biggest reason I became dissatisfied with my students' understanding of slope was because my own understanding had grown more sophisticated with each trip through the curriculum. In her article, Secondary Mathematics Teachers' Knowledge of Slope (1999), Stump investigated the definitions, understandings, and pedagogical content knowledge of 18 preservice and 21 inservice teachers. Nearly all the inservice teachers had degrees in math or math education, including eight with math or math ed masters degrees, and their teaching experience ranged from 1 to 32 years.
Stump's review of previous literature on slope revealed a number of descriptions, including ratios, tangent, and -- because of its applications in physics, calculus, and other real-world applications -- the key concept of linear functions and rates of change. Stump also found everyday ways of thinking about slope, such as the downward slant of a hill from top to bottom. These different, yet related, descriptions of slope had led to misunderstandings in previous studies with students. No one had yet tackled this kind of research with teachers, so Stump designed and administered a survey and conducted interviews to understand what her study participants understood about slope. Her questions, "What is slope?" and "What does slope represent?" elicited responses that were sorted into seven categories:
Both the preservice and the inservice teachers in Stump's study averaged about 2.5 representations per teacher in their definitions. The geometric ratio representation of slope was easily the most common for both groups (83% of preservice, 86% of inservice). but preservice teachers most commonly (61%) used algebraic ratio as a second representation, while inservice teachers commonly (81%) described a physical property. Descriptions of slope using the parametric, trigonometric, and calculus conceptions were rare or nonexistent.
- The category of geometric ratio included representations such as "\(\frac{\mbox{rise}}{\mbox{run}}\)" and "vertical change over horizontal change" and focused on slope as a geometric property.
- The category of algebraic ratio included representations such as "\(\frac{y_2 - y_1}{x_2 - x_1}\)" and "the change in y over the change in x, in which slope was defined by an algebraic formula.
- The words "slant", "steepness", "incline", "pitch", and "angle" were categorized as involving a physical property.
- Responses referring to slope as the rate of change between two variables were categorized as involving a functional property.
- The parametric coefficient category included references to m in the equation y = mx + b.
- A trigonometric conception of slope referred to the tangent of the angle of inclination.
- A calculus conception included mention of the concept of derivative. (p.129 )
Stump then gave the two groups six math questions, each designed to test different understandings of slope. Both the first question, about rate of growth, and the second question, finding a linear equation given its parameters, were answered correctly by 100% of the teachers in both groups. Questions about slope as speed, read from a graph, were answered correctly by about two-thirds to three-fourths of teachers in each group. The most dissimilar performance came on a question about angle of inclination, answered correctly by 33% of preservice teachers and 67% of inservice teachers.
Next Stump asked the teachers, "What mathematical concepts must students have experience with before they can truly understand slope?" (p. 132). By a wide margin, both groups said a geometric representation was most important, but only three teachers in each group mentioned experiences with functional relationships. Similarly, when asked for real-world contexts for understanding slope, both groups tended to choose a physical property instead of a functional property. About a quarter of the teachers in each group didn't mention either, naming algebraic or geometric representations instead (p. 133).
Stump's teacher interviews allowed her to dig more deeply into teachers' understandings about how students learn about slope. When asked about student difficulties, almost all the inservice teachers referred to a calculation procedure, saying "they put the x's over the y's" or "the order in which they subtract them" (p. 139). Preservice teachers predicted similar difficulties with symbol manipulation. One preservice teacher said:
My guess is that some might be frightened off as soon as you introduce a mathematical definition or a formula for a line, like the slope-intercept of the equation. As soon as some people see equations, they just go nuts, especially with symbols instead of numbers. ... Not because they don't understand what slope is, but because they are not making the connection between the intuitive and even the not-so-intuitive idea of taking the ratio of this to this. Not making the connection between that and the symbolic abstract equation on paper. That's just a guess. I haven't had experience with that. (pp. 139-140)In her discussion section, Stump acknowledges teachers' tendency to think of slope first as a geometric ratio, with a smaller majority commonly thinking of it as a physical property. Very few teachers -- less than 20% -- had a functional conception of slope. Stump continues:
Considering the importance of the study of functions for high school students, it is especially troubling that functional situations involving slope were missing from so many teachers' descriptions of their instructional practices. Their students may thus miss opportunities to make this important connection while forming their conceptions of slope. Rizzuti (1991) found that instruction that included multiple representations of functions allowed students to develop comprehensive and multi-faceted conceptions of functions. Based on the results of the present investigation, it is questionable whether the participating teachers could assist their students in developing such a rich conception of slope. (p. 141)Finally, Stump asks some important questions for further study, such as, "When textbooks connect various representations of slope, do teachers emphasise those connections for their students? Can teachers learn to make connections even if textbooks do not emphasise them?" (p. 141). I don't think we really know the answers to those questions, but I do absolutely agree with Stump's closing recommendation: "Both preservice and inservice mathematics teachers need opportunities to examine the concept of slope, to reflect on its definition, to construct connections among its various representations, and to investigate functional situations involving physical slope situations" (p. 142). It's good to see that kind of work being done, such as with Fred Peck and Michael Matassa's teaching experiment research and curriculum on slope they shared at ICME-12.
References
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