# Do you think teaching Division as "repeated subtraction" is confusing?

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My dd's 2nd grade Calvert Math (and no we are never using it again, though parts of it have been fun) introduces division as "repeated subtraction."

FOr multiplication, introducing it as "repeated addition" makes sense to me.

But when I think about division, I think about having a pile of 50 blocks, and then dividing them into 10 piles, and having 5 blocks in each pile. That's how I learned division, and that's how I (and Saxon) explained it to my ds. In this way, you are even using the term "divide" right from the beginning which most children (at least mine) at this age understand the concept of.

Of course my dd already understands the concept of division anyway, and of course I can just introduce it however way I want, regardless of what this book says (especially since Calvert math will sail off into the sunset forever in about 5 weeks!)

But I'm just curious...who else thinks it's confusing?

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I agree. My son's Math Mammoth had a few pages about this in the division chapter and it was hard for him to grasp. It wasn't introduced this way, however. I think that would be even more confusing. I think this was in there to encourage thinking about how it worked and why.

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:iagree:

I HATE teaching division as anything other than the opposite of multiplication. Period. Easy Peasy Rice and Cheesy. Why do all the curriculum make it so difficult?

It IS confusing. I'd skip anything that says that and teach it as the opposite of multiplication.

Blessings!

Dorinda

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Yep, I was just saying that about Math Mammoth, too. :D It's in 4B, I think?

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:iagree:

I HATE teaching division as anything other than the opposite of multiplication. Period. Easy Peasy Rice and Cheesy. Why do all the curriculum make it so difficult?

It IS confusing. I'd skip anything that says that and teach it as the opposite of multiplication.

Blessings!

Dorinda

Confused here....multiplication is simply repeated addition.

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Logically, if multiplication is "fast adding" and division is the opposite or inverse of multiplication, then division is "fast subtracting." However, I can't turn that "logic" into practical application; my brain breaks. I can't even remember how it was taught to me. I think of subdividing into groups, also. Hope someone can make the subtraction angle makes sense!

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I teach it as the inverse as well.

Although, conceptually, as multipication is addition so is division the subtraction.

It would depend on the type of learner that you have. It would depend on how you approached multiplication.

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Teaching division as repeated subtraction is a highly confusing way to teach it, and should be avoided. I'm surprised that anyone would try to teach it that way.

The reason it's so inherently confusing is that the thing you'd be repeatedly subtracting is the quotient, which you don't know until you've done the problem. This means that things get confusing for a kid especially badly when there's a fractional quotient.

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Teaching division as repeated subtraction is a highly confusing way to teach it, and should be avoided. I'm surprised that anyone would try to teach it that way.

The reason it's so inherently confusing is that the thing you'd be repeatedly subtracting is the quotient, which you don't know until you've done the problem. This means that things get confusing for a kid especially badly when there's a fractional quotient.

Exactly! Very good explanation! U r smart.

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Logically, if multiplication is "fast adding" and division is the opposite or inverse of multiplication, then division is "fast subtracting." However, I can't turn that "logic" into practical application; my brain breaks. I can't even remember how it was taught to me. I think of subdividing into groups, also. Hope someone can make the subtraction angle makes sense!

Teaching division as repeated subtraction is a highly confusing way to teach it, and should be avoided. I'm surprised that anyone would try to teach it that way.

The reason it's so inherently confusing is that the thing you'd be repeatedly subtracting is the quotient, which you don't know until you've done the problem. This means that things get confusing for a kid especially badly when there's a fractional quotient.

:iagree: to both of the above.

I've always thought of division as "sorting"- IOW, dividing. Easy peasy. Throw subtraction in there, though, and you're taking a simple arithmetic concept and teaching it as though it's calculus.

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Confused here....multiplication is simply repeated addition.

Yes. I wasn't refuting that. I was saying that division is the opposite of multiplication. Therefore, when teaching it, you teach it as, "If 5x5 is 25, than 25/5 is 5."

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You don't subtract the quotient though...

21/7:

.....21

-7..14

-7..7

-7..0

You subtracted 3 7's, so the answer is 3.

I agree with teaching it as the inverse of multplication, and MM does that, but has a short stint into repeated subtraction just to show that it works, but the kid doesn't have to focus on that method, and I wouldn't worry about it if it confused the kid.

