I've had this discussion with a couple of friends. I argued that teaching multiplication as repeated addition isn't a good idea because it doesn't help children differentiate between the two operations and see them as independent. My friends argued that it would otherwise be too hard and that multiplication as repeated addition yields a more intuitive approach.

Is there research on what, if it exists, is a good method for teaching multiplication and keeping it as an independent operation of addition?

• You may be interested in Keith Devlin's articles on the subject in his Devlin's Angle column (e.g. maa.org/external_archive/devlin/devlin_06_08.html).
– J W
Jun 19 '14 at 5:12
• Further Devlin's Angle columns (January 2004 - August 2011) can be found at maa.org/external_archive/devlin/devangle.html. More recent posts are at devlinsangle.blogspot.nl.
– J W
Jun 19 '14 at 5:28
• You 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 defined as repeated addition. Jun 19 '14 at 12:30
• @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. Jun 19 '14 at 19:38
• @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. Jun 20 '14 at 6:46

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:

1. Is teaching multiplication as repeated addition problematic?
2. Is the problem with teaching multiplication as repeated addition that it makes it difficult to differentiate the two operations (or see them as independent)?
3. Is multiplication too hard to teach without using repeated addition?
4. Is multiplication as repeated addition more intuitive?
5. 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).

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

1. 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.
2. 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.
3. 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.

