*[Photo by Alejandra Mavroski.]*

Myrtle called it The article that launched a thousand posts…, and counting comments on this and several other blogs, that may not be too much of an exaggeration. Yet the discussion feels incomplete — I have not been able to put into words all that I want to say. Thus, at the risk of once again revealing my mathematical ignorance, I am going to try another response to Keith Devlin’s multiplication articles.

Let me state up front that I speak as a teacher, not as a mathematician. I am not qualified, nor do I intend, to argue about the implications of Peano’s Axioms. My experience lies primarily in teaching K-10, from elementary arithmetic through basic algebra and geometry. I remember only snippets of my college math classes, back in the days when we worried more about nuclear winter than global warming.

I will start with a few things we can all agree on…

## We All Agree on Some Things

**Elementary students will use addition**to solve beginning multiplication problems. What else could they do? They have not yet learned multiplication.

**The multiplication table is initially built by repeated addition**, as our students add up (or count by) the numbers in each row and column. Even adults, when they get stumped by a math fact they’ve forgotten, will use repeated addition to figure out the answer.

**Repeated addition does give the correct answer**for any multiplication of whole numbers. Sometimes it is more trouble than it is worth — who wants to add up 957×842? But by using the Distributive Property, any multiplication of whole numbers can be reduced to a repeated addition calculation:

4 × 3 = (1 + 1 + 1 + 1) × 3 = 3 + 3 + 3 + 3

**Rational number multiplications can be calculated as addition of parts.**This is how the Egyptian scribes handled multiplication of fractions. But be careful! To calculate the parts, one needs multiplication — or rather, its inverse, division. So to use this as a definition, one must resort to circular reasoning.

**Finally, I think we can all agree that repeated addition is an important problem-solving tool.**Repeated addition can help students think their way through simple multiplication word problems. It doesn’t always work, but sometimes it is the quickest way to understand a situation.

## So What’s the Problem?

So what is wrong with the definition, “Multiplication is repeated addition”? Perhaps *wrong* is too strong a word — perhaps in some deep, theoretical sense, the statement is true, at least for the whole numbers — I’ll leave that argument to the mathematicians. (See, for instance, How multiplication is really defined in Peano arithmetic.) But speaking as a teacher, the phrase can definitely be misleading.

To define multiplication as repeated addition is to make multiplication a sub-species of addition.

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*. 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.

- The operation of addition has its identity element.
- The operation of multiplication has its identity element.

And they are not the same.

- Every number has its own additive inverse.
- Each number has a multiplicative inverse, too.

And they are not the same.

- Addition has its inverse operation, subtraction.
- Multiplication has its inverse operation, division.

And they are not the same, because the operations are not the same.

[Sidetrack: Oops! I forgot that zero does not have a multiplicative inverse. Jonathan pointed out my mistake.]

[Another side note: I find it interesting that repeated subtraction can be a useful tool in solving some division problems, just as repeated addition can be a useful tool in understanding multiplication. Subtraction plays an important role in the algorithm for long division. I suppose someone will argue that this is evidence multiplication is repeated addition after all.]

## Dimensional Reasoning

Dimensional analysis means looking at the dimensions (units of measurement) of a quantity to help you solve a science or engineering problem.

**Addition requires identical units.** The sum must always have the same units as the addends:

2 apples + 3 apples = 5 apples

2 apples + 3 oranges = ??

What does that second equation give you? Fruit salad? In order to add quantities with unlike units, we need to find a common denominator. Apples and oranges are both pieces of fruit, so…

2 apples + 3 oranges =

2 pieces of fruit + 3 pieces of fruit = 5 pieces of fruit

**Multiplication requires different units.** The product does not have the same units as either the multiplier or the multiplicand.

2 baskets × 3 apples per basket = 6 apples

How can we make multiplication come out the same as repeated addition? The only way to do it is to change the units.

3 cm + 3 cm = 6 cm

But…

2 cm × 3 cm = 6 cm^{2}

We need…

2 lengths × 3 cm per length = 6 cm

We do not normally think about dimensional analysis when we work with plain numbers in math class. But the fact remains that *multiplication changes things in a way that addition does not*.

Addition is one-dimensional, but multiplication is multi-dimensional.

This is why the rules for fraction addition and fraction multiplication are so different. When you add positive rational numbers, you always get a sum that is bigger than either addend. But when you multiply rational numbers, all bets are off — the product may be bigger, smaller, or somewhere in between the numbers.

## Language Does Matter

**Addition:** addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.

**Multiplication:** multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.

**multiplier**= “how many or how much”**multiplicand**= the size of the “unit” or “group”

*Different names indicate a difference in function.* The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

## A New Type of Number

In multiplication, we introduce a totally new type of number: the *multiplicand.* A strange, new concept sits at the heart of multiplication, something students have never seen before.

The multiplicand is a this-per-that ratio.

A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.

A ratio is a relationship number.

In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.

The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which *stays the same whether you buy a lot or a little.*

By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never fully understand multiplication.

## How Then Shall We Teach?

If we accept this argument, if we agree to no longer define basic multiplication as repeated addition, then what? How does that affect the way we teach?

Mainly, we need to change our focus from **how** to **why**.

