What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
denisegaskins.com
denisegaskins.com
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.
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!
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.
-----
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.
-------
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"?
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.
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.
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?
--------
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?
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.
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.
You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication?
This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.
That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square.
In fact, I have a high school textbook that makes that derivation.
That is one way to derive it, but there are others, including my personal favorite, the resolvents method (which also works for deriving cubic and quartic formulas).
Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from.
It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation. And there it was. The quadratic formula!
Only if you restrict to whole numbers. But multiplication does not just apply to whole numbers. How would you see multiplication by pi as repeated addition?
> how else would you define multiplication?
As a separate operation with its own properties. The subject article gives several examples of properties of multiplication that are simply different from properties of addition.
Multiplication by irrationals is just an infinite sum of multiplication by rationals, no? x times pi = x times 3, plus x times 1 / 10, plus....
I don't see there's any conceptual issue here. Mathematicians feel free to correct me.
Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.
First define a*b for integers by addition.
Define the the rational numbers as ordered pairs of integers where the last number may not be 0. Consider the rational number (a,b) to be equivalent to (c, d) if a*d = b*c
(Exercise for the reader: Prove that this is an equivalence relation)
Now define (a, b) + (c, d) = (ad + bc, bd), (a, b) * (c, d) = (ac, bd)
(Exercise for the reader: Prove that the subset of rationals (n, 1) behaves just like the integers)
So I still think conceptually it's fine to think of it as repeated addition? It might be algorithmically a bad way to do it ("increasingly perverse", taking limits when the number is irrational etc.). I don't know how computers actually implement multiplication - though wikipedia (https://en.wikipedia.org/wiki/Binary_multiplier) says it works via shifts and adds.
Or maybe, it's bad to teach it to kids this way? But then I think we need evidence from educationalists.
So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.
Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of 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:
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’s start with one, 1 + pi:
Add 1 to the 3 and keep the part to the right of the decimal point the same. Addition is commutative.
Basically, your argument is that multiplication is repeated addition as long as we're willing to adopt increasingly perverse interpretations of "repeated addition" as we expand the scope of the numbers we use. Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
Let’s make it clear: you cannot add pi or take pi times at all. You only can add to factors to the left of it (same as a+bi). That’s why you cannot multiply by pi by repeated addition, not because the numeric method is wrong.
Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?
For generalized reasoning, yes. For teaching, not sure. Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that? When they learn, they ask themselves why and it’s a rabbit hole. You should stop somewhere before turning into maths professor, if that’s not your goal.
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 repeated addition must cover all cases. You are saying it doesn't cover pi, which means it doesn't cover all cases. So the definition is wrong.
> For generalized reasoning, yes. For teaching, not sure.
Why not? What's wrong with telling kids, multiplication in general is a distinct primitive operation, we are teaching you how to multiply whole numbers using repeated addition, but be aware that this method will not generalize to all cases?
> Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that?
A student who has been told what I just described above is not in either of these positions, so you are arguing against a straw man.
> When they learn, they ask themselves why and it’s a rabbit hole.
No, it's natural human curiosity which should be encouraged, not stomped on. At some point they'll either realize that they're up against material they're not yet ready for, and put it aside for later, or they'll end up being a math prodigy. Both outcomes are better ones than them just being told "this is it, don't ask any questions".
Pi isn't particularly well understood by elementary school students either.
This part of math is so exciting and compelling, I feel like it should be more of a focus in schools.
Only for rational numbers. Doesn't work for real and complex numbers.
> It's even a useful thing to do, how else would you define multiplication?
Axiomatically, not algorithmically.
How exactly are you going to present multiplication of real numbers axiomatically without essentially including an axiom that bootstraps everything from repeated addition?
I suppose you can try defining the reals as "the unique complete ordered field" or the complex numbers as "the unique algebraically closed field of characteristic zero with cardinality c," but I don't think either of those are pedagogically useful to someone who is still learning what multiplication is.
The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, in particular define a particular "real number" to be the set of all rational numbers less than it, for example sqrt(2) is defined to be the set of all rationals p/q such that p^2/q^2 < 2. We can "recursively" define addition of these real numbers in terms of addition of rational numbers because to add two reals you "just" have to add all the rationals in their respective sets.
In general there is no sensible algorithm to do anything in the real numbers, since most real numbers aren't even computable (there is no way to represent an arbitrary real number on a Turing machine).
This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.
There are countable/computable/constructable subsets of the reals where multiplication has a finite algorithm and is it repeated addition.
One example is the algebraics, as well as extensions the including a few special constants like pi. These are the subsets of the reals most commonly used for math and science. So in a wide range of problems areas, multiplication is not just repeated addition.
Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots.
This metaphor is just training wheels. At some point you should lose it.
