There's a lot in this brief question, and I would like to try to give a brief answer, so I'm going to pick and choose what I respond to (and others might choose different things). Here are the parts of this question I see:

- Is teaching multiplication as repeated addition problematic?
- Is the problem with teaching multiplication as repeated addition that it makes it difficult to differentiate the two operations (or see them as independent)?
- Is multiplication too hard to teach without using repeated addition?
- Is multiplication as repeated addition more intuitive?
- Is there research on teaching methods for multiplication that do not use repeated addition?

Not all these are questions you are asking directly, but they are implicit (such as when your friends assert that repeated addition is more intuitive -- I have problematized it).

We could also well ask "Is multiplication repeated addition?"

Because if we're going to wonder whether we should teach multiplication *as* repeated addition, we should probably at least consider whether it *is* repeated addition. If it *is* repeated addition, then this isn't a problem to teach it that way. If it *isn't* repeated addition, then why would we teach it as repeated addition? Now, repeated addition may be a strategy students use in certain situations. But that's different from teaching multiplication as repeated addition.

OK, I'm going to draw heavily from Simon and Blume (1994) because it's such an interesting paper for many reasons and it references other resources that address multiplicative reasoning, and this will allow me to be a little lazy. The article is actually about elementary teachers understanding of area as a product of linear measures.

In this paper, Simon and Blume (1994) on page 474 reference earlier works by Kaput and by Schwartz which points out that multiplicative reasoning can result in the production of intensive quantities (that is, a quantity that is not counted or measured directly and is invariant with the scale of the system). This has implications in understanding proportion later. We see intensive quantities in division. Miles traveled divided by hours elapsed produces an entirely new quantity (speed).

The paper suggests "intensive" vs. "extensive" quantities may help us understand not just whether a student is solving a multiplication problem, but how sophisticated their approach is. This idea comes from Thompson (1994) who observed students solving problems relating to speed without conceiving of speed as an extrinsic quantity, revealing something about the sophistication of their conceptual understanding.

All that is simply to point out that there is some complexity to how learners may think about multiplication, and it is worth paying attention to not only because it is part of how multiplication makes sense.

The property of multiplicative reasoning to produce a quantity that is different from the factors in the problem can be considered a referent-shifting aspect of multiplication. Simon and Blume explain:

Schwartz
argues
that
multiplication and division are "referent transforming" operations,
because
they
take two
quantities
with different referents as
input
and
output
a third
quantity
whose referent is different from either of
the first two
(in Example
1: number of cookies x number of dollars/cookie
= number of
dollars).
Schwartz and
Kaput
further
point
out that the notion
of
multiplication as
repeated
addition is
problematic
because addition is referent
preserving,
whereas
multiplication
is referent
transforming.
Repetitions
of addition therefore cannot
yield
the referent that is
appropriate for the
product
in a
multiplicative situation.

So, yes to 1 -- there is reason to believe that repeated addition is a problematic way to teach multiplication: it may obscure the referent-transforming aspect of multiplication. To number 2, you may want to investigate the idea of independence further, but clearly there are other reasons that repeated addition has a different meaning than multiplication. In some sense: yes; failing to make clear the meaning of multiplication is a lost opportunity to separate the operations from one another.

I'll address #4 in a limited way by saying that we don't necessarily want to reinforce student intuitions. So the question of whether it is more intuitive or not may be moot. Is it helpful to our students in achieving the ability to reason multiplicatively?

I will address #3 and #5 together by directing you to a couple of other resources, but also by giving you an example.

Kouba and Franklin (1995) give a brief overview of their view of the research on introducing students to multiplication and division. Their conclusion is that students need a varied conceptual basis for multiplication and division. They include an example of students using objects they can touch in the process of scalar multiplication (which they may accomplish by using repeated addition as one of their own strategies).

However, as a teacher, how can we help students conceptualize multiplication in a way that is consistent with Kaput's and Schwartz's observation that multiplication involves a referent transformation? I look to an example in *Mathematics for elementary teachers* (Beckmann, 2010).

Dr. Beckmann gives a number of different examples for modeling multiplication, but in one example she uses an array of soft drink cans to show how the idea of cans-in-groups and then number of groups can be put in a useful representational structure. She points out that the rows or the columns can be used as the groups. I would also add that this lets us see that this is not just repeated addition of cans; this representation really does show referent transformation: cans per row * rows = cans also cans per column * columns = cans. This is conceptually different from saying "what's three times five cans?" One obvious difference is that the numbers have meaning in the array model, when we talk about them as a number of groups, or a size of a group.

And, if you want to discuss repeated addition, this model allows us to show why, in this case, repeated addition gives us the correct answer for the multiplication problem.

Repeated addition can be something you do, but it can be separate from a conception of what multiplication is.

