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

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

#211

Post by lhorie »

pdonis wrote: post #161
cjfd wrote: post #105

Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously. You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be def…

> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..."

My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had to do rote memorization of the tables), but hey they can recite tables now too.

Easing into multiplication via addition doesn't feel wrong IMHO. In fact, at later grades, I'd have to learn about associative/commutative properties anyways.

My daughter in particular has a somewhat peculiar story: she is in kinder and learning multiplication from a game[0] that her older brother started playing, and the way they introduced multiplication is by telling her that to calculate the area of a rectangle, she needs to literally count the number of blocks that compose it.

Is it "wrong"? I guess. Blocks aren't really proper units, but she gets that a rectangle made of 4x5 blocks has an area of 20 blocks, first by stumbling n' counting, then adding rows then eventually just memorizing the multiplication fact, so mission accomplished? Things build up from there, first without the grid, then into area of more complex polygons, spin off into word problems, etc.

IMHO what helps kids learn is just repeatedly seeing where the memorized mechanisms can be applied and where they can't. Eventually multiplication becomes second nature and the addition/subtraction crutches come off.

[0] https://www.prodigygame.com/

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

#212

Post by hinkley »

EvanAnderson wrote: post #210

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach mu…

The new new math isn't just multiplication is addition, it's addition and substraction.

39 x 40 = 40 x 40 - 40

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

#213

Post by majormajor »

pdonis wrote: post #199
zamadatix wrote: post #177

Earlier quoted context omitted.

The general concept is known as https://en.wikipedia.org/wiki/Lie-to-children and I think you'd be hard pressed to teach/learn/apply anything of significance without this "well actually it's more complicated than that" approach and recognizing how far you need to go with it to aptly do some task.

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 "who decides what you're smart enough to hear about now" question in its own way. Or "oh the teacher doesn't actually know what the difference is, or why this isn't 'true' 100%."

Despite the cutesy name, I don't think omission of detail is the same as lying. It's often impossible to tell 100% the truth. You probably don't even know it yourself!

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

#214

Post by RandallBrown »

pdonis wrote: post #196
wruza wrote: post #187

Earlier quoted context omitted.

How would you see multiplication by pi as repeated addition? By taking 3 times as usual and then another .14 approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28. (edit: if you’re confused by /100, read it as “two places next to a point”) If you’re asking how to do that exactly , first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:

> By taking 3 times as usual and then another .14 approximately First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition". Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times. > If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’…

> not in the elementary school student's understanding of "repeated addition".

Pi isn't particularly well understood by elementary school students either.

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

#215

Post by cryptica »

As someone with a pragmatic, visual mind, I rely entirely on this kind of reasoning in order to understand and apply math. For example:

- Multiplication is repeated addition.

- Power is repeated multiplication.

- The square of a number can be visualized as a geometric square when you duplicate a row of x items x number of times.

- The cube of a number can be visualized as a geometric cube when you duplicate a row of x items x number of times and then you take the resulting geometric square and duplicate it x times along the orthogonal axis.

- n to the power of p can be visualized as a tree with height p such that each branch splits up into n branches at each level.

- The logarithm base x of y is the height 'number of levels' of the tree when each branch splits into x branches at each level until the number of tips is equal to y.

- A factorial can be visualized as a tree whose branches split up in such a way that there is one fewer branch at each level until the branches cannot be split anymore.

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

#216

Post by munchbunny »

jschwartzi wrote: post #167
munchbunny wrote: post #125

Earlier quoted context omitted.

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'…

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 s…

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.

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

#217

Post by allturtles »

pdonis wrote: post #205
SilasX wrote: post #201

Earlier quoted context omitted.

Well, that would be true if all children were as smart and analytically adept as you.

I think this viewpoint is pernicious. A child doesn't have to be "smart" in order to deserve being told the truth. Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)

An abstraction/simplification/shorthand is not a lie. People don't say "multiplication is repeated addition" because they are trying to hide the truth for some selfish reason. It's a pedagogical strategy to help people learn a new abstraction by analogy to an old one. These kind of crutches are a necessity, you can't introduce all the complexity of the world to someone all at once. This applies to every subject - science, history, writing. Simple notions and shorthands are introduced first, and complexity and nuance added on later.

Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?

In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.

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

#218

Post by matheist »

EvanAnderson wrote: post #210

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach mu…

Math PhD here. You're a fantastic dedicated parent and you did no harm.

First, as to the merits of your approach --- I think of multiplication as scaling myself. That's totally valid.

Second, whether it's even possible to do harm here --- worst case scenario, the metaphor doesn't make sense to your kid and she doesn't use it in her own thinking about mathematics. Everyone has to develop their own intuitions --- like you had your epiphany --- and she'll develop hers even if you set her on a track that doesn't work, just as part of her learning and doing mathematics.

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

#219

Post by EvanAnderson »

hinkley wrote: post #212
EvanAnderson wrote: post #210

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach mu…

The new new math isn't just multiplication is addition, it's addition and substraction. 39 x 40 = 40 x 40 - 40

That seems like a reasonable way to do that computation, from a purely practical perspective. Simple "tricks" to arrive at the correct answer quickly don't at all seem like a bad idea. When we were doing addition of two digit integers and working with place value there was a lot of adding the ones separately from the tens, then adding those together. Later, when addition with carry / regrouping came up we used both strategies to solve the problems and demonstrated how they really were the same thing, just following a different "recipe" to get the result.

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

#220

Post by SilasX »

pdonis wrote: post #205
SilasX wrote: post #201

Earlier quoted context omitted.

Well, that would be true if all children were as smart and analytically adept as you.

I think this viewpoint is pernicious. A child doesn't have to be "smart" in order to deserve being told the truth. Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)

Along with what allturtles said, I don't think it's right to call it a lie, which, to me, implies moving someone's model away from the truth (on the basis of them trusting you to convey the correct one). An oversimplified, "wrong" model doesn't do that; it moves them from ignorance toward the correct model (i.e. increases their prediction accuracy).

And yes, even when I'm in the learner's shoes, I prefer that a teacher start with an approximate model, and then refine it as they go further. Starting with the full thing is barely comprehensible.

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