We can teach multiplication in much the same way that we do now, using manipulatives arranged in groups or rows, pictures of multiplication situations, and rectangular arrays of dots or blocks. But instead of drawing our student’s attention to the *process* of adding up the answer, we want to focus on the fact that the items are arranged in *equal sized groups*.

In other words, we teach our students to recognize the multiplicand:

- Teach children the useful word “per” and how to recognize a “this per that” unit.
- Have them label the quantities in their workbook: 3 cookies per student, 5 flowers per vase, 1 eye per alien, or whatever.

If we need a simple, elementary-level catchphrase to replace “multiplication is repeated addition,” how about this?

Multiplication is counting by this-per-that groups.

As with any such phrase, this statement fails to capture all that multiplication entails. The definition will have to be expanded as students learn about rational numbers. “Oh, look! We can count just part of a group, and we can measure with a unit that is not a whole number.” Our students will someday have to learn about real numbers, complex numbers, and matrix multiplication. Even so, the phrase captures an important aspect of many multiplication situations our students will meet in K-12: that there is a multiplicand, some “this per that” quantity.

This approach should be especially helpful to those frustrating students — you know, the ones with the blank stare — who read a word problem and then ask, “Do I add or multiply?”

## A Useful Tool

Completing the circle, I come back to the point of my first “repeated addition” post. I would like you to consider the teaching power of bar model diagrams to represent arithmetic operations. These diagrams are used in the Singapore Primary Math books, and according to one commenter they are popular in Russia and Australia. They are even beginning to show up in newer American textbooks, where they are sometimes called “tape diagrams.”

Here are some advantages of the bar diagram model:

- Bar diagrams chunkify the number line and make number relationships less abstract.

- They provide elementary students with a pictorial algebra that can help them think through complicated word problems.

- It is easy for students to see the inverse relationship between addition and subtraction, or between multiplication and division.

- Because they are based on the number line, the diagrams extend naturally to rational and real numbers, growing in application with your students’ growing understanding.

**Addition is “this AND that”:** putting two (or more) amounts together. This is the basic addition/subtraction diagram:

**Multiplication is “how many or how much OF the unit”:** measuring or counting parts of a given size. Here is the diagram for multiplication/division:

To learn more about modeling arithmetic problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:

And here is an example of the multiplication bar diagram in action:

## OK, Now It’s Your Turn

I’ve talked long enough. What do you think:

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

- Is it even necessary for teachers to
*define*multiplication? Or is the teacher’s job to provide plenty of examples of multiplication in action? Should we let the students intuit their own definition(s)?

- 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?

- Or do we already teach this way, only in different words?

- If you are an elementary teacher, how do you teach multiplication to your students?

- Are some students clueless because, no matter how we explain it, they just don’t pay attention?

- Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?

If you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book *Children Doing Mathematics.*

I remember being a young teenager and helping people do some simple software work on an Apple //e, I believe. I remember encountering a woman who simply didn’t know what… something like 3*9 was and I said… It’s 9 + 9 + 9…

She responded with something like “I never knew that.”

It was a strange moment. But suffice it to say that the main problem with math education is the stringency of it instead of just focusing on individual success at whatever level.

I think you have done a fantastic job of articulating an alternative vision to the incomplete “repeated addition” model of multiplication. I think we can all agree that that is not ALL it is.

I hope elementary math teachers (my background is in teaching 9-12 math) to keep two things in mind when designing their math instruction:

– hook the concept onto something the children already know

– leave room in their understanding (fancy word: “schema” ) for the rest of the real number system

(ok ideally the complex number system, but since I teach 11th grade, I’d seriously be happy with the reals. rationals, even, would be awesome.)

So what to do with multiplication? Repeated addition, it can be argued, is destructive if that’s the only way the kid understands it. Intuition (for a 3rd grader, we are talking about here), breaks down outside of natural numbers.

I really, really like a model of counting groups of things. Bravo on breaking this down.

How else to introduce it? Area of a rectangle will get you a long way, and is a model that can accommodate rationals, integers, polynomials, and factoring. Getting students to see one representation of quantity as length of a segment has other uses, too, is aligned with historical development of mathematics, and can be used to drive them into a nice contradiction when introducing irrationals.

Also useful is a sort of intermediate-grouping method. (Remember, the teacher is just introducing the idea of multiplication.) How many little paper cups full of water will it take to fill up a big garbage can? Kids get tired of counting little cups of water pretty quick, but you cleverly leave some intermediate sized containers laying around…understanding of multiplication that does not rely on naturals follows. I’ve also used this method to remediate 9th graders, but once that rigid schema is set, it’s hard to break it down.

I fear that part of the problem is the insistence that multiplication has to “be” any one particular thing.

There are many different tangible models for what multiplication is, and each model makes sense within some contexts but perhaps not all contexts.

Repeated addition makes good sense if the first multiplicand is a whole number: 3 x 1/5 = 1/5 + 1/5 + 1/5 is simple and straightforward. However, one has to do mental gymnastics to believe in this metaphor if the first multiplicand is negative, or a fraction, or irrational.

There are various counting metaphors for multiplication (I won’t link to the ubiquitous youtube video involving multiplication as counting intersections of lines); these work nicely for whole numbers, and can even be adapted (see youtube) to take place value into account. But again, adapting this to integers, fractions, or irrationals seems daunting at best.

