Last time we looked at the roles of multiplier and multiplicand from several perspectives. This time, I want to focus on one extended discussion about how children should be taught to think of multiplication.
Times, or multiplied by?
Cindy asked this in 2002:
Defining Multiplication I just read an article in Phi Delta Kappan (Feb. 2001) by Deirdre Dempsey and John Marshall titled "Dear Verity: Why Are All the Dictionaries Wrong?" They state that Euclid and some dictionaries define multiplication equations like 3x4 to mean "the number 3 four times." I was taught, I teach, and math textbooks say that 3x4 means 3 lots/groups of 4. An array would look like **** **** **** Because multiplication is commutative, it may not matter symbolically; however, as the authors mentioned in the article, taking 4 pills a day for 21 days is a lot different from taking 21 pills a day for 4 days. If I want my students to know what multiplication "is," what is the correct explanation?
At the time, I had no access to this magazine; later, after I started teaching at a community college, this question was brought up again and I was able to access it online through the library. I still can’t point you to an online source so anyone can read it. At any rate, at the time I could only answer Cindy’s specific questions, about definitions and about taking pills. I started by focusing on what “times” means grammatically (which, it turns out, the article agreed with, taking the use of the word “times” as problematic):
Your question is really not a math question, but a linguistic question, even though it involves symbols. Since, as you recognize, 3*4 and 4*3 have the same value, there is no mathematical value in the distinction. You are only asking about the underlying image when we say "three times four" in English. (It may have been different in Euclid's Greek.) And I submit that it can be reasonably interpreted either way. My first thought when analyzing the phrase is that originally '3 times' meant to repeat what follows three times; that is, it would mean the same as '4, taken 3 times'. So 3 times 4 means 4+4+4. But that seems awkward, considering the way we tend to say the phrase now. We might read through a calculation, saying "3, times 4 is 12, plus 2 is 14, ..." where each operation acts on the first number. We're really saying "3 multiplied by 4." Taken that way, we start with 3 and multiply it by 4, meaning that we repeat it 4 times. So 3*4 = 3+3+3+3. The difference here is entirely in the grammar: is '3 times' an adjective phrase modifying '4', or is 'times' a preposition equivalent to 'multiplied by', and 'times 4' a modifier of '3'?
I am saying here that we can read \(3 × 4\) either as “3 times 4” (3 groups) or as “3 multiplied by 4” (4 groups), which present different images. The expression as written does not imply anything about which number is the number of groups. (The authors of the article felt that it is better pedagogically to use the “multiplied by” form.)
But what was that about Euclid’s definition? I later found it in Book 7, Definition 15:
A number is said to multiply a number when the latter is added as many times as there are units in the former.
Of course, the Greeks had no symbolic notation for multiplication, so that says nothing about the meaning of \(3 × 4\); they wrote their math out in words, like “A multiplied by B makes C“, leaving no ambiguity about which is the multiplier. The article assumes that “\(×\)” is read as “multiplied by”, so that the second number is the multiplier. But that begs the question! If they had used our notation, might they not have seen \(3 × 4\) as 3 multiplying 4?
Anyway, I continued by considering the issue of pedagogy: Is there, and should there be, only one model of a given multiplication?
When we introduce children to multiplication, it's reasonable to start with one specific meaning, just so they have a concrete image to start with. But we can almost immediately point out that
****
****
****
can be seen equally well as 3 rows of 4, or as 4 columns of 3. Once you've drawn the figure, or once you've written 3*4, no one can really tell which interpretation you meant. The distinction has been abstracted out of the problem.