• This is a great answer. I agree that units (referents) make the best characteristic to differentiate between multiplication and repeated addition (although, it's only a manifestation, not the reason why they are distinct). However, is it known at what age children are able to grasp that $1\mathrm{m}$ is not the same as $1\mathrm{m}^2$? I remember from my school days that there where multiple students struggling with why you cannot add these two. Perhaps repeated addition is appropriate, and transformation to fractions is similar to transformation from factorial to the $\Gamma$ function. Jun 19 '14 at 9:19
• Multiplication by the number one is an identity operation. Multiplication by any a whole number is repeated addition based upon the induction that if $n$ is a whole number greater than one, $nx = nx+0 = nx-(x-x) = nx-1x+1x = (n-1)x+x$. Multiplication by things other than whole numbers (including dimensioned quantities) is a different operation but has a lot in common with whole-number multiplication. Jun 19 '14 at 18:57
• @JPBurke: I was responding to what's written above, not the material linked. Neither the material above, nor the linked question, made a distinction between whole number multiplication and any other sort, but I would suggest that the distinction is fundamental. Although some forms of multiplication have nothing to do with repeated addition, I would suggest that multiplying by a whole number anything that can be added is repeated addition. That's not just what multiplication means--it's what whole numbers mean. Two is the entity, which, when multiplied by any x that can be added... Jun 20 '14 at 1:33
• @JPBurke: I couldn't read the second article, which seems to discuss the points at issue, without spending $18. I think it may be worthwhile from a teaching perspective to note at some point that multiplying something by something other than a whole number may yield something whose type doesn't match either operand [e.g. if one multiplies an ohm by an ampere one one gets a volt; if one multiplies three whole jellybeans by 0.5, the result will not consist of whole jellybeans] but I can't see that as making sense until students understand things besides whole numbers. Jun 20 '14 at 14:26 • @supercat If going to make assertions about what is or isn't a remedy, I would like you to cite some research. Jun 20 '14 at 21:34 First, an answer to your title question: Is it advisable to avoid teaching “multiplication as repeated addition”? No, one should not avoid teaching multiplication in this way. There are other ways to teach it, carrying their own advantages and disadvantages, but such is the way that multiplication is defined for the natural numbers, and this (also building off of number chanting by$2$s and$3$s that occur even in pre-K) indicates that it is one of the ways in which multiplication should be taught. However, it is often advisable to discuss concepts in multiple ways, and I do not find multiplication to be an exception. Thus, being that this is a site for Mathematics Educators, and given that I have thought a fair bit about multiplication tables (see, e.g., here), I will sketch briefly an alternative way of teaching multiplication. Let us begin with a concrete object: a$10 \times 10$array of squares. We now proceed as follows. Pick a square. Designate this square as the bottom-right corner of a rectangle whose top-left corner is the top-left of our entire array. Count the number of discrete squares that make up the rectangle, and write the total inside of the chosen square. For example, if we choose the square that is$4$to the right and$6$down, then we form a rectangle of squares with$4$total columns and$6$total rows; counting up all of the squares in this rectangle, we find a total of$24$, and so we write$24$in that particular square. Do this for all of the individual squares. In our$10 \times 10$example, that will be a total of$100$individual squares, and, at the end, we will have filled out what many will recognize as the$10 \times 10$times table. (Side problem: For each pair$i, j \leq 10$, consider the total number of$i \times j$rectangles that can be formed using the individual squares in our$10 \times 10$array; demonstrate that a "times table" can be formed in this way, as well. For example: There is only$1$way to form a$10 \times 10$rectangle, i.e., the entire array; write$1$in the top-left corner. At the other end of the spectrum, there are$100$different$1 \times 1$rectangles possible, i.e., each individual square; write$100$in the bottom-right corner. The$i \times i$rectangles will fill out the NW/SE main diagonal, whereas the$i \times j$and$j \times i$for$i \neq j$rectangles come in pairs, just as, e.g.,$4 \times 6$and$6 \times 4$appear symmetrically about the main diagonal.) What are some of the pedagogical advantages to this approach? It is easy to get into: Draw an array of squares (perhaps a$5 \times 5$one for younger children) and just start filling them in by drawing the corresponding rectangles described above. Given the tedious nature of filling out so many squares, it encourages learners to look for short-cuts, possibly with scaffolding from the instructor. "Counting by$3$s" or noticing that the$i \times j$and$j \times i$rectangles have the same number of squares can both be incorporated. The former allows for the exploration of multiplication as repeated addition, though it is not explicitly defined as such; the latter can be used to demonstrate the commutative property of multiplication. Moreover, if we are to take the individual squares as "unit squares," we are essentially using an area model for multiplication. When we move on to, say, positive rational numbers, our model extends naturally to discuss what, e.g.,$1/2 \times 1/3$is. There are also natural ways to interpret the distributive law, which I leave as an exercise to the reader. In fact, I leave other methods of exploring this model for multiplication to the reader, too, as my answer length has grown somewhat unwieldy. As a final remark: Though the concern of the initial question stems from a desire to generalize to non-integers (non-naturals, even), I would like to post a cautionary note about not being too hasty to leave the standard times table. The link I posted earlier on in this post gives a number of examples of nonstandard problems that can be asked about the$10 \times 10$table, which I think has been viewed quite unfairly, historically speaking, as epitomizing rote learning, whereas it can serve instead as a fecund area for mathematical exploration. • I think you and I (and possibly others as well) are choosing to interpret "teaching multiplication as repeated addition" differently, with some consequences. I note the "as" and interpret it to mean that the two things are one and the same. My response would be different to someone who asked me whether students should use repeated addition (yes). I like your response. I like that you note the site is for mathematics educators, as well, because this detail seems to get lost as soon as strong opinions about mathematical content arise. Jun 20 '14 at 13:38 • When you write "it is often advisable to discuss concepts in multiple ways" I envision that we are all nodding with you. However, the truth is we should question it. It is not just good advice because we agree. In fact, it is not just good advice (as many people might assume) because of how sometimes multiple approaches are needed before multiplication "clicks". In fact, it is good advice because there are many human conceptions related to the mathematical concept of multiplication, and students need opportunities to grasp those concepts and form a robust understanding. I'm know you're aware. Jun 20 '14 at 13:43 • ... but I am not sure others are aware that these are more than opinions. Jun 20 '14 at 13:47 • Four years later... a downvote? But why?! Aug 10 '18 at 19:05 • To compensate, I hadn't upvoted yet... :) Aug 10 '18 at 21:38 We all want students to know how to compute and also to have the conceptual understanding of why their computational method works. I suggest that we can teach repeated addition as a computational strategy when multiplying with whole numbers. But there is more to the question can computation. JPBurke breaks the question into multiple questions, and I won’t repeat the ground he covers. I’ll address a different aspect: “What do students miss when multiplication is taught as repeated addition?” When students are taught multiplication as repeated addition, they are only being taught how to compute the answer. They are not taught the skills that are required for multiplicative reasoning, so they have great difficulty with ratio and proportion and rational numbers. Multiplicative reasoning requires a different way of thinking about units than additive reasoning. With addition, there is only one unit. With multiplication there are two units, and more importantly, a ratio relationship between the two units. This relationship can be thought of as a many-to-one correspondence, for example, 12 inches per foot, or$3 for every pound. Nunes and Bryant emphasize this ratio relationship as a basic element of multiplication in Children Doing Mathematics. Michael Goldenberg goes into more detail on the Nunes and Bryant arguments on his blog.(I am allowed only two links, link below) Park and Nunes did a study testing repeated addition versus correspondence as a basis for understanding multiplication. In multiplicative reasoning tasks, the students who were taught multiplication through correspondence significantly outperformed the children taught multiplication through repeated addition.