These repeated addition and combinatorial models can evolve into measuring arrays: thinking of 3×5 as the number of squares in a 3 by 5 rectangular array. This measurement model is well suited to working with fractions (indeed, fractions themselves first arise in the context of measurement, of comparison to units). Irrational quantities can also occur honestly in this context.

Another advantage of the measurement model is that it is well suited to proving properties of multiplication (in a way that is age appropriate to young learners). I would hope that 3rd graders would be dubious that 3+3+3+3+…+3 and 17+17+17 end up being equal, if the question is worded that way; I would also expect they would recognize that rotating a picture 90 degrees preserves area.

However, I wouldn’t dream of asking a 3rd grader to learn their single digit multiplication facts by drawing rectangles and measuring them.

None of these models (nor others I can think of offhand) are well-suited to deal directly with negative quantities. In my experience working with pre-service and in-service teachers, the most effective approaches to integer arithmetic tend to be more abstract and pre-algebraic. (Euler’s approach to this works as well as any: 3×3=9, 3×2 = 6, 3×1=3, 3×0=0, etc… )

So what is multiplication? Is it useful to insist that it be exactly one of these models? Why can’t it be all of them, and others besides, depending on the context to hand [in the same way that I’m a teacher, a researcher, a blogger, a father, a husband, a son, etc… depending on context]?

Knowing a variety of tools for arriving at a problem solution and being able to choose the appropriate tool in a given context (mental estimation, paper and pencil calculation, or calculator) is an important goal of K-6ish arithmetic instruction. Why shouldn’t being able to choose an apt model of the operation to hand also be a part of that?

Oh, I forgot to put that question in the “Your Turn” section. I’ll go edit to add it — Do we even NEED to define multiplication? Or is it enough just to provide plenty of examples and let the students intuit their own definition?

Denise,

I can’t let go of the higher math issues. I’ve been making a list of of things I personally want to learn more about. Abstract Algebra is at the top of my list. Second, I ‘d like to get a good book on the foundations of mathematics and figure out what it means for an axiom to be independent or a definition to be properly worded.

I don’t know that teachers standing up and saying what multiplication is makes much of a difference. I don’t think that simply hearing it from a teacher makes nearly the impact that using the principle does in exercises. I don’t think think that Singapore explicitly states that “multipication is repeated addtion” but the student will walk away with that impression after working through problems with arrays of objects and figuring out the areas of rectangles. Singapore deliberately sets the student up to think about it as repeated addition.

When you see how later mental math exercises are set up, it’s clear that the student is supposed to use the distributive law to solve the problem quickly. They don’t KNOW that it’s the distributive law, but they picked up on how that worked from the prior exercises involving repeated addition.

What teachers say tends to go in one ear and out the other, it’s the learning from doing the problems that makes the bigger impression.

As far as the dimensional analysis, it’s my experience that they won’t interpret adding up a bunch of rows with units. In fact, I have to work very hard to get them to include units in their work, they mentally leave it off and just work with pure numbers.

I only have three kids that I’ve worked with and in an exercise such as counting up groups, they don’t say 3 candies + 3 candies + 3 candies….or “3 groups of 3 candies each”

They will talk about it as “he has 3 and I have 3 and she has 3. We have 9.” And I can ask how do you know you have nine? “Because 3 plus 3 is six and six plus 3 is nine” I know the book wants them to make the distinction but it’s an uphill battle.

They don’t use units because they’ve assumed that the unit is irrelevant to the counting process. The unit is a special case in the physical world and they’ve already generalized the abstract concept that counting and adding works no matter what the units are. I do make them express units in word problems but they see that as irrelevant…and it doesn’t seem to cause any problems until they do problems involving rate. Then relevance is more obvious.

marvelous. lots to think about here.

the one thing i know we *shouldn’t* do

is what’s routinely done in the HS-for-college

“remedial” classes i’ve sort of specialized in:

*pretend* that the field axioms are important

by introducing ’em (and causing the diligent

to memorize their names) and then *ignore* ’em

for the whole rest of the course

(the “distributive law” will occasionally be invoked

by a student; the rest of the vocabulary

will be used only by instructors [if at all]).

one *wishes* for students with some idea

about developing a subject axiomatically;

this was classicly what (euclidean) geometry

was really “about”. these courses are

almost entirely gone now, as i understand.

and *given* such students, a course like the

standard “9th-grade-algebra, done worse”

that dominates the field would maybe be

perfectly appropriate. but a lot of these

poor saps can’t add fractions and *abstractions*

about algebra will be wasted on ’em until they

first learn to actually *do* some …

long live LPM!

As a 5th grade math teacher who was a struggling math student with limited number sense, I really loved your article. I really liked the breakdown of multiplier and multiplicand. It really makes complete sense. Because of your explanation, I do not think that multiplication should be called repeated addition, but rather introduced as a strategy. As stated, repeated addition is not the only way to solve a multiplication problem. It may only be a part of the way to solve it. For example, a student may solve a problem by remembering a larger multiplication fact and subtracting off what they do not need.

I think using ten frames and models to show the difference between 3×7 and 7×3 are super important. It will demonstrate that while you will get the same answer, it looks very different. It is not the same. Visual learners will cling to this idea and remember it much better.

