Republic of Mathematics blog

Multiplication of whole numbers, and how we think about it

Posted by: Gary Ernest Davis on: December 31, 2010

I want to take a – relatively minor – exception to Keith Devlin’s post, What Exactly is Multiplication?, on multiplication as repeated addition, and then largely agree with what he writes, with some amplification.

Multiplication is a defined notion for whole numbers and is provably commutative

Devlin claims that multiplication is not repeated addition and that “the mathematician’s concept of integer or real number multiplication is commutative: M x N = N x M. (That is one of the axioms.).”

I believe he is wrong on both counts.

First, both addition and multiplication are defined notions on the integers, rational numbers, and real numbers. For the positive integers, multiplication is defined recursively as m\times 1 = m \textrm{ and } m\times (n+1)=m\times n + m.

This makes multiplication exactly repeated addition.

Secondly, the commutative property m\times n = n\times m then follows as a logical consequence of this definition and is not an axiom.

It is an axiom for general structures such as rings or fields, where one posits a commutative operation of multiplication, but it is a provable property of the defined notion of multiplication of natural numbers.

Why we shouldn’t (only) think this way

One can prove properties of multiplication from the definition of multiplication of natural numbers – essentially defined as repeated addition – but, as Devlin points out very well, this is not a productive or creative way of thinking about multiplication.

The inductive definition of multiplication of whole numbers is like a lawyer’s definition: it’s designed for safety reasons, to ensure that we are able, at some low level, to prove statements we make about multiplication.

However, the legalistic inductive definition of multiplication is not really helpful in thinking in other ways about multiplication as an operation on whole numbers.

What is needed for younger students is a sound psychological – not simply logical – way of thinking about multiplication.

Concept definition and concept image

Keith Devlin is, to my mind, essentially blurring over the distinction between concept definition and concept image.

The inductive definition of multiplication is a concept definition and, as such, is a part of advanced mathematical thinking, as defined and elucidated by David Tall.

Defined notions are, in general, not especially suitable for younger students, who do not yet see the necessity for such rigor.

Much more importantly for students is what David Tall refers to as a cognitive root. This is a way of thinking about a concept that, while not formally correct, nor complete in all details, is sufficiently powerful to allow students to use the concept, and to not have to throw their ideas overboard as their learning advances:

“A cognitive root is a concept that:
(i) is a meaningful cognitive unit of core knowledge for the student at the beginning of the learning sequence,
(ii) allows initial development through a strategy of cognitive expansion rather than significant cognitive reconstruction,
(iii) contains the possibility of long-term meaning in later developments,
(iv) is robust enough to remain useful as more sophisticated understanding develops.”

An example of a sound cognitive root is the idea of a function as a machine with inputs and outputs.

A cognitive root is essentially a solid foundation for thinking about a concept that can be later modified, and made more rigorous, but which remains as a basis for understanding.

What is important for younger students is a solid concept image of multiplication, based on firm cognitive roots.

Devlin addresses this issue explicitly (but without the language of concept image and cognitive roots, when he writes:

“So what is my mental conception of multiplication? It’s a holistic amalgam of all the above and several variants I have not listed. That’s why I say multiplication is complex and multi-faceted. The dominant mental image I have is very definitely the continuous one of scaling, and I see all the others in terms of that. This means that my conception of scaling within this context is a very general one, that encompasses examples like my bags of apples. I can view the computation “3 bags each containing 5 apples gives 15 apples altogether” as “scaling” a bag of 5 apples by a factor of 3. In my experience, acquiring the concept of multiplication amounted to creating this mental amalgam – the amalgam that is my concept of multiplication.”

One problem we know the scaling concept of multiplication produces when fractions come into the picture is that scaling by (positive) whole numbers never produces a smaller result, yet multiplication by fractions less than 1 does produce a smaller result.

Students often take on board from the scaling notion of multiplication the idea that “multiplication makes bigger”. That is one danger of seeing multiplication as scaling, and is why this concept image is probably not a good candidate for a cognitive root.