And that is a good thing, not bad: in mathematics the ability to work abstractly and forget what the concrete problem was is a major advantage, because we can ignore details that don't affect the result, and rearrange the work to make it easier. If I took 4 pills a day for 21 days, and now I want ONLY to know how many pills I took, I can call it 4*21 (I just put the numbers in the order I saw them, without having to think about which is the multiplicand) and re-model the problem as 4 sets of 21, allowing me to add 21+21+21+21, which is easier than adding 21 4's. I can switch freely among different models, because I know that the outcome is equivalent. So I never bother to define whether 4*21 means 4 groups of 21 or 21 groups of 4, because that distinction would only be a hindrance. And if I were doing a calculation for which it made a difference whether I took 4 or 21 pills a day, such as finding the concentration of medication in my blood, I wouldn't be multiplying, because that would be the wrong abstraction!
So "what is multiplication?" It is a commutative operation that can be modeled in two symmetrical ways as repeated addition (when applied to whole numbers).
So, as adults, we don’t want to identify either the first or the second number as necessarily the multiplier, because that is a distinction that is meaningful only in the application, not in the calculation. But for children, it makes sense to start with a concrete model taking it one specific way, as long as we soon help them to see that two different models have the same result, so that they have control over how they do the calculation. (Even though children are described as concrete thinkers, they seem able to handle this “abstract” idea with no trouble, as we have often seen when kids ask about it.)
Doctor Tom also wrote an answer to the question of what multiplication “is”, by stating a mathematical definition of multiplication as an abstract operation, akin to what I have recently mentioned about axioms of arithmetic. It is probably not useful for Cindy’s purposes, but is interesting.
Interacting with the authors
Now, three years later one of the authors of the article cited, John Marshall, wrote to us, chiding me for commenting without having read the article (which, of course, I couldn’t do, so that my comments weren’t about the article at all), and defending his claims. Another three years later, the other author, Deirdre Dempsey, also discovered what I had written and wrote me; by that time I had been able to read the article, and had a better idea what it said.
One comment from co-author Marshall was this:
What I think you need to do is to explain to Cindy, and myself, just what part of the Webster’s 3rd Edition College Dictionary definition, for example, indicates there is a choice. Where in
“Multiplication: the process of finding the number or quantity (product) obtained by repeated additions of a specified number or quantity (multiplicand) a specified number of times (multiplier); symbolized in various ways (ex. 3×4=12 or 3·4=12, which means 3+3+3+3=12, to add the number three together four times).”,
does it make your point? It seems pretty clear to me. In 3×4=12 the 3 is the multiplicand and the 4 the multiplier. Just because the OPERATION of multiplication is commutative it doesn’t follow that the MEANING of multiplication is. AND IT IS MEANINGS WE ARE TALKING ABOUT.
Unfortunately for his argument, it is easy to find other dictionaries that say the opposite; for example, here is Webster’s Third New International:
Multiplication: A mathematical operation commonly indicated by ab, a·b, or a×b and having various significances according to the type of numbers involved, the simplest being in the case of positive integers where the process is that of repeating b as many times as there are units in a, or vice versa. [my bold]
Clearly they agree with me! (And they correctly note that for numbers other than positive integers, a different definition is needed than “repeated addition”; they are more mathematically astute.) Of course, dictionaries are not the final arbiter of mathematical meaning, as Marshall ought to know. The article quoted other dictionaries that left it properly ambiguous, and just rejected them because of an assumption that no definition can be ambiguous. (Mathematicians, or math educators, may not be the best lexicographers; the latter know all about ambiguity, while the former avoid it.)
He also explained the pill example:
You don’t seem to understand either that, in the pills issue that Cindy quotes, the problem is NOT asking for the total number of pills, but rather how would the pharmacist, or patient, interpret a prescription that used 4×21. It is about the meaning of 4×21 not the product.
When Deirdre Dempsey wrote after I had read the article, I said the following about that point. It turns out that the whole article was centered around students being asked to think of a situation represented by \(6 × 3\); the authors claimed this should be seen only as 3 groups of 6, and not as 6 groups of 3, though most people think it is the latter. The pill example claims that \(4 × 21\) specifically means “21 groups of 4”, and so could be used in a prescription to tell a patient what to take, and not just to calculate the total number of pills.
