Teaching Division

[medawyn]

DS8 is happily using Beast Academy, but we're approaching the division chapter, and I'm not thrilled with the speed it races through introducing the concept into division with remainders.  I feel like it leans too heavily on the notion that division is the inverse of multiplication without really building an understanding of division as its own operation.  Suggestions for supplementing?  I have (and will be using) Math Mammoth, but it too doesn't seem to have challenging enough problems before division with remainders, just lots of practice with basic facts.

@Not_a_Number, suggestions for the best ways to talk about division with my kid? I feel like my understanding of math started to fall apart around 3/4th grade because I didn't really internalize the concepts of multiplication and division.  I can (obviously) do the operations, but I stumble trying to teach because I doubt my ability to explain well outside of the algorithm.

FWIW he's fairly fluent in multiplication facts, can easily multiply numbers that are multiples of ten, and can multiply 2 or 3 digit numbers, even if it's a laborious process (haven't introduced that algorithm either).  He doesn't LIKE doing 2 or 3 digit numbers, but that's just because it's slow, I think, not because he doesn't grasp the concept.

[Not_a_Number]

Err, it probably won’t thrill you that I made the conscious decision to not do division with remainders, right? I thought it diluted the mental model.

I usually keep division simple and define, say, 15/3 as “how much one person gets if we split 15 things fairly between 3 people.” Then we work using that definition for a good long while, we figure out what else we can do with it and how it relates to multiplication, and we extend it slightly for fractions. But I don’t do remainders until later, because I think they confuse the model.

[medawyn]

26 minutes ago, Not_a_Number said:

Err, it probably won’t thrill you that I made the conscious decision to not do division with remainders, right? I thought it diluted the mental model.

I usually keep division simple and define, say, 15/3 as “how much one person gets if we split 15 things fairly between 3 people.” Then we work using that definition for a good long while, we figure out what else we can do with it and how it relates to multiplication, and we extend it slightly for fractions. But I don’t do remainders until later, because I think they confuse the model.

No, the remainders were what was giving me pause! It felt like it was muddying the waters much too soon.  I don't have confidence that he will routinely identify division in a word problem or instinctively recognize that he's splitting a number into equal pieces.  I suppose I'll have to make my own problems for him for a while and move into an unrelated chapter in math (because he'll go nuts if we work tons of the same kind of problems, and I'm not confident enough to make a more diverse worksheet). I'd like him to really dive into what happens when he divides numbers outside of the 1-12 facts before we play with remainders.

[Not_a_Number]

Just now, medawyn said:

No, the remainders were what was giving me pause! It felt like it was muddying the waters much too soon.  I don't have confidence that he will routinely identify division in a word problem or instinctively recognize that he's splitting a number into equal pieces.  I suppose I'll have to make my own problems for him for a while and move into an unrelated chapter in math (because he'll go nuts if we work tons of the same kind of problems, and I'm not confident enough to make a more diverse worksheet). I'd like him to really dive into what happens when he divides numbers outside of the 1-12 facts before we play with remainders.

I can tell you what our own division sequence was: 

1) Small divisions via "splitting into groups." So, you know, 20/4 is what one person gets if we split 20 things between 4 people. This stage is done with lots of mental strategies. Like, if we're splitting between 4 people, we could, for example, split into two groups and then split again. If we know how to split 20 between 4 people, then splitting 24 between 4 people is pretty easy -- each person gets one extra thing. And so on, so forth -- I don't tend to insist on specific strategies and keep things conversational. 

2) Bigger divisions via place value and splitting into groups. If place value is not fully internalized (I have found this to be the case for most kids in my homeschooling class), poker chips are very helpful to represent 1s, 10s, 100s. We would then use this opportunity to do lots of trading up and down. 

3) The observation that "splitting into groups" is the opposite of "taking copies of a group." That is, division is the opposite of multiplication. Using that fact to check that that one's "splitting" is giving correct answers via multiplying. If the earlier facts are memorized, I'd do this with bigger numbers, so that a kid had to actually access the meanings and not simply regurgitate. 

