Dawn Teuscher and Robert E. Reys' article, "Slope, Rate of Change, and Steepness: Do Students Understand These Concepts?" Teuscher and Reys bring up an issue that occurs with many students, in the fact that they don't understand the similarities and differences between slope, rate of change, and steepness when it comes to graphs of functions. They performed studies on tests that reflected questions that would appear on the A.P. Calculus exam, to see how students answered questions about slope, rate of change, and steepness. Unexpectedly, there were many students who believed that rate of change was the absolute value of the slope, or how "steep" a graph was, and gave incorrect intervals for answering questions about graphs and their rates of change. Teusher and Reys summed up the article by stating that teachers needed to do a better job of using the terms, and clarifying their meanings. This, in turn, will give students a better base for higher level mathematics, and prepare them more for future studies.
Teuscher and Reys bring up an excellent point that teachers need to better convey the concepts of slope, rate of change, and steepness, and explain their meanings, and how they are related. Doing so really will help students become better prepared for exams, like the A.P. exam, and for future, more advanced, mathematics. If students are allowed to continue with the incorrect ideas of slope and rate of change, when they take higher level classes, and subjects like physics, they will have an extremely difficult time, because they will have to change their views, and will not be able to connect their learning to the "knowledge" that they have about slope and rate of change. If teachers, however, explain these concepts thoroughly, it will make Calculus and Calculus based subjects much easier for students to grasp.