It seems to me that you have confused the idea of being able to represent a concrete problem as an arithmetic operation (which is an important indication of understanding) with the inverse process, being able to turn an operation into a single kind of concrete problem. The latter is not a valid problem to assign, unless you accept any situation that will yield the same operation. The fact is that an operation like 4×21 does not mean either 4 pills a day for 21 days OR 4 days taking 21 pills a day.
The specific point I dealt with in my original response was the implication that a prescription could be written saying to take 4×21 pills. I certainly hope no one would ever do that, because that phrase simply does not MEAN to take 4 pills a day for 21 days. The phrase “4 times 21” is not a description of a regimen, but an arithmetic operation whose meaning is “84”. All the dictionaries agree on that!
But the main point of the article, as I see it, was that children think concretely, and need to be taught concretely; they are not ready for abstraction. I concluded my answer with this:
Now, there is one point where I agree with you. Replacing the word “times” with “multiplied by” when multiplication is first introduced would be a good idea; it allows you to talk concretely about multiplication with a clear physical model, and it also avoids the unfortunate tendency of so many students to turn “times” into a verb (“I timesed 2 by 3 and got 6”). As long as you then move on quickly to talk about other models of multiplication and to emphasize the commutative property, I’m happy with that change.
The best reason for preferring “multiplied by”, I think, is that it fits into the model of other operations, where we start with a number and operate on it: \(a + b\) means “a increased by b“, \(a – b\) means “a decreased by b“, \(a \div b\) means “a divided by b“, so \(a \times b\) should mean “a multiplied by b“. That consistency is perfectly reasonable, especially in initial teaching. It just happens that it isn’t followed consistently in real life, and is not important in practice. To teach children that only one model is valid is incompatible with everyday usage, and leads to confused questions from students and their parents.
The problem with teaching only one way
Long after those interactions, in 2014, we received one of many examples of this happening in a classroom, which I quoted within my answer to yet another such question:
Defining Multiplication, Redux My 2nd grader failed her math quiz today because the teacher insists that there is an order as to how a multiplication sentence should be written. Here's an example: ## ## ## Addition Sentence: 2 + 2 + 2 = 6 Multiplication Sentence: 3 x 2 = 6 The teacher said that 2 x 3 = 6 would be wrong. Another example from the quiz shows a numbered scale with asterisks that represent jumps: <-0*--1---2---3*--4---5---6*--7---8---9*--10---11---12*--13---14---15*-> Multiplication Sentence: 5 x 3 = 15 The math teacher said that this is the only way that can be written, and that writing it as 3 x 5 = 15 is wrong. I have already spoken to the teacher and explained that these can be interpreted in two ways, but she insists that this is how 2nd graders should be taught! Unless I can prove her wrong with a written source of some kind, she will not consider my daughter's answers. Please help.
One of the things in education that trouble me most is when something is taught in a way that confuses the smart kids who really understand (and maybe have knowledge beyond what is being taught), and even costs them points. Intelligence should never be penalized. Nor should teachers take a position that parents can see makes no sense. But we get questions like this repeatedly.
