My first post of 2018 is a record of some rambling thoughts about remainders. I may or may not come to a final moral here, so consider yourself warned.
What has prompted these ramblings today was reading this excellent post by Kristin Gray about her own thoughts on division and remainders. In that post, I saw the following:
7÷2 = 3R1
For some reason, this bothered me. For some reason it’s always bothered me. Today I think I realised what the problem was: In my head “7÷2” is a number, and “=” indicates that two things are equal, but 7÷2 can’t be equal to 3R1 because 3R1 is not a number. It is only today that I realised that 3R1 isn’t a number.
How do I know 3R1 isn’t a number? Well firstly it’s two numbers. One is a number of groups and the other is a number of objects. I don’t even know how big the groups are — it could be 3 groups of 2 and one left over, or 3 groups it could be 3 groups of 7 and one left over, or 3 groups of 200 and one left over. I can hear people saying that actually all plain numbers could mean any number of different units, and a 7 could be 7 cm or 7 ducks or 7 groups of 200. But the 1 here is definitely 1 of something, while the 3 is some unknown size of groups of that same something. That seems like a totally different kind of unit issue than with a plain ordinary number.
Also, if it really is a number, then I should be able to place it on a number line, but where does it go? The 3 I certainly know where it goes, but what about the 1. Where does that live? It lives in a completely different land to where the 3 lives, and I can’t really put it on the number line until I know how big the groups are that the 3 represents.
It occurs to me that this is a good way of transitioning to a fraction sort of idea. The fact that the 1 is small relative to groups of size 200 and large relative to groups of size 2, and needing to encode this relative size would lead nicely to a need to write this as 7÷2=3+1/2. What an interesting idea.
My other really big issue is that the “=” sign in this context doesn’t work the way an “=” sign works. If 601÷200 = 3R1 and 7÷2=3R1 then usually the properties of “=” would mean that 601÷200 = 7÷2. But they aren’t equal. I suppose they both produce 3 groups with 1 left over, but that 1 is very different in size relative to the group in each situation! So they’re not really equal are they? Actually, this idea is going back to the same idea I had with the number line, where you need to encode the relative sizes.
My final problem is that if it really is a number, then surely you’d be able to do operations on it. But I don’t really know how you’d do that. You’d expect that if the 3R1 came from 7÷2, then 2*(3R1) would produce 7, but if it came from 601÷200, then what would 2*(3R1) even mean? I’ve been trying to figure it out, but to no avail.
It might be possible to do addition and subtraction, if you knew the groups were the same size. In that case 3R1 + 5R3 would be 3 groups and 1 plus 5 groups and 3, so it should be 8 groups and 4. So 3R1+5R3 should be 8R4. It seems you add the two numbers separately, which is actually super interesting. I’m still a bit worried about what would happen if the groups were size 4, say, because then 8R4 is actually the same as just 9. So now it seems like they are a lot more like numbers than I originally thought. This seems like a very interesting thing to investigate.
As you can see, I’m rather puzzled by remainders and where they stand numerically. I get that the idea of dividing a collection into groups requires us to have a concept of remainder. I just feel weird writing it in this way because these symbolic representations feel like they ought to make numerical and algebraic sense, but here they don’t.
In Kristin’s post, she floated the idea of the equation being not 7÷2=3R1 but instead 7=3*2+1. This second equation I feel completely comfortable with. It is 100% clear what the numbers are doing and the “=” really is acting as an equality here. I still wonder if there’s a more helpful way of representing the division-producing-a-remainder thing though.
And maybe that’s another issue I have with it, that this statement “7÷2=3R1” is about doing an operation and producing a result, as opposed to declaring a relationship, which is what I have come to believe the “=” sign is for. By using something that is not like a normal number and just encodes a description of an answer, are we reinforcing that “=” means “here is the answer”? I don’t know what to do with this question yet, or if it even really matters.
So there you go. I’ve rambled through a whole lot of thoughts and worries about remainders. I don’t have any conclusions or morals or recommendations. But it’s certainly helped me to try to write it all down. I hope it helped you to read it. I’d love to hear your own thoughts on it.