Those closely (or even not so closely) following the mathematics portion of the Internet may have run across a fuss last year over Keith Devlin’s article It Ain’t No Repeated Addition, contending that teaching children multiplication as repeated addition is both mathematically wrong and educationally destructive. Numerous furores arose (for example on this thread) and Mr. Devlin himself followed with two more columns, It’s Still Not Repeated Addition, and Multiplication and Those Pesky British Spellings. I have been reading every blog followup and planning an “ultimate” post judiciously examining all the evidence and delivering a final verdict.
This is not that post. There were subtle misunderstandings going on and I have concluded everyone is right (in some senses) and everyone is wrong (in some senses), which is a recipe for a complicated essay — don’t expect the final result for a while.
However, I have run across something that warrants sharing now. Keith Devlin cited a particular research paper, The development of the concept of multiplication. I read it, and was startled when I came across this:
Correspondence Group
The picture in the booklet was shown to the child. The experimenter said: “Amy’s Mum is making 2 pots of tomato soup. She wants to put 3 tomatoes in each pot of soup. How many tomatoes does she need?”
Repeated Addition Group
The picture in the booklet was shown to the child. The experimenter said: “Tom has three toy cars. Ann has three dolls. How many tyos do they have together?”
Stop for a moment before I give my take: I want to know if you see what I see.
…
Two different objects for two different children? How does this give a natural intuition of multiplication?
I can guarantee without even giving the experiment that the first model would be more successful than the second, but it’s more a matter of a badly-formed example than a true comparison of the concept they’re aiming at.
More importantly, it doesn’t match how I’ve seen multiplication depicted in education. Repeated addition tends to be more like:
Frank bought a $3 sandwich every day for four days. How much money has he spent total?
Repeated addition intuitively yet — one can think of the exact same problem with the “correspondence” model given above. If given without commentary, one would not know which model it would fit.
Yesterday, in the supermarket, a customer dumped a whole basket full of power bars on the counter. The checkout counter person started to scan them in, one by one. After seeing about seven of them scanned, with a lot more to go, the customer piped up: “they are all the same!” The checkout person laughed and started to count them.
(This was part of a thirteen part series on multiplication by Bert Speelpenning.)
That is, if Frank has bought 10 power bars worth $2 each, how much money has he spent? The motivation here is again entirely by repeated addition, but there’s no reason 10 power bars can’t match with pots and $2 can’t match with tomatoes. Furthermore, in terms of a mental arrangement, it’s much more common to have a geometric sort of layout:
That is, verbally one might describe the situation as repeated addition (due to the natural situation) but cognitively perhaps something else is going on. (Which consequently suggests some awareness in teaching the topic is warranted … but that’s a topic for my longer essay.)
I’m reminded of that article from Nature from last year which tried to assert abstract examples were better than concrete ones by making a badly-formed concrete example and then surprise — of course people are confused! I have gathered educational researchers occasionally make the questions of their experiments in a “value neutral” sense, that is, as long as the letter of the definition of the concept is followed, it’s valid. This ignores the other intrinsic qualities that make one example clearer than another. Even something as small as using boats as an example in a landlocked area vs. one next to the ocean can have a cognitive influence.
Something else I found curious about the paper is it doesn’t match with the recent efforts (spawned by the original Devlin essay) to rid multiplication of repeated addition. Joshua Fisher tried a model based on the geometry of a grid, while Maria Miller tried to follow Mr. Devlin’s suggestion of multiplication as a scaling factor. However, if the “correspondence” model given in the research paper truly avoids repeated addition, then (from my personal textbook knowledge) teachers don’t need to change their examples at all. Is the “addition” model given in the research is perhaps one that used to pop up in British textbooks? Were we all even arguing about the same thing?
If you’re curious: while it wasn’t up at the time of the controversy, someone has since uploaded the research paper to the Internet for all to read.
Filed under: Education, Mathematics
Thank you for a good addition to the discussion. I agree that second picture (the cars and dolls) is poorly designed to teach multiplication, even if one is using the repeated-addition model.
The biggest problem I see with the idea of “repeated addition” is that (depending of course on how it is taught) it feeds the confusion I see in my elementary students when they deal with word problems. They get a brain freeze. They stare at the problem, and then they ask, “Do I add or multiply?” Clueless.