In my answer, I tried to explain what the teacher is (I think) doing, and make dialogue possible:
I agree fully with you. But there is some reason behind the teacher's position; and a compromise is possible. Multiplication is commutative, so your point is correct; 2*3 and 3*2 are equal, and either could be used to represent the same calculation or situation. Mature thinkers don't need to pay any attention to the order, but use whichever order is most useful. On the other hand, it is reasonable for students to INITIALLY be introduced to multiplication using a single model, which in this teacher's curriculum is "this many of those" rather than "this repeated that many times." This simply allows everyone in the class to be on the same page when they give examples. But they should soon learn the commutative property so that, for example, if they want to find the cost of 50 items that each cost $2, they could just add two 50's rather than add fifty 2's! Interestingly, many years ago I answered a question about an article in a teacher's magazine that made the exact opposite claim: that 2*3 "really means" 2 multiplied BY 3 (2 taken 3 times): Defining Multiplication http://mathforum.org/library/drmath/view/61066.html Some time later, each of the authors of the article cited (which I didn't have access to at the time) wrote to me complaining that my comments were wrong. They didn't change my mind, but I did then get to read their full article, and it did clarify the point they were trying to make. My conclusion is that it does make sense to teach kids initially to read "2*3" as "2 multiplied by 3," and perhaps to ask them to use that interpretation in class to make examples of where a particular multiplication would be used. They should not, however, be taught that this is the only interpretation. If nothing else, you can refer your daughter's teacher to that 2001 article from the Phi Delta Kappan, if she has access to it (I had to get it online through my school library site), because it makes the opposite statement about the meaning of multiplication, showing that what your teacher does is not the ONLY way to teach the subject; and moreover that other educators have the opposite opinion. It is not the way students SHOULD be taught multiplication, just the way they ARE taught it in this curriculum. So here's the compromise I suggest: the teacher can teach one interpretation of the application of multiplication, but make it clear that it is not the ONLY answer, just the one they are using IN THIS CLASS, for now. Your daughter should be commended for seeing that it really doesn't make a difference; and rather than being marked wrong, can be encouraged to learn the way the teacher says to do it merely as a class convention, to help students who need more time to catch on.
It turns out that the Common Core standards (introduced long after the original article) appear to support the interpretation that \(2 × 3 = 3 + 3\), so the teachers in these cases may just be doing what they are told. Here is what the standard says:
CCSS.Math.Content.3.OA.A.1 Interpret products of whole numbers, e.g., interpret \(5 × 7\) as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as \(5 × 7\).
Now, all this actually says is that students should be able to interpret a product in terms of some physical model; the specific instance shown is just an example of one such model, and does not say that this is the only model. So with my understanding of the issue, I would say that they leave it open whether to introduce only one model, or which one to use. But I can see why teachers and curriculum writers might interpret it as requiring this one model. I wish that teachers, teacher educators, and curriculum writers could all have a sufficiently mature understanding of math beyond the level being taught to know what matters, and what doesn’t. But none of us can be expected to do everything right (I myself can’t claim to know all about elementary education), so we have to be gracious.
Unfortunately, the example the standards give is not the “multiplied by” model that the authors of the article convinced me is better as an initial model, but the “times” model that is probably most commonly taught already. Oh, well.
The problem I have with multiplication is that the base number is never considered when multiplied. You have two objects to start ## right? Multiply those two objects twice ## x ## ## = ######! 2×2=6 not 4. Same with zero, you have two objects ## multiply those two objects by 0, you still have the two objects you started with! 2×0=2 not 0 !! Everyone wonders why kids struggle with math? Because its wrong!!
In case it wasn’t clear ill state it another way. You have 3 and you multiply it 2 times it should look like this ; (3) the number you started with) multiplied twice (x2)= (3) 3 3=9 3×2=9. You multiply the number you started with (3) twice or 2 times 3once & 3twice or second time. 3×2= (3) the number you started with multiplied twice or two times 3 3 = (3)3 3=9. The base number has to be included in order for it to be multiplied. (3)x2(3,3)=9. (3) multiplied once 3 = 6. 3×1=6 you’ve multiplied 3 one time which is 3×1(3)=6. And if you start with 3 and you multiply it 0 times, the 3 doesn’t just magically disappear, its still there! 3×0=3
Hi, James.
Clearly, if your idea of multiplication always gives a different answer than what everyone else gets, you must be thinking of it incorrectly. The operation you are imagining is not what multiplication is.
The basic misunderstanding you have is discussed in Is Zero Really a Number?, which includes four answers to the question of how you can multiply by zero and get “nothing”. A key idea there is that, whereas in addition you start with two objects (of the same type) and combine them, in multiplication, you don’t. Instead, you have an object (the multiplicand) and a number by which it is multiplied (the multiplier) that tells you how many copies of the object you want. If the multiplier is zero, it doesn’t take away the object you had; it just tells you not to make any copies. So nothing “magically disappears”.