Together with the notion that "multiplication is repeated addition" comes the notion that numbers are quantities. Only some of them are, and this isn't really what makes them numbers. Now what exactly gets repeated, when you don't have quantities?
You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...
People are just bit sloppy, and say algorithm when they mean something slightly different.
See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion
Basically, you don't want an 'algorithm' here to produce the whole number.
All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).
Also, an irrational exponent is the product of component factors.
b = Prod(0,inf) a^[x_i * 10^(-i)] = a^x
So even with irrational numbers, operations can be decomposed - such as exponentiation into multiplication of integer exponents and roots.
Which makes sense, because in the sequence definition of reals you need a way to generate the resulting sequence from the two original sequences.
I think you’re trying to claim more than is true.
It's probably a tomato/tomato kind of thing, but I'm only objecting to the 'algorithm' part of parent's comment.
Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication.
To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.
Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case.
When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics.
Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method.
The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.
Obviously you’re not writing proofs in 3rd grade, but the emphasis should be on students being able to say why their answer is correct rather than just being able to produce the correct answer.
In Church arithmetic, multiplication is defined as function composition; that is, the dot operator `.` in Haskell-like languages. Addition is considerably more complex; it is defined as `lift (.)`. Exponentiation is even simpler; it is function application, so the `id` function - except that we write exponentiation backwards, so actually `flip id`.
\m n -> n (exp m) 1'
flip flip id . flip id . flip id
flip flip id . flip id . flip flip id . flip id . flip idDoesn't work in every context where multiplication is defined.
> It's even a useful thing to do, how else would you define multiplication?
Perhaps something along the lines of: A multiplication is an endomorphism on an additive semigroup.
With my six year old I told them to say "groups of" instead of "times". So 3 X 2 isn't read as "three times two" it is "three groups of two" which I think has been helpful.
The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant.
Furthermore, being able to compute/define multiplication through repeated addition doesn't prevent you from looking at the special properties of this new operator.
I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”.
I also happen to have an 8-year-old going through third grade right now, and when we were talking through his homework, it was quite clear that using simple concepts he already knew (addition & subtraction) to explain slightly more complex things that he was learning (multiplication and division) was really useful to him. As I recall it being to me.
[aside] I think the maths schedule is more advanced now than it was in my day anyway - he only did multiplication and division this year, but he also did algebra and simultaneous linear equations now, as in:
a + b + 8 = 24
a - b = 4
“Solve for a and b”
Pretty sure I only did that in senior school (11 and up), not at age 8. No powers as yet (presumably they’ll come after the multiplication/division stuff), so no quadratic formula, but still...
[/aside]
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's not particularly useful to talk about those other ways to think about multiplication until you actually need to.
It's too easy once you've mastered the concepts to forget how beginners look at them and struggle to understand them. I think the author isn't remembering what it's like to try to understand multiplication as a new concept - I certainly can't remember.
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.
(I do agree with your pedagogical position.)
I think the only difference might be I used "iterated" rather than "repeated" when helping him. Anyone who is just learning multiplication likely lacks the depth of experience necessary to make use of the "correct" jargon and abstract concepts as a starting point. "Repeated addition" is a useful aid in learning the operation to build that experience.
In fact, I would argue that multiplication being associative shows that this distinction is meaningless.
As for V=IR (I had to google U=RI, maybe U is the more modern version, but it was always V=IR when I were a lad), I don't really have a problem with ratios. When I was learning equations, the simple rule is "do unto one side whatever you do to the other", so ...
V = IR, divide by R -> V/R = I
I was happy with either representation, and I didn't think of it as multiplying, dividing, adding or subtracting, it's just "do the same thing" on each side. The problems I had were more "when do you apply Kirchoff's laws to figure something out, and when do you apply Ohm's law; that sort of thing you just get by experience, I think.
what I meant about the ratio part is that physicists don't care much about the computation they care about whenever you know one piece, you know the other will variate through a third factor in a multiplicative manner.. maybe it makes sense, but it's not at all a structural notion like (x) === iterate(+,n)
procedure product(multiplier, multiplicand)
acc := 0
for i = 1 to multiplier
acc := acc + multiplicand
end
return acc
end
> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
The point with bringing up dimensional analysis is that the above algorithm doesn't work because what does it mean to do `for i = 1 to 3 chairs/row`? You might think of it like
procedure product(multiplier, multiplicand)
acc := "0" # an "absolute" zero that cooperates with any dimension
each single_multiplier in multiplier
acc := acc + (multiplicand * single_multiplier)
end
return acc
end
One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use of the product operation "multiplication", at least in cases where there's only one product operation to use. EG Geometric Algebra has Inner, Outer, and Geometric products, so calling them "multiplication" seems less common IME.
If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue.
