## Question 5

### “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

Now this is when things get sticky.

It is a strong and presumptuous claim to say what an idea *is.*

In recent years, I have come to an understanding of why *repeated addition* is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it *is*.

You see, any rich enough mathematical idea has multiple meanings. What *is *subtraction? Is it the inverse of addition? Is it the distance between two points on a number line? Is it *takeaway*? Subtraction is all of these, sort of.

And what is a fraction? The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an *equivalence class* resulting from the equivalence relation,

if and only if

Is that what a fraction *is*?

No. But I am off task.

I suspect that my answer may vary from some others out there. (Although perhaps it will not.)

Repeated addition is shaky ground for establishing multiplication because it doesn’t capture the unique structure that multiplication represents.

There is additive structure, and there is multiplicative structure. Additive structure is about comparisons and changes involving the same units. Apples plus apples gives apples. Miles plus miles give miles.

Multiplicative structure is about comparisons and changes involving different units. *Hours* times *miles per hour* gives *miles. *Three different units; one of them a unit rate. Always.

These are related structures but they are different.

Multiplicative structure is captured better by this idea: means *A* groups of *B *(I am pretty sure I first ran across this particular characterization in Sybilla Beckmann’s textbook for math courses for elementary teachers). *A*, in this interpretation, is expressed in one unit. *B* is the unit rate (*things per group*). The product is expressed in a third unit.

This difference shows up in the following conversation between a mom and her daughter as they count the number of things in this array of meatballs.

Maya counted the top and bottom, 4 + 4 = 8. Then she counted L and R. 3 + 3 = 6. 8 + 6 = 14. 14 + 2 in the middle = 16. When I asked her why, she said, “Because you double count the corners when you count an array.”

She asked me to count so she could show me how. I counted 4 across the top and 3 down the side. “See, Mommy! You’re counting the corner one twice.”

Why *do* we count the corner one twice in this scenario? This seems to violate a fundamental principle of counting—*one-to-one correspondence*. One number word for each object, and one object for each number word.

The answer is that mom really did *not* count the corner meatball twice. The first time, she counted the *meatball* to establish that each row has 4 meatballs. The second time, she counted the *rows*. There are 3 rows, so there are 3 groups of 4 meatballs.

Much, much more on arrays in many places in my writing. Especially these:

Beyond the textbook wrap up (or *What does this have to do with mathematics?*)

Twister (on sister site, *Talking Math with Your Kids*)

Units is really the way to go to convince students that multiplication is not repeated addition. IF multiplication were repeated addition, then area would be in (units) and not (units)^2.

@K.: I’ve argued that myself over on the MIRABusters group and also in some public venues, but those who are wedded to the notion that anything a mathematics educator comes up with must be trivial or wrong just don’t seem to care about the rather obvious lessons to be drawn from units. Physics teachers would, I think, get it. I’m not sure the nay-saying mathematicians and engineers don’t get it: they just can’t ever publicly admit that this is an issue that causes problems for students, because it didn’t cause problems for them. And anything that didn’t cause them difficulties must be trivial: a difficulty only for non-mathy folks about whom they really couldn’t care less, no matter what lip-service they might pay to the contrary.

Welp, MPG pretty much put down my thoughts RE: it depends on what “is” is.

I suspect that a primary reason multiplication is explained as “just” repeated addition is the hope that the word makes it less threatening, but I prefer the approach taken in my programming class with new concepts: “Okay, we’ll start with this simple model… but the idea is a lot bigger ” then we go home and chew on it… and *then* move forward.

We also get even the simpler models explained in terms of the important big ideas. One of my recurring themes in tutoring the basic math folks is that adding is a simple relationship, and you have to add the same things… many of them are such avowed “plug and pray” students that they have minimal strategies for figuring out which operations to use.

“MULTIPLICATION IS JUST REPEATED ADDITION.” EXPLAIN WHY THIS STATEMENT IS FALSE, GIVING EXAMPLES.

Multiplication is not ONLY repeated addition on the integers because multiplication is ALSO repeated subtraction on the integers. Multiplication distributes over subtraction as well as addition. The proof by contradiction is as follows.

As +3 = 0 + 1 + 1 + 1 we get

2 x (+3) = 2 x (0 + 1 + 1 + 1)

= 0 + 2 + 2 + 2 and this is repeated addition

Now as -3 = 0 – 1 – 1 – 1 we get

2 x (-3) = 2 x (0 – 1 – 1 – 1)

= 0 – 2 – 2 – 2 and this is repeated subtraction

Therefore “Multiplication is NOT just repeated addition”.