To repeat, you don’t “have two objects to start with”; you don’t really even have one object to start with! You have, if you will, a template with which to make objects, and instructions for using that some number of times. In your illustration, you are forgetting to put the template back on the shelf when you are done! That’s why you end up having multiplied by 1 more than the given multiplier.
There are additional relevant ideas in the article referred to in the first line here, What is Multiplication? Multiplicand and Multiplier.
I hope that helps.
I’d like to make a try at explaining the point I think James Kerley may be attempting to get at.
It is common for people to describe multiplication by saying that \(p\times q\) means “\(q\) added to itself \(p\) times” (or maybe “\(p\) added to itself \(q\) times”, and also for people to describe exponentiation by saying that \(p^q\) means “\(p\) multiplied by itself \(q\) times” And the intended meanings of those descriptions generally seem perfectly clear to other people who already understand multiplication and exponentiation. But it seems to me that if you consider those descriptions literally, they are technically incorrect, or at least ambiguous.
Consider this: What do you think would be the result of adding 3 to 100 three times? Assuming you don’t pretend to misunderstand the intent of the question and say that it would be “103 every time,” I think you would agree that the result would be 109. More generally, the result of adding 3 to \(n\) three times would be \(n+9\). But by that reasoning, the result of adding 3 to itself (i.e., to 3) three times would be 12.
The similar description of exponentiation embodies a similar fencepost error. If \(n^2\) means the result of multiplying a number \(n\) by itself, isn’t it at least arguably a little odd to say that it’s also the result of multiplying \(n\) by itself twice?
Yes, what you say is essentially something I said in the last link I gave to James, referring to an old book:
Careful language can avoid mathematical misunderstandings. “Adding to itself” is an idiom, and taking it literally changes its meaning.
A nice thing about the “start with zero …” description is that it provides an explanation of why \(n\times 0 = 0\times n = 0\) for any \(n\): If you start with zero, then either adding any number zero times or adding zero any number of times still leaves you with zero.
Similarly, if we think of \(p^q\) as the result of starting with 1 and multiplying by \(p\), \(q\) times, we have an explanation of why \(p^0 = 1\).
On the other hand, while the aforementioned “explanations” may seem satisfactory to you or me, they may not seem compelling to someone who doesn’t already have a well-developed “number sense.” Such a person may not “get” the motivation for defining multiplication in a way that involves starting with 0 or for defining exponentiation in a way that involves starting with 1. (“So you’re telling me,” I can imagine someone saying, “that zero [one] is the additive [multiplicative] identity. Okay, I suppose it is. But so what?”) Or they might not see why “multiplying by \(p\), zero times” should mean something different from “multiplying by zero.”
Definitely. Number sense comes from multiple exposures to an idea, not from one perfect explanation. Teaching and logic are very different disciplines!
And that’s why, in Is Zero Really a Number?, which I also referred to, we gave four different approaches to multiplying by zero. I find that what works for one student doesn’t work for another, so I try to find one that works for the student in front of me. If there’s a whole class full, I give several answers. And then I wait for the questions.
Wouldn’t multiplication be better taught as an array where you would just look at it in both ways visually? However, in the case of 3×6, I would state it as I want it to be stated. 3 groups of 6 or if I want it to be 6 groups of 3, I would state it that way. An array tells you something different visually where you can look at it both ways, particularly as something 2 dimensional. It shows you what it is. You would point out the different ways of looking at it.
Ironically enough, the problem with your approach of this matter is that you are not thinking abstractly enough, which is what you reproach other of doing lol. (no offense intended)
When you start learning about things related to algebraic structures like ring theory, group theory, linear algebra, higher order logic, and even computer science to a certain extent, you quickly realize that multiplication is not always commutative and all of a sudden, understanding the difference in semantics between a × b and b × a becomes crucial.
