**The saga continues**

In an article dated September 2007 entitled What is conceptual understanding? Keith Devlin mentioned that he wished teachers would stop “saying that multiplication is repeated addition”. This led to enough response email to spawn a dedicated followup article in June 2008: It Ain’t No Repeated Addition. This led to an Internet firestorm and many more articles by Mr. Devlin (one, two, three, four, five) reinforcing the contention that teaching children multiplication as repeated addition is both mathematically wrong and educationally destructive.

I chipped in marginally on the educational question (The “Multiplication is Not Repeated Addition” Research) and solidified my opinion on the mathematical end (“Multiplication is Not Repeated Addition” Revisited) but I had yet to finalize what I thought of the educational issue: does teaching multiplication as repeated addition really hurt students, or is it as Mark Chu-Carroll puts it “a choice between teaching children no intuition, and teaching them a pretty good beginners intuition”?

Since then I have taught several mathematics classes for K-8 teachers as well as helped review most of the recent “mathematics for elementary educators” books, so I’m ready for my final (?) post on the issue. Given Mr. Devlin’s “final” article has been followed by four more, I don’t know how true this will be, but nevertheless–

**The short answer**

It is fine to expose students to multiplication as repeated addition if great care is taken to avoid the problem of identical mental models.

**The long answer**

The research I’ve seen does not indicate a significant problem with multiplication-as-repeated-addition at the immediate point of learning; the papers with that goal seem to be of similar type to the paper I already wrote about of not being an honest comparison.

The big issue comes later, with problems like this:

For instance, 12- to 15-year-old pupils were asked how to find the cost of 0.22 gallons of petrol if one gallon costs £1.20. (They were asked only to indicate the operation and not to perform the computation.) The most common answer was 1.20 ÷ 0.22. When the same question was asked with “easy” numbers, such as £2 for the price of a gallon and 5 gallons for the amount of petrol, the pupils answered correctly: 2 × 5. When interviewed, the pupils did not consider it incongruous for the needed operation to change when the numbers changed.

– From

The Role of Implicit Models in Solving Verbal Problems in Multiplication and Division

Yes, would could represent the above operation as repeated addition by considering 0.01 to be the unit. The mathematical possibility is irrelevant in terms of the clarity of the student’s mental model; multiplication as repeated addition is inadequate to the task.

Only one of the mathematics for elementary educator books I reviewed explicitly decreed multiplication was repeated addition and left it there; the rest to greater or lesser extents emphasized that students needed to be familiar with different ways of modeling multiplication, like the area model (“the area of a rectangle with side X and Y”) or the correspondence model (“for every X, there are Y things”). What the books fail to mention is it’s quite possible in early grades to “fake” using one model with another model.

**The problem of identical models**

A student fixed on multiplication as repeated addition can go a long way solely with that model, because with many problems it can be mentally handled in an identical way to other models. By way of example:

You have a brownie pan divided into 3 rows containing 5 brownies each row. How many brownies do you have to serve?

The most obvious model that applies here is an area model, but as given there’s no reason a student couldn’t simply apply repeated addition anyway: 5 for the first row + 5 for the second row + 5 for the third = 15 total, and done. This can proceed for several grades until a student reaches a problem like this:

You have a brownie pan divided into 3 rows containing 5 brownies each row. Unfortunately on the bottom row the bottom half of each brownie was burned and had to be cut off. In total area, how much brownie do you have to serve?

A fairly natural repeated addition can be rescued by considering the columns instead of the rows: . However, add a burn to the right edge

and the student is now lost. Yes, one could consider 1/4 of a brownie to be a “unit” but for a 5th grader this becomes too much, and situations start to arise like the aforementioned petrol problem where whole numbers and decimals used in exactly the same problem result in students providing different operations.

**The verdict**

I believe that sole fixation on any mental model is a bad idea, not just multiplication as repeated addition; a student who thinks of multiplication as the area of a rectangle would solve the petrol problem with greater difficulty than a student with a correspondence model. While teachers often (or least we train them to) pay attention to the “inner life” of the mathematical brain, they do not seem to be aware of the identical model problem until it’s too late.

