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Back in school, what I got taught during school was labeled 'math', but it was actually 'rote arithmetics.' This seems to also be the case of many other people. Some came to hate it and never came back to the field.

Since computers are unbeatable in this field (i.e. processing a logical structure), wouldn't it be the case that teachers concentrate more and more on teaching 'real mathematics'? They could explain how this or that algorithm works but not make the pupils solve mechanically lots of problem, nor test them for their ability to solve those problems.

Is this educational chance of concentrating on formalizing problems, proofs, and mathematical thinking being grabbed or is it just wishful thinking? Have computers changed the way math is taught and learned?

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We are indeed in a period of rapid change of the material that is rote-learned. –  André Nicolas Oct 27 '13 at 16:16
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The "rote" part is still important, as things like Gauss' solution to $\sum_{i=1}^n={n+1\choose 2}$ are reputed to have arisen from rote assignments, and students often gain "internalization" insight after doing several examples under the same mathematical principles. –  abiessu Oct 27 '13 at 16:18
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The latter. The history of "new math" illustrates that. It was supposed to bring in conceptual elements, but ended up with the memorization of new words such as "distributive." –  André Nicolas Oct 27 '13 at 16:50
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Mathematics (at school level) consists of a few grand ideas and a lot of small ideas. Surprisingly, all those small ideas are the most important part --- they make the few big ideas work. The computer will not change the ancient saying: "There is no royal road to mathematics." –  kjetil b halvorsen Oct 27 '13 at 17:14
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Since this question asks for opinions, I am going to make it CW –  robjohn Oct 27 '13 at 17:24

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up vote 2 down vote accepted

So far at least it appears that the computer has nine advantages. (To please Mr. Monk I'm tempted to make it an even 10 but I won't.)

  1. It can enable us to watch a lecture over and over, anywhere anytime. This does not mean that the lecture has any greater quality; but at least we can watch it more than once.

  2. During a lecture, it can spare us the need to watch a lecturer talk and write on the board. In graduate school we spend almost all of our class time copying down what was said and written on the board. It was stressful, boring, and not helpful.

  3. Presumably, as claimed in flipped learning, recorded lectures can leave class time to watch problems being solved, or ideally, to have questions answered and confusions clarified. The latter is of questionable value in our current state of teaching, since the ratio of confusion to explanation is enormous.

  4. It can provide us with innumerable practice exercises of any kind and any level. In high school algebra in the forties I longed for more exercises.

  5. It can show us step by step how to do something. This is a more concentrated focussed form of lecture.

  6. It can adapt its exercises and even instructions to emphasize areas where we make mistakes or are at a loss as to how to proceed.

  7. It can give us ready access to the profusion of questions and assertions on the internet, thus sort of conveniently substituting for physically accessing library books or tutors.

  8. Via email it can give us more access to our mentors, be they professors, assistants, or even fellow students, though I suspect the latter will be of little help. In my experience, in spite of its vocal advocates, collaboration is of very limited value. In my schooling, when I was lost, so was everyone else, and even more so when that was possible.

  9. Finally, the very presence of a computer can facilitate learning and practicing programming and using various remarkable mathematics applications.

(10. If I missed one, Mr. Monk will be relieved.)

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And we can all watch lectures delivered not by average lecturers, but by great lecturers. Also, a traditional lecture is seen only once, by only a few dozen people, so there is little motivation to perfect it; on the other hand a video lecture can be a permanent contribution, like a textbook, so the effort to make it a masterpiece could pay off more. –  littleO Oct 28 '13 at 0:35

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