There is another important difference in how students think about units with multiplication that Nunes and Bryant do not address. With additive reasoning, since there is only one unit, there is little attention paid to what is one whole. It is one, or what we could call the standard unit 1. With multiplicative reasoning, what is one whole or 100% can change with different elements in the problem situation. Davydov defines multiplication as counting with an intermediate, non-standard unit. For example, 3 x 7 can be considered as 3 units of 7, where seven is the new one whole. The seven is considered an intermediate unit because we want the count in standard units: 3 sevens is 21 ones. Thinking of units this way helps students be more flexible in thinking about what is one whole, and multiplicative reasoning requires that flexibility and more; it often requires actively looking for what is the whole or 100%. One last example, comparing 20 to 4 multiplicatively, 4 is the whole and 20 is 5 wholes or 500% of 4. This is essentially the quotative meaning of division, in which what is one whole can change with every division expression.

Multiplicative reasoning requires thinking of numbers not as individual fixed entities, but in relation to other numbers—that is, to see a correspondence of one number to another, or to see a number relative to one whole that is not the standard unit 1, but relative to a non-standard reference unit. When we teach multiplication, we should help students begin to think in terms of different units (scales) and the relationship between numbers or scales. Joan Moss says,

We know from extensive research that many people—adults, students, even teachers—find the rational-number system to be very difficult. Introduced in early elementary school, this number system requires that students reformulate their concept of number in a major way. They must go beyond whole-number ideas, in which a number expresses a fixed quantity, to understand numbers that are expressed in relationship to other numbers. These new proportional relationships are grounded in multiplicative reasoning that is quite different from the additive reasoning that characterizes whole numbers. (p.310)

When students are taught multiplication, we should begin to build the new knowledge that they need for multiplicative reasoning, and not just continue to reinforce additive reasoning.

Cited

Davydov, V.V. (1992). The psychological analysis of multiplication procedures. Focus on Learning Problems in Mathematics, vol 14, pp. 3-67.

Goldenberg, M.P. (13 March 2010). Tereza Nunes and Peter Bryant dole out the multiplicative harshness, Retrieved from Rational Mathematics Education blog: http://rationalmathed.blogspot.com/2010/03/terezinha-nunes-and-peter-bryant-dole.html

Moss, J. (2005). Pipes, tubes, and beakers: New Approaches to teaching the rational number system, in How Students Learn: History, Mathematics, and Science in the Classroom. Donovan, M.S. & Bransford, J.D. (eds), Washington, DC: National Academy Press. http://www.nap.edu/catalog/10126.html

Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Malden, MA: Blackwell Publishers.

Park, J-H & Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development, 16, 763-773.

Why do you feel the description of multiplying as repeated addition is problematic? Hopefully it's just a brief stage, but an important one. I'd even suggest that this is exactly the method we all use when multiplying larger numbers in our heads. I don't know 21 times 21, but can easily multiply 21 times 20, and then just add another 21.