One of my test questions for my K-8 teachers last semester was to write a word problem where multiplication as repeated addition **didn’t** work naturally. It’s possible with to force students out of their “comfort zone” and make sure they understand multiple mental models. The danger of multiplication as repeated addition is it can serve as a proxy for other models in early grades; as a “default model” the correspondence model is much more flexible and the one that should have greater weight. But it isn’t necessary to avoid multiplication-as-repeated-addition to such an extreme that mulplication tables are memorized without any notion of how they are derived in the first place (as was done in the early history of mathematics education).

**A resource to try**

Maria Droujkova’s site Natural Math has a page dedicated to different models of multiplication, and they also sell a poster. Here is a sample:

[Thanks to Alexander Bogomolny for sleuthing out the link.]

Filed under: Education, Mathematics

Alexander Bogomolny (@CutTheKnotMath), on November 22, 2011 at 9:22 am said:Have you seen this?

http://www.naturalmath.com/multmodels/3.html

Jason Dyer, on November 22, 2011 at 2:13 pm said:This was the one I was looking for (particularly the “wall chart”), thanks!

Keith, on November 22, 2011 at 3:38 pm said:Here you go: http://www.amazon.com/Multiplication-Models-Natural-Math-24X36/dp/0977693910/ref=sr_1_1?ie=UTF8&qid=1322001487&sr=8-1

The Educational Problem with Teaching Multiplication as Repeated Addition | Math | Scoop.it, on November 22, 2011 at 4:42 pm said:[...] The Educational Problem with Teaching Multiplication as Repeated Addition The saga continuesIn an article dated September 2007 entitled What is conceptual understanding? Keith Devlin mentioned that he wished teachers would stop "saying that multiplication is repeated… Source: numberwarrior.wordpress.com [...]

Robert Hansen, on November 22, 2011 at 8:08 pm said:I think Devlin dug himself a hole on this one and is just too stubborn to admit it. The major flaw in his analysis, and it is a big one from a pedagogical point of view, is that he misses the fact that mathematics (or any theory for that matter) grows in sophistication chronologically for a student, and not all at once. When you teach a 3rd grader multiplication the issue isn’t purity it is sophistication and it is only natural that you start with a theory of multiplication that involves repeated addition because that is the only experience the 3rd grader has to build upon and about as sophisticated as you can make it.

He made a statement that we should not teach something that is flat wrong. But almost all things are introduced that way, and they have to be if you want the student thinking. Under his rule we would not begin physics with newtonian physics because we know that it really doesn’t work that way, that there is relativity and quantum physics to take into account. But that would be a horrid first step for any student.

I think a rule of thumb, and this really comes from the arts, which math certainly falls under, is that students progress through these stages of sophistication in roughly the same order that mathematics has advanced through the ages. And the reason for this is that in parallel with the body of the subject the student must advance the state of the art of their thinking because that is exactly what mathematicians have done all along. So you can’t just dump on them mathematical notions all at once that are not inline with their current thinking state, unless you want a result that makes even rote learning look like a success story.

Bob

Jason Dyer, on November 23, 2011 at 5:55 am said:I think you are missing the argument somewhat if you are boiling it down to “Keith Devlin was wrong”. In fact, in most circumstances as things are done today he was right, but he was giving only a partial answer to what was going on: any student reliant on one mental model is going to have trouble, and multiplication as repeated addition is one of the most troublesome.

Robert Hansen, on November 23, 2011 at 7:04 am said:If we accept that virtually every successful student of arithmetic goes through the phase of multiplication as repeated addition then multiplication as repeated addition is not the problem. Multiplication as repeated addition isn’t “wrong” for the same reason it isn’t “right”. It is incomplete and no 3rd grader will ever start with a complete picture of multiplication. This is the beginning of their journey, not the end.

I do not think I am missing the argument. The argument is that these generalizations we begin with, and I should point out that very successful students begin with as well, are the cause of failure later on in many students. I disagree (strongly). Their failure is the cause of their failure. These generalizations and their progression is the art of math and thus Keith’s argument translates to “the math is causing the students to fail”, which I guess is now (after the translation) correct.

And some (maybe many) teachers don’t get that as well, I get that to. They misrepresent the generalizations to the students or misinterpret the students own generalizations. They don’t exude the thoughtfulness or skill that underlies mathematics. I don’t think you can fix that easily, other than getting different teachers. But let’s assume that we can do something about it (without getting different teachers). I don’t see how that something would involve removing the math. I think it must involve explaining this process better to marginal teachers who have misconceptions of what all these pieces do and how they make math. And maybe along the way, explaining math better as well.