The original article was contrasting addition and multiplication on the basis that addends are called the same while factors are supposed to be called differently, which not only makes no sense (it's just a naming convention), but it also breaks down when you have more than two factors: what is the "c" in a x b x c called? Or we're talking about non-associative operations now?
But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math.
Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!
Total technicality, but I could see myself using the term multiplicand/multiplier in my code if I had to implement e.g. a stack-based parser for arithmetic expressions.
I'm not sure what you mean, they are both operated on, it is a binary operator and commutative, there is literally no difference.
After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2.
To return to TFA I think the author is talking from a pedagogical standpoint, that teaching that multiplication is just a bunch of additions under a trench coat is not the best way to go. I'm not sure that I agree personally.
In particular this bit regarding multiplier/multiplicand makes zero sense to me:
>Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
Of course 3 rows and 2 rows aren't the same, but what does it have to do with the order of the multiplication? Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
*** **
*** **
**I will collect 12 apples 5 times. The 12 apples are the multiplicand and the 5 times is the multiplier.
The answer retains the units from the multiplicand.
The author has it all backwards. Basically that you can turn this arrangement by 90 degrees (and turning the chairs, if you do not modell them by points by 90 degrees) is the reason why multiplication is commutative. It is non trivial that 5+5+5+...5 (100 times) is the same as 100+100+100+100+100
The concept of a real closed field [0] (and its categorical second-order version, the Dedekind-complete ordered field) stands on its own, without multiplication being defined in terms of repeated addition. It is completely independent of whether you happen to encode the reals as Cauchy sequences, Dedekind cuts, or something else. The (equivalence classes of) Cauchy sequences are not the same thing as the real numbers, even if we sometimes abuse terminology in this way for expediency. The distinction becomes increasingly important as you delve into more exotic algebraic structures.
Another illustrative example is the Hessenberg product [1]. Even the ordinary product cannot really be reduced to "repeated addition", because you have to use the infinitary concept of a limit. And not just your everyday limit [2], but a limit on a proper-class sized domain [3]!
See also the category-theoretic product [4].
[0] https://en.wikipedia.org/wiki/Real_closed_field
[1] https://en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_ope...
[2] https://en.wikipedia.org/wiki/Limit_of_a_sequence
[3] https://en.wikipedia.org/wiki/Order_topology#Ordinal-indexed...
In general in a field I have addition and a multiplicative identity 1, as well as the distributive property, which means that for all elements x satisfying x = 1 + 1 + .... 1 for some number of ones I can write a.x = a.(1 + 1 + .... 1) = a + a + .... a, so there is always a subset for which multiplication works like repeated addition.
> there is always a subset for which multiplication works like repeated addition
A subset, yes.
[0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic)
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.
You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.
And that's my point, basically - By the time we're discussing real numbers, we need multiplication as an operator, because we're discussing ratios already.
While it's true, it obscures the fact that pi is defined as a fraction: Circumference/Diameter.
That this fraction cannot be represented as a numeric fraction is one of the great insights of early mathematics.
Irrationals break all our intuitions.
I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning about mathematics. However, this is true for a lot of basic properties that hold over the integers and not in other domains, so I'm not really convinced that it's actually wrong to teach kids about multiplication this way.
It literally looks nothing like multiplication when carried out manually.
The Jordan-Chevalley decomposition makes the difference precise -- the nilpotent part (off diagonal 1's in the jordan normal form)
A matrix is a function. Not all functions can be represented by matrices (although all smooth functions can be represented by a Taylor series that sums over matrices (and "the sum C=A+B" really means "the function C such that Cx = Ax+Bx for all x in the range of both A and B"))
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 and all of the more advanced concepts the author (and the related articles) discuss can be taught later.
But seriously, her first objection is
> To define multiplication as repeated addition is to make multiplication a sub-species of addition.
To me, that means the author is wrong from the get-go. Not all cases of multiplication might be easily seen as repeated addition, but to get hung up on a deliberate linguistic misunderstanding? And to add insult to injury, she finishes by proposing to see multiplication as the answer to "how many or how much OF the unit", and shows a diagram with _repeating_ units.
Oh, never mind, the answer is at the end: she's got books to sell, and probably tries to improve her sales by appealing to parents with children that can't multiply 83 by 17 when they're 8.
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 always, it only is for whole numbers. Others might be better served by telling them up front that multiplication is a separate operation, but can be "emulated" for whole numbers by repeated addition. (As a child I was in the latter group; I hated it when teachers told me one thing in one grade, and then said "well, what you were taught before isn't actually correct..." in a later grade. How many more times would they change the rules?)
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.