**In summary**

**There is reason to question the teaching of multiplication ***as* repeated addition on the basis of the other meanings and understandings of multiplication we want for our students. This is discussed in some of the research that highlight the differences between the reasoning that repeated addition produces and multiplicative reasoning.
**Students need a varied conceptual basis** to form an understanding of multiplication; examples of these can be found in the resources cited, along with representations that support them.
**Student use of repeated addition is one strategy**. I gave an example of how a representation could possibly be used to connect this strategy to another conceptual basis for multiplication.

**Cited:**

Beckmann, S. (2010). Mathematics for elementary teachers. New York: Pearson Addison-Wesley.

Kouba, V. L., & Franklin, K. (1995). **Research into Practice: Multiplication and Division: Sense Making and Meaning**. *Teaching Children Mathematics*, 1(9), 574–77.

Simon, M. A., & Blume, G. W. (1994). **Building and understanding multiplicative relationships: A study of prospective elementary teachers**. *Journal for Research in Mathematics Education*, 472–494.

Thompson, P. W. (1994). **The development of the concept of speed and its relationship to concepts of rate**. *The Development of Multiplicative Reasoning in the Learning of Mathematics*, 179–234.

2

You may be interested in Keith Devlin's articles on the subject in his Devlin's Angle column (e.g. http://www.maa.org/external_archive/devlin/devlin_06_08.html).

– J W – 2014-06-19T05:12:59.9031

Further Devlin's Angle columns (January 2004 - August 2011) can be found at https://www.maa.org/external_archive/devlin/devangle.html. More recent posts are at http://devlinsangle.blogspot.nl/.

– J W – 2014-06-19T05:28:38.663Not exactly the same identities as addition does, not always. In most spaces where you have both multiplication does not end up enjoying all the nice abelian group properties of addition. Furthermore, it works with number systems. How about when you try to teach students about matrix algebra? So matrix multiplication is repeated matrix addition? – Mark Fantini – 2014-06-19T07:34:03.440

And people learning it for the same time, using the concept that all multiplication is repeated addition, know that difference? – Mark Fantini – 2014-06-19T07:51:45.187

@LoopSpace As the comments aren't intended to be for lengthy discussions and I have no desire to engage in such, I leave with this remark: I'm interested in your opinion and possible research supporting it, but do post it as an answer, not as apparent attacks on comments. See my other comment at Burke's answer. Best wishes. – Mark Fantini – 2014-06-19T07:55:53.947

1You're right: I got wound up by the comments and answers. The issue stems from the lack of separation between multiplication of numbers and general products. When you relate "multiplication" and "repeated addition" I think you're talking about the former. Maybe you should be clearer in your question. – Loop Space – 2014-06-19T08:03:21.803

4You tag this as "secondary-education", and (if I'm not terribly mistaken) in that age range we got a (brief) introduction to Peano's axioms for arithmetic. There multiplication is

definedas repeated addition. – vonbrand – 2014-06-19T12:30:50.933@JW Devlin's https://www.maa.org/external_archive/devlin/devlin_01_11.html is another read on the topic. I am now convinced that the repeated adding is not ideal. The question remains, if this process isn't to be used even as a stepping stone, how is multiplication to be taught? If I am an 8 year old and remark that 3X4 is like adding 4 boxes of 3 identical items, how exactly should the teacher explain the error of my ways?

– JoeTaxpayer – 2014-06-19T18:46:47.1475@JoeTaxpayer They shouldn't. Devlin is wrong. Multiplication starts as repeated addition as vonbrand says. What is going wrong is the conflation of internal multiplication of integers and external scalar multiplication. What would be a good idea would be to separate these clearly in the students' minds. – Loop Space – 2014-06-19T19:38:50.357

@LoopSpace - I appreciate the support. But, given the strong opinions expressed both by Devlin and others here, I have an obligation to try to understand the other approach. So far, I see the strong objections, but no clear way to avoid the issue itself. – JoeTaxpayer – 2014-06-19T20:36:44.877

@BenjaminDickman I've changed the secondary education tag to primary. Sorry for the confusion. – Mark Fantini – 2014-06-19T21:03:06.340

3@JoeTaxpayer Devlin's article is interesting. He doesn't say what multiplication

is, he just says how he thinks of it. But how someone who's been through the entire system thinks of it is not necessarily how it ought to be taught. – Loop Space – 2014-06-20T06:46:43.1171Repeated addition and multiplication are not equivalent from a computational point of view. If I compute $11 \times 123$ by repeated addition I perform more than 3 times more operations than if I compute using a standard multiplication algorithm. This is why repeated addition fails in practice for multiplying moderately sized numbers (where what "moderate" means depends on one's experience). On the other hand, this failure is an opportunity for the teacher, because it exposes the need for something computationlly more effective than repeated addition. – Dan Fox – 2014-07-18T11:13:21.283