Comment by spacedcowboy | Hacker News Reader

Comment by spacedcowboy | Hacker News Reader

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I have a PhD in physics and more maths qualifications than I can shake a stick at; to me, multiplication is repeated addition.

I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”.

I also happen to have an 8-year-old going through third grade right now, and when we were talking through his homework, it was quite clear that using simple concepts he already knew (addition & subtraction) to explain slightly more complex things that he was learning (multiplication and division) was really useful to him. As I recall it being to me.

[aside] I think the maths schedule is more advanced now than it was in my day anyway - he only did multiplication and division this year, but he also did algebra and simultaneous linear equations now, as in:

a + b + 8 = 24

a - b = 4

“Solve for a and b”

Pretty sure I only did that in senior school (11 and up), not at age 8. No powers as yet (presumably they’ll come after the multiplication/division stuff), so no quadratic formula, but still...

[/aside]

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.

I 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.

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.

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.

Your 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.

Speaking 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.)

so the iterative pattern doesn't really hold when you go into abstract algebra ?

I have got almost exactly the same situation: PhD, physics, forgotten more math than most folks ever learn, etc., and with an 8 year old learning the same level of mathematics as you describe.

I think the only difference might be I used "iterated" rather than "repeated" when helping him. Anyone who is just learning multiplication likely lacks the depth of experience necessary to make use of the "correct" jargon and abstract concepts as a starting point. "Repeated addition" is a useful aid in learning the operation to build that experience.

Another physics PhD chiming in here. I have never before noted the difference between "multiplier" and "multiplicand". The whole article has me rolling my eyes.

In fact, I would argue that multiplication being associative shows that this distinction is meaningless.

Surely you knew subtrahend and minuend from vector subtraction? Or divisor:dividend?

Since you're a physician, do you think that helps for multiplicative relationships in real world laws ? It took me decades .. sadly, to be comfy with handling U = RI formulas. On the algebraic level it's stupid simple, but for real world physics the meaning is more bidirectional coupling of ratios and amplitudes and taps into a different part of my brain.

Physicist, not physician, but really - I haven't used my physics knowledge directly in a few decades now... I've been a software engineer for most of my life :)

As for V=IR (I had to google U=RI, maybe U is the more modern version, but it was always V=IR when I were a lad), I don't really have a problem with ratios. When I was learning equations, the simple rule is "do unto one side whatever you do to the other", so ...

V = IR, divide by R -> V/R = I

I was happy with either representation, and I didn't think of it as multiplying, dividing, adding or subtracting, it's just "do the same thing" on each side. The problems I had were more "when do you apply Kirchoff's laws to figure something out, and when do you apply Ohm's law; that sort of thing you just get by experience, I think.

hum sorry about the physician.. freudian slip

what I meant about the ratio part is that physicists don't care much about the computation they care about whenever you know one piece, you know the other will variate through a third factor in a multiplicative manner.. maybe it makes sense, but it's not at all a structural notion like (x) === iterate(+,n)

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