Not to suggest that credentials are everything, but, lacking any other information, they do provide useful signaling indicators, from http://en.wikipedia.org/wiki/Khan_Academy
"After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MS in electrical engineering and computer science), he pursued an MBA from Harvard Business School"
One would hope that his education background might have given him some insight into what "slope" was, and how to best explain it to someone in the 8th/9th grade.
I can _recall_ when it was introduced to me, and I didn't really grok it until a few years later - "rise over run = slope" was complex enough for my brain back then - I can just imagine if people started yacking about "Ratios of variables" - my head would have exploded.
> An effective math teacher will point out that “rise over run” isn’t the definition of slope at all but merely a way to calculate it.
Depends on the level. For introductory algebra, or classes before calculus, where the "slope" concept is used exclusively with linear functions, "rise over run" is a perfectly adequate definition.
For calculus, you can rigorously define slope as the limit of (f(x+h) - f(x)) / h as h -> 0 (or any of several other equivalent definitions). The subtleties of the proper definition are likely to be more confusing than enlightening to a beginner.
But even the rigorous definition is still a "rise" term f(x+h) - f(x) divided by a "run" term h. The phrase "rise over run" is an easy-to-remember mnemonic which actually does a really good job of capturing the underlying idea without going deep into more technical issues involving limits (calculus) or the guts of exactly what a real number is (real analysis).
Once the linear case is completely understood and the student has some basic facility with both mathematical reasoning and algebraic manipulation, THEN is the time to introduce the more general definition of "slope". Even then, mathematics teachers tend to use the word "slope" mainly in the linear case, and "derivative" in the general case.
As far as I know, this has been the standard way to teach high school / early college level mathematics for decades.
How often do you need signaling indicators, though? If someone has sufficient competence at a given field, they should be able to demonstrate that competence by delivering satisfactory work.
The proof should be in the level of understanding of the concept of slope acquired by students using various methods; it doesn't matter at all who is "right" with respect to some formal definition.
Totally agree with you that what matters is whether the students eventually grasp the material. What I like about Khan, is he is fundamentally interested in teaching students so they _understand_ the material. He doesn't get caught up in theories of pedagogy or mathematical models or precise correctness - he sees his job is to communicate a concept in a way that the student says, "Oh, I get it."
Now, with that said - there _is_ a danger in that model of teaching, in which the student gets lured into a zone of comfort. So there _absolutely_ is a place in teaching for instructors who are going to challenge, upset, and disturb the student - resulting in a form of stress that pushes to the student to new heights. This can be a very uncomfortable (and, indeed, upsetting/stressful) learning environment - but it does give students deeper, and sometimes much more meaningful insight into topics.
But that's not what Khan's about. He is the guy you go to when you just want to get over some hurdle about a topic that has frustrated you.
"One would hope that his education background might have given him some insight into what "slope" was, and how to best explain it to someone in the 8th/9th grade."
That leap you made there at the end of your sentence is the crux of the issue. Most educators argue that being an expert in a subject is not even remotely sufficient qualification to teach it effectively, especially to children.
"After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MS in electrical engineering and computer science), he pursued an MBA from Harvard Business School"
One would hope that his education background might have given him some insight into what "slope" was, and how to best explain it to someone in the 8th/9th grade.
I can _recall_ when it was introduced to me, and I didn't really grok it until a few years later - "rise over run = slope" was complex enough for my brain back then - I can just imagine if people started yacking about "Ratios of variables" - my head would have exploded.