Multiplicative thinking in my view is largely based around lots of units. Some things – for example, candies – come in bags, and having a certain number of bags of candies, each containing a fixed number of candies, is not a bad candidate for a cognitive root for multiplication.

When those unit “bags” are capable of being split we can see multiplication of fractions in much the same way as multiplication of whole numbers.

20 Responses to "Multiplication of whole numbers, and how we think about it"

This blog post, from someone who has studied mathematics education professionally, is why I am always reluctant to step into this area. As a mathematician, it never occurred to me that students would see “multiplication as scaling” as scaling by a positive integer (only). For me, the scaling concept always incorporated scaling up and scaling down, and indeed the scaling factor can be any positive real number. Moreover, if there was a stage in my life when I believed multiplication makes bigger, I no longer remember it. This one example highlights my oft-repeated point that it will take both mathematicians and mathematics educators to get our math ed system right! We mathematicians do in general understand the mathematics, but knowing the problems students have when they are learning it for the first time, and knowing how to help them, requires a whole different set of skills, knowledge, and experience.

Thanks for the comment Keith. I agree it takes a lot of dialog between people of good faith, who want mathematical education to shine and enrich students’ lives.

For the positive integers, multiplication is defined recursively as . . .

This makes multiplication exactly repeated addition.


1 + 2 + 3 + 4 . . .

Is that repeated addition? Yes. Is it multiplication . . .? No.

There is no ‘repeated addition’ that magically transforms into what we know as multiplication. It’s not about “concepts” and psychology. It’s just mathematics, pure and simple.

This is not complicated.

Joshua, I think perhaps I was not entirely clear about how I was using the term “repeated addition”.

It means repeatedly adding the same number, not adding different numbers.

From this definition, multiplication of whole numbers is repeated addition.

Let’s think about 4\times 3. The recursive definition tells us that:
4\times 3 = 4\times (2+1)=4\times 2+4 =
4\times (1+1)+4=4\times 1 + 4 + 4 = 4+4+4
so, lo and behold, in this example, multiplication by 4 is, in fact, repeated addition.

There is, of course, nothing special about this example.

You may not agree that’s what repeated addition means, but that’s what I intended it to mean, and I am grateful for an opportunity to clarify this point.

For further references to the use of the phrase “repeated addition” in school mathematics (which is what we are discussing) and its connection to multiplication see, for example:
(*) “Repeated Addition: Addition of equal groups; often used to model the concept of multiplication.”
(*) “Repeated addition is the idea that multiplication means ‘so many groups of’. If I have ‘5 lots of 3 bricks’, then the answer to ‘How many bricks are there altogether?’ is 3 x 5, or 3 + 3 + 3 + 3 + 3. We read 3 x 5 as ‘3 multiplied by 5’, since the 5 is operating on the 3.”

The psychology of multiplication IS somewhat complicated as the many people who have reflected on it have found.
I think Keith Devlin and I have tried to show how.

Concept definitions and concept images are not the same thing: not even for professional mathematicians.

It really is a question of psychology.

Bill Thurston, as I recall, stated that mathematics was, in fact, a branch of psychology.