I think, whatever model we teachers (or our textbooks) are using, we need to help our students notice this key feature of multiplication: There is a “this per that” relationship. 3 tomatoes per pot, or 3 toys per kid, or 3 dollars per sandwich… these are the things that make us think “multiplication” rather than “addition.” The tomatoes and pots picture makes this relationship clear, while the cars and dolls picture hides it.
It often happens as you describe with the power bars, that a situation can easily be viewed as addition or multiplication. Both perspectives are equally valid, but one is more efficient. If I can get my students to recognize that and to notice the “this per that” relationship when it appears in a problem, then I will feel like I’ve done a decent job teaching.
Sadly, elementary students aren’t the only ones to be unsure whether to add or multiply in word problems.
I’m having trouble seeing things like 12 inches x 12 inches as repeated addition…. what ever turns them into square inches…
It leads to the problem I mention here
The “inches * inches = square inches” point seems quite useful in explaining why the distinction is needed.
I think there is an intermediate step that is often overlooked when calculating area. At least, this is the way I try to explain it to my students: We start with length measurements, and then we need to define our unit of area before we can use these measurements to calculate the area of our rectangle. (I always start with rectangles. Does everyone do that?)
12 inches x 12 inches = 12 x 12 x (1 square inch)
Now we have 12 rows of 12 square inches, which can be seen as repeated addition if one really wanted to. But it is also easy to imagine what would happen if the calculation was 12.5 x 12.5 (we’d have extra partial rows), so we can stretch our understanding to rational and real numbers.
There is another significant failure in the multiplication as repeated addition: FRACTIONS. It should be no surprise that elementary students have difficulty in this area, since the addition of fractions and the multiplication of fractions do not lend themselves to similar models of teaching. I would be most interested if anyone knows of a model that uses the repeated addition to teach fractions which overcomes this problem.
To be fair, I don’t think anyone was arguing you could teach fractions with repeated addition; rather, they were using it as an initial model in the age range before students encounter fractions. Really there were two strands of debate:
a.) arguing that with Peano arithmetic, one can create multiplication on the natural numbers by repeated addition vs. arguing that thinking of multiplication as repeated addition is always wrong mathematically
b.) arguing that it was fine to teach multiplication as repeated addition as an educational introduction, and that it would be “augmented” the same way negative numbers etc. get gradually added to a child’s number system vs. arguing that it leaves harmful misconceptions to equate the two concepts that way at any point in a child’s education.
[...] Jason Dyer revisits last summer’s hot topic of controversy in The “Multiplication is Not Repeated Addition” Research. [...]
Nothing says summer like multiplication vs. repeated addition.
Look forward to your “ultimate” post.
One quick point, because I can’t resist:
It’s important, I think, to keep in mind Park and Nunes’ purpose in the study you cited (which I uploaded to Scribd). They wanted to see which understanding the origin of the concept of multiplication is grounded on–repeated addition or correspondence. They did not intend to test which understanding “worked better” to help students “learn” multiplication.
So, they didn’t “teach” (sorry for all the quotes) correspondence-as-multiplication or repeated-addition-as-multiplication (at least, that was not their intention). They “taught” correspondence by itself to one group and repeated addition by itself to another.
I disagree, then, that the repeated addition example from the study that you posted is a bad one. It shows exactly what it was supposed to show. It is an example of repeated addition–by itself.
“Nothing says summer like multiplication vs. repeated addition.”
LOL!
[...] Jason Dyer revisits last summer’s hot topic of controversy in The “Multiplication is Not Repeated Addition” Research. [...]
Forgive me for being woefully late in adding my thoughts to this article, but I’ve only just stumbled upon it.
I’m a computer scientist, by profession, but I’ve always taken an amateur interest in expanded mathematics, and I’ve always enjoyed the magic of numbers. As such, I find myself confused at the distinctions drawn between Addition and Multiplication in Mr. Devlin’s articles. Perhaps there’s something I’m missing (and if so, please help me understand it).
Mr. Devlin and others have specifically pointed to fractional arithmetic at being the point at which Multiplication ceases to perform as Repeated Addition. However, I’ll contend that the conceptual problems to which he alludes come as a result of the common notation used to describe the math, and are not inherent in the math itself. Let’s take, for example, the problem of three fifths multiplied by four sevenths. Most of us would not hesitate to quickly write this as [latex] \frac{3}{5}\cdot\frac{4}{7}[/latex]. This, of course, causes problems because it is conceptually difficult to add [latex]\frac{3}{5}[/latex] to itself [latex]\frac{4}{7}[/latex] times. The units simply don’t make sense.
