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(Migrated from the math stack exchange, where I received an apt-seeming suggestion to pose the question here, at the math-educators stack exchange)


I introduce my young kids to basic math concepts in the hopes that my armchair enthusiasm for math will rub off on them.

To this end, I've described multiplying as "a special kind of adding where you add a number to itself (however many times)."
In this regard, I've also described that $(n \times m)$ is always the same as $(m \times n)$, and I've explained why using items arranged in rows and columns, an egg carton being an easy illustrative example.
Putting this all together, I've been able to explain, for example, that $3 \times 4$ would be $(3 + 3 + 3 + 3)$ or $(4 + 4 + 4)$, i.e. "3 added to itself 4 times" or "4 added to itself 3 times."

I'm soon about to introduce fractions, and in my mind this is where my explanation, above, seems to falter.

Though my kids haven't formally learned fractions, they understand the concept of certain special, named fractions, such as a "half", and I can tie that to the "$\frac{1}{2}$"-style notation.
Therefore, I can reasonably explain that $(\frac{1}{2} \times 4)$ is "$(\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2})$"
I.e. this ties in nicely to my description of multiplying as "adding a number to itself (however many times)" because we're "adding $\frac{1}{2}$ to itself $4$ times."

But something falls apart when I try to circle back to the notion that $(n \times m)$ is always the same as $(m \times n)$.
I.e. "adding $\frac{1}{2}$ to itself $4$ times" is comprehensible...
but "adding $4$ to itself $\frac{1}{2}$ a time" doesn't quite make sense...at least grammatically.

Is my reluctance over the phrase "adding a number to itself a fractional number of times" merely a hangup over English grammar? Mathematically, is there any legitimacy to the notion of adding a number (to itself) $N$ times, where $N$ is not a whole number?


Update: Since my question generated so many "off-by-one" responses, I thought deeper about what exactly I'd said to my kids, and I think key is my phrase "however many times". In my original post, I kind of hand-waved away that phrase, but during discussion with my kids I would have explained in a manner that conveyed that "however many times" is $(n-1)$ or $(m-1)$ "instances" of $m$ or $n$ in the "addition chain". Still clunky, no doubt, but I wanted to clarify that I didn't mislead my kids to a state of "off-by-one" confusion. (I think both the clunky phrasing and the grammatically "off" addition by a fractional amount are nicely addressed by the accepted answer)

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    $\begingroup$ I think you'd be interested in what's called "Montessori Rechenstäbchen" in Germany. Google the term and see whether there's an equivalent in your location or whether one of the suppliers ship there for an affordable fee. It's your egg cartons, just better. Perfect for explaining fractions, commutative multiplication, squares and cubes. I had them as a child and I'm a huge fan. Montessori put an emphasis on hands-on, practical experience, and in my opinion that works very well. $\endgroup$ Jan 21, 2023 at 13:39
  • $\begingroup$ @Peter-ReinstateMonica It appears that “Montessori number rods” is the equivalent term in English. But there’s an interesting difference: the results I get in German seem to all have indentations on the rods (e.g. the length-4 rod is like four bevelled cubes joined together), but the ones in English are all smooth! $\endgroup$ Jan 22, 2023 at 2:47
  • $\begingroup$ @Tim Oh, mine were smooth. Not that it matters much, I think. $\endgroup$ Jan 22, 2023 at 2:58
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    $\begingroup$ @ilkkachu There’s discussion of this usage in the comments to Nick C’s answer. $\endgroup$ Jan 23, 2023 at 2:46
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    $\begingroup$ I would rephrase 4x2 not as "4 times 3 added to itself" but as "4 (sets) of 3". Then 1/2 of 3 makes sense and even 1/2 of 3/4 makes sense. $\endgroup$ Jan 24, 2023 at 17:40

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For the product $a\times b$, I intentionally don't use the phrase "add $a$ to itself $b$ times", but rather I prefer something like "start with zero and add $b$ (copies) of the number $a$". This makes sense for $\frac{1}{2}\times 4$, since we start with zero and write four copies of the number $\frac{1}{2}$, giving us $0+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2$. Similarly, it makes sense for $4\times\frac{1}{2}$, since we start with zero and add one half (copy) of the number $4$, giving us $0+2=2$.

So, my "solution" for your issue is to just phrase what is happening a bit differently. Instead of saying we are adding something to itself a fractional number of times, indicating that the second factor counts the number of times the addition happens, I use that second factor to count the quantity of the first factor we are adding. It's subtle, but I've found it helps arithmetic students.

[Note: I started doing this when teaching positive integer exponents, when it seemed that some students were misapplying the interpretation of $9^1$ as "multiply 9 by itself 1 time", thinking this could mean $9\times 9$ (since we have, indeed, multiplied 9 by itself just once). I began explaining $9^n$ along the lines of "start with 1, and multiply $n$ factors of $9$. This has the added benefit of providing an explanation of how $9^0$ equals $1$. Of course, we later just use exponent properties.