Then you can define an interface with abstract operations like addition and subtraction, then any given number system can try to implement this interface. Naturals, integers, rationals, reals, polynomials, rational functions, complex numbers, and many others can implement at least the semiring interface. But what's important, I think, is that there's no universal multiplication operation -- it's just a word that fills in for whatever is the right multiplication operation at the moment. It's some ad-hoc polymorphism.
Unless you're accepting the real line as being axiomatic (maybe from geometry, where you can define all the basic operations using ruler-and-compass constructions), all the definitions of multiplication for all the above number systems are going to be, eventually, based on the one for the naturals.
And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation.
But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever. The latter claim is the claim I'm arguing against.
> Unless you're accepting the real line as being axiomatic
Your computer can't manipulate real numbers. It can only implement finite operations on numbers with a finite bit size, which, as above, is a set of numbers isomorphic to the integers. So if you want to work with the real numbers themselves, abstractly, then you can't expect what the computer does to carry over.
What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences), and then you can embed the whole numbers inside and pretend that that was where they were the whole time.
I do get that you were giving different pedagogical approaches, and this was just one of them, but I think this one is at least as bad of a misrepresentation as saying "multiplication is repeated addition" without specifying "for whole numbers." That is, unless you're like the Greek geometers and accept a continuum from the start.
(There are other number systems that the integers sit inside, though, that don't themselves sit inside the real or complex numbers, which is a reason I would hesitate to think about whole numbers this way in general.)
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 repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences)
Ah, I see. As far as I know the axiomatic reasoning involved can go either way. But in any case, that wasn't what I was trying to get at with the term "emulating"; I'm sorry if my use of that term caused confusion, and I agree with you that telling a child something like "the whole numbers are inside the real numbers" without qualification would also be a misrepresentation. My point was simply that teaching a child a specific computational procedure, repeated addition, in order to get an answer to particular multiplication problems does not require telling the child that multiplication is repeated addition, without qualification. The two things are distinct, and I am fine with the former; I only object to the latter.
I also taught prime numbers by handing out (round) 1x1 pieces and told him to figure out which ones could be made into squares.
The teacher was surprised when - in the middle of a more convoluted explanation for primes - he just said "primes are numbers that can't be arranged as rectangles".
Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.
How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?
Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.
On the other hand knowing the relation is a good hint at how to explain exponentiation.
In the end teachers should be aware of what will be needed at higher grades to use tools such as it as a starting point but not ingrain it too much and try to push students onto thinking of multiplication as it's own functional primitive once they get the hang of basic tables and start moving on to long multiplication.
No it doesn't. In that example the "baskets" and "per baskets" (ie x/baskets) are cancelling each other. 2 baskets x 3 apples/baskets = 2 x 3 apples = 6 apples. Once you remove the unnecessary terms the actual multiplication is still basic repeated addition.
2 hours x 3 miles per hour = 2 x 3 miles = 3miles+3miles = 6 miles.
I think the real issue is about how to interpret the adjective "Repeated" when we turn to more general domains than the natural numbers. But that's not a huge issue, it's mostly a matter of terminology.
More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:
From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.
From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.
Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.
> Ironically, this article wants to be very normative about which mathematical intuition is better…
No, the article is not being normative about “which intuition is better,” this is an incorrect reading of the article.
The article is giving advice about how to teach multiplication. Advice that is apparently based on years of experience teaching multiplication.
It’s not uncommon for people to experience teachers who prescribe specific intuitions about math rather than accommodate different intuitions—but the author is not doing that. What she’s doing here is outlining the various ways in which one particular intuition may fail you.
When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:
What if it were that 2*2 = 5 ?
Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4
2*2 = (1+1) * (1+1) = 1+1+1+1 = 4
1. Associative
2. Commutative
3. Multiplicative Identity
4. Increasing ( If i,j > 1, then m(i,j) > i,j )
5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )
6. Uniqueness of Prime Factorization
An easy example is the odd numbers.
1,2,3,4,5, ... n
1,3,5,7,9, ... odd(n)
odd(n) = 2n-1
inverse_odd(x) = (x+1)/2
m(i,j) = inverse_odd( odd(i) * odd(j) ) = 2ij - i - j + 1
It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.
* addition: the cardinality of the disjoint union of two sets
* multiplication: the cardinality of the direct product of two sets
* exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a
I learned multiplication from counting squares of dots.
But agreed: I learned about exponentiation first via repeated multiplication.
https://mathoverflow.net/questions/360578/category-theory-an...
Even ignoring that as a historical accident (we invented set theory before category theory), those ideas are all natural, meaningful and interesting to examine in the context of sets even if one has never heard of a category.
Second, I find that claim highly dubious. To do metamathematics and talk about the relative strength of various axiomatic systems, you need to talk about large cardinals. So that, at least to me, seems like a good argument that set theory is foundational to mathematics in a way that category theory is not, since the latter has no non-eliminable place in the (study of) contemporary foundations of mathematics.
