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 to teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct...", and that a different approach is possible. One can argue about the pros and cons of each approach, but I don't think it is helpful to just dismiss one side of the argument as "pedantic falsehoods".
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!
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.
I really don't see people being confidently wrong about abstract mathematics as an issue. I certainly don't know the rigorous definitions of an integral, but I can apply the concepts of calculus to everyday life perfectly fine. People who care about rigorous math can do rigorous math, and I'm glad if they teach me an intuitive understanding that lets me live a happy, productive life.
Personally, I find layers of abstraction necessary for learning. Maybe there are people who don't, I suspect they would have to be prodigies though. Tell me how to add fractions practically, then teach me the principles when I need to know them. Framing that as a lie seems wrong to me, I'd call it "bounded knowledge".
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.
I'm not the one that used that word. If people are going to reference a concept that's been published in a book and uses that word, they should own it and be prepared to justify it. If they can't do that, they should not use the word in the first place.
To be clear, I'm not saying you are one of those people. But the poster I was responding to is.
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.
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.
Kids actually understand the point, and they can decide if they care enough to ask more questions or if it’s good enough for them.
We faced that when teaching divisions. Saying upfront we’d explain falsehoods for the sake of simplicity helped set aside the more difficult questions (infinity, etc.) that came right after. We just say it’s the complicated parts and move on.
The best aspect is they are more receptive to have their mental model broken afterwards, instead of clinging to what you explained as fully true.
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?)
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.
"Multiplication is repeated addition" is not "an abstraction/simplification/shorthand". Doing that for multiplication would be saying something like "multiplication is a distinct primitive operation, but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now." Is that really so hard?
> 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?
If you tell them everything you just said, no, you're not lying. But if you just tell them "always protect your queen", without explaining anything about why and without saying that there are some advanced situations where you might break this rule, yes, you're lying. It only takes a couple of sentences to add that extra information. Again, is that really so hard?
> 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.
Um, what? We're assuming they already know about addition of whole numbers. So it is simple to tell them "this addition thing that you learned, that's an example of an operation", and "those whole number thingies that you learned how to add, those are numbers". Once more, is that really so hard?
I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary.
People may prefer to consider the extended definition more "real", but I think the argument for going the other way is that usually the original limited form of something is more likely agreed upon by most people, whereas the generalization can be done in multiple ways which may owe something to history and culture, or context.
I feel like math is fundamentally different than physics, where the more advanced theory is objectively closer to correct. With math, it's more of an arbitrary aesthetic or social judgment. Nothing ever stops you from generalizing anything even more than anyone did yet, right?
Axiomatically, I think it can go either way. But one still has to recognize, as you do, that there are more cases than just the whole number case, and that what works for the whole number case might not work for other cases.
> the generalization can be done in multiple ways
There are certainly cases of this, but I don't think the case under discussion is one of them. There is only one generalization of the whole numbers under discussion here, the one from whole numbers to rationals to reals (and on to complex numbers if you want to take it that far, and still further on to matrices for some people in this discussion). There aren't multiple ways to do that: the rationals, reals, and complex numbers are all unique sets.
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.
It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "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".
Little kids don’t care about nuance when they’re still having difficulties with carries in addition. Your ideal world where we first explain children that base10 isn’t the only way to represent numbers and whatever other caveats simply doesn’t exist.
They don’t have the knowledge required yet to even understand the scenarios when “exceptions to the rule” apply.
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.
>"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.
I do. Some of the words might be changed, depending on what words have been used to describe the operation of addition and the set of counting numbers. But, as I think I've pointed out elsewhere in this thread, the very fact that the children know about addition and the counting numbers means they know what an operation is ("a thingie like addition") and what a set of numbers is ("a thingie like the counting numbers").
> you seem to reject the idea of giving the one caveat at the beginning of the course
I don't know where you're getting that from. I have already said the contrary--once you've said it, you don't need to repeat in every sentence.
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.
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.
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.