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Yes. I wasn't refuting that. I was saying that division is the opposite of multiplication. Therefore, when teaching it, you teach it as, "If 5x5 is 25, than 25/5 is 5."

I think I took it a step further.

My thinking: If multiplication is repeated addition, then division is repeated subtraction. Unless you taught multiplication in isolation, then you should not teach division in isolation. Does my train of thought make sense?

I don't find the repeated subtraction idea confusing....but I will have to wait to see if my children do. ;)

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I think I took it a step further.

My thinking: If multiplication is repeated addition, then division is repeated subtraction. Unless you taught multiplication in isolation, then you should not teach division in isolation. Does my train of thought make sense?

I don't find the repeated subtraction idea confusing....but I will have to wait to see if my children do. ;)

Exactly, every child is different. Whatever gets the job done simply and easily.:001_smile:

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The problem with slogans in teaching: Whether we are the teacher or the student, once we accept a slogan as dogma, we stop thinking.

"____________ is simply ____________ ."

Fill in the blanks however you want, and you have a BAD statement for a teacher to make.

"Division is simply repeated subtraction?"

As to the original question: I agree with the other commenters that "repeated subtraction" is a poor way to introduce division. On the other hand, it IS a great mental math technique to have in your toolbox for solving certain math problems. And the slogan does encapsulate ONE way of looking at division.

Let's consider the types of story problem (or real life) situations our student might meet which require division. First, we might have some amount of stuff that must be shared evenly among a certain number of whatevers, and we need to find out how much stuff each whatever will receive. Second, we may have some amount of stuff that must be measured out in chunks of a certain size, and we need to find out how many chunks we can make.

The latter situation looks much like subtracting the size of the chunk over and over until we run out of stuff. For instance, we might need 3/4 yard of fabric to make a certain type of pillow cover -- so how many pillows could we cover with 6 yards of fabric?

Also, as another commenter has pointed out, this understanding of division is at the heart of the standard long-division process.

Conclusion: Don't teach with the slogan. But do, as a teacher, think about what might have inspired the slogan and how it might help you develop a deeper, more flexible understanding of division.

This slogan has the SAME problem as the statement about division (it encapsulates ONE way of looking at one very limited application of multiplication), but because the multiplication slogan is so familiar to us, we teachers don't recognize the problem. We have a familiar, comfy slogan, and we don't think deeply enough to realize the problem this can cause for our students.

If we train our students to think "multiplication is repeated addition", then we have no cause to complain when those same students can't solve story problems or when they get confused trying to remember the fraction rules.

Consider:

(2/3) x (5/6) = (2 x 5) / (3 x 6)

but

(2/3) + (5/6) is NOT = (2 + 5) / (3 + 6)

Why not?

Isn't multiplication just a special type of addition? So WHY are the rules so different?

The Fibonacci Series is created by repeated addition of the two previous numbers. Is that multiplication? We can form the square numbers by adding up the odd numbers: 3^2=1+3+5, and 4^2=1+3+5+7, and 5^2=1+3+5+7+9. That's definitely repeated addition, and squaring a number is a sort of multiplication...

The problem with the definition "multiplication is repeated addition" is that it leave unstated the MOST IMPORTANT difference between the two operations. That's why so many students are reduced to staring blankly at a story problem, asking, "Do I add or multiply?" We haven't given them any way to recognize the difference.

For more examples of how not understanding the difference between addition and multiplication makes learning fraction rules difficult:

For a more thorough exploration of the "repeated addition" debate:

Then how SHOULD we teach multiplication?

If we accept this argument, if we agree to no longer define basic multiplication as "repeated addition", then what? How does that affect the way we teach?

Mainly, we need to change our focus from how to why.

We can teach multiplication in much the same way that we do now, using manipulatives arranged in groups or rows, pictures of multiplication situations, and rectangular arrays of dots or blocks. But instead of drawing our student’s attention to the process of adding up the answer, we want to focus on the fact that the items are arranged in equal sized groups.

In other words, we teach our students to recognize the multiplicand:

• Teach children the useful word “per” and how to recognize a “this per that” unit.
• Have them label the quantities in their workbook: 3 cookies per student, 5 flowers per vase, 1 eye per alien, or whatever.
• If your story problem has a "this per that" quantity, then it must be a multiplication or division problem. You may be able to solve it with an addition or subtraction approach (especially if the numbers are small), but the heart of the problem is multiplicative.