Here's a list of some approaches.
https://mathoverflow.net/questions/310004/categorifications-...
I think it was Gerolamo Cardano who in a book of worked exercises came up with a problem like "obtain two numbers that add to 10 and multiply to 40" and arrived at the solution had to involve square roots of -1. Then he goes "this makes no sense, but let's calculate nevertheless..."
from what i recall that's the way to go:
You can use this to extend the definition of exponentials as "repeated addition" from allowing only natural numbers expoonents to allowing positive rational numbers. Afterwards you can "complete" this to allow real exponents.
But this requires some hard maths to do rigorously.
This same pattern can be seen driving discoveries in math. - Integrals are generalizations of Riemann sums. - Fractional exponents generalize taking square roots. - The gamma function sorta generalizes factorials. It's okay to start with something not-quite-right and explore how it generalizes.
Multiplication has started as repeated addition. That's where the idea came from.
> It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication.
Nobody. Literally nobody said that
Of course, you'll need to forget a bit the idea of repeated multiplication when you get into the rationals/reals/complex numbers, but even there it kinda makes sense
So no, I think this is the kind of teacher that makes the students even more confused and prone to hating math
It is just as conceivable that people were faced with a problem like 'each person needs 2 apples, we have 5 people, so we need 10 apples'?
In this case repeated addition is a perfectly fine algorithm to calculate the product but the product itself is not defined as repeated addition, it's the solution to a particular type of problem.
The sentence "2 for him and 2 for her and 2 for him and 2 for him and 2 for her" is unnatural and not the way people generally think (unless they're just tallying up but then we're not even talking about multiplication anymore, a tally rarely consists of all equal numbers)
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 multiplication when we got to that point (which we did about a year later).
I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.)
I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her.
With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller.
It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?
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.
Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.
The gist of my statement was something like: Adding repeatedly is a fine method to solve this kind of problem but it doesn't for all classes of numbers. We can't say, like we can w/ addition and subtraction, that multiplication implies a specific direction of movement (towards or away from zero) on the number line.
39 x 40 = 40 x 40 - 40
https://www.feynmanlectures.caltech.edu/I_22.html
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.
- 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.
I've always run into the following problem: I try to motivate multiplication as repeated addition, which does help with intuition, but then things totally fall apart when we move on from integers into fractional values.
1/2 * 1/2 -> 1/4.
Sure you can teach someone to simply multiple the numerator and denominator, but it doesn't necessarily help them make clear sense of what's going on.
There's a standard way of lifting each type of number to the next type, and this lift is compatible with all the basic operations (the lift is a "homomorphism"), so it's easy to pretend that the real numbers (or complex numbers if you want) are the universal system.
So with your example of going to fractional values, you're right, repeated addition falls apart -- but I'd say that's because it's not the definition for multiplication of rational numbers! Multiplying numerators and denominators is the usual definition, but that gives about as much intuition as does the definition for multiplying natural numbers. Sort of "the point" of multiplication of naturals, I think, is that it represents how many things you have if you arrange them in an n by m grid. Rational numbers show up in geometry with similar shapes (scaling), and for a few reasons you'd want multiplication to represent by how much something scales after a composition of scalings; maybe "the point" of rational number multiplication (at least algebraically) is that you can defer dividing until later, i.e. (a/b) * (c/d) is (a*c)/b / d.
Take number n and multiply it by number x/y. To do this, you have to split number n into y parts and take x number of them. So to multiply 8 by 3/4, you split 8 into 4 parts (2 + 2 + 2 + 2) and then take 3 of those parts. This ends up being 2 + 2 + 2 = 6.
For multiplying two fractions, you have to extend it to n/m * x/y. Since you can multiply the top and bottom of a fraction by the same number, you can write n/m as ny/my. Then you can have n/my be your “equal part”, and take x of them. So 1/2 * 1/2, you take 2/4 and split it into 1/4 + 1/4 and 1 of them, so the answer is 1/4.
To me at least, this makes sense as an extension of multiplication is repeated addition. It’s when you get to irrationals that it starts to fall apart, and even then the intuitions the above way of thinking led me to have served me well.
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?
You are tilting at windmills
> Is it even necessary for teachers to define multiplication?
No, at least not in formal terms
> Or is the teacher’s job to provide plenty of examples of multiplication in action?
I think so
> Should we let the students intuit their own definition(s)?
Yes
> Will it help students if we change our focus from “how to get the answer” and teach them to identify the multiplicand, the “this per that” unit? Or will that introduce new difficulties I haven’t considered?
It will introduce new difficulties
> Or do we already teach this way, only in different words?
We don't, thankfully
> If you are an elementary teacher, how do you teach multiplication to your students?