In response to JPBurke's comment, I suggest a viewing of Vi Hart's How I feel about logarithms, in which she starts off suggesting that for integer math, even adding 3 and 5 is really a series of repeated plus ones. She then goes on to support my view that multiplication is adding, at least for integers. Respectfully, I understand comments to be brief requests for clarification or short remarks. Unfortunately, they have a minute long edit time, and I intended to return to add the link once I found it.

• Because it creates trouble with nonintegers. Jun 19 '14 at 2:12
• Is this an answer, or is it more appropriately a comment? The simplest, direct answer to "Why do you feel the description of multiplying as repeated addition is problematic?" would be that multiplication is not repeated addition. Repeated addition is a process that can result in numerical answers equal to multiplication in some cases, but that doesn't make it the same thing. That basis alone (i.e. that it is incorrect) should make us question whether we ought to teach that they are the same thing. No? Jun 19 '14 at 4:35
• @Fantini maybe, but it is the definition for integers... Jun 19 '14 at 8:09
• @vonbrand It depends, you can define it as a size of the Cartesian product. Jun 19 '14 at 8:55
• @dtldarek Except that to realise it as a number then you have to order the cartesian product and lexicographical ordering gives you multiplication as repeated addition. Jun 19 '14 at 19:39

If we're talking about young children (elementary school) being taught multiplication for the first time, there is no alternative to presenting it as repeated addition. Let's assume they have addition down. They're just not going to be able to grok any other concept when just being introduced to it, given their limited toolkit.

Once they have down the concept of "2 x 3 is 2 + 2 + 2", and can generalize it to "m x n is m + m + ..." (for specific integers), you can start expanding to multiplying multidigit integers (first, rote digit-by-digit and add subresults, and then the concept of shift-multiply digit-add). At some point they'll be ready to handle negative numbers and decimal numbers and even fractions, but not right off the bat.

• "there is no alternative to presenting it as repeated addition" This is not actually true. In Russia, using curricula associated with the ideas of Vygotsky and Davidov, children's first exposure to multiplication is based on measurement. Clearly, area is length (any length, not just whole numbers) times width (any width). There is no sense of repeated addition in this. I don't know enough to be sure this is better for students, but it certainly may be. Jun 20 '14 at 4:43
• @SueVanHattum Why is it that area = length x width? If you analyse why that statement is true, it's very hard to avoid counting squares and that brings us back to counting, and thence to multiplication-is-repeated-addition. Jun 20 '14 at 6:48
• @SueVanHattum - You are correct. On the subject of what is better for students, whatever we do to get students to understand multiplicative relationships rather than relying on repeated addition is better. You see the consequences later when talking about slope. For some students and even teachers, slope is "this many" rise for "this many" run, rather than seeing the continuous equivalence class of ratios that makes up a rate. I would also conjecture it is related to people having difficulty determining when proportional vs. linear vs. additive reasoning applies to given contexts. Jun 20 '14 at 13:21
• @JPBurke: Discrete numbers are different from real numbers, and discrete multiplication is thus different from real multiplication. If a truck containing ten packages arrives every hour starting at midnight, how many packages will have arrived by 7:42am? Even though that time is 7.7 hours after midnight, there will have been eight deliveries of ten packages, with the last having arrived at 7:00am. Jun 24 '14 at 18:51

[note: links below in reference section]

Let’s put this issue in the larger context of how we teach math. Phil Daro says Japanese teachers have the goal of teaching math concepts while American teachers have the goal of teaching how to compute answers. But all teachers will say that their goal is to teach students to understand math concepts as well as to compute correct answers. The evidence, such as the TIMSS video’s that Phil Daro reviewed, says that teachers believe they are teaching math concepts but are only teaching how to compute answers. They don’t in practice distinguish between conceptual knowledge and procedural knowledge.

Why do we confuse a computational procedure with the concept? Why are we still stuck on the idea that if students can compute the answer to a problem then they understand the concepts involved? I can think of two reasons. One, that is how math has been taught. We think math involves memorizing computational procedures and computing the answers to problems, and we don’t understand much else. That is why we can multiply but we can’t explain what multiplication is. We don’t think of it as anything other than how to compute the answer.