Edited by letsplaymath
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Keep in mind that there are two "types" of division: partitive and group. We use the same algorithm and arrive at the same answer for both (of course), but it may help when you think about repeated subtraction.

However, repeated subtraction makes sense if you consider group division. For example, I have 8 cookies and a serving size is 2 cookies. How many children can receive a serving of cookies? Here, you can subtract your 2 cookies over and over again until you realize that there are 4 groups of 2 cookies in your original 8, so your answer is 4 children.

So, I think repeated subtraction can make sense as one way to introduce division.

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I think it is a nice mind bender for depth.

In the OPs example: If I have 50 blocks and want to divide them into 10 piles, every time I put one in each of the piles I subtract 10 from 50. I repeat that until I run out of blocks. So yes, 50 / 10 = 5 is the same as saying 50 minus 10 five times = 0 and notice I do not need to know how many times I can subtract 10 before hand (I could subtract repeatedly and count how many times I subtracted).

This is actually used whenever doing long division -- and is visible if you guess a quotient less than the real one (such as multi-digit divisor), you can correct it by subtracting more multiples of the divisor.

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Actually, you could subtract the four. Imagine handing out the cookies one at a time. You hand out 4 cookies, one to each student, and you have 4 left. Subtract those (hand them out), and your cookies are gone. You subtracted twice, and each child got 2 cookies.

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Confused here....multiplication is simply repeated addition.

Try doing that with decimals. :001_smile:

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Remainders make more sense when division is initially shown as repeated subtraction. Conceptually speaking, division is tricky because it can be viewed in more than one way. I can't speak for other programs, but MEP deals with remainders from the beginning and uses a multi-pronged approach to division:

* dividing into a specific number of groups/sets to find how many members per group

* dividing into groups a certain size to find how many groups

* repeated subtraction until all you have left is zero or remainder (i.e. R=0), though this is not the major focus

That division "undoes" multiplication is emphasized in the function chains and in the "check your work" boxes.

Edited by nmoira
clarity
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My daughter did not find the "repeated subtraction" method of division confusing, but she already understood the concept of division before we discussed that way of looking at it. I had her do a few problems that way, and she also tried a few other division strategies with each problem just for comparision purposes.

So 164/4 could be solved a bunch of different ways:

By repeatedly subtracting 4:

164-4=160, 160-4=156, 156-4=152, 152-4=148, .....

and seeing 4 had to be subtracted 41 times to get to 0

By repeatedly subtracting a multiple of 4 (aka "chunking"), using 40:

164-40=124, 124-40=84, 84-40=44, 44-40=4, and 4-4=1.

So she subtracted the following number of 4s: 10+10+10+10+1, which =41

By the place-value method:

100/4=25, 60/4=15, 4/4=1, and 25+15+1=41

By dividing in half 2 times:

164/2=82, 82/2=41

By long division:

4 goes into 16 4 times, and into 4 1 time, for a quotient of 41

Finally she used the calculator to "check her answer" and saw it was still 41.

In the end it was quite obvious that repeated subtraction worked, but it is not an efficient strategy for dividing large numbers.

Anyway, I can now say she knows this method, and when her teacher gets around to division by repeated subtraction, she'll know what the teacher is talking about. :001_smile:

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Goodness, if elementary math is this confusing what am I going to do once my dc reach middle and high school levels? If we ever get there! geesh!

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The reason it's so inherently confusing is that the thing you'd be repeatedly subtracting is the quotient, which you don't know until you've done the problem. This means that things get confusing for a kid especially badly when there's a fractional quotient.

I haven't read all the responses, so this might have already been noted, but you wouldn't be subtracting the quotient, you'd be subtracting the divisor. Very simply, you're discovering the quotient by seeing how many groups (i.e., divisors) can be subtracted from the dividend. If you have a problem like 45/5, you'd start with 45 and keep subtracting groups of 5. After 9 subtractions, you're down to 0, so the quotient is 9. My ds actually found it quite interesting to see how repeated subtraction relates to division and how the process of dividing normally is so much faster than all the subtractions. (When I taught this to my ds, I put 45 beans on the table and took 5 away at a time. Really, it's the same as taking the 45 beans and putting them into groups of 5, but instead of leaving the groups on the table, each group is taken away as it is counted.) I explained to my ds that we could sit and spend a lot of time subtracting one group at a time, or we could just use the division algorithm. With enthusiasm, he fully agreed the algorithm was much more efficient. (I used the same process when teaching that multiplication is the same as, but much faster than, repeated addition.)