4) The observation that "splitting into groups of 5 objects" gives the same answer as "splitting between 5 people." This is one of these things that's really obvious if you think of division as a purely abstract thing, but is REALLY not obvious if you actually start with a "splitting into groups" model. This one is best integrated with a good understanding of WHY multiplication is commutative: that is, WHY 4 copies of 5 is the same as 5 copies of 4. 

5) Extending the definition of division into integer fractions and using the intuition we've built up to understand when fractions are equal and how to add them. 

I will say that this sequence for us was interleaved with other stuff and took a few years. We then continued on with the same definitions for fraction multiplication as well. We're now well into algebra (DD8 can solve linear equations with fraction coefficients, can expand any variable multiplications like (x+2y)^3, and can solve quadratics via factoring) and still has not worked on remainders much. She can absolutely answer questions in which a remainder is called for, like "If we had 120 cookies and we split them between 11 people, how much does each one get if we aren't splitting cookies?" but it's not what she would call division. I just asked her this question, by the way, and I asked her whether this was division, and she told me it wasn't exactly -- division is when we figure out what one person gets when we split everything, whereas this question plays by different rules 🙂 . And she told me that each person got 10 cookies and that 10 were left over 😄. 

[medawyn]

2 hours ago, Not_a_Number said:

I can tell you what our own division sequence was: 

1) Small divisions via "splitting into groups." So, you know, 20/4 is what one person gets if we split 20 things between 4 people. This stage is done with lots of mental strategies. Like, if we're splitting between 4 people, we could, for example, split into two groups and then split again. If we know how to split 20 between 4 people, then splitting 24 between 4 people is pretty easy -- each person gets one extra thing. And so on, so forth -- I don't tend to insist on specific strategies and keep things conversational. 

We've done lots of this orally.  I've probably never really called it division.  He's particularly adept at anything in groups of 4 or 6 due to our family size.  Wouldn't want to have anything not shared equally!  Last night I asked him how many ounces were in 3/4 of a pound, and he asked if I was quizzing him in math or if I needed to know.  I said both, lol.  But he was able to tell me 12 easily.  He informed me that 16 was a perfect square, so of course there were four groups of four in sixteen, so 3/4 of a pound was obviously 12.  There was a strongly implied "duh, mom" in there, but since he didn't vocalize it, I didn't follow up.  

2 hours ago, Not_a_Number said:

2) Bigger divisions via place value and splitting into groups. If place value is not fully internalized (I have found this to be the case for most kids in my homeschooling class), poker chips are very helpful to represent 1s, 10s, 100s. We would then use this opportunity to do lots of trading up and down. 

This is where I think I need to start with him to check his understanding.  I'm very confident about his place value; he does almost all his mental math with various place value strategies.  I just need to work with big enough numbers that I'm confident he's actually dividing not relying on multiplication inversions.

2 hours ago, Not_a_Number said:

 

 

2 hours ago, Not_a_Number said:

3) The observation that "splitting into groups" is the opposite of "taking copies of a group." That is, division is the opposite of multiplication. Using that fact to check that that one's "splitting" is giving correct answers via multiplying. If the earlier facts are memorized, I'd do this with bigger numbers, so that a kid had to actually access the meanings and not simply regurgitate. 

 

2 hours ago, Not_a_Number said:

4) The observation that "splitting into groups of 5 objects" gives the same answer as "splitting between 5 people." This is one of these things that's really obvious if you think of division as a purely abstract thing, but is REALLY not obvious if you actually start with a "splitting into groups" model. This one is best integrated with a good understanding of WHY multiplication is commutative: that is, WHY 4 copies of 5 is the same as 5 copies of 4. 

5) Extending the definition of division into integer fractions and using the intuition we've built up to understand when fractions are equal and how to add them. 