I will let elementary teacher answer
> Are some students clueless because, no matter how we explain it, they just don’t pay attention?
While we can't always blame teachers when students don't pay attention, if it is systematic, maybe there is a problem with your teaching.
> Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?
I will let teachers respond
Ok, I hope you are still there and didn't just click the "downvote" button.
I think the author raises good points for at least high-school level students in STEM fields. But in reality, the important thing in elementary school is that we want people to know how to count. 5 apples at $0.50 each is $2.50, this kind of thing.
Dimensional analysis is very important in engineering, and I definitely think is should be taught, as well as everything in the article, but at college level, not in elementary schools. Teaching such abstract concepts too soon is a recipe for disaster, I know because it has been tried. The expectation was that it will make better engineers in the future, the reality is that while it may have helped a tiny minority get college degrees, it mostly produced kids who couldn't do simple arithmetic.
The argument they are making is kind of strange. But not 'wrong' per se. If you mix 2 types of number systems you get a different number system out of it. It is a similar argument I make about fractions. It is basically physically impossible to have a fraction of some things. If I have a car and divide it in half. I no longer have a car. I have two of something else. They are clearly not a car because they can not act like a car anymore.
Wtf, why not? Why wouldn't you want people to draw connections between different parts of math. The whole point of math is to find the patterns and interconnections.
And also”3 cm + 3 cm = 6 cm” as 3x+3x=6x and removing the numbers it become 3x/3+3x/3=6x/3 => x+x=2x it makes more sense. And”2 cm × 3 cm = 6 cm^2″ can be then seen as 2x*3x=6x^2.
Multiplication doesn’t change anything. We must see what we add, in this case the unit and calculate accordingly both for addition as for multiplication.
If you want to define multiplication as repeated addition (and addition as repeated invocation of the succesor function), by all means do so. Its not wrong.
Sure you could say it doesn't work for complex numbers, but usually when you introduce complex numbers you say something like, multiplication works as normal except i×i=-1. I hardly see that as a problem with the original definition of multiplication. Its more about the definition of i than the definition of multiplication.
There is the symbol '*'. And when there's two integers it does one thing, and you can think about it as repeated addition. But when there's something different, it works differently. But now we're just going to concentrate on integers.
In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless.
Of course, we should start teaching immediately that you can't add apples and pears. Maybe we should just do this explicitly there, that 5 times 3 apples is fine, as 5 is scalar. Explain that immediately and then add the unit analysis in physics.
"Scalar" is too broad; irrational numbers are scalars, but multiplication by an irrational number is not repeated addition.
https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
Multiplication is repeated addition.
Yes, I know about roundoff errors, overflows, etc.
This is the first video in the series Thinking Mathematically (after the introductory video, "Why think Mathematically?") which I put on YouTube under a channel of the same name. It proceeds through the sets of numbers, N -> Z -> Q -> R -> C and yes it's for beginners. Would love some feedback:
https://www.youtube.com/watch?v=Wd1lzqR3xW0
The other videos you'll find on that channel are:
1. Why think mathematically?
2. Numbers and Algebra
3. Sets and Infinity
4. Logic and Probability
https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
(PS: I am generalizing this approach in our upcoming app, https://teaching.app)
Because you can indeed multiply through repeated addition (which she acknowledges), repeated addition is one valid way to look at multiplication. But it might not be the maximally helpful framework for students.
Why mix concepts of Linear Algebra/Physics with basic Arithmetics for no good reason?
The multiplication, that is just repeated addition, is very possible in dimensional analysis and is called scalar multiplication:
The outright worst part is this weird complication:
2 baskets × 3 apples per basket = 6 apples
of 2 x 3 apples = 6 apples.
Just why? This is just a scalar operation.
Staying in strictly mathematical concepts, the operation described in OP are closer to the cross product of orthogonal vectors, then multiplication in my opinion.
[4 0 0] × [0 3 0] = [0 0 12]
https://en.wikipedia.org/wiki/Scalar_multiplication
https://en.wikipedia.org/wiki/Cross_product#Geometric_meanin...
The problem with "multiplication is repeated addition" is that the concept breaks down once you move on from integers.
2 X 3: "add 2 together, three times" - works well
½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.
I can see why teachers who aren't that comfortable with math might be getting confused about this though. Repeated addition is a reasonable algorithm when you've got one operand that is a relatively small integer.
If you spend a long time teaching small integer multiplication you may forget that "You can solve these multiplication problems with repeated addition" does not mean "Multiplication is repeated addition."
On the contrary, continuing to conceptualize fractional multiplication exactly as your "1/2 X 1/2" as addition example served me well. My wife—with a higher measured IQ than mine, FWIW—evidently can't see it that way. She basically stopped learning math when multiplication of fractions was introduced (it's interesting that one can continue to earn decent math grades, regardless) as all the results felt arbitrary and magical to her, and still do in middle-age. This is actually a very common point for students to stop following WTF is going on in their math classes and never really get back on track (yes, all the way down in, what, 2nd or 3rd grade) from what I've seen; factoring is another, later on.
