Wiggins question #5

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,

\frac{a}{b}=\frac{c}{d} if and only if a\cdot d=b\cdot c

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: A\times B 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.

Image from The New York Times.

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)



15 responses to “Wiggins question #5

  1. Michael Paul Goldenberg

    I think the key issue is the word “just,” combined, as you say, with “is”; I think some mathematicians, most notably Keith Devlin, would ask not what multiplication IS but rather what multiplication does. And in that regard, the fact that you can get the answer to classes of questions involving multiplication by performing repeated addition is true but misleading. He prefers the notion of scaling, and he mentions the work of Terezinha Nunes and Peter Bryant as providing research support for the notion that we do a disservice to children by insisting that “Multiplicatioin IS repeated addition” (MIRA). He further asserts that he sees problems rearing their heads (based on the MIRA viewpoint) with Stanford students in calculus courses. He wrote a series of articles on this issue a few years ago in his Devlin’s Angle column that are worth looking at.

    Honestly, I’m surprised that Mr. Wiggins would come down on the what I call the MIRA-busters’ side of the debate, simply because its hardly the common viewpoint inside or outside mathematics/ mathematics education. As the founder of a secret discussion group called “MIRAbusters,” I suppose my position is pretty clear. However, I still draw juice from the” multiplication can be thought of as repeated addition” and “division can be thought of as repeated subtraction” metaphors because, well, I’ve found them very fruitful for the students I work with. But then, I rarely work with K-3 students, so my feelings of guilt are minimal.

    If I were teaching younger students, I think that as things stand, I’d start with area as a model for multiplication. But if I were teaching from a Davydov-oriented curriculum that puts measurement ahead of counting in K-2, I’d definitely want to start with scaling or at least get to that model/metaphor pretty quickly, well before the “repeated addition/subtraction” ideas.

    • I didn’t say I believed it. I asked for a good argument as to why it is wrong. And the obvious counterexamples involve fractions less than one.

      • Michael Paul Goldenberg

        Uh, where did anyone, least of all I, suggest that you believed it, Grant? As for “obvious” counter-examples, I don’t think you’ll get really far starting with that one at the point where students are first learning multiplication as something distinct from repeated addition. Remember: they don’t know fractions yet.

  2. 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.

    • Michael Paul Goldenberg

      @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.

  3. 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 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.

    • Michael Paul Goldenberg

      I would be shocked if any of the folks (including mathematicians) who support the idea that the first model kids should be given for multiplication is ‘repeated addition’ (and for some of such folks, that model suffices in perpetuity) would be moved one iota by pointing out that it can be thought of as repeated subtraction also. If anything, they’d shrug. Pointing that out doesn’t get multiplicative reasoning separated in any way from additive thinking. And the issue is just semantics if there’s really no fundamental mathematical and/or conceptual difference between adding/subtracting and multiplying/dividing.

  5. 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.

  6. 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.

  7. 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?

    • Michael Paul Goldenberg

      As I suggested in my first comment: given that 99.9+% of US curricular materials for K-5 are grounded in counting, rather than in measurement, the area model of multiplication is probably best suited to conveying something “other” about multiplication compared with addition.

      You also get (free) that rotating a rectangle 90 degrees clockwise doesn’t change its area, which strongly supports the commutative property of multiplication (e.g., a 3 x 6 rectangle must have the same area as a 6 x 3 rectangle). No need to refer to adding, though kids will inevitably use counting as they build up their mastery of multiplication facts for one-digit whole numbers.

      If by some miracle you’re in the tiny little slice of schools using a measurement-based K-5 approach a la V. V. Davydov (they exist but are extremely rare in English), then I suspect scaling as a model for multiplication might be built into the texts or be relatively easy to sell to students. But that point is almost assuredly moot.

  8. ” 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?

  9. 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.

    Click to access 1301.6602.pdf

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