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    $\begingroup$ So you're computing a binomial coefficient like $\binom{20}{8}$ and you've got $\dfrac{20\cdot 19 \cdot 18 \cdots 13}{8\cdot 7 \cdot 6 \cdots 1}$ and the student reads the dots between numbers as a verb in the imperative mood commanding the reader to multiply. This seems like a case where (1) efficiency is worth it, and (2) maybe knowing that the denominator must cancel out completely helps conceptual understanding. $\endgroup$
    – Michael Hardy
    Commented Nov 5, 2010 at 22:48
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    $\begingroup$ +1 for this answer. @Steven-Gubkin, excellent answer for the specific situation of teaching soon-to-be-teachers, and possibly for others also. I'll have to take a look at this book and remember to recommend it to those who are willing and able and interested. Also, another +1 if I could give it to you for recognizing what an exceptional circumstance that class is, and for your well-presented and apropos response, particularly your commentary on the role of teachers to "direct and clarify" and let the students learn and master the material. $\endgroup$
    – sleepless in beantown
    Commented Nov 6, 2010 at 9:30
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    $\begingroup$ One remark on a popular geometry illustration repeated in this book relates to the "proof" of SSS congruence using a rigid model triangle of straws joined along a thread. This actually proves "rigidity", i.e. lack of infinitesimal deformations. This implies the moduli space is discrete, not necessarily a singleton. Indeed the same model shows SSA implies rigidity, but of course not congruence; i.e. this moduli space is in general 2 point. Thinking about these questions motivated me to read about Cauchy's rigidity theorem for convex polyhedra, and Connelly's non -convex counterexamples. $\endgroup$
    – roy smith
    Commented Feb 1, 2011 at 19:14
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    $\begingroup$ @roy Actually I had a very strange experience. I taught two recitations of the same class. For some reason one class was always really full of energy, fun, laughter, and insight, whereas in the first class everyone was very quite, reserved, and obviously didn't care about the material. Evaluations followed suit: One class gave the best reviews I have ever received, and in the other they were just terrible. I still really don't know what happened, even after pondering it for over a year now. $\endgroup$
    – Steven Gubkin
    Commented Feb 1, 2011 at 23:19
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    $\begingroup$ @Steven: Evaluations are problematic. As a nervy young teacher with two identical sections of precalculus, I tried an experiment. I handed out the evaluations in one class attached to the back of the final exam on exam day. In the other class I passed out champagne and glasses, followed by the evaluations, on the last day of class. Guess which class liked my teaching infinitely more than the other? (I eventually lost that position.) $\endgroup$
    – roy smith
    Commented Feb 3, 2011 at 0:38