It is no secret that some students extrapolate what they know to what they don’t know much more easily than others. They progress through the layers of sophistication more naturally. Personally, what I have found is that the best hope of getting a student through these phases when they are not progressing naturally on their own, is to drag them. You just have to do a lot of math and slowly they begin to pick up the generalizations and progression. But even that does not guarantee success.

gasstationwithoutpumps, on November 22, 2011 at 9:45 pm said:But multiplication in the integers is repeated addition: see the Peano axioms: http://en.wikipedia.org/wiki/Peano_axioms

Not only is repeated addition a decent way to teach 3rd graders, but math majors will come back to that definition in number theory.

There are other understandings of multiplication (over the reals, the complex numbers, polynomials, quaternions, matrices, … ), but that doesn’t make multiplication as repeated addition in the integers wrong.

Keith Devlin, on November 23, 2011 at 12:52 am said:False. The Peano axiom definition of multiplication gives you a function that is definitely NOT repeated addition. Many people, including some prominent mathematicians in years past, have misunderstood the Peano axioms definition. See my Novmber 20 profkeithdevlin blog for details: http://profkeithdevlin.wordpress.com/

Keith Devlin, on November 23, 2011 at 12:58 am said:Sorry, I referenced the wrong blog. The explanation of the Peano axioms definition of multiplication is in my November 1 Devlin’s Angle blog for the MAA: http://devlinsangle.blogspot.com/ (But Jason’s readers might like my algebra article as well, and likely far fewer will find it disturbing!

Keith, on November 23, 2011 at 8:03 am said:Very nice article. Thank you for posting!

Robert Hansen, on November 24, 2011 at 10:42 am said:I too think spreadsheets should be a bigger part of the curriculum for many students, but not because I think they help in algebraic reasoning. Excel is ubiquitous in virtually all business settings and the users use it more regularly and put far more effort into it than they ever did with their algebra schoolwork during their school days. Yet I don’t see any algebraic renaissance occurring. What this tells me is that there is a demand for practical and useful business math, done with spreadsheets, much larger than the demand for algebra.

A bit more regarding this comparison…

http://k12sense.wordpress.com/

erlines, on November 23, 2011 at 5:04 am said:Multiplication as repeated addition is only only one of the many over-generalizations students make in their study of whole numbers. But, what’s the big deal? Of course, I’m coming from math teaching point of view. Here’s my contribution to the discussion: http://math4teaching.com/2010/04/01/math-war-over-multiplication/

Robert Hansen, on November 23, 2011 at 5:32 am said:I agree with Devlin that calling Peano’s definition “repetition” is incorrect. Recursion and induction are different mechanisms than repetition, and not just subtly different. I doubt you would call the factorial function repeated multiplication. Some (probably most) of the problem is as Devlin states in his blog, people just don’t understand recursion and induction and teachers know this all too well and the fact that they do not linger on these topics very long is intentional, or they waste a good portion of the term trying to get every student to see the light.

So we can conclude (humorously)…

1. Multiplication did not exist before Peano.

2. Multiplication is too complicated a topic for 3rd graders, they should be learning analysis instead.

gasstationwithoutpumps, on November 23, 2011 at 9:06 am said:Well, the recursion in the Peano axioms is tail recursion, which is formally equivalent to iteration (in fact, the formal sematics of iteration is usually given by a recursive definition), so I don’t think that it is incorrect to call it repetition.

But since Devlin and Dyer are convinced that “repeated addition” is evil, and must never be taught, despite it usefulness at some levels, there will be no convincing them. It is like try to teach a LISP purist that iteration is often easier for people to understand than recursion and a better place to start beginners. Sure, recursion is a more “elementary” concept in the mathematical sense, but that does not make it simpler to understand or to teach.

Note: I may be arguing the wrong point here. If Jason is saying that we can’t stop with “repeated addition”, because it doesn’t work on the reals or the complex numbers, I agree entirely. Other mental models are needed for understanding multiplication. Stopping with just “repeated addition” is not good.