I will say that this sequence for us was interleaved with other stuff and took a few years. We then continued on with the same definitions for fraction multiplication as well. We're now well into algebra (DD8 can solve linear equations with fraction coefficients, can expand any variable multiplications like (x+2y)^3, and can solve quadratics via factoring) and still has not worked on remainders much. She can absolutely answer questions in which a remainder is called for, like "If we had 120 cookies and we split them between 11 people, how much does each one get if we aren't splitting cookies?" but it's not what she would call division. I just asked her this question, by the way, and I asked her whether this was division, and she told me it wasn't exactly -- division is when we figure out what one person gets when we split everything, whereas this question plays by different rules 🙂 . And she told me that each person got 10 cookies and that 10 were left over 😄. 

This is all VERY helpful.  It suggests that my instincts are right to work on this more and in different ways before jumping into remainders.  He's very aware that remainders of some sort exist, because his favorite weekend hobby is to look at football scores in the newspaper and calculate how many touchdowns each team gets.  Since sometimes there are field goals (3 pts), safeties (2 pts) or missed extra pts (6 pts), he's pretty efficient at breaking a score down into groups of 7 and then debating with his dad how the team scored the leftover points.  But I'd like his mental model of division be be more robust before we formally work remainders.

[Not_a_Number]

4 minutes ago, medawyn said:

We've done lots of this orally.  I've probably never really called it division.  He's particularly adept at anything in groups of 4 or 6 due to our family size.  Wouldn't want to have anything not shared equally!  Last night I asked him how many ounces were in 3/4 of a pound, and he asked if I was quizzing him in math or if I needed to know.  I said both, lol.  But he was able to tell me 12 easily.  He informed me that 16 was a perfect square, so of course there were four groups of four in sixteen, so 3/4 of a pound was obviously 12.  There was a strongly implied "duh, mom" in there, but since he didn't vocalize it, I didn't follow up.  

This is where I think I need to start with him to check his understanding.  I'm very confident about his place value; he does almost all his mental math with various place value strategies.  I just need to work with big enough numbers that I'm confident he's actually dividing not relying on multiplication inversions.

 

 

This is all VERY helpful.  It suggests that my instincts are right to work on this more and in different ways before jumping into remainders.  He's very aware that remainders of some sort exist, because his favorite weekend hobby is to look at football scores in the newspaper and calculate how many touchdowns each team gets.  Since sometimes there are field goals (3 pts), safeties (2 pts) or missed extra pts (6 pts), he's pretty efficient at breaking a score down into groups of 7 and then debating with his dad how the team scored the leftover points.  But I'd like his mental model of division be be more robust before we formally work remainders.

It sounds like he’s intuitive enough with it that he ought to do all of this easily! I’m sure I’m the only one on here who did it in this order, so you’ll probably have lots of people say that he’ll be fine if you do remainders now (which is probably true.) All I know is that my personal strong preference is for making the model robust first.

[medawyn]

54 minutes ago, Not_a_Number said:

It sounds like he’s intuitive enough with it that he ought to do all of this easily! I’m sure I’m the only one on here who did it in this order, so you’ll probably have lots of people say that he’ll be fine if you do remainders now (which is probably true.) All I know is that my personal strong preference is for making the model robust first.

I think my biggest problem is that HE is intuitive, and I AM NOT.  So I'm always retreading material to ensure that it really is entrenched, since in my math education, I was moved on after very superficial understanding.  I think we'll pause and work with bigger numbers so I can really get an idea how how consistent his mental model is before jumping back into the BA chapter that introduces division.  Our next chapter is measurement, and he hates that, so we'll split our time working through both topics, and we'll both be happier.

[Not_a_Number]

2 hours ago, medawyn said:

I think my biggest problem is that HE is intuitive, and I AM NOT.  So I'm always retreading material to ensure that it really is entrenched, since in my math education, I was moved on after very superficial understanding.  I think we'll pause and work with bigger numbers so I can really get an idea how how consistent his mental model is before jumping back into the BA chapter that introduces division.  Our next chapter is measurement, and he hates that, so we'll split our time working through both topics, and we'll both be happier.

I think that the fact that you don't fully understand this math but WANT to and are working hard to understand it can be a real superpower, actually. People who've long since learned the material don't notice the discrepancies, because they've already integrated all the different kinds of understanding. But it was obvious to YOU that remainders are a different model, because you were learning alongside him, and it feels different to YOU. 