Your example of multiplying pills and days is ambiguous. On one hand if you are only interested in the *product* (i.e. the result of the multiplication) then we are in the domain of arithmetic and the commutative law holds. However if you are interested in the distribution of those pills pay day then the problem is more of a group/ring theory problem and commutativity does not hold.
I may be misunderstanding you but from reading your article it sounds like you are denying the fact that these two POVs and problems exist and saying that only the arithmetic interpretation is valid and that noncommutative multiplication is fake or something. If that were the case linear algebra would not work and you would not be reading this message being rendered by your GPU.
AFAIK we have an operator to explicitly refer to the product (·) when commutativity holds and times should be reserved for non commutative multiplications. Of course this is almost never respected so I understand why everyone is confused all the time.
When teaching children I think it’s important to quickly make them understand that not all multiplications are commutative and using different operators like I said might help. However I don’t think most elementary school math teachers understand this to begin with so it’s a lost cause.
Teaching them that multiplication is always commutative is a big lie, and it would just make the smart students, that really understand, confused…
I do agree with you however that the word “times” is horribly ambiguous and implies commutativity.
Hi, Leonard.
Yes, you seem to have misunderstood; you are looking at far too high a level of abstraction.
This discussion is entirely in the context of elementary education, and therefore about arithmetic only. In that context, multiplication is definitely commutative. Saying that is not a lie, and I don’t think it confuses the smart students when they, much later, discover that we can, amazingly, extend the concept to new operations on new kinds of objects, where it isn’t commutative.
Furthermore, one of my main points is to distinguish the “product” and “process” points of view. And the pills example is not mine, but comes from the original question, and the article it referred to.
(We discussed something like what you are talking about in a later post in the series, What is Multiplication … Really?, including non-commutative kinds of multiplication.)
So I think we probably agree more or less, except that it absolutely is not useful to teach young children that there are non-commutative forms of multiplication. (Even their teachers probably are unprepared to understand that, whereas high school math teachers should.) I suppose I also don’t think the word “times” implies commutativity; in fact, I think it makes it all the more surprising that multiplication turns out to be commutative!
I fully agree about the value of encouraging, or at least allowing, students to think in different (correct) ways about multiplication or other mathematical concepts, as opposed to teaching that one way is “right” and that some other way that always gives the same answer is “wrong” or forbidden.
I have some thought experiments to consider. Unfortunately, I probably won’t have the opportunity to carry them out in practice, so I can only speculate about the results.
First, imagine that you could engage one of those persnickety teachers in conversation at a moment when they weren’t focused on being pedantic about something such as whether \(5\times 3\) means \(3+3+3+3+3\) or \(5+5+5\), and when they weren’t on guard about being caught out in such pedantry, and you were to ask them to show how they would teach calculating “1451 times 12” by long multiplication. Do you suppose they would write the calculation something like this
$$\begin{array}{r}
1451 \\\times 12 \\\hline
2902 \\1451{\color{White}0}\\\hline17412
\end{array}$$
? Or would you expect them to write something more like this
$$\begin{array}{r}
12 \\\times 1451 \\\hline
12\\60{\color{White}0}\\48{\color{White}{00}}\\12{\color{White}{000}}\\\hline17412
\end{array}$$
? My guess is that they would most likely work the problem as in the first alternative above, although that corresponds more nearly to calculating twelve 1451s (or 1451 “multiplied by” 12) than to calculating 1451 twelves. (If anyone reading this doesn’t understand why I’m saying the first method corresponds to calculating twelve 1451s, note that the partial products 2902 and 14510 are multiples of 1451 and not multiples of 12.)