It's much more productive pedagogically to get an intuition for slope and area than it is to get an intuition for compactness and the infinite intersection of open sets, but slope and area are A LIE.
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?
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 of scope for that class was enough.
> Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
My high school physics teacher had no problem explaining classical mechanics including the caveats. The caveats took only a few minutes early in the semester. What's the problem?
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 invalidated Rutherford's model.
The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Same thing in physics. We started with debate about nature of light and examined things like lumniferous aether and Michaelson Morley experiment, double slit experiment etc. Then Maxwell and electromagnetism leading to Heinrich Hertz and the photoelectric effect, which led to Planck and Einstein.
I certainly never thought I was being lied to. We were treading along the same path as those who had came before.
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.
Someone who thinks they understand physics without considering experimental accuracy doesn't understand physics.
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.
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.
That isn't a crutch, that is how large numbers can be multiplied in one's head!
90 * 21 is (90 * 20) + 90
Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally.
Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14.
Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memorized and I'll add my way from there. I don't remember 7 * 6 but I know its 7 * 7 - 7.
Yeah, that’s interesting how we do things in our head like that.
I would frequently break that down as 9*10 + 9*4 in my head.
But I might also do it as 10*14 - 10 - 4.
I’ll try different ways in my head until I can solve one of them easily.
This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system.
Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying.
Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together.
18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108
vs.
18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108
Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.
They might not be. If they're just being taught how to multiply in particular cases (whole numbers) using repeated addition, then they're not being lied to. If they're being told straight up that, while multiplication is not identical to repeated addition, they can ease into multiplication by learning repeated addition, even better.
They're only being lied to if they're being told, authoritatively, that multiplication is repeated addition, no ifs, ands, or buts. And yes, I have had teachers like that, and I suspect many others have too.
But there is a way to teach mathematics from axioms that is intuitive: geometry. Euclid's Elements was the key mathematics textbook for about 2000 years.
Of course, you simply can't cover as much ground if you must derive everything.
But I do wonder if the collapse in public discourse is partly because of faith-based mathematics. i.e. taught as: it's true because we tell you it's true, not because you can see for yourself that it's true.
Then, after GCSEs, during A-levels, we were told (by the same teachers, in the same classrooms) to forget this model, and that the situation was actually more complicated, with s, p, d, f orbitals etc.
I realise that this is was a simplification and not necessarily an outright lie, and can understand why they did it this way. But it was the first time I realised I’d been deliberately taught something that was incomplete or inaccurate.
I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.
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 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.
I have no problem with discovering lies I believe because I told them to myself (and I think most of the lies anyone believes are lies they've told themselves). What I have a problem with is discovering that someone else deliberately lied to me. But probably not everyone feels the same way I do about such things.
Are you sure it's pedantic? As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again - because it isn't helpful as you incorporate fractions, and negative numbers.
The standard manual method of multiplication of large numbers relies on leveraging heavily:
(1) the fact that my multiplication is equivalent to repeated addition for nonnegative integers,
(2) the fact that shifting digit positions are equivalent to multiplication by the base (usually 10) or it's multiplicative inverse, depending on direction, and
(3) memorization of multiplication tables for single digits in the base.
So, no, I don't think the “multiplication is repeated addition” thing is something people are exposed to and then never use.
Why, prey tell, did you make a distinction between positive and negative integers? Is it that repeated addition starts to fail as conceptual model there?
>the fact that my multiplication is equivalent to repeated addition for nonnegative integers, ..
Your claim seems to be that in some abstract way any algorithmic approach to multiplication relies on the fact that multiplying non-negative integers (there's that distinction again .. nasty stuff those negative integers are for your argument) can be rewritten as a repeated addition ... but that's not how people (not even children learning multiplication) do it. That's not how computers do it either.
So no, algorithmically multiplying numbers is not the same thing as 'using' the concept of 'multiplication is repeated addition' in any sense, colloquial or otherwise. Also it starts feeling like you're begging the question when you claim that.
They're not saying "we should be teaching kindergarteners using Euclid" they're suggesting that focusing on the process of doing multiplication hides a lot of very useful insights about why someone would do multiplication in the first place.