However, that is not because the Multiplication involved is suddenly no longer Repeated Addition. It is due to the fact that our notational system utilizes explicit shortcuts. Yes, we can calculate the expression much more quickly by multiplying numerator to numerator and denominator to denominator, but that does not change the fact that we are essentially adding together twenty sets of 21, where our unit is one thirty-fifth.
Now, I can understand a desire to find the best method of teaching multiplication to students, especially if one feels the current method does not prepare them adequately for higher math; however, it seems disingenuous to me to claim that Multiplication is not mathematically the same as Repeated Addition, based on the arbitrary use of certain units and notation. Is there really any case in mathematics in which Multiplication cannot be solved through the use of Repeated Addition?
I do have an extended answer to your query, so if you’ll be patient I will be posting Is Multiplication Ever Mathematically Equivalent to Repeated Addition? where I untangle what Mr. Devlin is being all fussy about. (Short answer: he considers multiplication to be not multiplication unless it is done in an integral domain. There is at least some logic to this but it requires an explicit philosophical standpoint which he never makes clear.)
My humble opinion is that Devlin made a huge fool out of himself and for some reason the mathematical community decided to pretend it never happened.
Multiplication on real numbers IS repeated addition. It’s so MATHEMATICALLY. It’s the standard way to DEFINE multiplication on integers in foundations of mathematics. And it can be easily extended to all real numbers (infinite sums).
Furthermore, kids first learn multiplication on natural numbers, NOT fractions or real numbers. Teaching kids multiplication over natural numbers WHILE telling them multiplication is not repeated addition is IDIOTIC. The kids would never trust another word the teacher says.
What Devlin in effect advocates is that we should LIE to kids initially so that we can PRETEND later on that some conceptual problems do not exist.
Well he should start by outlawing the term MULTIPLY since in ordinary English it usually means to become more numerous (e.g. the Biblical go forth and multiply).
Why Devlin gets away with his crap is a mystery to me. I guess it’s because he wrote 50 books on far too many topics than he can truly understand.
It’s more complicated than that.
Messages like this mean I _really_ need to finish my next post, which I’ll try to do this week.
In short (although he unfortunately never makes this clear) while Devlin acknowledges the existence of Peano Arithmetic et. al. and building multiplication off of them, that the number system is not the one we use until we reach the level of integral domain and by then our definition has to be changed.
In fact, problems start happening as soon as one adds 0. Here’s a quote from a number theory book:
The idea we followed in the last chapter was that muliplication was repeated addition, and so 2 * 3 told us when to add 3 2’s together. The idea easily covers 0 * 3. It tells us to add 3 0s together. But, using the definition above, (0 + 0) + 0 = 0 + 0 = 0, so 0 * 3 ought to be 0.
There is more of a problem with 3 * 0 however. If we apply our basic idea, we find ourselves adding together 0 3s. We have no 3s (or anything else) to work with, so we certainly can’t get a number for an answer. Our original motivation isn’t sufficient to cover this case.
…
Now, in this case, the modification is slight, but things start to get worse later.
What’s going on, essentially, is a difference of philosophy of mathematics. Either standpoints has possible arguments. I promise I will finish my full exposition soon.
What’s going on is that Devlin PRETENDS to have the objective truth on this and PRETENDS that anyone who disagrees is factually wrong.
No amount of philosophizing is going to change that.
And on real numbers multiplication IS repeated addition in a way that most mathematicians would recognize as valid. That you have to extend the concept of sum to empty or infinite sums is obvious, but so what? This is entirely common in mathematics. And once again, it already occurs with the term MULTIPLY, the meaning of which needs to be EXTENDED to products like 3×1 (or 1×3?) by kids.
Devlin is utterly lost, as evidenced by his three confused articles. Why else would people like you need to go to such lengths to try to find some remote specks of sense in his diatribe?
I’m going to guess you’re not seriously suggesting 3.5 * 2.6 or
are modeled by repeated addition, but rather you just mean natural numbers, right? (Note of course we are only talking about addition and multiplication then; as soon as we add division we get 3.5.)