Instead of thinking of arithmetic operations as ‘producing’ results, would it be useful to think of them instead as descriptions of types of partitions? In other words, 7 + 5 doesn’t ‘make’ 12; rather, 12 already IS 7+5. A quantity of 12 units already contains a group of 7 units and a group of 5 units. In fact, it contains 12C5 such partitions. 12 can also be partitioned into 3 groups of 4, and when we partition a quantity in such a way, into a set of equally sized parts, we have then expressed the quantity as a product. In this way of thinking, a product might well be equivalent in value to a repeated sum, but that’s not what we’re thinking about when we construct this product. The 3 and the 4 didn’t turn into 12 through some kind of operation. Rather, it is simply a fact that a quantity of 12 already contains 3 quantities of 4 units. It’s one of the properties of any quantity of 12 regardless of the type of unit. Different kinds of quantities composed of different types of units can be partitioned in different kinds of ways. Some units cannot themselves be further partitioned, but others can be. If we can partition a unit 1 into x equally sized parts of 1/x, what if we partition it into two parts x and (1-x) such that x/1 = (1-x)/x? That forces us into issues of incommensurability. It might be stretching the meaning of ‘partition’, but could we also talk about the ways to partition zero? Instead of thinking that opposites ‘annihilate’ each other to ‘produce’ zero, we can say that the quantity zero already contains the partition x and -x. So, would it make sense to think of all arithmetic operations as descriptions of the ways we can partition quantities? Now, to address the main issue here, how might a partition represent a scaling? The best I can do with that so far is to think in terms of similar triangles. Consider a line segment representing a distance d. Now consider some other point not on that segment (or its extension). Let t and r be the two segments joining that point to the endpoints of d. Partition both d and t into n equal segments, and visualize a sequence of n segments parallel to r forming the body of the triangle. We have illustrated a scaling. Partitioning is involved, but it’s a little different here.

This is just something I’ve been wondering about.

– Michel

Michel, thanks for this highly creative comment.

I like the idea of multiplicative partitioning.

This general idea reminds me of two different ways of thinking about division of whole numbers – quotitive and partitive. One leads to a productive way of thinking about division of fractions, the other doesn’t.

Which interpretation of (1/2) ÷ (1/3) is not productive?

(a) How many 1/3-cup servings does a half cup of rice make?

(b) If half a cup is 1/3 of a serving, then what is the serving size?

Also, do you think that students should learn the words “partitive” and “quotative”, or is this just for teachers? I can never keep these terms straight.

The direct partitive version of (1/2)÷ (1/3) doesn’t really make sense to me: if I have 1/2 of something ( a cup. e.g.) and split it into 1/3 equal parts, how much will be in each part?. This is more or less a direct translation from the whole number case: e.g. if i have 20 objects and split them into 5 equal groups, how many will be in each group? This is a partitioning (=sharing) problem, and not really a counting problem. One can translate it into your version, but that does not seem to me to be a direct translation, or a direct extension, of the mental model of partitive division of whole numbers (or am I mistaken?)

On the other hand, the quotitive version of (1/2)÷ (1/3) is: how many pieces of size 1/3 are there in a piece of size 1/2? and that is a counting problem : counting by 1/3 units.

A more general way of thinking of partitive division is to think of it as an operation to determine the per-unit amount. Multiplication, partitive division and quotitive division are all related to the proportional relationship. If AxB=C, then, we have the situation, A is to 1, C is to B. Obviously, if C is unknown, we have a multiplication situation. If B is unknown, we have a quotitive situation. Finally, if A is unknown, we have a partitive situation. I think a major reason for teaching multiplication and division of fractions and decimal numbers is (in addition to having computational procedures) is to help students to develop this unified view of multiplication and division from a proportional perspective, in my opinion.

Are we going around again with this topic? What fun. :)

Speaking as a elementary/middle school teacher, the most frequent problem I see is this: Students read a word problem and then ask, “Do I add or multiply?” Drives me crazy when they do that!

Somehow, we need to teach multiplication in a way that distinguishes it from addition in the students’ minds. Whichever side wins the theoretical “multiplication on whole numbers is or is not repeated addition” debate, we must communicate to our students the distinction between “multiplier/scale factor” and “multiplicand/unit/equal-sized group.”

As Keith Devlin wrote, “the nature of the units is a major distinction between addition and multiplication.” You say the same thing in your response to Joshua Fisher above, even though your comment is arguing FOR the repeated-addition understanding. I think this is significant, that both sides can agree on a distinction which will help children recognize when multiplication is appropriate and when it is not.

It seems to me that the thing that goes unstated in 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.

So far, IMO, this debate has produced two practical pedagogical suggestions:

(1) Teach children to recognize “this per that” types of numbers.

