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

https://denisegaskins.com/2008/07/28/whats-wrong-with-repeated-addition/

by harperlee • 5 years ago

152 410 5 years ago

harperlee2• 5 years ago

I just found this interesting article. If you are interested, it is a review of the following "conversation of articles" (which are reachable from the OP but I guess it's useful enough to list):

  - 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)

alberto_ol• 5 years ago

I wanted to post the same links, you can find the same articles and others about that subject by googling "Keith Devlin addition multiplication"

Jtsummers2• 5 years ago

Clickable:

- 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.)

threatofrain3• 5 years ago

The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?

cturner• 5 years ago

Both examples use the same operator, but we resolve them to different functions. Mentally, we use the tuple of argument types to select the appropriate function. Addition does not provide equivalent functions for all multiplication cases. Hence, multiplication is not repeated addition.

salawat• 5 years ago

You need to stop conflating the tyoe mumbo jumbo with multiplication.

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.

d--b• 5 years ago

Arguably, what they're saying is that 2 x 3 should be understood as a unit-less "scaler" 2 that multiplies a number (of things) "3".

caddemon• 5 years ago

Yeah I don't think they are the same. You can of course also have 2 cm * 3, which still fits with the repeated addition idea, as long as you frame it as 2 cm + 2 cm + 2 cm. I'm not an educator by any stretch, but that seems like an intuitive way to introduce units without causing confusion about previous number teachings.

vishnugupta• 5 years ago

but is 2 cm × 3 cm the same thing as 2 × 3?

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.,)

harperlee2• 5 years ago

Another way to say it is that 2cm * 3cm is shorthand for (2)(1cm)(i) * (3)(1cm)(j). We are complecting 3 semantics: measure and unit (which compose a module, in vector terms) and “meaning” (dimension, in vector terms).

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.

insert_coin• 5 years ago

The author is confusing math with physics.

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)

jmchuster• 5 years ago

I like how 3blue1brown presents it in terms of animated number line translations

https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365

tromp• 5 years ago

In some settings, "repeated addition" is the only sensible way to think of multiplication. Such as multiplying an elliptic curve point with a scalar.

sumtechguy• 5 years ago

Multiplication is a shorthand for 'repeated addition'. Much like power is short hand for 'repeated multiplication'. Even with imaginary numbers. You can create an in infinite series of additions that looks a lot like the imaginary number.

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.

t0r0nat0r2• 5 years ago

I don’t think the aruments made here are compelling. Multiplication can be defined in this way. Saying that multiplication is repeated addition is not the same as saying that multiplication IS addition. Addition relates to multiplication as multiplication relates to powers.

Agingcoder• 5 years ago

I don't understand.

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.

drewcoo• 5 years ago

Agreed. Multiplier and multiplicand are different words but the commutative property says their values can swap equivalently, so . . . what was the author's point again?

tomxor3• 5 years ago

This was exactly my thought, I don't understand why those two word exist since it's commutative. Perhaps all this confusion is purely semantic.

gugagore4• 5 years ago

If, for a moment, you conceptualize of multiplication on non-negative whole numbers as repeated addition, then this is the algorithm:

  procedure product(multiplier, multiplicand)
      acc := 0
      for i = 1 to multiplier
          acc := acc + multiplicand 
      end
      return acc
  end

Swapping the arguments is a different computation. But after thinking about it, you realize that you get the same answer all the same. That's the point of

(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

But then what is `(multiplicand * single_multiplier)` ?

andi999• 5 years ago

(2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

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

midjji• 5 years ago

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

horsawlarway3• 5 years ago

pi + pi + pi + (.141592... of pi) ~= 9.8696

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.

eigenket15• 5 years ago

You first define multiplication of natural numbers to be repeated addition, then define multiplication of rationals in terms of multiplication and addition of naturals, then define multiplication of reals in terms of multiplication of Cauchy sequences of rationals ;)

thrower1232• 5 years ago

3.14159 + 3.14159 + 3.14159 + (3.14159*0.14159)

scarmig• 5 years ago

How would you compute \pi+\pi using addition?

