What’s Wrong with “Multiplication Is Repeated Addition”? (2008) • Hacker News Board

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#321

Post by hc-taway » Wed, Mar 03, 2021, 9:57 PM UTC

jmull wrote: post #183

To me the "So What’s the Problem?" section has a lot of irrelevant stuff. The problem with "multiplication is repeated addition" is that the concept breaks down once you move on from integers. 2 X 3: "add 2 together, three times" - works well ½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore. I can see why teachers who aren't that comfortable with m…

> ½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.

On the contrary, continuing to conceptualize fractional multiplication exactly as your "1/2 X 1/2" as addition example served me well. My wife—with a higher measured IQ than mine, FWIW—evidently can't see it that way. She basically stopped learning math when multiplication of fractions was introduced (it's interesting that one can continue to earn decent math grades, regardless) as all the results felt arbitrary and magical to her, and still do in middle-age. This is actually a very common point for students to stop following WTF is going on in their math classes and never really get back on track (yes, all the way down in, what, 2nd or 3rd grade) from what I've seen; factoring is another, later on.

"1/2, added 1/2 times" is precisely how I think of it. If I can't just pattern-match or rule-follow my way to a solution (because I've forgotten the rules, say) that's still my line of reasoning to figure out what to do to get the solution.

I do exactly the same sort of thing to come back to my senses if I get lost or forget exactly what is happening in fractional division. "How many times does 1/2 fit in 1/4? 1/2. How many times does 1/4 fit in 1/2? 2." Even if I have to manipulate some things to figure out the result with uglier fractions, that's absolutely how I think about what I'm doing as I do it, and it keeps me focused on the ultimate purpose of the calculation. I imagine that's also wrong, according to the author.

In short I can vouch that yes, that exact thing was very helpful to this particular person, including for multiplication of fractions. I truly don't know how else I might have understood those problems, to avoid joining the ranks of the mathematically-lost as early as lower elementary school.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#322

Post by musicale » Wed, Mar 03, 2021, 10:03 PM UTC

Calculator multiplication can be performed by what is essentially repeated addition.

Probably the author would argue that what calculators do is neither addition nor multiplication because of precision loss. In practice it works pretty well though.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#323

Post by pdonis » Wed, Mar 03, 2021, 10:10 PM UTC

pessimizer wrote: post #313

pdonis wrote: post #299

Earlier quoted context omitted.

> Explaining the structure of an atom without having to explain atomic orbitals and standing waves. My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out…

Why would you explain classical mechanics at all? It's an inaccurate simplification.

It's much simpler to compute answers with classical mechanics, and the answers are accurate enough for many practical purposes. As the saying goes, all models are wrong but some are useful.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#324

Post by kortilla » Wed, Mar 03, 2021, 10:12 PM UTC

roenxi wrote: post #308

majormajor wrote: post #213

Earlier quoted context omitted.

The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there, even when explaining, say, Kubernetes to adults! It can be a confidence crusher. There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to…

> It can be a confidence crusher. If confidence gets priority over truth in explanations, then society will churn out people who are confidently wrong. This is a bad idea even if everybody does that and even if it is the traditional approach. If people were honest that they don't know something then the world at large would be a lot nicer to live in.

The confidence is in the ability to learn a topic, not confidence about the knowledge.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#325

Post by spacedcowboy » Wed, Mar 03, 2021, 10:14 PM UTC

agumonkey wrote: post #262

spacedcowboy wrote: post #104

Earlier quoted context omitted.

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…

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.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#326

Post by pdonis » Wed, Mar 03, 2021, 10:14 PM UTC

SilasX wrote: post #312

pdonis wrote: post #303

Earlier quoted context omitted.

You appear to have a much more impoverished view of kids and their ability to learn than I do. My experience (not to mention my memory of how I was myself as a kid) is that kids grasp the fact that there can be more to a subject than adults are able to teach them at a particular time and place, so they're ok with adults honestly admitting that. But they do not like adults telling them categorical statements that late…

There's a difference between prefacing a course with "oh hey these are simplifications that you'll improve upon in higher grades" vs loading down literally every claim with that long chain of caveats.

> loading down literally every claim with that long chain of caveats.

I have never proposed doing the latter, so you are attacking a straw man. Once it's understood that you're teaching a simplified, approximate model, you don't have to repeat in every sentence that you're teaching a simplified, approximate model. You just have to not say it's "the Truth", without approximation and without qualification.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#327

Post by eigenket » Wed, Mar 03, 2021, 10:14 PM UTC

SAI_Peregrinus wrote: post #249

eigenket wrote: post #111

Earlier quoted context omitted.