"1/2, added 1/2 times" is precisely how I think of it. If I can't just pattern-match or rule-follow my way to a solution (because I've forgotten the rules, say) that's still my line of reasoning to figure out what to do to get the solution.
I do exactly the same sort of thing to come back to my senses if I get lost or forget exactly what is happening in fractional division. "How many times does 1/2 fit in 1/4? 1/2. How many times does 1/4 fit in 1/2? 2." Even if I have to manipulate some things to figure out the result with uglier fractions, that's absolutely how I think about what I'm doing as I do it, and it keeps me focused on the ultimate purpose of the calculation. I imagine that's also wrong, according to the author.
In short I can vouch that yes, that exact thing was very helpful to this particular person, including for multiplication of fractions. I truly don't know how else I might have understood those problems, to avoid joining the ranks of the mathematically-lost as early as lower elementary school.
Or just re-arrange the terms to be (1*1)÷(2*2). No need to add half, half a time, just understand that multiplication distributes over fractions.
That said i think the concept would make sense to most school children as a way of extending the intuitive concept of multiplication of natural numbers.
Measure theory (for your volume) and the structure of dimensional quantities [1] are non-trivial topics, and while probably not giving good answers, I think the article is certainly asking good questions.
[1] https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...
multiplication is an overloaded operation, or in more modern terms, it is doing multiple dispatch, depending on whether the input is a whole number or integer or rational or real or complex
- https://www.maa.org/external_archive/devlin/devlin_09_07.html
- https://www.maa.org/external_archive/devlin/devlin_06_08.html
- https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/
- (the link I posted that creates this thread)- https://www.maa.org/external_archive/devlin/devlin_09_07.htm...
- https://www.maa.org/external_archive/devlin/devlin_06_08.htm...
- https://denisegaskins.com/2008/07/01/if-it-aint-repeated-add...
(Sorry, it just bothers me when people don't use clickable links here.)
2 apples + 3 apples = 5 apples
This confuses a lot of people when they later learn algebra and believe that it is equivalent to:
2a + 3a = 5a
Both of these equations look correct at face value but the critical difference is that an apple is not a variable. Do we mean the weight of the apple? Or the price of the apple? Or its energy content? Apple is too concrete to be used as a variable and it's not quantifiable. The word 'apples' invites people to think of the variable as a concrete object instead of a placeholder for a quantifiable property of an object.
The fundamental theorem of arithmetic says for any non-zero natural number x there is a unique finite sequence of integers a_i such that x = 2^a_0 . 3^a_1 . 5^a_3 ... p_n^a_n and that given x the sequence a_i is computable (similarly given a_i you can easily construct their x).
This basically gives you a (bijective) mapping between integers and tuples of integers x <-> (a_0, a_1, ... a_n) which lets you build up more complicated structures and eventually construct the sentence you need for an incompleteness theorem (if you're as smart as Gödel).
We went to very different universities it seems. Neither Peano arithmetic nor decidability were ever mentioned
This made me think too, in particular the teacher's example. So; here's what I think it should be calculated.
3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm).
What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm".
However 3 cm + 3 cm != 2 cm X 3 cm as the author states which is what leads them to wrong conclusion. In other words 2 cm X 3 cm leads to an entirely new thing in geometry which is "area". So while 3 cm + 3 cm is still length (and hence it should be translated to 2 X 3 cm i.e., one scalar unitless number and other number with unit) 2 cm x 3 cm is not length. And the resultant area is correctly written as 6 cm^2; by multiplying units as well as the numbers. Now, how does one multiply numbers (2 X 3)? It's indeed by repeated addition!
The author, however is onto something. Which is when you multiply a something with unit (like 3Kg, 2Cm) with a unitless number that resultant product is still something (that weight, length) that you began with. However when you multiply two things both of which have units the resultant product is something entirely new (i.e., area as opposed to length, pressure as opposed to weight etc.,)
This is why in high school physics you are taught to always write down the units of each number: typing them helps avoid a big amount of errors.
Note that “typing” and “vector terms” are not essential, just helps me express it with the HN crowd easily.
Math: 3x3 = 3^2 physics: cmxcm = cm^2
3cm x 2cm = (3x2)(cm x cm) = (math)(physics)
The context gives the meaning.
Or in computers, how many bits would the representation of a matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory is linear)
We don't define it based on what machines do with it, we define it based on what WE do with it. In the context of teaching, dimensional analysis comes far later in the process of education.