Multiplication may INVOLVE either repeated addition OR subtraction on the integers. The sign of the multiplier determines the case. Q.E.D.

Jonathan Crabtree

P.S. On the reals, multiplication is simply a variation of the multiplicand by the ratio of the multiplier to unity. This is called proportion, which some confuse with the idea of re-scaling discrete measurements.

Hi Michael

I answered the question exactly as asked. No more, no less.

There are major and fundamental difference between adding and subtracting relating to the cognitive capacity of children needing concrete instantiations of mathematical principles.

If you have three apples you cannot add a negative apple and then arrive at two apples. The idea you can do away with subtraction and division is only valid upon entering the realm of abstraction, which is NOT what elementary teachers do at the tender age kids learn basic arithmetic.

Removing the addition signs from multiplication entirely just points out the flaw in the logic of those promoting MIRA on the integers.

Having just had an A student wondering whether to add or multiply when confronted with temperature lapse of 6.5 degrees per kilometer, over a distance of 13 kilometers. Understanding that we can figure out repeated addition with multiplication would be extremely useful for this student.

So, as an elementary teacher how do you break down the purpose of multiplication to the simplest form if not building from the repeated addition concept first?

I’m trying to wrap my head around how to do it conceptually for say, grade 3. Ideas?

” But that is a far cry from making claims about what it is.”

I’m a bit confused. Isn’t the question only about what it is not? That is, haven’t you answered the question the minute you show that multiplication is something other than repeated addition?

Quote: “The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an equivalence class resulting from the equivalence relation”

Since the days of Aristotle, a definition must be expressed in terms of things which are prior to, and better known than, the things defined.

That was one of the 16th century motivators for the MIRA meme when explaining discrete multiplication on the naturals. Of course once you explain multiplication via ‘repeated addition’ on the naturals, then exponentiation can be explained via ‘repeated multiplication’.

Variation that was continuous in nature was always discussed under the banner of geometry, not arithmetic. Thus many mathematical manuscripts and books had the word ‘proportion’ or ‘geometry’ in the title alongside ‘arithmetica’.

What people struggle with is the shift from discrete arithmetical ideas to continuous geometrical ideas. Thus direct instruction in the manner of Davydov (as Michael mentioned) with a focus on similarity of measurement changes will be key. As you halved your stick so will I. As you doubled your stick, so will I.

Soon the multiplier and divisor can be seen as variations from 1 and so as the multiplier varies from 1, so must the multiplicand. As 1 varies to become the divisor, so must the dividend. It can become a ‘follow the leader’ game or function copying game. Whatever is done to the unit of measurement to make the multiplier, you do the same to the multiplicand.

So for example, two multiplied by negative three can be explained via proportion (equality of ratios) as follows.

Q. What did you do to 1 to make -3?

A. I trebled it and changed its sign/side.

Q. So what do you do to 2?

A. I treble it and change its sign/side, so that makes it -6!

All new curriculums, whether they be American, British or Australian can only benefit from a better appreciation of proportional thinking, provided ratio is not confused with fraction.

So for kids learning what multiplication does on the naturals, without addition, simply revert to tallies and objects. One duck multiplied by three means one duck for every | in | | |. So as 3 = | | | we have a one-to-many proportional correspondence, so as | is to | | | then duck is to duck duck duck. Many-to-one proportional correspondences can also be played for division. So as ‘2’ is to 1, ‘duck duck’ is to duck.

Proportional ‘copy-cat’ change based on simple counts and measures will unlock both discrete and continuous understanding of multiplication and the fact that multiplication can make fewer and ‘shorter’ as well as more and ‘longer’.

The blending of the realms of discrete and continuous into the reals is a recent phenomena. Yet the various successor logics from Grassmann, Dedekind, Peano from the 19th century, just kept the focus on the discrete positive integers. Descartes from the 17th century (and Hilbert) gave us simple models for multiplying lines without changing the unit via geometry that extends to the reals, yet not many people today appreciate the power of proportional thinking.

The paper, “A Geometric Approach to Defining Multiplication” by Peter McLoughlin and Maria Droujkova is worth a look.

http://arxiv.org/pdf/1301.6602.pdf