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I learned so much from this thread. Before reading it, I thought division being repeated subtraction was too confusing, and I wouldn't even know how to process that. But some posters here did an excellent job at explaining it:)

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Try doing that with decimals. :001_smile:

Certainly not as easy, but definitely doable.

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Letsplaymath- I agree with your emphasis on equal sized groups. Thank you for taking the time to give that explanation. I fear we were a bit too simplistic at the beginning of this thread. :)

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Even "equal groups" is being a little bit too simplistic, since most things in life are continuous rather than discrete. When teaching elementary students, we do tend to think in terms of discrete items and groups, but in real life "this per that" units are everywhere. For instance, we can travel 100 miles at 65 miles per hour (or more?), and get 19.48 miles per gallon of gas, which costs us \$3.49 per gallon and rising.

Being able to work with "this per that" units will be especially important in high school science. Better that our children get used to noticing and working with them while the problems are still relatively simple.

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I've changed my mind. I was worried about confusion in dealing with fractional quotients, dividing by a fraction, etc. but I think now that showing these things as repeated subtraction could actually be clear.:smash:

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Thanks Denise!!!! That was super helpful. I think this might really help my 4th grader and I'll try tO start that way with my 2nd grader.

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I've always taught it as dividing into equal groups and showed that with manipulatives. That seemed to make the most sense to kids.

Timez Attack is a software program that helps kids memorize math facts like multiplication and division. They do a wonderful job of showing the division concept (dividing into groups) with their graphics. My youngest daughter loved that program when she was younger.

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Keep in mind that there are two "types" of division: partitive and group. We use the same algorithm and arrive at the same answer for both (of course), but it may help when you think about repeated subtraction.

However, repeated subtraction makes sense if you consider group division. For example, I have 8 cookies and a serving size is 2 cookies. How many children can receive a serving of cookies? Here, you can subtract your 2 cookies over and over again until you realize that there are 4 groups of 2 cookies in your original 8, so your answer is 4 children.

So, I think repeated subtraction can make sense as one way to introduce division.

Thank you Kate for the excellent explanation of the two primary ways of thinking of division. Understanding group division, and especially how remainders relate to it, can help in understanding and solving a number of problems.

Here is one that I have seen referenced as something most American adults get wrong:

(my approximation of a problem I don't remember the details of)

There is a group of 48 soldiers who need to be transported from point A to point B in buses. Each bus holds 15 soldiers. How many buses will be needed?

Apparently, even among those who successfully infer that division can be used to solve this problem, and correctly set up the operation of 48/15, most end up with the wrong answer to the original question--typically they answer 3.2 or 3 remainder 3. They correctly worked the division problem, but failed to think through the implications of that answer for the question at hand. 3.2 buses, or 3 remainder 3 buses, is meaningless. The answer of course is staring them in the face, but most people don't stop to consider what that .2 or remainder 3 really represents. But if you think of division as repeated subtraction the real answer can become clear: the first bus pulls up, 15 soldiers climb on, 33 are left waiting. The second bus pulls up, 15 more climb on, 18 are left behind. The third bus pulls in, 15 climb aboard, there are still 3 soldiers waiting. Obviously we need one more bus!

I personally like to give my kids as complete a conceptual understanding of mathematical operations as possible, and I see this as one useful way of understanding division.

--Sarah

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Keep in mind that there are two "types" of division: partitive and group. We use the same algorithm and arrive at the same answer for both (of course), but it may help when you think about repeated subtraction.

However, repeated subtraction makes sense if you consider group division. For example, I have 8 cookies and a serving size is 2 cookies. How many children can receive a serving of cookies? Here, you can subtract your 2 cookies over and over again until you realize that there are 4 groups of 2 cookies in your original 8, so your answer is 4 children.

So, I think repeated subtraction can make sense as one way to introduce division.

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• 9 years later...

Division should not be taught as repeated subtraction, for it is repeated grouping or composition. We should be teaching students that we are not taking away but composing as many equal groups that one is contained in another on a basic level, for in the end, it's essential for students to interpret the meaning of their solutions. The measurement model of division is the way to go for it can better help students make sense of division with fractions, integers(pos and neg) etc.

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Multiplication and division are opposites. It is logical to use the reverse process for one that was used for the other.