If he is saying that because it does not work for all types of multiplication, it should not be taught, then I disagree strongly. Teaching only abstract algebra (groups, rings, and fields) may cover the essential properties of multiplication for all types of multiplication, but is pedagogically pointless until students have several forms of multiplication firmly in their heads. Repeated addition seems to me to be a perfectly valid way to explain positive integer multiplication.

Robert Hansen, on November 23, 2011 at 2:28 pm said:Yeah, I don’t think the point is Peano. Devlin was just countering the argument that the Peano definition was repetitive, it was a tangent the original article. And Peano isn’t even about the meaning we are talking to here. It is a mathematical definition and proof that multiplication exists and maps all of the possible combinations of the natural numbers to some other natural number.

The point here is how does a successful student of mathematics, a person, progress through the meaning of multiplication (and all the other things as well). What are the mechanisms and analogies they devise as they are introduced to multiplication in different contexts. I don’t disagree with Josh when he says that the meaning of multiplication does not ends with repeated addition. My disagreements are with two things so far discussed…

1. The Devlin Conjecture – That we should not tell students ever that multiplication is repeated addition.

2. That instead of starting with what the 3rd grader already knows, addition, we should baffle and overwhelm them prematurely with 9 different and exotic models of multiplication because they impress adult math teachers.

This is a progression. There are steps. I have dozens of math books and workbooks lying around the house. Many I grabbed because I liked only one page. It is very important to have a variety of material handy, but when the time is right.

I think before we keep debating the solutions we should go back and reaffirm what the problem was. As far as I recall, Devlin made a point that if you were to stop someone on the street and ask them what multiplication is they would say “repeated addition”. Heck, excepting the fact that we have been in this discussion, I would bet that would be the first thing out of most of our mouths. And I think we all know what multiplication is. So I would go back and reexamine what you mean when a student doesn’t know what multiplication is. I don’t think it is because they can’t explain it or because the first words out of their mouths (assuming they are not 3rd graders) is that it is repeated addition. It is whether or not they can apply it in the various contexts that you would expect them to be able to apply it. Can they work with area. Can they calculate a tip. Can they understand proportion. And you won’t know that until you get there.

Robert Hansen, on November 23, 2011 at 6:10 am said:“Multiplication as repeated addition is only only one of the many over-generalizations students make in their study of whole numbers.”

Learning math, I mean really learning math, is all about generalization and then later refining those generalizations, again, and again, and again. It is recursive.:)

I can’t for the life of me imagine a teacher that does not or cannot join the student in that process. Such a view point would seem to support the notion that mathematics is not an art but some lifeless man made body of knowledge, like history.

Weekly Picks « Mathblogging.org — the Blog, on November 30, 2011 at 9:10 pm said:[...] The Number Warrior shared a perspective on the “multiplication is not repeated addition”-saga. [...]

The Educational Problem with Teaching Multiplication as Repeated Addition | Supporting Problem Based Instruction in Mathematics | Scoop.it, on January 12, 2012 at 5:13 pm said:[...] jQuery("#errors*").hide(); window.location= data.themeInternalUrl; } }); } numberwarrior.wordpress.com – Today, 6:13 [...]

Cameron, on January 26, 2012 at 1:41 pm said:We had a big discussion in my math ed class last night about what it meant to teach multiplication multiplicatively-glad so many are interested in the topic! The difference between understanding multiplication as repeated addition and multiplicatively was challenging. It involved if the students can imagine the result of multiplying without carrying out the action of adding. Non-trivial concept! My professor suggested reading Steffe’s 1988 work to sort it out.

Joshua Fisher, on February 29, 2012 at 12:43 pm said:I’m sorry to have not noticed this very nice post for so long. Kudos. I’d love for a writer to address a more fundamental question that has been kicked up by this discussion:

If well-meaning, educated, intelligent adults in the business of mathematics education can be confused about what is at present a third-grade concept in the United States, what other topics might a group of teachers, mathematicians, and other stakeholders productively examine?

PUFM 1.5 Multiplication, Part 1 « Let's Play Math!, on July 16, 2012 at 3:37 am said:[...] of you who dislike conflict. This week’s topic inevitably draws us into a simmering Internet controversy. Thinking my way through such disputes helps me to grow as a teacher, to re-think on a deeper level [...]