(2) Teach multiple representations of multiplication from the very beginning.

It will be interesting to see what new ideas come out this time around.

Denise, I think I am NOT arguing for children understanding multiplication as repeated addition.

What I do argue, and Keith Devlin does not agree, is that the formal definition of multiplication in the set of (Peano) natural numbers IS what natural number multiplication is, and thereby produces multiplication as repeated addition. That is a formal consequence of the definition of multiplication.

As I understand it, where Keith differs is that for him multiplication is a much broader and richer notion than the formal definition, and the formal definition is only a model of his richer notion.

Since I do not – and potentially cannot – know what this broader notion of multiplication is, I must disagree; that is, for me, the essence of multiplication of whole numbers is captured by the formal inductive definition.

BUT, I would not be teaching the formal definition of natural number multiplication to school children and I would not teach it as repeated addition.

I agree – as per the discussion of cognitive roots – and pretty much in agreement with Keith, that we need broad and useful mental models of multiplication (long before we ever, if ever, need a formal definition).

I have to take issue with the idea that it is something about ‘scaling’ that would lead students to believe that multiplication always makes things bigger. That seems flawed from several perspectives. First, there is the common use of scaling in the phrases “reduced to scale,” “scale model,” etc. that suggests quite the opposite (except when being used humorously to illustrate someone’s comic literalness: see, for example, ZOOLANDER). I see nothing inherent in ‘scaling’ that limits the direction, but if anything, my first thought would be reducing, not increasing.

That said, in my experience, it is how we teach addition and then multiplication-is-repeated-addition in this country (I won’t comment on Australia or anywhere else) that leads students to mistakes about how those operations work – namely that each always results in increase.

Of course, we know that there are examples students encounter fairly early that should disabuse them of that notion, namely addition of and multiplication by zero, but not enough is made of these examples, I fear, to get it across to many US students that adding may merely keep things the same and that multiplication may in fact reduce in magnitude. Once negative integers are thrown into the mix, many American kids are just gob-smacked by the notion that adding can ‘make things smaller’; that multiplication by x, 0 < x < 1 can do so is unclear or at least not an instantly accessible idea to many, many high school students.

It's precisely the notion of multiplication as scaling that appeals to students' real world experience AND which does NOT depend on their 'common sense' ideas of how numbers work (particularly when the student has not yet bumped into negative or rational numbers, or barely so. Thus, I imagine it's clear, I'm very much in Keith Devlin's corner on this argument.

Further, I don't think he need apologize for not being a "mathematics educator," at least in the sense of not having taught K-12. I'd say that he's done a lot more deep thinking about the concepts involved and how to model them than have many K-12 teachers I know. It's not that being a mathematician qualifies anyone automatically as an expert on K-12 teaching, but rather that being someone who does what Keith Devlin does makes one well-positioned to weigh in on the topic under discussion without apology. Would that some mathematics educators who don't have a very deep knowledge of mathematics be as quick to apologize for their lack thereof when taking issue with Keith (and I have personal experience both with taking him to task and later apologizing, publicly and privately, for my temerity).

It is a common observation in mathematics education research that many students – and, sadly, teachers – will say “multiplication makes bigger”. This mental model comes into stark relief when students meet multiplication of fractions. Check, even anecdotally, how many people are puzzled by squaring making numbers between 0 and 1 smaller, not bigger.

As I had recently tried to say, multiplication is repeated addition of the SAME number; in the other direction, division is repeated subtraction of the same number. Initial understanding comes from that(or those) idea(s).

As for the surprising results of learning about negative numbers, this sort of thing becomes clear in a good, well-taught introductory Algebra course, hopefully in 9th grade, high school.

Gary, you seem to have misinterpreted my point. I am well aware that many students think that multiplication (and addition) always make things bigger. I thought I’d made that clear in my previous comment. My issue with your claim is as to what the SOURCE of that mistake is. I don’t believe it’s the “multiplication as scaling” idea, but rather “multiplication is repeated addition” that leads them astray.