I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "algorithmic" description of real addition or multiplication is probably a bad idea.

I wasn't even attempting to give an "algorithmic" description. Just a formulaic one, that depends on the axiom of infinity (and accepting transfinite induction as a valid process). Since even at least one of the usual process for constructing the Reals (Dedekind cuts) needs this I don't feel it's much of a stretch in reasoning. And the Surreals have a nicer recursive formula for multiplication that eventually turns i…

You can get algorithmic explicit constructions of (for example) integer addition, rational addition and multiplication and even addition and multiplication of algebraic numbers on a Turing machine (all algebraic numbers are computable). All of these sets are infinite. The reals are particularly "badly behaved" even as far as infinite sets go.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#328

Post by pdonis » Wed, Mar 03, 2021, 10:23 PM UTC

majormajor wrote: post #213

pdonis wrote: post #199

Earlier quoted context omitted.

There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.

The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there, even when explaining, say, Kubernetes to adults! It can be a confidence crusher. There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to…

> The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there

The qualifications are there; that's just a fact. Being told about them, or at least about their existence if not every detail of them, up front seems better to me than finding out about them later on when your mental model is solidified around the simplified version that you then find out doesn't always work.

> I don't think omission of detail is the same as lying.

Saying "multiplication is repeated addition", without qualification and without any caveats, is not "omission of detail". It's a false, categorical statement, i.e., lying.

As for where the line is where you stop giving details, obviously that will depend on the circumstances. A teacher who says "we don't have time to talk about that during class today, but yes, there is much more detail here that you can look into on your own" is not lying and is not saying the child is "not smart enough" to take in all the detail now. (Bonus points if the teacher says "see me after class and I'll give you some pointers on where to go for more information".) A parent who says something similar because they have to get dinner ready and the child needs to do the rest of their homework before bed is also not lying and not saying the child is "not smart enough". Limitations of time are a fact of life, and children need to deal with it just like the rest of us.

A teacher who just says "we're not talking about that", or who doesn't even know about the qualifications, or who gets snippy when a child asks a natural question, is obviously not doing the child any good; but that is because of the teacher fixating on a simplified model and treating it as "the Truth", so doing more of that won't fix it.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#329

Post by SilasX » Wed, Mar 03, 2021, 10:24 PM UTC

pdonis wrote: post #326

SilasX wrote: post #312

Earlier quoted context omitted.

There's a difference between prefacing a course with "oh hey these are simplifications that you'll improve upon in higher grades" vs loading down literally every claim with that long chain of caveats.

> loading down literally every claim with that long chain of caveats. I have never proposed doing the latter, so you are attacking a straw man. Once it's understood that you're teaching a simplified, approximate model, you don't have to repeat in every sentence that you're teaching a simplified, approximate model. You just have to not say it's "the Truth", without approximation and without qualification.

I'm relying on these examples you gave of how to do it:

>"multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with"

>"repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later"

If you disagree that that's "long" or would feel that way in having to do it in every sentence, we can have a great discussion about that, but it is not a strawman -- you seem to reject the idea of giving the one caveat at the beginning of the course, and instead want to make each sentence rigorous.

If you recognize that your complicated sentences are probably not ideal for teaching math to second graders, then I think we're in agreement.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#330

Post by pdonis » Wed, Mar 03, 2021, 10:29 PM UTC

kmill wrote: post #319

pdonis wrote: post #293

Earlier quoted context omitted.

> At some point you need a `Nat.mul` that's concretely defined And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation. But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever…

My computer can manipulate and prove things about real numbers just fine. I think it can do at least everything I can do normally, though much more laboriously. What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that…

> My computer can manipulate and prove things about real numbers just fine.

If you mean it can, with appropriate software, do symbolic manipulations of general formulas that are valid for real numbers, yes, of course. But that's not the same as doing specific concrete computations with them.

> What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences)

Ah, I see. As far as I know the axiomatic reasoning involved can go either way. But in any case, that wasn't what I was trying to get at with the term "emulating"; I'm sorry if my use of that term caused confusion, and I agree with you that telling a child something like "the whole numbers are inside the real numbers" without qualification would also be a misrepresentation. My point was simply that teaching a child a specific computational procedure, repeated addition, in order to get an answer to particular multiplication problems does not require telling the child that multiplication is repeated addition, without qualification. The two things are distinct, and I am fine with the former; I only object to the latter.