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

denisegaskins.com

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

#391

Post by pdonis »

bigger_cheese wrote: post #371
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…

I remember my High School chemistry lesson from the 90's clearly explained we were working with historical models, we talked about nature of the models developments and the experiments that led to them. We started with the JJ Thomson "plumb pudding model" Then we learnt about Rutherford and the gold foil experiment which invalidated Thomson's model. Then we learnt about emission spectrum and the Bohr model which inva…

> I certainly never thought I was being lied to.

That's because you weren't. You said:

> The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.

Which is exactly what I think should be told to students when a simplified model is being taught. But the people I've been responding to are advocating for telling students the simplified model as if it were exactly correct and covered all cases. That is what I am saying is lying.

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

#392

Post by thaumasiotes »

guram11 wrote: post #387
thaumasiotes wrote: post #374

Earlier quoted context omitted.

If you want to work up to it conceptually, then I'd say consider the meaning of something like 5·3 + 2·7 We add 3 five times, and then we add 7 twice. It is then easy to extend this to 5·3 + 2·7 + 0·4 and say, OK, add 3 five times, and then add 7 twice, and then add 4 zero times. And you get 29. At that point you notice that when you say 5·3 means "add 3 five times", you forgot to say what you were adding it to. You'…

> exponentiation distributes over multiplication things that irks me a**b * a**c == a**(b + c) != a**(b * c) for exponentiation to distribute over multiplication, it must become addition, thus reinforcing the multiplication IS repeated addition?

Distributivity of exponentiation over multiplication looks like

  (ab)^x = (a^x)(b^x)
You're talking about "being exponentiated", not exponentiation. This one isn't a commutative operation.

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

#393

Post by guram11 »

thaumasiotes wrote: post #392
guram11 wrote: post #387

Earlier quoted context omitted.

> exponentiation distributes over multiplication things that irks me a**b * a**c == a**(b + c) != a**(b * c) for exponentiation to distribute over multiplication, it must become addition, thus reinforcing the multiplication IS repeated addition?

Distributivity of exponentiation over multiplication looks like (ab)^x = (a^x)(b^x) You're talking about "being exponentiated", not exponentiation. This one isn't a commutative operation.

> commutativity

yes my mistake

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

#394

Post by cyphar »

pdonis wrote: post #390
cyphar wrote: post #382

Earlier quoted context omitted.

> Do you have a specific example of where you think you need to lie to teach physics? If we are defining "lie" to mean "simplifying the model to allow students to understand what's going on" (which is what we're talking about) then basically every field of physics required some simplifications in order to teach. For instance, it would be "lying" to teach Newtonian dynamics and Newton's theory of gravity at all withou…

> If we are defining "lie" to mean "simplifying the model to allow students to understand what's going on" (which is what we're talking about) No, it isn't what we're talking about. We're talking about making categorical statements that are false. Telling students that you're teaching them a simplified model and there are complications that will be added back later is not lying. But telling them categorically that th…

> No, it isn't what we're talking about. We're talking about making categorical statements that are false.

That's not what we're talking about. Describing multiplication as repeated addition as a way of teaching a new concept to elementary school children is simply not lying.

Your issue is that the fact it is a simplified model is not being explicitly told to students, which is being called lying. While it would be false to say that a simplified model is the complete picture, teachers aren't standing up in class and saying "this is all there is to this subject, no need to learn any more!".

My main issue with your suggestion is that young students simply aren't going to remember a side comment at the beginning of their classes that this isn't the full picture -- so you will either have to repeat this regularly (which will be demoralising and confusing) or there really is no strong benefit to doing so.

You say that the "categorical statement" is not being repeated with each sentence -- but the thing is that when students use or discuss the tools they've learned, they're reinforcing their mental model of how the tool works. So in a way, the simplified model is being repeatedly reinforced in their mind. If you want to counteract it, saying once at the beginning of each class that introduces a new concept "this isn't the full picture" won't help most students overcome the issues they hit when they learn their old model doesn't work for more complicated problems.

> I have never said they should be told that.