The biggest problem with math as it was taught to me in grade school is that it was focused on rote memorization of a procedure. Which is fine if you need to calculate how long to cut 2x4s or stair stringers but it doesn't help you at all if you need to understand why that works the way it does. And god help you if nobody ever shows you the exact method of calculation for those two things in the first place. You're set up to fail if you only focus on process.
I just finished studying Galois Fields, which LITERALLY redefines multiplication and addition operators to study new forms of math (IE fields in particular)
In all fields and rings, multiplication and addition are related by the distributive property.
A(b + c) is equal to Ab + Ac, in literally every form of math you can every think of. A and b could be matrices, vectors, polynomials, prime numbers, integers, rational, complex, real, or Galois polynomials over a weird modulus ring thingy inside of a vector inside of a matrix. Doesn't matter, addition and multiplication are and always defined in that manner.
The thing is that as you can write m (let it be a positive integer) as m=1+...+1 (m-times), you can write n·m=n·(1+...+1), invoke the distributive property for · wrt + and express it as: n·m=n+...+n (m-times), so it looks like "repeated addition" for integers in this case. But it's not a good idea to let ourselves get carried away, we still have two binary operations going on. At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.
n * 0 = n * (1 + (-1)) = n + (-n) = 0.
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The 0-element in a Galois Field works identically btw. In GF(5), the 0 element is 5 (5 mod 5 == 0).
n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0 mod5.
For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0.
2 * (1 + 4) == 2 + 8 == 2 + 3 (mod 5) == 5 mod 5 == 0. Etc. etc.
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This property literally holds in all fields and rings (but not groups).
You don't think it's easier to say "n·5 = n·(5+0) = n·5 + n·0"?
All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is:
[ 0 0
0 0 ]
[ 1 0
0 1 ]
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As such, the concept of "5" does not necessarily exist in all possible Ring-systems. "5" exists in GF(5) for example, but not really in GF(3). Case in point, what does "5" mean in 2x2 Matrix Linear Algebra over GF(2)?
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"-A" is called the additive inverse of A, which also exists in all rings. A - A = 0.
In GF(5), -1 is 4 for example. In 2x2 Linear Algebra, -1 is [-1 0; 0 -1]. In Real Numbers, -1 is... well... -1.
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Anyway, the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings. (And then later, I did an example in the GF(5) system as a specific example).
Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared.
> the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings.
To repeat myself:
n·5 = n·(5+0) [definition of 0]
n·(5+0) = n·5 + n·0 [multiplication is distributive over addition]
n·0 = 0 [definition of 0]
> You don't think it's easier to say...
No. I disagree, your way of thinking is harder for me to think. :-)
You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, although we tweaked the formulas to look like the simplest ways for our own brains.
n·0 = 0 [definition of 0]
n·(1 + -1) = 0 [definition of additive inverse]
n·(1 + -1) = n + (-n) [multiplication is distributive over addition]
n + (-n) = 0 [definition of additive inverse] n·0 = 0 [we seek to show this]
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n·0 = n·(1 + -1) [definition of additive inverse]
n·(1 + -1) = n + (-n) [multiplication is distributive...]
n + (-n) = 0 [definition of additive inverse]
QED
I'll happily take your advice however! Proper mathematical rigor is always something I appreciate, even if its something I'm not very much practiced at.
It boggles my mind that you've been studying Galois theory yet somehow try to reduce the algebraic structures associated to two binary operations to playing with one of them.
Nonetheless, in a ring (and all fields are rings), multiplication must and always is related to addition, through the distributed property (which I argue, the distributed property IS the mathematical term for "repeated addition").
Without the distributed property, you have no ring. You at best only have a group. Therefore, all multiplication operators ever defined (or more precisely, all rings) must have multiplication related to addition: (A * (B+C) == AB+AC)
All this is trivial, if you consider a ring, you get a·0=0 as a property, if your starting point is the Peano axioms for the arithmetic of natural numbers that's one of them, for the latter seeing it as "repeated addition" makes no sense, for the former, well you have a ring, you have two binary operations, not one, and of course you have some form of distributive property or else you'd be studying this set with just one binary operation at a time.