I was taught to memorize in math. I remembered math routines rather than learning the reasoning behind WHY I did what I did when solving problems. I regrouped (as we called it “carried a number”) because my teacher said that’s how you do it. I “dropped a zero” because that’s just what you did. As I got further along in my education, math became increasingly difficult because I was lacking this type of number sense. I knew place value as far as how to identify the ones, tens, hundreds, etc., but I didn’t understand that it realated to everything else. I divided fractions by multiplying by their reciprocals because, again, that’s what my teacher said to do. Your article was so helpful in affirming just how important the WHY is when successfully teaching math to students.

That’s why I tell people that following someone else’s rules isn’t math. It’s just being obedient. But when

you do the thinking, and when you understand why something works, that’s math.Okay, Denise

I scanned this for you from a first grade worksheet:

It’s not the only way that the information is presented, but this is one way it’s done in Singapore. The other exercises reinforce the idea that multiplication is iterative addition by presenting illustrations of groups of items that would correspond to a particular multiplication problem. The student is asked to come up with the product not based on rote memorization of the multipication table, but by adding the groups in the illustration. They will be expected to have automatic recall of multiplication facts by end of second/third grade.

Is the contention with the teacher saying multiplication is iterative addition? Or is it that the student walks away with that impression that it is from having done particular kinds of assignments?

If we don’t want teachers literally stating that multiplication is iterative addition, then wouldn’t it also make sense to avoid exercises that might lead students to jump to this erroneous conclusion?

For me, the idea is to examine the concepts of multiplication and addition rather than parrot what I’ve heard and the way I learned it. I want to clarify my own thinking — and since I have a blog, I’m doing it “out loud.”

I don’t know that it really makes much difference how we explain multiplication to 1st and 2nd graders. They are going to understand it in terms of addition no matter what we say. But I think it may affect how I teach my upper-elementary and middle school kids.

At the least, it has given me an idea of something to try the next time I run into one of those clueless students whose first response to a word problem is to give up.

I have a suggestion for improving your multiplication diagram. Instead of putting the “units” next to each other horizontally you should stack them vertically, and make them the same size and shape as the “whole amount” in the addition diagram. This reinforces the notion that multiplication introduces dimensionality in a way that addition does not.

That’s true, and it would lead to the idea of multiplication as area, too. But it loses the connection to the number line, and I think it would be more difficult to use for the pictorial algebra (like at the Thinking Blocks site). I am not sure which is the better choice — it probably depends on your students and your situation.

Hmmm .. from a geometrical (rather than number-theory, or calculus viewpoint), we DEFINE area as : space enclosed in a 2D shape.

In order to find HOW MUCH area is enclosed in a particular shape, we count the number of UNIT areas contained therein.

Later, (not surprisingly) we find that in a regular shape (eg rectangle), we can calculate the area by multiplying length by width.

Eg, dimensions of 5 units by 3 units = 15 square units.

But this work ONLY because we have 5 rows squares with three squares in each row.

ie, AREA is: multiplication as ONLY repeated addition.

Just wanted to say thanks for your most recent post on multiplication.

What a great read!

Your ideas about ways to teach the two concepts are really nice. I think those ideas can be fiddled with, perhaps refined, perhaps left as is–maybe even, if we play with them enough and/or think about them enough, we may see that some other topic BEFORE multiplication can be adjusted to make it easier to learn the distinction. Perhaps we can take a look at a topic AFTER multiplication and fiddle with it to make the learning smoother. That is what I love about these discussions. Eventually, ideally, they can snowball and lead to sweeping improvements in the way we teach math. Certainly what is clear is that the pedagogy has to change to fit the math, not the other way around.

What continues to strike me is how people just can’t imagine that we can introduce children to the “meaning” of multiplication without starting to find products. “If a third grader doesn’t have the solution to 3 x 2 memorized, then how can he or she find the solution without repeated addition?” they ask.

Why do we need to start out immediately by finding solutions?

Tom has 3 race cars and Sally has 3 race cars. How many race cars do they have in all? For that kind of problem, children, we ADD the numbers 3 and 3–Tom’s 3 race cars and Sally’s 3 race cars. Model with me. Take three counters to show Tom’s 3 race cars. Take 3 counters to show Sally’s 3 race cars, and combine them with Tom’s three race cars to show how many in all. Very good. You just ADDED the numbers 3 and 3. You can write the result as 3 + 3.

A year later, two years later, or that very same day:

Sally has 2 buckets. She will put 3 race cars into each bucket. How many race cars will she have in all? For that kind of problem, students, we MULTIPLY the numbers 2 and 3–the 2 buckets and the 3 race cars in each bucket. Model with me. Draw two big circles to show Sally’s two buckets. Put three counters into each circle to show that Sally will put 3 race cars into each bucket. This also shows how many in all. Very good. You just MULTIPLIED the numbers 2 and 3. You can write the result as 2 x 3.

Repeat until students have made a clear distinction between multiplicative situations and additive situations.

It’s not perfect; it can be improved upon, or, perhaps, changed completely. But, as I said, the pedagogy needs to change for the math, not the other way around. What’s important–and perhaps Devlin was a little insensitive to this by saying, “In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition”–is that good practices in education take a little time to materialize, not only because teachers’ hands are essentially tied by state standards, but also because any good practice has (or at least should have) implications for a number of other practices.