You misunderstood what I said -- I said that constantly being told that what they're learning is "not the real theory" (or however you want to phrase) is demoralising in of itself -- it needlessly gives the impression to students that they're not smart enough to understand "the real theory". I never claimed that you said that teachers should say that explicitly.

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

#395

Post by cyphar »

pdonis wrote: post #285
WalterBright wrote: post #256

Earlier quoted context omitted.

Teaching Newtonian Mechanics to students is also a lie. But I don't believe starting with General Relativity is going to do anything but confuse students. I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.

> Teaching Newtonian Mechanics to students is also a lie. Not if you tell the students that it's an approximate model that works well in the domain they're currently studying, but doesn't work well in a more expanded domain. Of course if you insist on acting like an authority and telling students that Newtonian Mechanics is "the Truth", then yes, you are lying to them. But you don't have to tell them that to teach th…

> Of course if you insist on acting like an authority and telling students that Newtonian Mechanics is "the Truth", then yes, you are lying to them. But you don't have to tell them that to teach them Newtonian Mechanics.

Do you think teachers really say "Newtonian Mechanics is the Truth" in class? Maybe it's different in the US but in Australia I've never had an interaction with a teacher like that.

Or is your issue that teachers are omitting that information? If so, that's quite a different thing to discuss.

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

#396

Post by Double_Cast »

tomxor wrote: post #376
Double_Cast wrote: post #292

Earlier quoted context omitted.

Suppose your friend Alice arranges your wedding. You ask her to arrange the lawn chairs in an arrangement of 2 rows and 3 columns. But she misinterprets your request as 3 rows and 2 columns. Oops. Now a particular family can't all sit in a single row without rearranging the chairs. If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" ma…

Except we are talking about scalar multiplication, that was the premise, not matrices.

The point isn't "matrix multiplication != scalar multiplication". The point is "the process of evaluating a reducible expression inherently discards information about the original expression", which is a fact about evalution rather than any specific operator. The fact that "the information discarded is of little consequence to the compressed result" is a quirk specific to scalar multiplication. Thus, the commutative property distracts from explicitly modeling the "AST" so that the student understands what multiplication represents under the hood, beyond the rote memorization of scalar multiplication tables.

Perhaps an analogous situation would be: Suppose a teacher wanted to introduce the notion of limits to a calculus curriculum. "That makes zero sense. The only things a student needs to know are the shortcuts for each parent function, e.g. that (d/dx x^2) reduces to (2x) via handwavey magic." But what if an engineer needs to integrate over an arbitrary curve? Can students solve the problem without being comfortable with Riemann Sums? Maybe 1st-year calc students should rederive the shortcuts from scratch? "Except we're talking about a math course, not an engineering course."

> In particular this bit regarding multiplier/multiplicand makes zero sense to me.

> Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.

It's bizarre to simias (and you, I assume) because y'all can't imagine performing the operation without thunking. (Don't get me wrong, I think "repeated addition" is the best method. I'm just attempting to explain the opposite perspective so that it feels less bizarre.)

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

#397

Post by kmill »

pdonis wrote: post #388
kmill wrote: post #367

Earlier quoted context omitted.

> 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. No, I don't mean computer algebra systems, if that's what you mean. In something like Lean, you can set up an actual construction of the system of real numbers (in mathlib it's the Cauchy completion of…

> In something like Lean, you can set up an actual construction of the system of real numbers I'm not familiar with Lean, but from what I can gather, it's a theorem proving system. That is included in what I was referring to. Proving general theorems is still not the same as doing specific concrete computations.

Yes, it's a theorem proving system, and it's based on foundations that are powerful enough to represent all of (formal) math. Including something like Lean excludes math itself from being able to do "specific concrete computations," which is absurd.

