If you compare Karim and Sal’s definitions to Stump’s list, you’ll likely judge that while both have been correct, neither has been complete. We could stop here and declare this duel a draw, but to do so would foolishly ignore that there is much more to teaching and learning mathematics than knowing what belongs in a textbook glossary. Indeed, research suggests that a robust understanding of slope requires (a) the versatility of knowing all seven interpretations (although only the first five would be appropriate for a beginning algebra student); (b) the flexibility that comes from understanding the logical connections between the interpretations; and (c) the adaptability of knowing which interpretation best applies to a particular problem.
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.
This returns us to Karim’s original point: There exists better mathematics education than what we currently find in the Khan Academy. Such an education would teach slope through guided problem solving and be focused on the key concept of rate of change. These practices are recommended by researchers and organizations such as the NCTM, and lend credence to Karim’s argument for conceptualizing slope primarily as a rate. However, even within this best practice, there is nuance. For instance, researchers have devoted considerable effort to understanding how students construct the concept of rate of change, and they have found, for example, that certain problem contexts elicit this understanding better than others.
The Gates Foundation's implicit position has become "learning science is bunk (with the possible exception of Gates funded research started in the past five years)." That's always been my interpretation of their support for Khan Academy; I'm glad the disconnect between the research base in math education and KA is becoming more widely understood. You could also see this as an embrace of disruptive innovation -- get a cheaper, inferior product out to everyone quickly instead of slower, more expensive projects. It is tricky strategy as philanthropy though.
It is also consistent with Gates' habit of acting as if nobody ever had the idea of research in education before they came around. I suspect Bill Gates believes this is true, maybe the rest of the staff convinced themselves as well.
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