And, of course, good ideas are worth waiting for. Thanks again.

A very well written post – and responses.

One of the areas of my own research interest (I am a math educator at a university) is to examine the elementary school mathematics curriculum. One thing I noticed from my analysis is that they have this notion of “expanding the meaning” of various mathematical ideas. So, subtraction may be introduced as an operation to represent take-away situations, but then they try to “expand” the meaning of subtraction to include comparison. In the same way, they introduce multiplication as equal groups, then when they expand the numbers to decimals/fractions, they expand the meaning of multiplication to a more proportional (i.e., per-one etc.).

The important part is that they are “expanding” the meaning, not replacing it.

I think it is also important to think about the role of mathematical notations. Statements such as 3+4=7 or 4×6=24 isn’t just computation exercises. They are supposed to communicate our own thinking or represent a situation. Thus, if we are only interested in knowing the total number, we can use repeated addition, skip counting or any other methods. However, by expressing 4×6, we can more easily tell that there are 4 groups of 6 objects – no need to count the number of 6’s to figure that out.

We need to emphasize the importance of mathematical notations from early on, in my opinion. I think it is an important part of the foundation toward algebra.

Finally, Japanese textbooks use the diagram called double number line, which isn’t quite the same as the bar model Singapore math uses. It is basically a two number line “hinged” together at 0 but have different (proportional) scaling. Just like the bar models, it can be a very powerful thinking tools for children in any proportional situations – multiplication is a proportional situation.

No. After now carefully examining this issue and the mathematics that underlies it, I think it is just fine to define multiplication as repeated addition, and, in fact, Devlin’s argument makes more sense drawing the exact opposite conclusion. If anything, you should define it as repeated addition on the whole numbers, and then extend that definition by defining subtraction and division. That is the analog to how it is actually done, mathematically speaking. While you should not try to teach 2nd graders Peano’s Axioms, this idea that it is all up for grabs, then, and that you shouldn’t try to remain as faithful as possible to the actual real development is absurd. Defining multiplication as repeated addition doesn’t make it a subspecies of addition. Handling it exclusively that way does. Obviously, a teacher that defines multiplication that way is not trying to turn all multiplication problems into addition problems — they are simply trying to explain multiplication.

This units business is completely overblown. What is e^(5 meters)? Yes, the concept of a square meter is meaningful and there is this analogue to multiplication of numbers when dealing with units that has some mileage, but all of that is a purely scientific heuristic. I am not even sure if there really is a formal way to stipulate what it really means to actually “multiply units”. What a square meter is isn’t just anything. It has a specific geometric interpretation and you cannot just start multiplying units together willy nilly and preserve that interpretation. Say you multiply the lengths of two line segments sitting end to end for no particular reason. Well, you can interpret that by moving a line segment and interpreting it is a rectangle. But, that is all informal not mathematical. Sometimes it makes sense, sometimes it doesn’t. It always depends on the broader scientific context it appears in. And, it is not some kind of absolute fact that is completely generalizable or otherwise just always valid like a mathematical fact would be. So “multiplying units” is an equivocation on “multiply” here. Yes, it is different than multiplying numbers, and, in general, science is different from math. But, that is all beside the point. This is (supposed to be) *math* class not physics.

Also, counting by groups is repeated addition. A diagram where you have 5 units by 7 units and you show how there are 35 little boxes in the big rectangle is repeated addition. None of this is getting away from repeated addition. If you really want to get away from that, you will have to use a straight edge and compass constructions and nothing else. And even then, I don’t know if you can really truly escape seeing how it is also just repeated addition right off the bat (how your diagram can be seen as consisting of a bunch of little congruent squares).

Look, Devlin’s just wrong. His argument is that we commonly teach multiplication as repeated addition when it isn’t. The math teachers have gotten the math wrong in a fundamental way which is why mathematicians need to get more involved in K12 education. And, for crying out loud, at least stop teaching that multiplication is repeated addition when it is simply just not that at all! That’s his argument. Okay, now, I am not trying to pull rank on anyone. Although, people seem to paradoxically try to do that to me all the time. I am just talking about the mathematics. Heuristics have nothing to do with this. Science problems have nothing to do with this. Pedagogy has nothing to do with this. And, let me be clear about what the issue here is. It isn’t whether or not teaching repeated addition is pedagogically acceptable. It is whether or not repeated addition could ever possibly be pedagogically acceptable. Devlin is saying that it cannot because it simply is not mathematically correct, in the first place. Teaching it is like teaching that 2+2=5 — no matter how well you can teach that to students, it will never be acceptable to teach that!

The problem is that Devlin is just wrong about the math. You can easily look this up all over the internet. You can ask other mathematicians about how multiplication is defined on the natural numbers. In fact, Devlin, himself, knows how it is defined and cannot and has not denied that it is defined as repeated addition in that case. How can it be so unacceptable to define multiplication as repeated addition in the first grade where you are specifically talking about multiplication on the natural numbers and nothing but that and where repeated addition is the most expedient way to do explain it and it is how the official derivation, itself, goes??