On the note of fractional exponents (which could come next in the conversation), while I've never actually said this, I suppose "starting with 1 and multiplying a half of a factor of $9$ could be justified for $9^{\frac{1}{2}}$, though I don't think I'd want to have to also invent a convenient way to explain "half of a factor" as "the square root".]

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    $\begingroup$ I like the phrasing you're proposing; thank you. Perhaps by the time my kids are old enough to start learning about exponents, and specifically fractional exponents, they'll be old enough to not need Daddy-simplified analogies anymore. :) $\endgroup$
    – StoneThrow
    Jan 20, 2023 at 21:03
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    $\begingroup$ It's interesting that you phrase n * m as "add n m times to zero". Natural language use clearly has it the other times around: Spoken aloud, it's "n times m", not the other way around. ("I went 5 times to school this week", or "to school 5 times", but not "to school times 5".) This is also expressed in the usual notation of coefficients: If we have x 5 times, we write "5x", not "x5". I suspect you do the commutation silently already in your head: Yes, n*m is also "n m times", but only because we know it's commutative. $\endgroup$ Jan 21, 2023 at 13:47
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    $\begingroup$ The mathematical phrase "times $n$" has entered the English lexicon to mean $n$-fold amplification. For example, a quick Google news search turned up "This is pretrial publicity times ten," from an article in the Rapid City Journal. $\endgroup$ Jan 21, 2023 at 15:56
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    $\begingroup$ ...but people who use that phrase are no doubt implicitly invoking the commutative law. :-) $\endgroup$ Jan 21, 2023 at 16:04
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    $\begingroup$ Is this terminology really a problem? The students already seem to understand that you're starting from 0 when you translate 4xN to N+N+N+N. $\endgroup$
    – Barmar
    Jan 21, 2023 at 16:41
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Frame challenge: I think your verbiage "adding (whole number) to itself (whole number) times" is misleading and incorrect and exhibits an off-by-one error. Think about the example $(4×1)$. Would you call that "adding 4 to itself 1 time"? No, because you're not doing any addition at all. The result of "adding 4 to itself 1 time" should be the result of one addition, i.e. $(4+4)$, equivalent to $(4×2)$.

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    $\begingroup$ I agree that verbiage can be confusing. I think it's Terrence Howard's core confusion in his argument that 1 x 1 = 2. twitter.com/terrencehoward/status/925754491881877507 $\endgroup$ Jan 21, 2023 at 18:07
  • $\begingroup$ Since my question generated so many "off-by-one" responses, I thought deeper about what exactly I'd said to my kids, and I think key is my phrase "however many times", which during discussion with my kids I'm sure I would have explained in a manner that conveyed that "however many times" is "(n-1)" or "(m-1)" instances of m or n in the addition chain". Still clunky, no doubt, but I wanted to clarify that I didn't mislead my kids to a state of "off-by-one" confusion. $\endgroup$
    – StoneThrow
    Jan 25, 2023 at 16:25
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The Common Core State Standards "definition" of multiplication is that $A \times B$ represents the number of units when you have $A$ groups, each $1$ group containing $B$ units.

For instance $4 \times \frac{1}{2}$ could represent the number of pounds of flour you get from $4$ bags when each $1$ bag contains $\frac{1}{2}$ of a pound. Note that this is "backwards" from the way you are interpreting multiplication. Now I am aware that multiplication is commutative, but this is not obvious from either your definition or the CCSS definition so it needs to be argued.

Similarly $\frac{1}{2} \times 4$ could represent the number of pounds of flour you get from $\frac{1}{2}$ of a bag when each $1$ bag contains $4$ pounds.

The CCSS adopted this definition (rather than a "repeated addition" perspective) exactly to solve this issue: it provides a unified mental model which applies equally well across all number system representations. We can equally well make sense of $\frac{5}{4} \times 1.7$: If each bag of flour has 1.7 pounds of flour, how many pounds of flour would be in $\frac{5}{4}$ of a bag?

I will make two more observations:

(1) The "repeated addition" perspective is really limited to integers. The "proof" would be something like $A \times B = A \times (1 + 1 + 1 + ... + 1) = A +A+A + ... + A$. In other words, multiplication by an integer $B$ is related to repeated addition because of the distributive property.

(2) The CCSS definition is very much related to unit conversion and measurement. One could say that $A\times B$ represents the following situation: you measure an object using $\textrm{unit}_1$ find that the object is $A \textrm{ unit}_1$ . When you measure $\textrm{unit}_1$ using $\textrm{unit}_2$ you find that each $1$ $\textrm{unit}_1$ is $B \textrm{ unit}_2$. Then $A \times B$ is defined to be the number of $\textrm{unit}_2$ you would obtain when measuring the same object.