Irrationals break all our intuitions.

spacedcowboy4• 5 years ago

I have a PhD in physics and more maths qualifications than I can shake a stick at; to me, multiplication is repeated addition.

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]

munchbunny2• 5 years ago

I think the useful distinction is that, when you teach multiplication as a mechanical computation (arithmetic) it's useful to talk about it as repeated addition.

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.

sidlls• 5 years ago

I have got almost exactly the same situation: PhD, physics, forgotten more math than most folks ever learn, etc., and with an 8 year old learning the same level of mathematics as you describe.

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.

agumonkey3• 5 years ago

Since you're a physician, do you think that helps for multiplicative relationships in real world laws ? It took me decades .. sadly, to be comfy with handling U = RI formulas. On the algebraic level it's stupid simple, but for real world physics the meaning is more bidirectional coupling of ratios and amplitudes and taps into a different part of my brain.

bmc75053• 5 years ago

Does anyone have an intuition for why Presburger arithmetic is decidable, but once you introduce x, it is not? If + can be rewritten as repeated succ, (which is decidable) and x can be rewritten as repeated +, then it seems logical that x should be decidable, but as any undergraduate CS student knows, Peano arithmetic is undecidable. What is so special about x exactly?

wizzwizz4• 5 years ago

It's a variable number of addition operations.

bmc75053• 5 years ago

Isn’t + a variable number of succ’s? Why does x break decidability, but + does not? It seems to me you should be able to define a second order Presburger system with + replacing succ, and x replacing + to restore decidability. Or is there some issue with countability or assumption that is broken?

eigenket15• 5 years ago

Its probably worth pointing out that the system with only multiplication and no addition is also decidable (Skolem arithmetic). Things get interesting when you have both multiplication and addition defined and the reason for that is basically the fundamental theorem of arithmetic which gives you unique prime factorisation and directly the Gödel encoding (writing out statements about integers as integers).

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).

bmc75053• 5 years ago

That’s a great explanation, thank you!

IAmLiterallyAB• 5 years ago

but as any undergraduate CS student knows, Peano arithmetic is undecidable

We went to very different universities it seems. Neither Peano arithmetic nor decidability were ever mentioned

mensetmanusman• 5 years ago

It may be wrong, but it is good enough to run all of our computation at the transistor level :)

The fascinating thing is that now these transistors are helping make new math proofs beyond human comprehension.... how does that come out of addition?

flohofwoe• 5 years ago

AFAIK hardware multipliers are not quite as dumb and work more like this though:

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

;)

Garlef8• 5 years ago

My favourite way of thinking about these operations comes from category theory:

* 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

Rerarom• 5 years ago

This is just set theory, not category theory.

Garlef8• 5 years ago

Neither is more foundational than the other. So I think the use of "just" is not justified here.

https://mathoverflow.net/questions/360578/category-theory-an...

t0r0nat0r2• 5 years ago

This seems a bit restricted. How do you go about extending it to real and complex numbers?

Garlef8• 5 years ago

From what I know: This is still an open question.

Here's a list of some approaches.

https://mathoverflow.net/questions/310004/categorifications-...

AzzieElbab2• 5 years ago

That is exactly how I explained multiplication to my 5 years old, right after getting him into programming by playing with profunctor optics

Garlef8• 5 years ago

I learned adding from counting disjoint unions of dots.

I learned multiplication from counting squares of dots.

But agreed: I learned about exponentiation first via repeated multiplication.

YeGoblynQueenne• 5 years ago

You waited 5 years to teach programming to your 5 year old?

threatofrain3• 5 years ago

Can this perspective accommodate fractional exponents, like 2^(0.5)?