I have tediously and carefully taught my very accelerated DD8 in a way that does not leave gaps and that strives to be extremely consistent. I have been beyond pleased with the outcome -- I'm sure she would have excelled with any curriculum I'd have handed her, but I don't think she would have attained as deep an understanding without the mental models we've worked hard to build up.

By the way, the reason I was able to do so is because I've worked with enough kids by now to realize where the common issues with mental models appear. If I had just been myself without any teaching experience, I doubt I would have known where the steps that might trip one up are. It all feels too fluent to me 🙂 . 

[Eilonwy]

10 hours ago, Not_a_Number said:

But I don’t do remainders until later, because I think they confuse the model.

Do you choose questions with no remainders to avoid them initially, and then gradually work into fractions where you can split things into pieces anyway?

Edited March 20, 2021 by Eilonwy

[Not_a_Number]

2 minutes ago, Eilonwy said:

Do you choose questions with no remainders to avoid them initially, and them gradually work into fractions where you can split things into pieces anyway?

I never did any remainders at all. We did questions with no remainders for a good long while, then we did fractions. Then at some point I checked whether DD8 could do remainders, and she could 😛 . I mean, she used them to calculate fractions bigger than 1 anyway, since she found it more intuitive to first split off the integer amounts... we never did division except as "splitting into groups/splitting between containers" so that was always how she thought of it. 

Edited March 20, 2021 by Not_a_Number

[Publia]

Regarding supplementing, Publius is the math instructor in our house and he creates his own math lessons for our children. These are his thoughts on introducing division and division with remainder, in case they’re helpful. 🙂

A key idea in our discussion of division was to understand division as a counting. For example, in this perspective we would consider 12/3 to mean “how many times must we count 3 to get 12?”  The student who is comfortable counting groups has no difficulty with this approach of understanding *division as an expression of counting* and this approach also reinforces the idea of counting groups.

Notice also that with this approach it easy to explain the term "denominator"---literally, this means *what we are counting*. It is the *denomination* in our division. In this perspective, we are interested in expressing a given quantity (the numerator) as a counting of the denominator.

We did a lot of work circling around this idea of division as an expression of counting and also incorporating other elements such as visualizing division and patterns in division. At some point of comfort we moved onto bigger divisions by breaking these down into smaller divisions using distributivity (similar to what we did in multiplication).

We then moved into division with remainder because this is easy to step into from the division-as-a-counting perspective. For example, when the student is comfortable with an expression like 24/8, it is natural that at some point you (or the student) will say, okay but what if we want to count 8 into 25 instead? What is 25/8? The answer is 3 with a remainder of 1.

At this point we considered what does it mean to say "3 with a remainder of 1"? This is the fun part of the discussion (and it is a good preview for the fractions discussion). We can make the statement more clear with a purely mathematical expression of what is going on.

For this, we focused on counting.  We are simply counting 25 with 8. We get most of the way with 3 eights; we get to 24. But we haven't finished counting. We've still got 1 left to count and we are still counting with 8.

So we asked, how do we express the idea of counting 1 with 8? What is 1 in terms of 8? We know that 8 * 1 = 8. So one is (what we will call) an eighth of eight.  1 = (1/8)*8. This lets us rewrite the division 25/8 without the remainder language or, you could say, with the remainder language more clear:

25/8 = 3 + 1/8.

After counting three eights, there is still this piece of eight that we must count for the full 25 that we want.

Now we can think of division in two steps: first a counting of the whole denominator: for example counting three eights to get to 24. And then, if necessary, a counting of a part of the denominator to complete the counting of the whole numerator.

Notice also from 25/8 = 3 + 1/8 we can see the complete counting of 25 by 8 like this:

25 = 3*8 + (1/8)*8 = 24 + 1.

I think this is a worthwhile discussion for two reasons. First it enables the student to make sense of divisions like 25/8 as well as 24/8. And also it is a great case featuring the practical need to count a part of a unit (here 8), so it is a good preview for the fractions discussion.

