The dangers of teaching in the digital era

I just finished up my rather experimental introductory category theory class. I’m sure it’s probably a bad idea to teach basic category theory to 16-year olds, but it actually went pretty well. My goal was to get the students to understand the idea of a universal construction by discussing products, coproducts, initial, and final objects in some very simple examples. And I think most people in the class understood what was going on.

Yesterday I was talking about products in categories coming from posets. That is, the objects are elements of your poset, and there exists exactly one morphism X \rightarrow Y when X \leq Y. The first example I did was the positive integers under divides. So I took two objects, 7 and 5, and asked what their product was. Eventually they get the right answer, so I write the answer on the board. Suddenly their are flashes going off around me from the students digital cameras. At this point I realize I’ve just written:

7 \times 5 = 1 and 300 \times 6 = 6.

7 thoughts on “The dangers of teaching in the digital era

  1. Great! Why shouldn’t 16 year olds learn category theory? I think it should be introduced in elementary school – along with kindergarten QM!

  2. For a while the US tried to teach children set theory. The result was about a 10-year cohort that is probably still unable to add.

    I’m not saying that category theory shouldn’t be taught to adults. Just that this sort of thing should wait for graduate school. Sort of like a drinking age for abstract nonsense. Along the same line my undergraduate class in QM taught me almost nothing about the subject; it was just an exercise in mathematics. The graduate classes were easier, from a conceptual point of view.

    A good balance would be to study the usual elements of undergraduate mathematics, but to have Kea occasionally give a glimpse of the generalizations. Same with QM.

  3. I’m afraid I do not have any primary sources to back me up here, but I have heard from several education specialists that the new math curriculum of the Sputnik era was rather successful in its initial trials, when it was taught by expert teachers who knew the material well. Apparently, the planners did not consider how badly this could fail when the curriculum was pushed out to schools where the teachers could not be trained as rigorously. It is tempting to interpret this failure as proof that children cannot handle abstraction, but instead, I think it is evidence that teacher training is essential, and that curriculum designers should carefully consider the costs of retraining when suggesting radical changes.

    At any rate, it is not clear to me how the problems with teaching set theory to primary school children at large are relevant to the question at hand, which is whether it is suitable to teach category theory to a self-selected group of talented and motivated high school students that have a strong background in mathematics. Noah is close to an ideal instructor here – he’s no slouch at category theory, and he has several years of experience teaching math at summer programs for high school students. Also, I have some empirical data, since I am visiting Mathcamp this week, and I can say with some confidence that the students loved it.

  4. Teaching first graders set theory….if done correctly, should not interfere with their ability to do basic arithmetic.

    As I write this, I’m teaching my first grader set theory…well, at least it’s simplest concepts.

  5. The key phrase there is “done correctly.” If they were taught things like how to add with the usual amount of class time, and then extra class time were added so they could learn some set theory, that would be harmless (though as Scott mentioned, given the number of 1st grade teachers who understand set theory, perhaps not very useful). But my understanding of the new math is that this is not what happened at all….

  6. I have a son in 3rd grade and a daughter in kindergarten and I would like to teach them set theory. Kanishka, do you have any suggestions for learning material?

    I am a member of the cohort that was taught set theory instead of memorizing addition, subtraction and multiplication tables. I usually end up typing 1 digit by 2 digit multiplications into a computer, rather than trying to do them in my head or on paper. This is pretty much the extent of the negative consequences. On the other hand, I have seen mistakes made by engineers and PhD physicists (that were not taught set theory at an early age) that could have cost millions of $. These mistakes usually boiled down to a fundamental lack of understanding of very basic mathematical concepts, which you learn in set theory.

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