Next, as a variation on the thought experiment above, suppose that instead of asking about “1451 times 12”, you asked about this word problem: “The Willis (formerly Sears) Tower in Chicago has a height of 1451 feet (excluding the tall broadcasting antennas on the roof). How tall is the tower (excluding antennas) in inches?” Do you think they would write the required value as something like this
\[1451\,\hbox{feet} \times \frac{12\,\hbox{inches}}{1\,\hbox{foot}}\]
(or perhaps \(\hbox{“}1451\,\hbox{feet} \times 12\,\hbox{inches/foot}\hbox{”}\) or \(\hbox{“}1451\,\hbox{ft} \times 12\,\hbox{in/ft}\hbox{”}\) or just \(\hbox{“}1451\times 12\hbox{”}\))? Or do you think they would more likely write
\[\frac{12\,\hbox{inches}}{1\,\hbox{foot}} \times 1451\,\hbox{feet}\]
(or some other form with the “12” to the left of the \(\hbox{“}\mathord{\times}\hbox{”}\))? I think they would more likely write something of the first kind – that is, with the “12” to the right of the \(\hbox{“}\mathord{\times}\hbox{”}\). Additionally, I think they would most likely carry out the calculation as in my first alternative from the previous paragraph, even though I think the word problem more naturally reads as asking for the value of 1451 twelves (or 12 “multiplied by” 1451) than as asking about twelve 1451s.
As I said, I don’t expect that I (or most people reading this) will have opportunities to carry out experiments like the ones described above. Perhaps, however, someone could carry out a sort of proxy for such an experiment by seeking out textbooks, teacher training materials, articles in education journals, etc., that address teaching multiplication as repeated addition, including especially any in which the authors specifically recommended teaching \(m\times n\) as \(n+n+\cdots+n\) but not as \(m+m+\cdots+m\) (or vice versa). One might then look for materials by the same authors on the topic of long multiplication and judge whether or not they are entirely consistent as to how they decide which factor to treat as the multiplicand and which to treat as the multiplier. I have not attempted such an investigation; if anyone reading this does, I’d be interested in learning the results!
My final thought experiment would be even more difficult to carry out in practice.
Imagine that you could visit a bunch of primary school classrooms and identify three groups of students:
(1) students who have learned that the addition sentence \(3+3+3+3+3=15\) corresponds to the multiplication sentence \(5\times 3 = 15\) and that writing it as \(3\times 5=15\) (or writing \(5+5+5=15\) as \(5\times 3=15\)) is somehow wrong;
(2) students who have learned, to the contrary, that the addition sentence \(3+3+3+3+3=15\) corresponds to the multiplication sentence \(3\times 5 = 15\) and that writing it as \(5\times 3=15\) (or writing \(5+5+5=15\) as \(3\times 5=15\)) is wrong; and
(3) students who find the commutativity of multiplication practically as natural as breathing (i.e., so natural that they hardly realize when they’re using it) and who would find it puzzling – and perhaps even irksome – if their teacher made a fuss over whether \(3+3+3+3+3\) corresponds to \(5\times 3\) (only) or to \(3\times 5\) (only).
(Of course there may well be many students who, for various reasons, wouldn’t neatly fit any of these three descriptions, but for the sake of this thought experiment, I will say no more about them here.)
Now imagine those same students years later taking a course in which they have just been taught how to do matrix multiplication and shown that it is generally not commutative. Out of groups 1, 2, and 3 above, in which group would you guess that the students, on average, would most reliably and/or most quickly perceive that the familiar algebraic identity \((A+B)(A-B) = A^2-B^2\) does not necessarily hold when \(A\) and \(B\) are matrices? One might reason that I have just described the exact sort of situation in which it would be a big disadvantage to have a habit of relying on the commutativity of multiplication while hardly realizing it. Nonetheless, I suspect that if an experiment of this sort could actually be carried out, the smart money would be on \(\hbox{group~3}\).
Alas, I can’t think offhand of a practical way to perform even a rough proxy for this last thought experiment, and thereby to gain objective evidence either supporting or debunking my intuition about the likely result. More generally, it seems to me that there are many, many cases in which it could be quite difficult to obtain solid evidence re the relative merits or disadvantages, especially in the long term, of different proposed methods of teaching various ideas in elementary mathematics. To elaborate further would be to enter on a vast topic that goes far beyond the subject of this blog page, so I will stop here.