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

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

#381

Post by TrevorJ » Thu, Mar 04, 2021, 1:41 AM UTC

zamadatix wrote: post #177

pdonis wrote: post #161

Earlier quoted context omitted.

> 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…

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.

It costs nothing to tell someone: "I'm going to teach you one way of thinking about this, but there are other ways you will learn later".

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

#382

Post by cyphar » Thu, Mar 04, 2021, 2:05 AM UTC

pdonis wrote: post #282

wcarey wrote: post #275

Earlier quoted context omitted.

How would you teach physics without lying to children?

Um, by not lying to them? Do you have a specific example of where you think you need to lie to teach physics?

> 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 without explaining relativity -- Newton's theory of gravity is "wrong" after all. It would be "lying" to teach electromagnetism without first explaining quantum mechanics. However it would be impossible to teach anything to a high-school or undergraduate student if you first had to explain everything else in order to teach F=ma or Gauss's law.

I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to or that they're being told they're "not smart enough to understand the full theory". Physics was developed over many thousands of years, it's not really reasonable to expect a student to be able to learn it in one shot. Physics courses are often structured so that you learn concepts in historical order so that you learn how concepts were developed and what motivated future discoveries -- that is also a very useful thing to learn in addition to the actual physics.

I believe the same applies to mathematics education.

Source: I majored in physics. Every year we had an electromagnetism course and we learned that the model we had learned last year was too simplified to work with. Same goes for thermodynamics.

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

#383

Post by wcarey » Thu, Mar 04, 2021, 2:17 AM UTC

pdonis wrote: post #282

wcarey wrote: post #275

Earlier quoted context omitted.

How would you teach physics without lying to children?

Um, by not lying to them? Do you have a specific example of where you think you need to lie to teach physics?

Well, I'd be curious what your first ten sentences to a group of, say, 11th graders would be about Newtonian gravitation, and how those would compare to what you'd say about the same subject to 8th graders.

It seems elsewhere in the thread that you don't consider it lying if one gives a general disclaimer that models aren't perfect, but I wonder whether or not that general disclaimer wouldn't inoculate the idea that multiplication is repeated addition? It's an imperfect model that super useful for all the numbers available to students when they learn what multiplication is. And I wonder further whether there's not an implicit disclaimer that what you're learning isn't the final word inherent in the structure of the educational system?

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

#384

Post by wcarey » Thu, Mar 04, 2021, 2:25 AM UTC

pdonis wrote: post #171

D13Fd wrote: post #42

I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition. For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases a…

> I'm at a loss how you would teach it without explaining that it is repeated addition. How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition? Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't a…

The approach you describe definitely does not work for all children. The vast majority (that I've taught) are better served by a scaffold of increasingly complex metaphors and definitions. The fact that the metaphor of repeated addition wears thin actually highlights the idea that different sets of mathematical objects have different operators for most students in a way that just telling them that ahead of time does not.

The changing of the rules is good preparation for scientific study where our understanding actually changes over time and we have to internalize new models regularly. I taught my chemistry students that quarks only combine in pairs or triplets, until one year Fermilab released evidence that quarks combine in quartets and quintets while I was teaching the unit. How could I avoid telling my students that what I'd taught them before was incorrect? Literally no one knew. So the practice of revising our ideas has utility.

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

#385

Post by Koshkin » Thu, Mar 04, 2021, 2:38 AM UTC

In 2п, п can be though of as added to itself; on the other hand, it is not immediately obvious how to think about 2 being added to itself п times.

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

#386

Post by Koshkin » Thu, Mar 04, 2021, 2:43 AM UTC

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…

Better served by the notion of cardinality of the categorical product.

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

#387

Post by guram11 » Thu, Mar 04, 2021, 3:08 AM UTC

thaumasiotes wrote: post #374

mnl wrote: post #363

Earlier quoted context omitted.

Could you sketch a proof starting from some definition of the operation product of integers as "repeated addition" without using the distributive property, which would imply that we already have another binary operation besides the sum? I was thinking about the Peano axioms when I wrote that (hence the working backwards thing). Obviously if you start with a ring, you don't have to impose it, you get that as a propert…

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?

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

#388

Post by pdonis » Thu, Mar 04, 2021, 4:54 AM UTC

kmill wrote: post #367

pdonis wrote: post #330

Earlier quoted context omitted.

> 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 repea…

> 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.

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

#389

Post by pdonis » Thu, Mar 04, 2021, 4:58 AM UTC

smolder wrote: post #373

pdonis wrote: post #298

Earlier quoted context omitted.

> They don't need this information when learning that 3 groups of 5 fruit are 15 fruits. They also don't need the "information" that multiplication is repeated addition, period, full stop. But that's what others in this discussion appear to be trying to argue for.

I'm really not sure how you could arrive at an understanding of multiplication without first going through the exercise of adding up X groups of Y to yield X * Y. We don't need to declare to children that the precise definition of multiplication is repeated addition. If that's what you're arguing, we agree. If it's that 3*5 shouldn't be explained as 5+5+5, I'm not sure how kids would be expected to learn the subject.…

> I'm really not sure how you could arrive at an understanding of multiplication without first going through the exercise of adding up X groups of Y to yield X Y.*

Arriving at an understanding of one particular special case of multiplication might indeed be a necessary step towards arriving at an understanding of multiplication in general, but that doesn't mean the two are the same.

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

#390

Post by pdonis » Thu, Mar 04, 2021, 5:03 AM UTC

cyphar wrote: post #382

pdonis wrote: post #282

Earlier quoted context omitted.

Um, by not lying to them? Do you have a specific example of where you think you need to lie to teach physics?

> 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 the simplified model is true and not even talking about the fact that it's a simplified model and there are complications being left out, is lying. And the latter is what the posters I've been responding to are attempting to defend.

> I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to

I have never said that is necessary. Of course it's not. Once you've explained at the outset that you're teaching them a simplified model and there are complications being left out, obviously you're not going to repeat it with every single sentence. Any more than the people who are advocating telling students categorically "multiplication is repeated addition" with no mention of all the complexities lurking underneath are advocating repeating that with every single sentence.

> or that they're being told they're "not smart enough to understand the full theory"

I have never said they should be told that. Teaching the simplified model could be due to nothing more than limited teacher time and knowledge. There's nothing wrong with the teacher honestly telling the students that. Bonus points for encouraging them to investigate further on their own if they're interested.