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    $\begingroup$ With a little mental flexibility, repeated addition is not really restricted to integers. If you say e.g. "3.5 * 5", you can do the same distribution as with an integer: $A \times B = A \times (1 + 1 + 1 + 1/2) = 1 \times A +1 \times A + 1 \times A + \frac{1}{2} \times A$ $\endgroup$ Jan 21, 2023 at 13:53
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    $\begingroup$ You can also, for children, imagine "cutting the last A off" to the desired fraction: Say, we are adding 3.5 Us (for purely typographic reasons): "U+U+U +J" :-). I find that also helpful in imagining logarithmic laws: Sure, $\log_{2}x = 3*\log_{8}x$, because if you write x as "2*2*2*2*2...", you can group the 2s so that you have "(2*2*2)(2*2*2)...". Now if you want to take a base whose logarithm is not an integer, you imagine similar parenthesized groups, just with some fraction "cut off" of the last number. The idea is similar: How often does the smaller base "fit" into the larger one. $\endgroup$ Jan 21, 2023 at 14:04
  • $\begingroup$ @Peter-ReinstateMonica I think the issue is really with the "adding it to itself" terminology. This implies you always start with at least one. You could, instead, read $A \times B$ as "Sum together $A$ copies of $B$". This is kind of intermediate between the OP's phrasing and the CCSS definition. It also accounts for situations where the number of copies is less than $1$. $\endgroup$ Jan 21, 2023 at 14:49
  • $\begingroup$ Yes, of course, the OP's wording had an off-by-one-error. That goes without saying. $\endgroup$ Jan 21, 2023 at 21:11
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Putting this all together, I've been able to explain, for example, that 3×4 would be (3+3+3+3) or (4+4+4), i.e. "3 added to itself 4 times" or "4 added to itself 3 times."

No, 3 added to itself 4 times is something like: 3+3, 3+3, 3+3, 3+3. To describe 4 times 3, you should say: four threes all added up. Then there is no problem explaining half times 4, since it is just half a four (all added up).

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    $\begingroup$ This is the closest to the correct answer in my opinion. The reason we use the word "times" in "4 times 3" is that it is "3 (4 times)", i.e. 3+3+3+3. The modern translation of "4 times 3" should be something like "4 threes" like you say. $\endgroup$ Jan 26, 2023 at 0:20
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I think it is important to consider your audience. Are these kids going to immediately jump to asking about "adding 4 to itself $\frac{1}{2}$ times?" I'd guess not. I think this is a perfectly fine definition of multiplication for positive integers. Let them discover extensions to larger sets of numbers once they have mastered this.

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  • $\begingroup$ Most children I know would jump to any occasion to say "You told us this, and now you're telling us that, but when we put this and that together it doesn't work". $\endgroup$
    – Stef
    Jan 22, 2023 at 17:06
  • $\begingroup$ I just think no one learning multiplication for the first time is going to appreciate the subtle differences. I prefer to introduce the most intuitive, easy-to-state definition, and then when that's been understood, you start talk about what to do with fractions. $\endgroup$ Jan 23, 2023 at 19:01
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You can try going back to the way you may have first taught them the concept of addition: combining collections of things -- "if you have 3 apples, and I give you 4 apples, how many apples do you have?". So 4 means a collection like

X X X X

When you multiply, you duplicate it that many times. So 4x3 looks like:

X X X X
X X X X
X X X X

Fractions are similar, but you instead of duplicating the whole collection, you stop short. 1/2 means you stop half-way through the collection.

X X

This allows them to understand mixed fractions like 3 1/2. This has 3 complete repetitions, followed by a partial repetition.

X X X X
X X X X
X X X X
X X
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Go back to your egg-carton analogy. Then, move it to the area of a rectangle as number of unit squares Cutting the rectangle in half, either horizontally or vertically, produces the same answer.

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I think you have to teach progressively. Even with bright kids, advanced kids. Even when not really establishing mastery, but giving exposure. So, let the kids have more exposure to fractions themselves, writing, rationalizing, adding subtracting, etc. before worrying about this conundrum.

So instead of concerning this conundrum and how to contort yourself to push it with a kid that is very young, I would concern yourself more with getting other parts of their number sense mastered. And mastered does NOT mean "heard the rule", it means going for walks and asking them little (easy) questions like 5+8, 13+ 7. (Or whatever level they are on...I ask an 11 year old relative of mine, first degree in x questions as we walk/scooter. Kids enjoy hanging with you and showing off a little with stuff of a little difficulty, but still within their grasp.)

Also, it is very natural to think of half times four as 1/2 * 4. Once they are more familiar with the "easy way" (fraction first), you can introduce the reverse. Hopefully after they've also had experience with 3 * 2 = 2 * 3 from extensive single digit quizzes, learning the times table, and even hearing the rule, etc.

I would also do some writing of the numbers. Both verbal drill and written drill are helpful for learning things. Multiple modes...

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Half times something is the same as half of it. When students are used to that, and also used to the distributive law, then you can put them together like $2\frac{1}{2}\times a$ is $a + a + \frac{1}{2} \times a$.

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