I'd like to see how "repeated addition" works in polynomial rings.
Consider the following polynomial: x0 * b^0 + x1 * b^1 + x2 * b^2 ... xn * b^n, where "n" goes to both positive infinity and negative infinity.
When "b = 10" and when "x" can be numbers from [0-9], we have the so called base-10 set of real numbers, do we not? IIRC, if b = sqrt(-1) * 10, we then have the set of complex numbers (a non-intuitive result. I may have made a mistake somewhere, but I assure you there's a surprising property along those lines).
That's the funny thing about real numbers and complex-numbers. Real numbers and even complex-numbers ARE polynomials, and therefore a polynomial ring. 3.1415926 == 3 * 10^0 + 1 * 10 ^-1 + 4 * 10 ^-2 ...
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I'm using a lot of words here. But all I'm saying is once again: Pi * 3 == 3.14... * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3 + ... == 9.42...
We can evaluate 3 * Pi by splitting Pi up into a set of additions (3 + 0.1 + 0.04 + 0.001...), even if that set of additions is infinite. Then evaluate 3*(each component). This is possible because Pi is easily represented as a polynomial X0 * 10^0 + X1 * 10^-1 + ... Xn * 10^-n.
There's a reason why polynomial multiplication is usually called "Carry-free multiplication". Because Real-numbers are just polynomials where you have that annoying "carry the one" property to keep track of. Remove the "carry the one" property (in say: Galois extension fields), and all the math still works.
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Hmmmm... I probably could have said all that in fewer words. TL;DR: "Real numbers ARE a polynomial ring". (And complex numbers are probably a polynomial ring, I just forgot how to prove that factoid)
> I'd like to see how "repeated addition" works in polynomial rings.
EDIT: Just thought of a cute and simple retort. You ever do a CRC32 check? There ya go.
I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 * (1-x^3)) ??
The polynomial itself gives us the means at which we logically split up the multiplication into component parts.
Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we move onto polynomials, we do the same exact thing. EDIT: remember, ALL REAL NUMBERS ARE POLYNOMIALS with a base of 10.
Or to put it another way: when x == 10, your polynomial of (1 + x + x^2) * (1-x^3) == 111 * (-999). That is to say: real numbers are simply polynomials where "x" has been defined to be a particular number, instead of an abstract entity. We call that number the radix-base.
If you instead defined the base to be x = 16 (hexadecimal numbers), you'd get 111 * (-FFF), which you'll find will satisfy similar properties. Now leave x-undefined (since it could be 10 or 16), and what do you get?
Polynomial math. Or so called "Carry-less multiplication" (https://en.wikipedia.org/wiki/Carry-less_product). We don't have a ring yet though: we still need to perform a modulus on all those polynomials to return to a proper ring (and if the modulus is irreducable, we have a Galois field). But we can already see how polynomials and the Reals are so closely related.
Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.
That's not what this blogpost is arguing about. This blogpost is arguing that "Multiplication is Repeated Addition" is unhelpful at the elementary school level and stops being true at some point.
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My argument is otherwise. "Multiplication is Repeated Addition" is clearly helpful in grade school. Almost everybody I know has learned Multiplication through that method.
Secondly: I cannot think of a single instance where its not true. Yes, I've had to use linear-combinations to extend it out to polynomials, but clearly the property holds even in polynomial-land.
Its not useful to teach multiplication of polynomials with "Repeated Addition". But the advice is "not wrong", in fact, polynomial multiplication continues to see many similarities with Real and Complex multiplication. Especially if we consider a "Basis" to be analogs to the thing that's repeatedly-added.
Are you kidding? This is the exact opposite of the truth; the nature of multiplication as repeated addition is the entire reason why multiplying by 0 gives the additive identity. It's exactly the same as how exponentiating by 0 gives the multiplicative identity, since exponentiation is just repeated multiplication. And this is so fundamental that 1 is frequently referred to by this property, as "the empty product".