I wrote on this issue again… http://homeschoolmath.blogspot.com/2008/08/its-been-very-good-and-educational-for.html

I get the feeling that some people are thinking of the issue in slightly different terms than others.

Some only think how it’s defined. But Denise above is thinking how do students end up viewing it.

Don’t we all agree that multiplication is considered a different operation from addition?

Are we talking about that, or are we simply talking about how it technically gets defined?

[Edited to add: How funny! Apparently you and I were typing the same thing at the same time, Maria.]

I find it fascinating how differently we all think. The things that don’t interest you, Adrian, are exactly the things that do interest me: the physics and the pedagogy, and the heuristics of word problems. And the things that interest you and Myrtle — the abstract, college-level mathematical definitions — simply make me shrug.

You may well be right about the definitions. I can’t say. But it seems to me that the thing that goes

unstatedin the definition “mathematics is repeated addition” is exactly the thing our students need to focus on: the fact that we are replicating equal-sized amounts.None of us would define squaring numbers as repeated addition, would we? But for the natural numbers, we can always find their squares by adding a series of odd numbers. If that’s not repeated addition, I don’t know what is — yet it is not what we mean when we say “repeated addition.” We leave unstated the very thing that we do mean: that there is a multiplicand, a “this per that” ratio that is invariant within the problem.

From my point of view, that is the issue.

I want to teach in a way that will help the students who get that deer-in-the-headlights expression when faced with a word problem. I want to lay a foundation for understanding the “math monsters” that always stump these students: fractions, ratios, and proportions. And it seems to me that this change in emphasis — from

How do we get the answer?toWhy is this a multiplication problem?— is the sort of thing that will help.Good response, Denise and Maria.

I may have stated this before, but that thought never stopped me from saying something (laugh).

Anyway, mathematics IS abstract, and it is this fact that makes mathematics so powerful. But, as far as elementary mathematics education is concerned, what we need to keep in mind is not the adjective “abstract,” but the verb, “to abstract.” We have to help children abstract mathematical ideas from concrete situations and activities. So, it IS extremely important for us (teachers – and students, too) to pay attention to units as most of numbers we come across in our everyday life is either measured quantities (including counting) or the result of arithmetic operation on measured quantities.

Very interesting post! As a 3rd grade teacher, where my kids are only just learning multiplication, I try to enforce the “always check by adding” policy. And for anything beyond 2-digit numbers, I have them add 14 5 times or 22 8 times, etc.

Great post. So much to think about. As always, though, I think we set up a lot of false dichotomies for ourselves. Yes, I do think this boils down to what the meaning of “is” is.

What I mean is that one definition does not have to exclude the others. If I define a being human as, say, “of or characteristic of people as opposed to God or animals or machines, esp. in being susceptible to weaknesses,” and you define it as, “of or characteristic of people’s better qualities, such as kindness or sensitivity ,” they can both be correct.

What I believe it comes down to is “intent.”

What is it about the operation of addition that you are trying to awaken the student to in that particular lesson?

If you are trying to show some of the beauty of mathematics, then go for the loftier definition. If you are trying to go for simple manipulation of numbers, go for “repeated addition.”

The key (OK, not that I’d really know what the key is, but heck, it’s pretty close to the key) is to

consistently make the student aware that the story does not end with the lesson.Each lesson should be taken as a step towards deeper understanding.If your point is that we should not leave students with the impression that multiplication is

simply and onlyrepeated addition, I’m with you.Cum gran salis ~

Brian (a.k.a. Professor Homunculus at MathMojo.com )

Human – featherless biped (n’est-ce pas?)

Human – of or belonging to the genus Homo (nicht wahr?)

Denise,

I’m coming to this way too late but I did want to commend you for opening up an excellent discussions of ‘first principles.’ Math educators need to grapple with this and many other thorny issues of content and pedagogy. Normally I would weigh in on this topic with passion but, instead, I’m going to be silly. I hope you will forgive me. I’m sure you’ve read the following but I love wordplay and this does seem relevant if only obliquely…

Dave Marain

: : : : And so it was to be, that after the waters receded, Noah commanded all the animals to “Go forth and multiply.”

: : : : The ark quickly emptied, except for two small snakes, who stayed behind. When Noah asked them why, they replied, “We can’t multiply. We’re adders.”

: : : : Noah, being the resourceful man he was, immediately got busy cutting down trees and building a large table with the unfinished lumber therefrom.

: : : : And he saw that it was good.

: : : : The snakes were overjoyed when Noah picked them up and placed them on it. Noah and the snakes both knew that even adders could multiply on a log table.

I love that one! 😀 But I think I will have to tweak the format so it will display properly. On my screen, it has the choppy look of a multiple-forwarded email.

Joshua has posted an interview with Keith Devlin here. Relevant quote:

I may expound on this (yet another addition/multiplication post?) if things calm down around here. As it is, I lost my internet connection once again — the joys of antique equipment! — and I have been so busy with last minute things to do before school starts that blogging has become temporarily impossible.

I’ve been on Devlin’s side on this debate of multiplication vs repeated addition; however, I must disagree with him about the multiplication-makes-things-bigger belief. I think it has little to do with how we conceptualize multiplication. Rather, it is simply an over generalization based on multiplication of whole numbers. After all, every multiplication situation except multiplying by 0 or 1 will result in a product that is larger than the multiplicand. Moreover, the VAST majority of multiplication problems children encounter initially are only with whole numbers AND very rarely they have to multiply by 0 or 1. I think Devlin is correct to say that this belief is generated by “something the child observe him or herself” no matter how s/he thinks about multiplication.

