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

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

#421

Post by djur » Fri, Mar 05, 2021, 3:30 PM UTC

roenxi wrote: post #306

djur wrote: post #294

Earlier quoted context omitted.

Explaining the structure of an atom without having to explain atomic orbitals and standing waves. Explaining classical mechanics without including a bunch of caveats about relativistic speeds.

"When they performed [this experiment] at [experimental accuracy] they observed [that result] which makes them think [atomic orbitals and standing waves]." Someone who thinks they understand physics without considering experimental accuracy doesn't understand physics.

The goal of primary education isn't to understand any particular thing, it's to provide the intellectual training to understand anything. Someone doesn't need to leave middle school or even high school "understanding physics", but with a base for future understanding. The Bohr model is more than sufficient to help understand chemical reactions.

People seem to be interpreting "lie-to-children" as meaning "[you must] lie[ ]to[ ]children". I don't think there's anyone who thinks that children should be actively deceived into not knowing about quantum mechanics, just that it's OK to use simplified explanations as one step in the process of learning.

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

#422

Post by cannabis_sam » Sat, Mar 06, 2021, 1:54 PM UTC

betterunix2 wrote: post #317

cannabis_sam wrote: post #243

Earlier quoted context omitted.

> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?

When you multiply two negative numbers.

As long as we have

  -1 * -1 = 1

can’t we recover the interpretation of multiplication as repeated addition? As in

  -x * -y = -1 * -1 * x * y = 1 * x * y = x * y

Or is the issue that people are claiming a stronger relationship than an interpretation?

(I guess we would need

  x * 0 = 0

as well)

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

#423

Post by kortilla » Sun, Mar 07, 2021, 1:25 AM UTC

pdonis wrote: post #332

kortilla wrote: post #324

Earlier quoted context omitted.

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

It seems to me that the way to build people's confidence in their ability to learn is to allow them to learn. That means not making oversimplified statements because you don't think they're "ready" for more details. It means giving them the details, and letting them decide when they've had enough for now. (Of course one's time, say in a classroom, will be limited, so at some point one has to say "we don't have time t…

That doesn’t work for lots of people (including myself). If you’re spending a lot of time on details that don’t actually matter for the topic at hand, it takes away mental bandwidth that should be spent on the topic being covered.

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

#424

Post by betterunix2 » Sun, Mar 07, 2021, 2:14 PM UTC

cannabis_sam wrote: post #422

betterunix2 wrote: post #317

Earlier quoted context omitted.

When you multiply two negative numbers.

As long as we have -1 * -1 = 1 can’t we recover the interpretation of multiplication as repeated addition? As in -x * -y = -1 * -1 * x * y = 1 * x * y = x * y Or is the issue that people are claiming a stronger relationship than an interpretation? (I guess we would need x * 0 = 0 as well)

Well, first, why should (-1) * (-1)= 1? Also, you take for granted that -x = (-1) * x, and why should that be the case?

You had asked when the idea that multiplication is repeated addition breaks down. What does (-1) * (-1) mean if multiplication is repeated addition?

Really all these statements, including x * 0 = 0, are about the relationship between addition and multiplication. 0 is the additive identity, so why should we expect anything special about it when it comes to multiplication? The answer is that addition and multiplication are related by the distributive law:

0 * x + x = 0 * x + 1 * x = (0+1) * x = x

Thus (0 * x) is an identity element for addition, but the additive identity is unique (this can be proved using only the properties of addition) and therefore 0 * x = 0. From here we can show:

(-1)*x + x = (-1)*x + 1* x = (-1 + 1) * x = 0 * x = 0

Thus (-1)*x is an additive inverse of x, and again, we can show that additive inverses are unique and therefore (-1)*x = -x. Now we have everything needed for your proof.

Again, the idea that multiplication is repeated addition plays no role here. What actually matters is the distributive law, the meaning of "0", "1", and of "negative," and that multiplication and addition are closed (i.e. the sum or product of two numbers is a number). Those properties are part of the definition of addition and multiplication (along with the associative and commutative laws, and some form of cancellation for multiplication / multiplicative inverses) and can be taken as axioms.

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

#425

Post by cannabis_sam » Thu, Mar 11, 2021, 5:10 AM UTC

betterunix2 wrote: post #424

cannabis_sam wrote: post #422

Earlier quoted context omitted.

As long as we have -1 * -1 = 1 can’t we recover the interpretation of multiplication as repeated addition? As in -x * -y = -1 * -1 * x * y = 1 * x * y = x * y Or is the issue that people are claiming a stronger relationship than an interpretation? (I guess we would need x * 0 = 0 as well)

Well, first, why should (-1) * (-1)= 1? Also, you take for granted that -x = (-1) * x, and why should that be the case? You had asked when the idea that multiplication is repeated addition breaks down. What does (-1) * (-1) mean if multiplication is repeated addition? Really all these statements, including x * 0 = 0, are about the relationship between addition and multiplication. 0 is the additive identity, so why sh…

The original commenteer I responded to (not you), said:

>Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related.

So thank you for explaining the actual relationship!