A senior capstone presentation on Oregon math standards for rate of change and slope got me thinking. At the end of the talk I commented that the concept of rate of change is not dependent on the concept of slope but was somehow more fundamental. Afterward I decided to put my money where my mouth was and design a series of exercises that “naturally” developed the concept of rate of change without ever mentioning the dreaded S-word. About halfway through the process I started to get depressed. What I was developing were exercises to teach interpolation and extrapolation. And my personal understanding of rate of change – as a step in interpolation/extrapolation – comes from the fact that I grew up using logarithm tables.
Now don’t get me wrong, I teach rate of change using slope and I teach fairly well including my “world famous Hatton wrist-twist” method. I was depressed because I had forgotten two fundamental axioms of teaching: 1.Students can and will have a different model for any particular concept in their individual brains. and 2. The predominant model for a particular concept changes over time. Slope is the modern way of thinking about rate of change. In my favorite old college algebra text, College Algebra (1902) by G.A. Wentworth there is no mention of rate of change or slope. (I downloaded the text and did a search.) Wentworth develops derivatives using ratios of increments. He uses very few graphs and none for straight lines as such.
So where am I now? I know that for a subset of my students graphs are anathema. The slope concept as a visual paradigm does not work for them. For these students I try to use a kinetic model with steps one right and two up for example. This is sort of the way Newton thought of derivatives – two points moving at different speeds. In addition I have found that most students can get the algorithm/formula for rate of change (slope – wrist-twist) but still can’t solve an a unfamiliar problem. For instance they can’t interpolate or use proper units like miles pear hour or gallons per minute. Maybe there is a better conceptual model for rate of change – a more modern analogy. I wonder what that would be?
I have attached the exercises that I developed though I probably will never use them.