Usually when you talk about slope we're assuming "slope of a line". In that case, Khan's definition is fine. If you're trying to really rigorously define "slope", which is fairly silly since you should just start using the term "gradient" which is more rigorously defined, then I could see this going two ways:
1. You could argue that if you're talking about slope, you mean slope of a line. If you want to talk about the generalized notion of slope of a line, you should use terms like gradient, derivative, etc. If Khan had taken this stance, I would have been fine with his defense.
2. You go with the fully rigorous definition of slope as basically being a synonym for gradient. This is what mathworld actually does. This is where Khan's reputation gets really knocked, in my opinion. He quotes mathworld (http://mathworld.wolfram.com/Slope.html) but only selectively. What he failed to mention are these key points:
- The very first sentence defines slope this way: "A quantity which gives the inclination of a curve or line with respect to another curve or line."
- The sentence he did quote begins with "For a line in the xy plane..."
For the vast majority of his students, including those other points when they're first learning about slopes would do more harm than good. He shouldn't be judged on his rigor, but on how effectively he enables learning and understanding. They wouldn't know what to do with gradients.
Sure, but I'm not talking about how effective his teaching is. I'm simply addressing the technical definition of slope.
I am perfectly fine with Khan teaching it as "rise/run". I thought criticizing this was silly on Karim's part. But then Khan was the one who came back to argue that his definition was correct and Karim's was wrong. I think Khan's wrong here and shouldn't have even engaged in such a trivial dispute...
1. You could argue that if you're talking about slope, you mean slope of a line. If you want to talk about the generalized notion of slope of a line, you should use terms like gradient, derivative, etc. If Khan had taken this stance, I would have been fine with his defense.
2. You go with the fully rigorous definition of slope as basically being a synonym for gradient. This is what mathworld actually does. This is where Khan's reputation gets really knocked, in my opinion. He quotes mathworld (http://mathworld.wolfram.com/Slope.html) but only selectively. What he failed to mention are these key points:
- The very first sentence defines slope this way: "A quantity which gives the inclination of a curve or line with respect to another curve or line." - The sentence he did quote begins with "For a line in the xy plane..."