Additionally it lets you get into the cyclic remainder patterns. For example, you can discuss how the range of possible remainders for division by 8 will be 0 through 7 and so on. It's a neat pattern for numbers and you can think of ways to use this pattern practically. For example, if you wanted to sort numbers (or things designated by numbers) into 8 different categories.

[Eilonwy]

22 hours ago, Not_a_Number said:

She can absolutely answer questions in which a remainder is called for, like "If we had 120 cookies and we split them between 11 people, how much does each one get if we aren't splitting cookies?" but it's not what she would call division.

This is really interesting, and yes, division with remainders does seem to take a detour from the division of model of dividing x items evenly into y groups. DS9 completed that particular chapter in BA recently, and I think he does still need more reinforcement of what division is. 

Edited March 20, 2021 by Eilonwy

[Not_a_Number]

51 minutes ago, Publia said:

We then moved into division with remainder because this is easy to step into from the division-as-a-counting perspective. For example, when the student is comfortable with an expression like 24/8, it is natural that at some point you (or the student) will say, okay but what if we want to count 8 into 25 instead? What is 25/8? The answer is 3 with a remainder of 1.

Except that I think that 25/8 is 3 1/8, full stop. What ARE remainders, anyway? Are we actually counting how many 8s there are in 25, or are we not? Why did we decide to stop at 24 and just leave that 1 out? What's wrong with that extra 1 that we can't split it? 

I'm not actually saying I don't understand the answers to those questions, but if a kid is still struggling with the concept of division, I worry that taking a detour into a totally new definition might confuse a kid. Whereas if you just move straight to fractions, it's not particularly hard to circle back to remainders when need be. 

[Publia]

From DH 😉 Just to clarify, I am not advocating the terminology "with remainder." To the contrary, part of the problem on this topic is the ambiguity of that term but I think we can avoid that issue in a worthwhile way. What I described in the prior post was offered as an option to enable students to make sense of divisions "with or without remainder" (and without even having to use that term). It might come in handy. For example in a discussion of division a student asks---but what about 16/3?

A few additional comments.

- Certainly I agree that the teacher is in the best position to know the best ordering of topics and their timing. Working through fractions and then circling back to division clearly works.

- I think it is beneficial to the student to see the same topic treated from different perspectives early and to become familiar with the idea that one perspective might be useful in one context and another in a different context. The different perspectives often lead to interesting insights and I think that this versatility is good to practice early on. To the extent that a different perspective did cause confusion, that would be an opportunity to make the student's model more robust.

- As I mentioned in the prior post, since we were coming at this from the perspective of counting the denominator, the discussion flowed naturally to the idea of fractions because we were confronted with the need to count a part of the unit. For example to count 1 with 8 (so we needed the idea of counting pieces of eights 🙂 ). So division here offered a way to get the idea of fraction rolling.

[EKS]

On 3/19/2021 at 7:29 AM, Not_a_Number said:

Err, it probably won’t thrill you that I made the conscious decision to not do division with remainders, right? I thought it diluted the mental model.

I think this might be a better approach.  When you start out by introducing division with remainders, it sets the stage for divorcing division from fractions in the student's mind, and then you end up having an uphill battle to reintegrate the two.

[Not_a_Number]

42 minutes ago, Publia said:

From DH 😉 Just to clarify, I am not advocating the terminology "with remainder." To the contrary, part of the problem on this topic is the ambiguity of that term but I think we can avoid that issue in a worthwhile way. What I described in the prior post was offered as an option to enable students to make sense of divisions "with or without remainder" (and without even having to use that term). It might come in handy. For example in a discussion of division a student asks---but what about 16/3?

A few additional comments.

- Certainly I agree that the teacher is in the best position to know the best ordering of topics and their timing. Working through fractions and then circling back to division clearly works.

- I think it is beneficial to the student to see the same topic treated from different perspectives early and to become familiar with the idea that one perspective might be useful in one context and another in a different context. The different perspectives often lead to interesting insights and I think that this versatility is good to practice early on. To the extent that a different perspective did cause confusion, that would be an opportunity to make the student's model more robust.