(Extra thought. Or maybe you are implicitly doing scaling, considering multiplying 3.5 as adding something 3 and a half times. But that isn’t quite repeated addition, although I see how one might semantically make the switch. Hmm.)
I agree the three articles were somewhat muddled, but as I’ve pointed out, he is not insane, simply he requires an integral domain before he consider the process being worked with “arithmetic”. That he was never totally clear about this was unfortunate, and I do think his insistence on considering more primitive arithmetic (that’s a technical word, not a insult) simply invalid is frustrating.
Of course 3.5 * 2.6 or pi * pi can be modeled by repeated addition.
Multiplication on fractions in particular is typically DEFINED by repeated addition. One third is the number q that added 3 times equals 1. One third times 5 is the number q that added 3 times equals 5. Two thirds times five fourths is the number q added twice, where q is the number that added 3 times equals five fourths. See http://math.berkeley.edu/~wu/NMPfractions4.pdf for a *mathematican’s* definition of multiplication on rational numbers that is modeled on repeated addition.
As for irrational numbers, they are limits of sequences of rational numbers, and their products are limits of sequences of products of rational numbers… in others words of repeated addition.
So yes you can model multiplication on real numbers by repeated addition and this is indeed a natural way to extend the concept from natural to real numbers.
Of course students would make no sense out of this until they truly understand irrational numbers, which means truly understanding limits of infinite sequences. That is why the geometrical interpretation as an area is preferable, though only AFTER students encounter the REAL LINE. And of course it can be shown easily for natural numbers, and fairly easily for fractions, that this is equivalent to repeated addition.
As to scaling, please give me PRECISE mathematical definition of scaling to be used A) in 2nd grade and B) in middle school. Understandably the definition in elementary school can be less abstract than in middle school, but must suffice for intuitively clear proofs of multiplicative properties — the way repeated addition and rectangular areas suffice at these levels (respectively).
I’m a little clearer on your standpoint here, thank you.
The issue that you’re not realizing is that Devlin is essentially ceding your mathematical point but _still claiming multiplication is not repeated addition_.
That is, he would argue while one “builds up” with addition, and forms a “natural” extension as you say, this extension is not *identical* to repeated addition.
If this seems like a strange nitpick, well, you might be right. But again, I said, it is a philosophical standpoint; he is claiming even though an extension might be “natural” it is still substantially different enough to deny isomorphism.
Basically: do you view structures like “integral domain” as of the whole cloth, and any “primitive” versions of that as side effects, or do you consider mathematics to be “built up” from the primitive concepts?
As to scaling, please give me PRECISE mathematical definition of scaling to be used A) in 2nd grade and B) in middle school.
Now, all this is separate from the educational issue. I agree the “scaling factor” thing can be dodgy (see Maria’s post linked in the main text of this post) and this very post addresses another one of the replacements candidates which seems to be a bit muddled.
(Also, nice article. Thanks for linking to that.)
And keep in mind there’s a world of a difference between saying multiplication MUST be viewed ONLY as repeated addition, and saying that multiplication CAN be viewed, among other things, as repeated addition.
Teachers who are unwilling to present multiplication ALSO as, say, Cartesian product, or geometrical area, or scaling, certainly should be criticized. But this is NOT what Devlin has done. He says ANY teacher, even in 2nd grade, who presents multiplication as repeated addition, is an ignoramus spreading falsehoods.
Again, no amount of philosophizing is going to make Devlin any less of a fool than he already made himself in this matter.
I should point out the subtlety that he *thinks students should discover the particular case of addition being repeated multiplication in the context of a more general definition of multiplication* and not that he wants the thought excised altogether.
Mind you, there are reasons to believe he is wrong in this, but there are opposing reasons as well that don’t involve just being ornery.
I should leave you with, though: in essence we are just talking about sequencing: should we take a general case and then find interesting things about the special cases, or should we start with special cases which may involve more intuition and work our way up?
Jason, every irrational number is the limit of a sequence of rational numbers. It is so in a trivial manner: see Cauchy sequences and thinks decimal expansion.
As to “integral domain” I think it’s a red herring here and really has little to do with Devlin’s rather silly and demeaning attack on so many teachers.
Chaitin’s Omega?
Sorry, I was thinking of computibility here. You’re right otherwise.
If you want to distill his argument into “natural extensions are not isomorphisms”, it doesn’t matter. If you want to think he’s being silly about that, fine. But that needs to be the target of counter-argument.