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taiyo comments on Open Thread June 2010, Part 4 - Less Wrong
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Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud. You multiply by the number up to the top right of the letter then reduce that number by 1. Or you do the reverse in the reverse order. You know, like you put on your socks then your shoes but have to take off your shoes then take off your socks.
Sometimes drawing a picture helps prime an intuitive understanding of the physics. You start with a graph of velocity vs time. That is the 'acceleration'. See... it is getting faster each second. Now, use a pencil and progressively color in under the line. that's the distance that is getting covered. See how later on more when it is going faster more distance is being traveled at one time and we have to shade in more area? Now, remember how we can find the area of a triangle? Well, will you look at that... the maths came out the same!
I teach calculus often. Students don't get hung up on mechanical things like (x^3)' = 3x^2. They instead get hung up on what
has to do with the derivative as a rate of change or as a slope of a tangent line. And from the perspective of a calculus student who has gone through the standard run of American school math, I can understand. It does require a level up in mathematical sophistication.
That's the problem. See that bunch of symbols? That isn't the best way to teach stuff. It is like trying to teach them math while speaking a foreign language (even if technically we are saving the greek till next month). To teach that concept you start with the kind of picture I was previously describing, have them practice that till they get it then progress to diagrams that change once in the middle, etc.
Perhaps the students here were prepared differently but the average student started getting problems with calculus when it reached a point slightly beyond what you require for the basic physics we were talking about here. ie. they would be able to do 1. and but have no chance at all with 2:
I'm not claiming that working from the definition of derivative is the best way to present the topic. But it is certainly necessary to present the definition if the calculus is being taught in math course. Part of doing math is being rigorous. Doing derivatives without the definition is just calling on a black box.
On the other hand, once one has the intuition for the concept in hand through more tangible things like pictures, graphs, velociraptors, etc., the definition falls out so naturally that it ceases to be something which is memorized and is something that can be produced ``on the fly''.
A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan "brought to you by apostrophe, the letters LIM and an arrow" but you can go on to tell them "this is how impressive mathematcians say you should write this stuff that you already understand" and they'll get it.
I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn't easy.
That is close to saying "this stuff is hard". How about first showing the students the diagram that that definition is a direct transcription of, and then getting the formula from it?
(Actually, many would struggle with 1. due to difficulty with comprehension and abstract problem solving. They could handle the calculus but need someone to hold their hand with the actual thinking part. That's what we really fail to teach effectively.)