The pedants answer is that there is more than one multiplication operator taught. It's most basic incarnation is, for all intents and purposes, repeated addition.
It's more advanced forms, which include extra auditing for dimensional analysis isn't merely repeated addition. Though I'd challenge that too.
Dimensional analysis is an extra aspect stapled onto and seperate from the arithmetic operation. Therefore makes up an additional level of auditing seperately as a whole, and therefore inappropriate to define as an inherent property of the operator. You multiply numbers.
People need to stop trying to teach unrelated things bundled by some programmer or language writer to kids/adults. It doesn't make one seem smart. It just confuses people by blurring orthogonal aspects of problem solving, which puts people off of it.
Probably the author would argue that what calculators do is neither addition nor multiplication because of precision loss. In practice it works pretty well though.
1. IS A CLICKBAIT
2. Course you can decopose multiplications in a series of repeated additions.
3. Extra points for the flawed argument of adding different, well you can't because they are different.
x^2 = x * x = x + x ... + x (x times)
Therefore,
d/dx(x^2) =
d/dx(x * x) =
d/dx (x + x + ... + x (x times)) =
d/dx(x) + d/dx(x) + ... + d/dx(x) (x times) =
1 + 1 + ... + 1 (x times) =
x
E.g. why don't we ever have something like kg^s?
d/dx (f) = k*f
Example: population growth is exponential.
d/dt(p) = (ln(2)/r)*p
p(t) = p0 * 2^(t/r)
p : population
p0: population
t : days
r : days (time it takes for the population to double)
t/r : scalarThe fascinating thing is that now these transistors are helping make new math proofs beyond human comprehension.... how does that come out of addition?
5*11 = 5*(8+2+1) = 5*8 + 5*2 + 5*1 = (5<<3) + (5<<1) + (5<<0)
...and then doing all this in parallel.
Instead of:
5*11 = 5+5+5+5+5+5+5+5+5+5+5
;)
The reservation I'd have with the statement "Multiplication is repeated addition" is that the use of the word "is" implies an identification, or a necessary derivation of one from the other. The operations aren't identical, and one is not necessarily derived from the other. As operations, they're distinct.
(The issue also has nothing per se to do with units. When you introduce units of length or area, you're using numbers in a particular applied context. Outside of that context, numbers don't have units.)
In everyday usage "addition" and "multiplication" are conventional designations. Conventions always have ambiguities and edge cases.
The most common formal structure in which you have both addition and multiplication is in a ring. A ring is a set with two operations, which we call "addition" and "multiplication" --- but those are just names. We could just as well have used "foo" and "bar". We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention/convenience. (Note also that those operations are not unique.)
Rings include the integers, rationals, reals, complex numbers - but also (for instance) polynomials with real coefficients, 42 x 42 matrices with real entries, quaternions, or finite fields.
Now consider what happens as you list the axioms for a ring. You say that addition is associative, addition has an identity element (conventionally denoted "0"), every element has an additive inverse, and addition is commutative. That's addition.
Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted "1").
At the moment, you have two independently defined operations, which have nothing to do with one another. "Multiplication" is therefore not identical to (or defined as) repeated addition. But it's not too useful to have two independent unrelated operations. You connect the two operations by introducting the distributive axiom: For all a, b, c in your ring,
a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c.
a * (b + c) = a * b + a * c.
Once the distributive property is available to connect the operations, it explains why you can "think of" multiplication as repeated addition: For instance,
3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1 = 3 + 3.
Or consider one of the standard illustrations given to kids in grade school:
3 * 2 is $ $ $ which is [ $ $ $ ]
$ $ $ [ $ $ $ ]
But what happened here? We relied on our physical intuition to "know" that putting the original 6 dollars into 2 bags in the second step didn't change the number of dollars. But formally, it is
3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1,
(BTW regrouping the dollars again into 3 groups of 2 is a standard way of motivating commutativity of multiplication, since 3 groups of 2 is visibly "the same as" 2 groups of 3.)
(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don't know math history well enough to say, but historical derivation doesn't imply identity or logical derivation.)
So I think saying "Multiplication is repeated addition" is a little sloppy in the use of the word "is", but we can agree to disagree about how much sloppiness is okay. The statement is fine as a way of giving students one (of many) ways to think about multiplication. (Pictures are important, too!)
Here's something to think about. (I don't have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as "7 i added to itself (3 + 4 i) times". If you don't like the sound of that ... why? You might say "You can't have '3 + 4 i' things." Well, in the real world discrete "things" come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that "-3 things" means "I'm missing 3 things" or "you owe me 3 things". Nothing stops us from making agreements about the use of words. So maybe we could "agree" that "(3 + 4 i) times" means exactly an occurrence of the expression "3 + 4 i" in a context like this one. Is there any harm in that? :-)