Garlef8• 5 years ago

No (At least not yet: open research question). But neither can "repeated multiplication". How would you multiply 2 with itself "a half" times?

forinti• 5 years ago

The more methods one studies, the better one understands. There is no one true method.

rustybolt• 5 years ago

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.

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.

yakubin5• 5 years ago

You can definitely see multiplication as repeated addition.

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.

WallWextra• 5 years ago

If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.

SAI_Peregrinus4• 5 years ago

Multiplication of the Surreals is a recursive operation using sums (addition and subtraction on the left and right sets). Since the Reals are a strict subfield of the Surreals one can define multiplication of the reals using only the same recursive formula and restricting both operands to be Reals.

xirbeosbwo1234• 5 years ago

Only for rational numbers. Doesn't work for real and complex numbers.

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.

DanielMcLaury2• 5 years ago

Multiplication of real and complex numbers is typically defined by starting from repeated addition, extending this notion to rationals, and then extending that notion to reals by taking limits.

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.

rafaelturk2• 5 years ago

Yep, it is a clickbait

parenthesis• 5 years ago

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.

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.

sudosysgen2• 5 years ago

We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.

betterunix25• 5 years ago

"That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square."

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).

baryphonic• 5 years ago

Funny, my teachers in middle and high school frowned on using the quadratic formula until we could demonstrate we knew how to complete the square.

Frost1x• 5 years ago

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.

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.

jordan_curve• 5 years ago

To make this more explicit, mathematics (even at the primary school level) is arguably more about “proving” things than it is about computation.

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.

scatters2• 5 years ago

how else would you define multiplication?

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`.

shawnz• 5 years ago

Interesting, what about tetration then?

swiley• 5 years ago

Anyone that teaches this way should find something else to do. Memorizing a formula is easy but if you want to do well later you must be able to complete the square.

daveFNbuck• 5 years ago

I never learned to complete the square, and I managed to do well enough later to get a degree in math.

pdonis37• 5 years ago

You can definitely see multiplication as repeated addition

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.

dash23• 5 years ago

Multiplication by rational fractions p/q is just multiplication by whole p, followed by multiplication by 1/q. 1/q is just the number that when multiplied by q gives 1. So you can cash that all out as repeated 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.

wruza2• 5 years ago

How would you see multiplication by pi as repeated 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:

whatgoodisaroad2• 5 years ago

Exactly. Probably every discovery in math starts out with something that's only close to the truth (like multiplication-is-repeated-addition) but generalizing it to open up whole new areas.

This part of math is so exciting and compelling, I feel like it should be more of a focus in schools.

thrwn_frthr_awy• 5 years ago

how else would you define multiplication?

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.

alentist10• 5 years ago

You can definitely see multiplication as repeated addition.

Doesn'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.

raverbashing• 5 years ago

I'll tell you what's wrong: the mentality of the author of this text

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

contravariant2• 5 years ago

Now that's an interesting point, did it start as repeated addition?

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.

cyborgx7• 5 years ago

Your problem just reinforces the notion that multiplication is repeated addition. "So we need 2 for him and 2 for her and 2 for him and 2 for him and 2 for her. 2+2+2+2+2 = 5 x 2 = 10"

y7• 5 years ago

I think the author's argument becomes more clear when you consider multiplication in rings other than the integers, for example (square) matrices. The product of two matrices A*B does not correspond to repeated addition, and it is not commutative (A*B does not equal B*A in general).

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.

prionassembly8• 5 years ago

The better line of thought here might be "A*B is not multiplication, it's function composition".

It literally looks nothing like multiplication when carried out manually.

eigenket15• 5 years ago

It looks exactly like scalar multiplication when your matrices are diagonal.

mantap• 5 years ago

The mathematical term is "product". It's a product.

jmull2• 5 years ago

It's just that matrix multiplication is not the same thing as scaler multiplication. I think you need to understand that no matter how you define or think of scaler multiplication.

anticensor• 5 years ago

Hadamard multiplication of matrices is a scalar multiplication.

https://en.wikipedia.org/wiki/Hadamard_product_(matrices)

D13Fd• 5 years ago

I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition.