If it’s all right with y’all, I’m going to respond to the latest comments here in the comments section of my interview with Devlin.

(Do stop by and leave a comment–whatever you like. I’m guessing that there’s about a 95% chance that Devlin himself will respond.)

There is also a discussion going on at mathfest:

And Keith Devlin revisits the topic one more time, giving us a reading list of math-ed research that will fill up my spare time for the rest of this school year:

I wish my library loan system had access to a copy of Children Doing Mathematics. I found excerpts at Google Books (here, if you’re in the U.S.), but that was only enough to whet my appetite.

Drat. Looks like Devlin is dropping out of the conversation. Oh well. Now I can go refill my chill-pill prescription.

Adrian says:

“It isn’t whether or not teaching repeated addition is pedagogically acceptable. It is whether or not repeated addition could ever possibly be pedagogically acceptable. Devlin is saying that it cannot because it simply is not mathematically correct, in the first place.”

Thank you Adrian. That’s putting the finger on what bugs me.

In the other blog entry on this topic I tried nailing down the very same issue in another way.

Mathematics has the answer for the question of what “IS” is, at least for functions. Function “F” *is* function “G” (or, F = G, which is as close as you will get) when they have the same domains and co-domains and have equal values for every element of the domain.

So, if we are talking *mathematics*, integer multiplication *is* repeated addition (with other species of multiplication being likewise with varying levels of convolutedness.) And when I say Devlin is wrong it means mathematically he is wrong.

Now if you are talking cognitive psychology instead of mathematics (which is what Devlin has veered off into in his latest musings) well, then, that’s a horse of a different color. But – your guess might just also be as good as Devlin’s GUESS, because he has no special authority in those NON-mathematical areas, even if he does quote a smattering of studies. Actually I have no dog in that fight, as they say. It is an interesting question.

My preference is for the multiple models approach. Let the student conceptually understand multiplication by abstracting out for themselves what is common among all the different ways of illustrating multiplication.

“To define multiplication as repeated addition is to make multiplication a sub-species of addition.”

Ooooh! I see now – we are not arguing what the meaning of *is* is – we are really arguing the meaning of “subspecies”.

One can *define* PI as the sum of a certain infinite series, or as the limit of a certain infinite sequence, or even as the root of a certain equation. That in

no waymakes PI a “subspecies” of sequence, or of series, or equation.Once again, Devlin sets a trap – if you accept his premise (the subspecies bit…) then you have a problem. Reject the premise, no problem.

Umm… The “subspecies” bit was not Devlin. That was me, trying to come to an understanding of the pedagogy of multiplication.

Denise — sorry about that then – although I think your phraseology here does pretty much capture what Devlin is trying to say. I think he may be afraid that defining multiplication by means of repeated addition leads to a confusion in the student that other properties of addition are *inherited* by multiplication (which of course is not true.) (Thus I appreciated your pointing all the properties of multiplcation that are distinct from similar properties for addition – very good!) However, we don’t really know that it is usually the case that students habitually get confused on these matters because of the repeated-addition model, and neither does Devlin.

I’m all for teaching some of the more “abstract” properties as early as possible – especially the notion of “inverse operations”. I think simply pointing out both the similarities (they both have identity elements; they both have inverse operations) and the differences (the identity elements and the inverse operations are different for multiplication and addition) is enough to dispel potential confusion in most students.

I agree, inverse operations should be taught from the very beginning. I’ve written about that before, for addition and subtraction, and the same principle holds for multiplication and division.

The confusion Devlin mentions, that “multiplication makes things bigger,” may not stem from defining multiplication as repeated addition. But it is certainly a prevalent misconception, which I think could be ameliorated by teaching multiplication by fractions from the very beginning, too — as Miquon Math does

in the first semester of first grade.Of course, to teach that way would require a different explanation of multiplication. “Repeated addition” doesn’t work so well when one deals with rational numbers from the start.

I find some resistance in myself to the number bar concept – and maybe its because it seems so one-dimensional. As you point out, multiplication lends itself to 2 (or higher) dimensional understandings. What’s better – to introduce the number bar first and ask “how many?”, “how much?”, or to introduce the 2-dimensional grid of dots (perhaps fat dots that can be subdivided) and say – 3-1/2 copies of 5 rows of pies? Does it matter which is introduced first? How do we decide which works better?

Good questions! I would like to know the answers. Or whether it matters at all — like you, I think multiple representations are a good idea.

The number bars are not used in Singapore math until about 3rd grade. The transition from pictures to bars happens gradually throughout second grade, with the pictures being placed into rows and then boxes drawn around them. They also introduce area in 3rd grade, but the bars are preferred for their ability to help students think through multi-step word problems.

Miquon, on the other hand, uses bars from the beginning, since much of that program is based on Cuisenaire rods. They use other models as well, but bars are primary.

I put in my two cents worth a few comments ago, and already regret it.

This is a great blog and a lively discussion. I’m sorry my (truly) humble thoughts have may have been misunderstood by some. As they are not particularly insightful, and not at all important, I guess I’ll recuse myself from further input.