Second, carrying out procedures does involve concepts--procedural concepts, not concepts of the underlying mathematics. So the term “conceptual understanding” is ambiguous and easily interpreted as understanding how to use a computational procedure correctly. This is what Liping Ma describes in her book Knowing and Teaching Elementary Mathematics on pages 36-37. Some of the teachers in her study had a strong belief in “teaching mathematics for understanding.” In their explanations and teaching practices, however, “no conceptual learning was evidenced at all.” What the teachers wanted the students to “understand” was the correct steps in carrying out the computational procedure.

The larger issue, then, is to make a clear distinction between what Richard Skemp calls relational understanding and instrumental understanding. Both involve “concepts” and “understanding,” but mean different things. The distinction is an issue for every topic in math, not just multiplication. Phil Daro says if there was one thing he could change in our educational system, it would be to go beyond answer getting and teach relational understanding. For this to happen, research by James Spillane says that teachers need to see a different way of teaching. They need to make their teaching public and collaborate. Elizabeth Green’s book Building a Better Teacher has many examples of the power of teacher collaboration.

Cited

Green, E. (2014) Building a better teacher: How teaching works (and how to teach it to everyone). New York: W. W. Norton & Company.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum.

Skemp, R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77:20-26. Also at http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf

Spillane, J. P. (1999). External Reform Initiatives and Teachers' Efforts To Reconstruct Their Practice: The Mediating Role of Teachers' Zones of Enactment. Journal of Curriculum Studies, 31(2): 143-75.

Teach it whatever way the students can understand the concept you are getting at right now, and be explicit that it is a model of multiplication -- that multiplication itself is this other thing they will learn more about later.

Models change as new ideas are encountered. It is a normal part of the process of teaching: to provide a simple model sufficient for a trivial example case initially, and then blast that model and replace it with a more detailed one as the student advances. Its the only way we are capable of learning anything complex anyway.

If you are teaching young children, multiplication usually involves only positive integers. In this case thinking of multiplication as iterative counting is sufficient. Obviously, this model falls apart entirely once you get into more interesting cases of multiplication and functional composition -- but that's not the point. The point is to teach at the level of abstraction that is appropriate to the student's level and the problem at hand.

If your students are bright and analytically mature you can teach them about abstraction itself and how this particular model of multiplication is a necessary simplification. But do not, for example, dive into foundational mathematics with 2nd graders in an effort to get them to grasp the principle of abstract reduction. After all, that defeats the whole purpose of abstract reduction!

Edit; observation:

After reading some more of the comments and (long) posts on this topic, it is clear to me that preliminary mathematics educators and electrical engineering / comp-sci educators do not talk much. There are strong parallels between teaching a first-year engineering student something hopelessly complex by way of successively accurate models of the same thing and teaching primary school students something hopelessly complex by the same method. Peano arithmetic (where addition can be thought of as successive "pile swapping by ones") provides a foundation for deeper concepts of addition of things that aren't unitary in the same way that teaching multiplication by "successive adding" provides a foundation for deeper concepts of multiplication of entities higher up on the numerical type order. Fast forward and you see the same thing in, for example, SICP where thinking of a language interpreter as a direct implementation of Lambda calculus provides a foundation for thinking of machines in a stateful (and therefore explosively complex) way.

I write this observation because while I see engineers embracing the concept of teaching by abstraction, I see basic math educators arguing over whether its a good idea or not to do this. I find these to be remarkably parallel cases with strikingly opposite interpretations.

Second edit

A talented member of the Erlang community, Anthony Ramine, linked http://worrydream.com/AlligatorEggs/ , a light introduction to the untyped lambda calculus today. That led me to http://worrydream.com/#!/SomeThoughtsOnTeaching . I think both are highly relevant to the discussion, because the issue here is not multiplication, it is models of pedagogy and their underlying motivations.

