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mathwonk
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Please take this with a grain of salt, as everyone's experience and opinion of teaching and learning differ, but in my opinion, (at the moment), the primary problem with teaching is how to teach more than one person at a time. I.e. every student's background information, motivation, and speed of learning is different, so it is very challenging to keep the attention of a class of more than one, and present useful material without going too slow or too fast. If you are the fastest learner in a class of 35, or the most conscientious, it is quite likely you will almost always be bored and wonder why more challenging content is not offered.
Even with one student it is not trivial to teach significant information. Just think of this forum, where each student has the freedom to start his own thread devoted to his own specific question, and the answers are directed precisely at him/her alone. Experts here, even in combination, still often struggle to make themselves clear and eradicate misunderstandings.
Now place yourself in a high school classroom with 40 kids, some or most often ill prepared and uninterested, and try to design an effective program that will satisfy most of them, hopefully including the most gifted.
Perhaps for this reason the most accomplished young students i have met were mostly home schooled, essentially individually. This unfortunately may deny them the socialization benefits and comradeship of a standard school atmosphere.
What to do? I do not know the answer, after some 50+ years of teaching, with my most successful experiences limited to very small groups of highly gifted and motivated students.
Of course I may be unusually challenged, as I can share one teaching technique I have used that absolutely guarantees failure: once while I was giving an explanation of a clever trick for proving Taylor's theorem in an honors class, (the argument from Courant, and reproduced in Spivak), a very bright and motivated student in the front row leapt ahead mentally and shouted out the point of the still incompletely explained trick. Delighted, and somewhat embarrassed to continue, since I concluded the point was now obvious, I complimented him and stopped the explanation right there, thus guaranteeing that exactly one student in the class would understand it. Never do this. Plod right ahead with the explanation in full. The silent majority will thank you, as no doubt everyone else here already knows.
Even with one student it is not trivial to teach significant information. Just think of this forum, where each student has the freedom to start his own thread devoted to his own specific question, and the answers are directed precisely at him/her alone. Experts here, even in combination, still often struggle to make themselves clear and eradicate misunderstandings.
Now place yourself in a high school classroom with 40 kids, some or most often ill prepared and uninterested, and try to design an effective program that will satisfy most of them, hopefully including the most gifted.
Perhaps for this reason the most accomplished young students i have met were mostly home schooled, essentially individually. This unfortunately may deny them the socialization benefits and comradeship of a standard school atmosphere.
What to do? I do not know the answer, after some 50+ years of teaching, with my most successful experiences limited to very small groups of highly gifted and motivated students.
Of course I may be unusually challenged, as I can share one teaching technique I have used that absolutely guarantees failure: once while I was giving an explanation of a clever trick for proving Taylor's theorem in an honors class, (the argument from Courant, and reproduced in Spivak), a very bright and motivated student in the front row leapt ahead mentally and shouted out the point of the still incompletely explained trick. Delighted, and somewhat embarrassed to continue, since I concluded the point was now obvious, I complimented him and stopped the explanation right there, thus guaranteeing that exactly one student in the class would understand it. Never do this. Plod right ahead with the explanation in full. The silent majority will thank you, as no doubt everyone else here already knows.
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