I enjoy most of the comments here and have learned much from them. I’m thankful for the post, as it’s gotten me to read the original posts from Devlin’s Angle, and agree or disagree with the guy, you’ve got to admit that he is one of the most insightful guys and thoughtful writers around.

I gotta go, though. There are kids who need to be taught some math.

Brian Foley (a.k.a. Professor Homunculus at MathMojo.com )

Certainly, go teach those kids — that’s the reason we’re all here, right? But please don’t recuse yourself from the discussion. I’m still trying to wrestle through these ideas myself, and I appreciate all input!

I had forgotten that way back in ’06 I had discussed a study talking about “best models” for multiplication.

The first post is here. The second is here.

The paper that I cite in the first post is free for download, so go check it out. It’s a fun little paper in that it combines “complexity science” with sitting down and chatting with a group of teachers.

This was the money quote for me:

One key point established in the discussion was that the main conclusion of the activity was not that there are so many figurative aspects of multiplication to be taught, but that there are many physical actions out of which multiplication arises.You spent a lot of words to explain something incorrectly. For all mathematics is multidemsional addition and all Addition is unit based by default. By definition the word add includes the concept of units units. Substraction, Multiplication and Division are only renamed forms of addition.

For example: Subtraction is addition of vectors. the positive(+) and negative(-) signs indicate direction and are used to show which direction the units are to added. Multiplication and Division are only ways of grouping addition of units.

Numbers and letters are only symbols that represent concepts. Addition is a concept that can be described in various ways with one of them being named multiplication.

Coming back to this topic after many ages… Here’s a post that sheds light on one teacher’s experience with student (mis)understanding of addition and multiplication:

Middle School Math – What is Hard?

I have studied repeated addition thoroughly and find that it is the best “tool” to teach students multiplication and other math objectives. Repeated addition of unit ones corresponds to the whole numbers, the natural numbers (counting numbers), and the cardinal numbers.

Each natural number names a sum starting at (0+1 = 1). Thus 0+1 names the sum of zero and one. Thus 0+1 + 1 = 2, 0+1+1+1 = 3. Repeated addition can be noted by 0+1+1+1, …, +1 = sum N. The unit ones can be grouped into uniform units composed of x amount of unit ones which can noted as G(x)/G =Product P = 0+1+1+1, …, +1 = Sum N. You can use items like cookies to count and match the count to a sum and group them (divide) them into groups. This concept is realistically meaningful and much easier for the students to understand since they can create their own addition/subtraction and multiplication/division tables. Repeated addition also helps introduce the concepts of prime numbers in the form of G(x)/G + 1 = PN, the whole and its parts (fractions), and percent. Repeated addition and subtraction are instrumental in setting the stage for introducing the concept of the number line. Repeated addition is “linear in nature” since it follows an “action process” which contains order and amount. It can be used to introduce multidimensional concepts such square and cubic units since the numbers are products written to the first power and the units are written in “square or cubic” notation. BUT…TOM…LINE…I DO NOT SEE ANYTHING WRONG WITH REPEATED ADDITION USED TO TEACH MULTIPLICATION, DIVISION, AND EVEN SUBTRACTION!

meh. I really feel I must have overlooked something, but I don’t see what it is. First I would argue that scaling is just a special case of addition and so we can still synthesize scaling with addition. Mainly we don’t because it provides no utility for a human to do it, though that is how it would be done on a binary computer.

More troubling it seems to me like we have forgotten the utility of the slide rule. Basically, if you change the base, you can turn any multiplication problem into an addition problem. That is the whole point of the slide rule.

Lets take an example of fractions though: 1/3 x 1/2 = 1/6

All we need to do is follow the lead of the slide rule and change the base to 1/6. The result is equivalent to repeated addition: 2base1/6 x 3base1/6 = 6base1/6 = 2base1/6+2base1/6+2base1/6

I would think anyone familiar with radians would notice the same feature with base Pi. So where is the exception that disproves this small pivot that makes it easy to synthesize multiplication with addition? I don’t think there is one. In any binary computer the multiplier is little more than a cascade of adders. To suggest there is some mathematical operation that cannot be synthesized from addition/NAND universal gating implies that such an operation cannot be performed on a digital computer.

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 2x*3y/x=6y removing the numbers and the equation becomes clear xy/x=y or we the numbers for them selves 2*3=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.

Adrian,

I am a mathematician and I think Denise is correct. The following is from the end of my essay “The Fourth Operation”:

http://ariwatch.com/VS/Algorithms/TheFourthOperation.htm#Afterword

In practice many numbers are not pure numbers but quantities, that is, they measure something in a certain unit such as meters or kilograms. We will say a pure number has the “null unit” so that all numbers have units. When adding two numbers both must be in the same unit. “You can’t add apples and oranges” as the saying goes. When multiplying there is no such restriction, each number can have any unit. When neither number is pure the result is in a new unit. For example, two meters times three meters is six square meters, two watts times three hours is six watt-hours.

Multiplication can be considered iterated addition only when one of the numbers – the number measuring the degree of iteration – is pure. Adding five apples to itself three times makes 15 apples. But three apples or three oranges as a count for adding makes no sense.