• While I agree about the lack of communication, first-year engineering students (ought to) have a very different maturity level than primary school students. Jun 20 '14 at 14:49
• I would define the multiplication of anything for which addition is defined and associative, by a whole number, as repeated application of that thing's addition operator. Many things may be meaningfully multiplied by things other than whole numbers, and such multiplication often does not represent repeated addition, but for any x that defines an associative addition operator, 2x = x+x [using "2" to represent the result of adding the Universal Multiplicative Identity to itself]. Can you offer any counterexamples? Jun 20 '14 at 16:27
• @supercat that is exactly how multiplication by an integer is defined in rings Jun 21 '14 at 0:57
• One example of how an idea of "levels" contributed to a model of human mathematical thinking comes from the van Hiele levels. This is a venerable (1959) example (a classic!) which should not be taken as current theory, but shows that abstraction (in a theory about levels) has been part of math education thought for quite a while. The intro puts it in perspective. Van Hiele, P. M. (1984). A child’s thought and geometry. English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele. Washington DC: NSF. geometryandmeasurement.pbworks.com/f/VanHiele.pdf Jun 21 '14 at 1:11
• @vonbrand: Not just rings--it works for groups; if one multiplies only by counting numbers (positive integers), it works for monoids; for whole numbers, it even works for semigroups with no additive identity element. Jun 21 '14 at 2:07

I try to avoid (and advocate that teachers try to avoid) lying to students. Stating explicitly that "Multiplication IS repeated addition" (MIRA) is a lie and an easily avoided one. First, the area model can be presented with sufficiently small integers that counting isn't necessary - subitizing (instant recognition of "how many" without counting - suffices. For most humans that's somewhere in the 5- 9 range, making areas up to about 8 feasible candidates.

When getting into the relationship between counting and addition, do we say "addition is 'fast counting' or 'repeated counting'? Probably not. We can think about addition that way, but it's a lot more and different, even for integers, which is evident to most people when the addends are at all "large." And while "repeated addition" may get us the correct product for a given multiplication problem, it simply isn't true that multiplication is the same thing. It's a different operation in a fundamentally different way than subtraction is a different operation from addition. One of many reasons multiplication is qualitatively different from addition is suggested by the fact that we need to worry about units and/or common denominators in order to add; no such things need concern us with multiplication. If that doesn't puzzle you, you likely shouldn't teach mathematics at any level.

To avoid lying to kids, it's reasonable to state things like "You can think about multiplication as repeated addition, but it's both different from and more than that." Or perhaps, "One way to think about multiplication is as repeated addition, but that only gives part of the story." I'm sure we can come up with other reasonable and TRUE things to say. But why would we want to tell children, "addition always makes things larger"? It's simply not true. Same for the other operations. You simply cannot truthfully claim that any of the four basic arithmetic operations ALWAYS yields a larger or smaller result than the initial input. Period. So why claim it? So doing only serves to confuse many kids down the line. More modest claims can be made honestly. It's also a lie to aver that "You can't subtract a larger number from a smaller one." It's true that you can't do that with strictly non-negative numbers. But even some young elementary students are aware of the existence of negative numbers and some actually have a degree of understanding of and/or facility with them. Why lie? It's legitimate to tell someone that "With the numbers we're currently using, there is no answer to 7 - 9." But doing that already suggests that there might be numbers within which the answer to that problem exists, and kids can begin thinking about what that could be on their own.

Being doctrinaire on these issues, insisting upon a narrow and false claim because "kids can't handle the truth" or "that's the way I learned" or "that's how it's always been done" simply doesn't make sense. We are supposed to be a species capable of growth and change, of learning from past mistakes, etc., but when it comes to teaching elementary mathematics, at least in this country, we resort to almost blind religious defenses of outright lies. Frankly, that's disappointing and disturbing.