If Calvert taught multiplication as repeated addition, then I can see why it described division as repeated subtraction (since addition and subtraction are also opposites).

Of course, it is possible (and I think the most likely explanation) that there is a mathematical error in calling it "repeated addition/subtraction" in the first place, and Calvert's explanation of division simply exposed that at some level to you and your kid. The answer is to call it what it actually is, "repeated grouping/ungrouping", show what you mean, and carry on calling them opposites after that.

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zombie 🧟‍♀️…  though an interesting question

Edited by Eilonwy
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On 2/29/2012 at 3:32 PM, Blessings2all said:

Timez Attack is a software program that helps kids memorize math facts like multiplication and division. They do a wonderful job of showing the division concept (dividing into groups) with their graphics. My youngest daughter loved that program when she was younger.

Dang, and I got my hopes up that Timez Attack was still out there for general use!   That was such a great program, but I believe it is now used in schools with a license?     I was going to ask about it!

But now that Common Core has been in use for several years, and this is still a strategy for division, it would be interesting to hear how it is accepted now and if liked or not.

BTW, I am not related to the original poster, nor did I copy her signature!😉

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It is confusing and not helpful.

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I think the usefulness really depends on if your main mental model of division is that x/y means you are splitting x into y equal groups and the answer is how many one group gets (which is what I use mainly), or that it means splitting x into some number of groups of size y, and the answer is how many groups. If it’s the second model, then repeated subtraction could be quite useful and intuitive.  If it’s the first model, it’s not much help at all.

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I've definitely never taught it that way. I don't really like "counting" methods for either multiplication or division, frankly -- kids get stuck doing them and then they aren't flexible. I far prefer teaching the mental model of "splitting into groups" and then slowly making kids realize it's also the opposite of multiplication and other shortcuts.

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...

Edited by Nothingtoseehere
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57 minutes ago, BaseballandHockey said:

Why wouldn't you just subtract the divisor?

You would.  You would subtract the divisor and count the number of times it takes to get to zero.

I've never taught it this way, nor have I taught multiplication as repeated addition (specifically), but it isn't at all confusing.

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1 hour ago, Not_a_Number said:

I've definitely never taught it that way. I don't really like "counting" methods for either multiplication or division, frankly -- kids get stuck doing them and then they aren't flexible. I far prefer teaching the mental model of "splitting into groups" and then slowly making kids realize it's also the opposite of multiplication and other shortcuts.

But how are kids supposed to know what size groups to split into without (at least indirectly) subtracting.

The very first time I hand a 3 year old a pile of M+Ms and tell him to split it evenly between himself and his siblings, all of my kids have initially attempted random splitting...with less than stellar results. We count the piles and realize they are not fair and equal.

Through Socratic dialogue, we realize that we should split more systematically. At first by giving each child 1 M+M over and over until the pile is gone. Eventually, that gets tedious, and we realize that it is more efficient to hand out 2 M+Ms to each child at a time. And then we start getting clever and using the pile size to gauge how many M+Ms we should hand out to each child at a time...if the pile is huge we can hand out 10 at a time, if it is very small then we should only hand out 1 at a time, etc.

That right there is the physical manifestation of repeated subtraction. Hand out some M+Ms evenly, which by default subtracts them from the pile, and then do it again and again until your pile is gone.

All of my kiddos have then been able to transition to doing the same thing symbolically. For example, to divide 132 by 3 they could subtract 3s over and over, but they are well versed in assessing pile sizes, so instead they can start by repeatedly subtracting 30 (3 groups of 10) from the dividend and adding 10 to the quotient each time. After doing this four times, they have a working quotient of 40 and a remaining dividend of 12, which they recognize as 4 more groups of 3. For a total quotient of 44.

To me, teaching division as repeated subtraction (not always subtraction of the divisor itself, but also of multiples of the divisor), is the only way that makes sense for young children. My young children do not know the 3's times table up to 44. So they can't look at 132 and immediately jump to splitting into 3 groups of 44. They also don't know the long division algorithm...specifically because I don't teach that until they fully understand the partial quotients method which is repeated subtraction.

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30 minutes ago, wendyroo said:

But how are kids supposed to know what size groups to split into without (at least indirectly) subtracting.

I do do handing out into equal piles. I don't mind subtracting indirectly, I'd just never point out that it's subtracting.

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