A study conducted in the United States, investigated secondary mathematics teachers’ concept definitions, mathematical understanding, and pedagogical content knowledge of slope. The data were collected from a survey of 18 preservice and 21 inservice teachers and interviews of 8 teachers. Geometric ratios dominated teachers’ concept definitions of slope. Problems involving the recognition of parameters, the interpretation of graphs, and rate of change challenged teachers’ thinking. You can read the study in Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144.
In the study, Stump asked the question “What is slope?” and “What does slope represent?” and these elicited responses that she sorted into seven categories:
1. Geometric ratio: rise/run; vertical change over horizontal change
2. Algebraic ratio: (y2-y1)/(x2-x1); change in y over change in x
3. Physical property: “steepness”, “inclination”
4. A functional property: rate of change between two variables
5. Parametric coefficient: in y = mx + b; m is a parameter and is also interpreted as the slope of the line.
6. Trigonometric conception: slope is the tangent ratio of the opposite side over the adjacent side
7. Calculus conception: the derivative as the slope of the tangent to the curve
All these different conceptions are in the secondary school curriculum. Here in the Philippines, the first five are all in the in Grade 8 of the new k-12 mathematics curriculum. Putting them all in this grade meant it has the potential to make strong the connection between its algebraic, geometric and real-life conception. This will be embedded in the chapters on linear function, equations of lines, systems of equation. Very challenging.