For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases and all of the more advanced concepts the author (and the related articles) discuss can be taught later.

prionassembly8• 5 years ago

You can teach (some intuition for) multiplication without numbers by multiplying lengths to get areas. Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows.

Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.

magicalhippo2• 5 years ago

Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows.

How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?

whizzter• 5 years ago

Having been teaching programming I can see an immediate problem with it, whilst it works great as a starter tying it too hard in the mind will require more un-doing when it comes to more advanced concepts where it isn't an appropriate level.

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.

tgv• 5 years ago

You start by teaching rings, duh.

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.

fifilura• 5 years ago

Lego is a great tool for this.

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".

eigenket15• 5 years ago

*rectangles (hopefully)

pdonis37• 5 years ago

I'm at a loss how you would teach it without explaining that it is repeated addition.

How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition?

Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't 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?)

kmill6• 5 years ago

Maybe I've been doing too much computer-assisted proofs recently, but I'm not understanding what all the fuss is. At some point you need a `Nat.mul` that's concretely defined, and probably the only reasonable definition is recursively using repeated addition (and if your natural numbers are in unary, which is theoretically the simplest, then even `Nat.add` is going to be defined recursively by repeated incrementing! like taking a bead from a pile one at a time and putting it into another).

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.

wcarey3• 5 years ago

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

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

Double_Cast3• 5 years ago

I tend to think of multiplication as a transformation. "repeated additional" almost makes it sounds like we're merely telescoping an interval within a single dimension. Whereas "transformation" (to me), evokes more of a Cartesian Product or Quadrature sort of mental image.

rafaelturk2• 5 years ago

As software developers we can defenetly point the many flaws in this article:

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.

EGreg• 5 years ago

A few years ago, I was teaching a class and decided to record a short series on the fundamentals of mathematics, for complete "beginners" - whether they be kids or adults.

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

Here is the channel with all the videos. They might be useful to share with people who you want to understand these concepts from the ground up:

https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA

(PS: I am generalizing this approach in our upcoming app, https://teaching.app)

fullshark• 5 years ago

Imagine explaining the content of this post to a five year old instead of that multiplication is repeated addition.

thothamon• 5 years ago

I think the author is trying to say that _from a teaching point of view_, thinking of multiplication as a kind of addition is suboptimal.

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.

GuB-42• 5 years ago

Is there really a difference between multiplication and repeated addition, or am I tilting at windmills here?

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.

glitchc• 5 years ago

A bit meandering and muddled in some respects, but agree with the gist: Multiplication is not just repeated addition.

scythe• 5 years ago

The nicest formal definition of multiplication is "the operation that distributes over addition". When defining operations on abelian groups that might be called "multiplication", it's the distributive property that makes the term useful, and brings us into the categories of ring, semiring, [weird math prefix here]-ring, etc.

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?

cjfd• 5 years ago

Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously.

You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be 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.

jschwartzi2• 5 years ago

Except this isn't a pedantic falsehood. Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. Understanding ratio is key to understanding a lot of the physical sciences and you can build a really good intuitive understanding of a lot of simple physical concepts if you can just do dimensional analysis. But if you think of multiplication and division as a kind of addition then it all breaks down and you can't reason about much at all with those tools. In fact you can't even get over the hurdle of doing dimensional analysis in chemistry or physics in the first place. It doesn't make any sense. And you'll have been set up to fail by your teachers.

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.

dragontamer13• 5 years ago

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

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.

613style• 5 years ago

We can't really talk about what multiplication "is" or "isn't" independently of context. It's an operation on two objects and the context its used in is necessary for defining the operation.

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.

cannabis_sam4• 5 years ago

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

My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?

aaplok3• 5 years ago

Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name.

