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I think the useful distinction is that, when you teach multiplication as a mechanical computation (arithmetic) it's useful to talk about it as repeated addition
by munchbunny 6y ago
I think the useful distinction is that, when you teach multiplication as a mechanical computation (arithmetic) it's useful to talk about it as repeated addition.
As you get to negative numbers, rational/irrational numbers, complex numbers, matrices, etc. it becomes more useful to think about multiplication in more abstract ways, among which repeated addition is still often a useful way to look at it.
I also think it's not particularly useful to talk about those other ways to think about multiplication until you actually need to.
It's too easy once you've mastered the concepts to forget how beginners look at them and struggle to understand them. I think the author isn't remembering what it's like to try to understand multiplication as a new concept - I certainly can't remember.
- spacedcowboy 6y agoYour last two paragraphs are, I believe, the crux of my own argument. Sure, matrices aren’t even commutative, and complex numbers have their own quirks because of i^2 == -1, but these concepts build on the earlier axioms the kid has learnt. Our entire education system is built on “lies-to-children”, and as you progress they point out that what you comfortably believed was a gross simplification. This is no different.
- eternalban 6y agoSpeaking of “lies to children”, and since you are a physicist, it was only in my last year studying EE that Feynman’s QED came my way and just imagine my surprise on finding that photons do not travel in a straight line. (I do agree with your pedagogical position.)
- jschwartzi 6y agoI think the heart of the issue is whether it's more useful to teach children how to multiply two abstract numbers together as a kind of "mathematical procedure" that they need to memorize, or whether it's more useful to teach children that if they measure two sides of a square with a measuring tape, they can "multiply" the measurement and that the result is now in "square inches" rather than regular inches. And the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and other people believe that the procedural memorization does kids a disservice by divorcing mathematical thinking from the concrete world entirely.
- munchbunny 6y agoAnd the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and other people believe that the procedural memorization does kids a disservice by divorcing mathematical thinking from the concrete world entirely. In my very limited teaching experience, I think the answer differs based on the student. At the individual level I don't think there's much controversy. Just align with the student's learning style. At scale, I have no idea and do not have the data/experience to have a well-formed opinion. I was perfectly happy to focus on mechanical mastery of the multiplication rituals well before I had any concrete reasons to use them. I know plenty of others didn't work that way.
- agumonkey 6y agoso the iterative pattern doesn't really hold when you go into abstract algebra ?
- deleted 6y ago[deleted]