Here are some examples to give you an idea of concrete calculations. Lean is not a CAS, so it needs some help in the form of tactic proofs (which are unreadable without an IDE, but their structure sort of mirrors work you'd do on paper):

  import data.real.basic
  import data.real.sqrt
  import tactic

  -- that 2^2 - 2 = 0
  example : (real.sqrt 2)^2 - 2 = 0
  := by norm_num

  -- that if x solves x^2 - 2x + 1, then x = 1.
  example (x : ℝ) (h : x^2 - 2*x + 1 = 0) : x = 1
  := begin
    have : x^2 - 2*x + 1 = (x - 1)*(x - 1), ring,
    rw this at h,
    cases eq_zero_or_eq_zero_of_mul_eq_zero h with h' h',
    repeat {
      convert_to x - 1 + 1 = 1,
      norm_num,
      rw h',
      norm_num,
    },
  end
These were just some quick examples, but what I'm trying to show you is that in Lean you are working with the real numbers. They're just as real as the ones defined in a real analysis textbook, maybe moreso. They're just more tedious to deal with because the computer needs every calculation to come with some sort of proof of correctness, which thankfully can be filled in automatically sometimes with tactics like norm_num.

(A meta-comment: I'm not sure why if you're not familiar with Lean you'd tell me what it can and cannot do. Maybe someone told you at some point "theorem prover systems prove general theorems" or "theorem provers can only prove things about computable reals" and you had no reason to believe that they weren't telling you the whole story? Don't take this too seriously, I'm just riffing on your original point.)

p.s. Looking back to the beginning of this thread, I realize my saying "I'm not understanding what all the fuss is" could be misinterpreted -- I didn't meant to refer to you in particular. This was meant to be in the context of all the comments and the original article, and somehow I ended up replying to yours. Maybe one thing that caught my eye was the idea that there was much to unlearn here, but I ended up not addressing that point because I didn't have anything cogent to say. I think the sibling comment put it well, that there's value in learning to unlearn.

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

#398

Post by jobigoud »

pdonis wrote: post #295
wruza wrote: post #224

Earlier quoted context omitted.

But what is that number? You only described an algorithm that relies on a vague definition of “part to the right”. See, you can’t even write it down (unlike 1 + 3.14 = 4.14), because pi is not really a number, in a sense. It is an infinite calculation that happens to converge between 3.14 and 3.15. You can never get rid of “pi” in your calculations unless it cancels out naturally, so for practical reasons it’s 3.14 a…

> what is that number? Pi minus 3. If you are saying you can't actually perform that operation, great! That means you are agreeing with me. See below. > you cannot multiply by pi by repeated addition Thank you for agreeing with my main point. See below. > not because the numeric method is wrong. If someone is going to claim that multiplication is repeated addition, then their definition of multiplication in terms of…

I think what they are saying is that if we can't say that multiplication is repeated addition due to the reason you propose, then addition itself is ill-defined as a concept, which is a bigger problem.

That by using your argument we move the whole debate one level down the fundamental operations and now you no longer have a way to simply explain addition or subtraction.

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

#399

Post by codethief »

alentist wrote: post #365
codethief wrote: post #353

Earlier quoted context omitted.

Exactly. I've scrolled through hundreds of comments here now and it really is beyond me how the question of whether you can define x·y in terms of addition for x and y being arbitrary reals is even a matter of debate.

The question is not whether you can define it in terms of addition in some abstract way, but whether you can define it in terms of repeated addition , i.e. something of the form x + x + … + x.

> The question is not whether you can define it in terms of addition in some abstract way

This is splitting hairs. The reals themselves are defined "in some abstract way". For instance, what does addition of reals even mean? How do you add two arbitrary real numbers? Exactly, by adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) and the only abstraction involved here is the abstraction due to the way the reals are constructed in the first place.

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

#400

Post by aaplok »

betterunix2 wrote: post #316
aaplok wrote: post #289

Earlier quoted context omitted.

Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name. The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation…

"Multiplication is literally repeated addition" How do you repeat something a negative number of times? What does it mean to multiply two negatives, if multiplication is repeated addition? This is not an overly generalized concept, we teach negative numbers to elementary school students.

I don't know what you mean by "overly". Multiplication by negative numbers is a generalisation of the earlier concept of multiplication by positive numbers. In a sense even multiplying by 1 is a generalisation (at least if you read Euclid he doesn't really define multiplying by 1).

I don't claim that we should not teach generalisations to children. What I claim is that of we teach generalisations we at least should be aware that they are generalisations, and maybe even tell them about it.

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