The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation 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.

pdonis37• 5 years ago

there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously

I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides 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".

zamadatix• 5 years ago

The general concept is known as https://en.wikipedia.org/wiki/Lie-to-children and I think you'd be hard pressed to teach/learn/apply anything of significance without this "well actually it's more complicated than that" approach and recognizing how far you need to go with it to aptly do some task.

JosephRedfern2• 5 years ago

I never got over being lied to about electron shells and the 2-8-8 "rule".

curiousllama• 5 years ago

Idk discovering all the lies I believe is one of the most fun parts of learning for me.

lhorie• 5 years ago

teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..."

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.

[0] https://www.prodigygame.com/

hyperpallium2• 5 years ago

Pre-university mathematics is mostly like this.

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.

WalterBright3• 5 years ago

Teaching Newtonian Mechanics to students is also a lie. But I don't believe starting with General Relativity is going to do anything but confuse students.

I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.

tvanantwerp• 5 years ago

I still encounter this as a grown adult. I'm working through a course on a programming framework I'm unfamiliar with right now, and several times have gone through a half-hour setup to encounter "Now that I've shown you the awful way to do this, here's a built-in way that's far simpler and less error-prone!" Extremely irritating.

macspoofing2• 5 years ago

I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods

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.

dragonwriter• 5 years ago

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.

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.

Koshkin2• 5 years ago

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

mansoor_• 5 years ago

He makes a good case for multiplication being a special subset of (repeated) addition, whilst semantically trying to argue otherwise :D

jeff-davis2• 5 years ago

Is addition a shortcut for counting, or is that wrong, too?

andrewprock2• 5 years ago

tl;dr: "Nothing"

LJD_E• 5 years ago

I think the author mixes variables with numbers. If you add units in to the mix then you have to think like it variables so for example that’s taken “2 baskets × 3 apples per basket = 6 apples” would correspond to 2x3y/x=6y removing the numbers and the equation becomes clear xy/x=y or we the numbers for them selves 23=6, That is because numbers are like their own variable but with connection to other numbers(variables). So if we would define 1 as x then 2 would be 1+x or we could call it y but y would still be 1+x in relation to 1.

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.

chrisandchips• 5 years ago

I strongly believe that helping younger students gain strong intuition for these operators pays dividends towards their later success in maths.

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.

kmill6• 5 years ago

I think it's sort of an illusion (though certainly a useful one) that numbers are all part of the same system. Multiplication of natural numbers is defined to be repeated addition, and I'm not really sure how you could define natural number multiplication in any other way. From the natural numbers you can go on to define the integers, rationals, then reals, and each has its own definition of multiplication, though they each depend on the definition of multiplication from the previous system.

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.

tsian2• 5 years ago

As someone with an almost solely intuitive understanding of mathematics, I still find the "repeated addition" idea useful when it comes to fractions. If 4 * 4 is 4 repeated 4 times (4 + 4 + 4 + 4). Then 1/2 * 1/2 is a half repeated half times, or a half of a half, which is a quarter. The numbers get hard to work with but the intuitive idea is still there.

NineStarPoint• 5 years ago

Fractional numbers make sense to me as an extension, but it also requires an intuition of division on the same lines.

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.

OlleTO2• 5 years ago

Proof: The derivative of x^2 is x.

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) =

michaelwu• 5 years ago

It's actually 2x...

OlleTO2• 5 years ago

Yes, because multiplication isn't actually repeated addition :)

maweki• 5 years ago

I think what the author is missing is the word "scalar". I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.

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.

pdonis37• 5 years ago

I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.

"Scalar" is too broad; irrational numbers are scalars, but multiplication by an irrational number is not repeated addition.

whatgoodisaroad2• 5 years ago

I like this article, but I feel like it misses the value of teaching analysis to students. It's true that it's not-quite-right to say multiplication is repeated addition, but it's also a really nice demonstration of the analytical/generalization approach anyone could use to "invent" multiplication on their own.