• In what way is the notion that 2x is x+x a "lie"? How else would one define multiplication by 2? If one has a row of five apples and multiplies by three, one may nicely arrange the apples in a 5x3 grid, but that entails starting with a group of five apples and then adding two more groups of five apples below it. Only multiplication by whole numbers represents repeated addition, but I would dispute any claim that multiplication by whole numbers represents anything other than repeated addition? How else would one define it? Jun 20 '14 at 16:18
• I agree with @supercat that integer multiplication is just repeat addition (ask Peano). Sure, multiplication also comes up between rationals and reals, where your area explanation is required, no question about it. Jun 20 '14 at 17:34
• @vonbrand: I think the area explanation can be tied into the repeated-addition explanation very nicely by showing what happens if you start with a row of squares, add another identical row below it, and then another row, etc. BTW, I said "whole number" multiplication rather than "integer", since some algebraic structures are closed under addition, but include elements without additive inverses. Jun 20 '14 at 18:11
• repeated addition plus associativity plus commutative is more than enough to multiply fractions, as a matter of fact, so that area is not "required", concatenation and division of segments is ok, if one wants a geometrical model on the line. Jan 4 '15 at 17:34

I think that there is some merit to teaching properties of the integers before extending them to rational numbers. This well help students understand important properties such as the remainders that come with whole-number division, prime decomposition, modular arithmetic, etc. More generally, it will help students understand the ring structure of the integers. Moreover, it may help exponentiation make more sense as repeated multiplication. In these contexts, multiplication is repeated addition. See, for example, the work done by Stephen Campbell and Rina Zazkis.

This isn't to say that students shouldn't also learn rational number operations, just as we teach students to extend division to the rational numbers.

• Hello MathTeacher. You should expand this answer. Dec 27 '14 at 2:16
• @MarkFantini See, e.g., the latter half of my answer to MESE 6031 for a recent paper on related topics (specifically: it has citations to the work of Campbell, Zazkis, and others). Dec 27 '14 at 11:21
1. I would probably introduce multiplication as area of a rectangle because it extends easily to mixed numbers and decimals.
2. But multiplication is also an additive power and should be dealt with as such, perhaps initially with a different symbol. Also, eventually, one can/should introduce some kind of logarithmic idea, that is the correspondence between $3a = a + a + a$ and the multiplicative power $a^{3} = a\times a\times a$
3. Multiplication is also an operation between a vector space (e.g. basket) and its dual (price list). Initially $3\, apples\, \times 5\, \frac{cents}{apple} = 15\, cents$ is something that can be dealt with very early.
4. But in fact, $<3\, apples,\,2\, bananas,\, 5\, carrots>\otimes<5\, \frac{cents}{apple},\,7\, \frac{cents}{banana},\,2\, \frac{cents}{carrot}>$ is just as simple.

Regards --schremmer

There are numbers and there are magnitudes (3 vs. 3cm). Multiplication of numbers is repeated addition (i.e. 3x5 = 5+5+5), multiplication of magnitudes is not (i. e. 3N x 5cm is not an addition). The problem is that standard mathematical education does not explain what are magnitudes and how to calculate with them (although it is obvious, at least to physicists). Math education pretends that there is no calculation with magnitudes and instead of mathematically rigorous (400 m / 50 sec) / 10 sec = 8 m/sec*2 use mathematically incorrect (400 / 50 ) / 10 = 8 m/sec*2 or non-explanatory (400 / 50 ) / 10 = 8. This is just ignorance. C.f.

https://www.dropbox.com/home/zs-mm-share/web_dostupni_seminari_i_predavanja/fizika?preview=Magnitudes+dbk.docx

So, we should appreciate number/magnitude distinction and put magnitudes and calculations with them back into math curricula (where they belong).

No, the highway of multiplication is not repeated addition, but its on-ramp is, just as the highway of the product of topological spaces is not simply their Cartesian product, but its on-ramp consists of studying the Cartesian product of finitely-many topological spaces. These can be considered to be caterpillar-versus-butterfly situations, or as logarithmic (inverted 'L') situations, the rapid-rise portion corresponding to the initial state, and the slow-rise portion corresponding to the final state. Artifact-evolution follows the same inverted 'L' pattern. For example, the changeover from the biplane to the monoplane follows this pattern.

So, no, don't avoid teaching multiplication as repeated addition, but point out, when the time is right, that multiplication eventually takes on a life of its own.