Please reread my comment and respond to what I actually stated. You might want to address what I had to say about the common uses of scaling as making things SMALLER, too. Does research actually support your viewpoint that multiplication as scaling is the culprit? If so, I’d like to see it. I’ve been a bit out of the loop in mainstream math ed research, but I can’t say I recall that notion being documented.


my apologies for misunderstanding what you were saying. You are right – I have no empirical evidence that scaling is likely to be a cause of the “multiplication makes bigger” mental model. I doubt that it’s repeated addition per se that’s the issue: rather, it seems to me more likely to be that we deal with multiplication of whole numbers long before fractions. For positive whole numbers it’s true that “multiplication makes bigger”.

Thanks for your input.

That is exactly how logical sense can best be made about this scaling. Students have enough opportunity once they begin learning about ratios and fractions that multiplying by a fraction gives something smaller than what one started with. When necessary, the teacher can plainly teach this.

There seems to be a major viewpoint on teaching/learning basic arithmetic missing from this conversation, one that I think would suggest that students are perfectly capable of learning from the beginning that operations are not one-directional. That is the measurement-based approach developed by V.V. Davydov and colleagues, building on the ideas of Vygotsky, in the Soviet Union back when there was one. Similar approaches have been used here (see the MEASURE UP! curriculum out of the University of Hawaii Curriculum Research and Development Group), the work of Barbara Dougherty at Iowa State, the work of Jean Schmittau at SUNY@Binghamton, and the work of Susan Addington with preservice student teachers at CSU-SB).

As for the comment that “the teacher can plainly teach this,” seems to me that therein lies so much of the problem with US mathematics education: relying on teachers to plainly teach something that is conceptually challenging for many students. I believe US teachers have been “plainly teaching” a great deal in mathematics (not ALL of it wrong-headed) for more than a century. The wide-spread fear and loathing of mathematics as well as the relatively shallow understanding of it has been part of the legacy of such teaching as the dominant pedagogy of the nation.

Keith’s idea is to approach multiplication as the second operation on real numbers, with the distributivity law as the only reference to addition. (In this approach multiplication 3 x 5 would become a way of calculating 5+5+5 rather than the opposite.) Can it be so done from the very beginning? I think it can. A constructivist way of building a new concept image is experimenting. It is done in elementary geometry; why not in arithmetics? As in geometry pencil, ruler, squared paper, other materials, and dynamic geometry software (Cabri) are widely used, for experimenting in arithmetics we have elctronic calculators. Let’s tell children do 2 x 10, 3 x 10, 4 x 10 etc. and tell each time what has been added to 10. Then 1 x 10, 1.1 x 10, 1.2 x 10, 1.3 x 10 etc. with the same question. Then 0 x 10, 0.1 x 10, 0.2 x 10, 0.3 x 10 etc. (What has been subtracted?); 1 x 100, 1.1 x 100, 1.12 x 100, 1.13 x 100 etc.; 1 x 100, 11 x 100, 112 x 100, etc. and so on. After some time of such work the teacher would ask the students to tell their discoveries. In this way they would discover both the meaning of the decimal point and digits, and also the function of the key x.
I havn’t done this in a classroom. I havn’t seen it being done. It’s my “thought experiment” as Hans Freudenthal called such an imagined lesson. Who would dare to try it?

Would it help to try to bring a bout a rephrasing of the commonplace “multiplication is repeated addition” by “repeated addition is a form of multiplication”. That would open the door to “scaling is a form of multiplication” and perhaps even raise questions about other possible forms of multiplication. In terms of ‘cognitive roots’ or what used to be called ‘canonical images’, such a reversal would enable students to take on board things like matrix multiplication as ‘multiplication’.

At CERME last year I met Jorge Soto-Andrade from Chile who proposed that for mathematicians, multiplication is composition, hence the multiplication (composition) of functions. So commutativity is a property of particular forms of multiplication, but not multiplication itself

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