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.

sandworm101• 5 years ago

> Multiplication requires different units. > 2 baskets × 3 apples per basket = 6 apples

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.

cuspycode• 5 years ago

Yes, that was the most glaring mistake in the article, in my eyes. The runner up strange argument was the one about multiplication and addition having different identities and inverses. But why would the operations "Addition" and "Repeated Addition" have exactly the same properties? The operations are not the same.

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.

jmull2• 5 years ago

To me the "So What’s the Problem?" section has a lot of irrelevant stuff.

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."

bawolff3• 5 years ago

½ X ½:

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.

hc-taway• 5 years ago

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.

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.

cryptica2• 5 years ago

It doesn't make sense to say:

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.

tarkin2• 5 years ago

Why not just say:

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.

fchu• 5 years ago

Learning isn't possible without unlearning whatever temporary construct we used as a crutch, but it doesn't mean using that temporary construct is wrong even if it's not technically correct.

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.

klodolph• 5 years ago

The author is speaking from personal experiences teaching children K-10 (ages 5-16). It sounds like she is documenting her experience teaching these various temporary constructs, and that her experience is that teaching multiplication as repeated addition is a less effective way of teaching it.

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.

bikenaga• 5 years ago

I think it's okay to say: "It's useful to visualize (or think of) multiplication as repeated addition, in contexts where that makes sense." Properly qualified the statement seems true and unbojectionable, and I think it's a useful way to help students understand multiplication. (It's not the only way.)

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.

(scythe pointed out the importance of the distributive law in another reply.) The distributve axiom is huge! Consider

  a * (b + c) = a * b + a * c.

From left to right, it says that "you can multiply out"; from right to left, it says that "you can take out a common factor", which are both standard operations in algebra.

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.

So you say "3 times 2" can be thought of as "3 added to itself 2 times". It does not say that multiplication is repeated addition, if "is" means "defined as" or "derived from",

Or consider one of the standard illustrations given to kids in grade school:

  3 * 2 is   $ $ $   which is   [ $ $ $ ]
             $ $ $              [ $ $ $ ]

This is 2 groups of 3, i.e. 3 + 3. So 3 * 2 = 3 + 3.

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,

which is the distributive law again.

(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? :-)

scelerat• 5 years ago

One of the best concise guides I've encountered to the relationship between the various algebraic operations -- and how they circle back to one another -- is chapter 22 "Algebra" of volume I of the Feynman Lectures on Physics. For some reason this tied everything together in my head in a way that made so much sense, when the pedagogy I head leading up to it simply did not supply me the same vantage point.

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.

EvanAnderson4• 5 years ago

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may.

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?

hinkley• 5 years ago

The new new math isn't just multiplication is addition, it's addition and substraction.

39 x 40 = 40 x 40 - 40

EvanAnderson4• 5 years ago

That seems like a reasonable way to do that computation, from a purely practical perspective. Simple "tricks" to arrive at the correct answer quickly don't at all seem like a bad idea. When we were doing addition of two digit integers and working with place value there was a lot of adding the ones separately from the tens, then adding those together. Later, when addition with carry / regrouping came up we used both strategies to solve the problems and demonstrated how they really were the same thing, just following a different "recipe" to get the result.

matheist• 5 years ago

Math PhD here. You're a fantastic dedicated parent and you did no harm.

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.

EvanAnderson4• 5 years ago

I appreciate the reply. I didn't feel too terribly off-base, but seeing the discourse here gave me cause to question.

maxerickson2• 5 years ago

I suspect that the problem isn't introducing a given model, rather the problem is then not moving on to additional models.

Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.

EvanAnderson4• 5 years ago

I will say what we did work thru some multiplication of integers as repeated addition and scaling, just to show that the answers "come out the same". I used the scaling by a fraction to show that the repeated addition method doesn't work on all classes of numbers. (We've talked some about the difference between integers and real numbers.)