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 property. I think I mentioned that in a later post.
5·3 + 2·7
5·3 + 2·7 + 0·4
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're adding it to the identity, 0.
And finally we say that starting from 0 and adding something zero times leaves you where you started, at 0, so we can observe that 0·n = 0.
If you formalize that, you'll end up either deriving or postulating the distributive property, depending on what you start with. But it isn't arbitrary; it's not a coincidence that (as I mentioned above) exponentiation by 0 gives the multiplicative identity, and (as I haven't mentioned yet) exponentiation distributes over multiplication the same way multiplication distributes over addition.
(Asymmetry does start creeping in; aggregate exponentiation isn't as nice since exponentiation doesn't have the nice properties that addition and multiplication do.)
things that irks me
a**b * a**c == a**(b + c) != a**(b * c)
(ab)^x = (a^x)(b^x)
yes my mistake
Why is this not a good idea?
Can’t we just accept/postulate that the additive identity is different from the multiplicative identity?
And still define a relationship between the addition and multiplication?
Maybe I misunderstand the issue..
(I’m not trying to be pedantic, but my math background has some holes :)
That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.
Am I missing something here?
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This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related.
Pi * ([ 1 0 ; + [ 1 0 ;
0 1 ] 2 1 ])
Pi * [ 1 0 ; + Pi * [ 1 0 ;
0 1 ] 2 1 ]
[ Pi 0 ; + [ Pi 0 ;
0 Pi ] 2Pi Pi ]
[ 2Pi 0 ;
2Pi 2Pi ]
Pi * ( 1 0 + 0 0 + 1 0 + 0 0 + 0 0 )
0 0 0 1 0 0 2 0 0 1
3 * pi = pi + pi + pi
but how do you represent the other distribution, where 3 is added together pi times?
3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...)
Aka: 9.4245...
You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)
You can't. Pi is irrational.
You just need to break it up infinitely times. We usually call the sequence 31415926...
I've responded to this elsewhere in the thread: I don't think adopting increasingly perverse interpretations of "repeated addition" as you try to include more and more numbers is a useful way to teach multiplication.
Sorry, I should have said substitute b = pi or c = pi. Or see the sibling post by iio8999.
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 is expected to leave that property intact.
This is not unique to multiplication. Modern mathematics is all about extracting fundamental properties that make something true, to generalize results while keeping the original simpler statement true. Starting from the generalisations without explaining the intellectual process it took us to get there means just giving facts without explaining them.
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 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.
Repeated addition is just a technique for multiplication. It is one of many techniques we teach children, and we should leave it at that -- a technique. When we get to multiplication by negative numbers or fractions, we teach children other techniques, and we do not feel any need to try to "define" multiplication in terms of those techniques. Why should we "define" multiplication in terms of repeated addition, and then do circles around ourselves trying to generalize that "definition?" We can just say that multiplication is one operation we can do on numbers; addition is another, and they are related by the distributive law (and it from the distributive law that we can derive the various techniques we use for multiplication).
I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.
Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").
That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.
Though it can be helpful to think of multiplication as scaling or rotation in certain contexts, or as repeated addition in others, none of those are a universal truth.
To say it's not related to addition at all is also too broad to be true. They're related in many useful ways that other commenters have pointed out, in addition to the obvious way that in some situations you can define one in terms of the other. Even at the cutting edge, our inability to prove the Goldbach Conjecture might have something to do with not fully understanding the deepest relationships between the operations.
My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?
-1 * -1 = 1
-x * -y = -1 * -1 * x * y = 1 * x * y = x * y
(I guess we would need
x * 0 = 0
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.
>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!
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 defined in any or all of these contexts if one enjoys doing so. There are far ranging context to define an object and less far ranging. I am quite sure that in grade school it is quite unhelpful to look for the furthest ranging context in which one could define multiplication. E.g, let us start in grade 1 with the definition of an algebra and derive everything from that.