- As I mentioned in the prior post, since we were coming at this from the perspective of counting the denominator, the discussion flowed naturally to the idea of fractions because we were confronted with the need to count a part of the unit. For example to count 1 with 8 (so we needed the idea of counting pieces of eights 🙂 ). So division here offered a way to get the idea of fraction rolling.

Yeah, we might be saying the same thing in different ways here! We did deal with the idea of remainders via fractions, and it turned out to be very robust.

However, we worked on division with integer answers for a good long while before starting fractions and weren’t sorry. There turned out to be LOTS of intuitions to build before we had to split things up into non-integers. It was excellent training for place value, the different models of division (partitive and quotitive), distributivity, the idea that it’s the inverse of multiplication... there was a LOT there. And spending the time on it did make fractions a lot easier, since DD8 had very robust intuitions.

 

35 minutes ago, EKS said:

I think this might be a better approach.  When you start out by introducing division with remainders, it sets the stage for divorcing division from fractions in the student's mind, and then you end up having an uphill battle to reintegrate the two.

Exactly. You put that very clearly. I wanted one unified model of division that felt really robust, since I knew how bad kids’ models of fractions usually were in college. And I’ll say that DD8 has found fractions easy and intuitive by using my sequence. And along the way, she figured out how to use remainders anyway, because they came in handy... she just doesn’t think of that as a completed division problem.

Edited March 20, 2021 by Not_a_Number

[medawyn]

4 hours ago, EKS said:

I think this might be a better approach.  When you start out by introducing division with remainders, it sets the stage for divorcing division from fractions in the student's mind, and then you end up having an uphill battle to reintegrate the two.

 

3 hours ago, Not_a_Number said:

 

 

Exactly. You put that very clearly. I wanted one unified model of division that felt really robust, since I knew how bad kids’ models of fractions usually were in college. And I’ll say that DD8 has found fractions easy and intuitive by using my sequence. And along the way, she figured out how to use remainders anyway, because they came in handy... she just doesn’t think of that as a completed division problem.

So this morning, DS8 comes downstairs to inform me that he is going to be very good at division and could I please quiz him but "I don't want any 1/3s or 1/4s or anything." I inquired further, because it was 5:45 am, and I did not want to start my day by giving the wrong math problems before coffee. He explained first that he really liked splitting numbers into groups, because it was a neat way to think about how many ways you could split up a number but, "Mom, if you tell me something like 37/6, then I can't really finish the problem.  Because 36 is a perfect square and there are 6 groups of 6 in 36, but then you have one leftover.  I'd have to cut it up and give every group 1/6.  You can cut up cookies, but it would be really hard to cut up a marble, and I'm having groups of marbles in my head."

Okay then.  I think I can panic less about introducing division and work him towards problems like 216/8.  We'll just forget remainders for the time being, but maybe we'll start chatting about fractions (eep).

[Not_a_Number]

7 minutes ago, medawyn said:

So this morning, DS8 comes downstairs to inform me that he is going to be very good at division and could I please quiz him but "I don't want any 1/3s or 1/4s or anything." I inquired further, because it was 5:45 am, and I did not want to start my day by giving the wrong math problems before coffee. He explained first that he really liked splitting numbers into groups, because it was a neat way to think about how many ways you could split up a number but, "Mom, if you tell me something like 37/6, then I can't really finish the problem.  Because 36 is a perfect square and there are 6 groups of 6 in 36, but then you have one leftover.  I'd have to cut it up and give every group 1/6.  You can cut up cookies, but it would be really hard to cut up a marble, and I'm having groups of marbles in my head."

Okay then.  I think I can panic less about introducing division and work him towards problems like 216/8.  We'll just forget remainders for the time being, but maybe we'll start chatting about fractions (eep).

Hahahah, I think he's doing fine, yes 😄. I'd do bigger problems and talk about fractions, as you say, but it sounds like he pretty much gets fractions already. Just tell him fractions are divisions and I think he'll sail on ahead. And you can discuss with him when fractions are equal, if you want... or even start on working on fractions "officially." He sounds ready to me.