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.

cryptica2• 5 years ago

As someone with a pragmatic, visual mind, I rely entirely on this kind of reasoning in order to understand and apply math. For example:

- 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.

devinplatt• 5 years ago

(Unrelated to pedagogy, but related to when multiplication is not repeated addition.)

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

So I sought to define some form of multiplication that was as close to regular multiplication as possible, but without the distributive property. I ended up defining this generalized form of multiplication as a function m:NxN -> N (N = the natural numbers 1,2,3,4,...) with the following properties

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

Indeed, for the odd numbers m(2,2) = 5! The second odd number (3) multiplied with the second odd number (3) equals the fifth odd number (9).

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.

ubasu• 5 years ago

To put it in terms of what the audience here can relate to:

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

mxfh• 5 years ago

This is more confusing then it helps.

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...

amelius2• 5 years ago

By the way, why doesn't exponentiation have units in physics?

E.g. why don't we ever have something like kg^s?

kmill6• 5 years ago

I think it would have to play well with Taylor series -- for example e^s would be 1 + s + s^2/2 + s^3/6 + s^4/24 + ... -- and that would be mixing units. That's not to say it's meaningless, but rather you'd have to accept measurements like 4s + 5s^2, with incompatible units hanging around. (There's something similar in geometric algebra, I think, where expressions contain terms of different dimensions added together.)

amelius2• 5 years ago

Interesting point. But I think that the Taylor series is just a mathematical tool, and as such it hides the units that exist in these expressions. From a physics point of view, every term in the Taylor series needs to be augmented with a unit that follows from the definition of the Taylor series.

rovolo• 5 years ago

This is purely a guess, but I think it's because all the exponents are scalar. You get exponents when the derivative of a function is proportional to itself:

    d/dx (f) = k*f

So exponentials will be scalars because the exponential part is scaling the original value.

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 : scalar

thysultan• 5 years ago

Philosophical mathematicians trying to confuse the plebs.

WalterBright3• 5 years ago

I distinctly remember when I realized that the times in "4 times 3" is 3 added 4 times. That was the end of trying to memorize the times tables. If I forgot what "6 times 9" was I just thought "6 times 10 minus 6" and there it was.

WalterBright3• 5 years ago

I know how addition is done in hardware. I've programmed multiplication and division on computers that only had shift and add instructions. I've implemented IEEE 754 floating point packages, too.

https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...

Multiplication is repeated addition.

Yes, I know about roundoff errors, overflows, etc.

bitwize• 5 years ago

In mathematics, sometimes a wrong approach can be used as a teaching tool until you get the concept, at which the correct definition may be disclosed. Example: derivative calculus is easier to teach with infinitesimals, even though the derivative is correctly defined in terms of limits.

bawolff3• 5 years ago

Is that really how we want our students to think? Multiplication is not a mere sub-species of addition. Multiplication is its own animal, an independent operation.

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.

ducaale• 5 years ago

https://www.youtube.com/watch?v=F_0yfvm0UoU

bawolff3• 5 years ago

I feel like people here have a weird definition of "truth". Platonism aside, there is no outside ground truth on what "addition" or "multiplication" is. It is an abstract symbol defined solely by what it does to numbers. As long as your definition describes its behaviour, your definition is no more or less right than any other definition.

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.

yarg2• 5 years ago

It's not circular reasoning - it's bootstrapping the rational implementation atop the integer implementation.

yarg2• 5 years ago

The euclidean algorithm can compute the GCD of the factors of the naive rational form, repeated subtraction (down to zero) of the GCD from the denominator and numerator can be used to find them in their reduced forms.

musicale• 5 years ago

Calculator multiplication can be performed by what is essentially repeated addition.

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.

ac42• 5 years ago

Many commenters have already said it: the article is probably not helpful when teaching kids. Or is it? I recall having asked myself these questions at school: if lines are infinitely thin, how is it repeated addition that defines surface areas? What is "inside" the variables we use as units?

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...

Koshkin2• 5 years ago

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

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

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