Inside the Build Math Minds Facebook group, we had a post last month asking for a way to teach subtraction with borrowing to a 2nd grader.
I’m Christina Tondevold, The Recovering Traditionalist, and today we are going to take a look at Why Subtraction With Borrowing is Bad in our quest to Build our own Math Minds so we can Build the Math Minds of our kiddos.
Watch the video or read the transcript below:
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Build Math Minds Facebook group
Investigating an Understanding of Place Value video by Marilyn Burns
T-Shirt Factory unit by Cathy Fosnot
Full Contexts for Learning Mathematics kit by Cathy Fosnot
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I’m the creator of the Build Math Minds Facebook group and I’m also known as The Recovering Traditionalist. Now, I call myself that because I used to teach math very traditionally. I would stand in front of the class and give them the step-by-step directions and procedures to solve certain math problems.
We would do the problems together, and then I’d give them the set of problems to work on by themselves. I quickly learned that that did not work for a lot of my kids, but I didn’t know what else to do. It wasn’t until I went back and got my master’s degree that I was introduced to Cognitively Guided Instruction and lots of other insights that changed the way I personally thought about math problems and the way that I taught it to my students.
So I know what it’s like when we are that teacher who’s doing the best we can with the knowledge we have right now.
We never really sit down and plan a lesson thinking, “I’m gonna make this as bad and as horrible for my students as possible.”
That is not what goes through our mind. We always have the best intentions for our students, but as we learn more, we change our teaching.
One thing I used to teach was subtraction with borrowing. I didn’t know at the time how it wasn’t the best thing for my students…but at that time I was doing the best I could with the knowledge I had at the time.
I know that I would probably be a teacher right now who didn’t know anything was wrong with having 2nd graders do subtraction with borrowing, if I hadn’t had the experiences that I had many years ago when I went back and got my master’s degree. I would still be one of those who was doing that algorithm and teaching kids those steps.
So today, as our hot topic, we’re going to talk about why subtraction with borrowing is bad and what else we could do when the textbook and people around you are using it. No matter what, even if we all say, we shouldn’t be doing subtraction with borrowing with young kids, kids are still going to be exposed to it whether it’s from another teacher, a parent, a sibling whatever it might be. So let’s first talk about why it’s bad, and then we’ll talk a little bit about some alternatives.
The big concern here for me is that our textbooks are not aligning with our standards.
If you actually go into the standards, you will find that multiple grade levels talk about having kids use strategies to help them when they are adding and subtracting. So 1st grade has it, 2nd grade has it, 3rd grade has it. There is not a mention of the algorithm, which is where that borrowing comes in. The algorithm does not appear until third grade…and it just says algorithms. The standard algorithm, which we all associate with the borrowing or better known now as regrouping, happens in 4th grade.
Now, unfortunately, all the textbooks still have the traditional/standards algorithm for subtraction at 2nd grade. They get away with it because they say it’s a strategy “based on place value.”. Kids need to be using strategies based on place value in 2nd grade, but is that the standard algorithm really?
Does the algorithm really emphasize place value? I say no. In fact, it de-emphasizes place value. It puts no emphasis on place value whatsoever and makes it harder for kids to develop their place value understanding. So let’s take a look at this 57 – 19.
The traditional way that we used to teach this, I know we use better words and everything now, but we would tell kids, “you can’t take 9 from 7. So you have to come over and borrow 1 from the 5.” That’s not the terms we use now, but there may be people who are still using that.
First thing, you can take 9 from 7. That’s why kids struggle so much with negative numbers. You may not want to teach negatives, but at least say, “if we did take 9 from 7, we would go into the negatives.” At least put it in their brain that it’s possible to take a larger amount from a smaller amount.
Also, there was no mention of what the numbers really are. When I take the 1 from the 5, the 1 is really a 10. The 5 is really a 50, but we don’t say it in that way, or we didn’t when we grew up. I know that we do better now with explaining it a little better for kids, but still, the way that it looks if all you look at is just the symbols, that 7 magically becomes 17 somehow, but yet the 5 becomes a 4. It doesn’t help kids really see any of the place value understandings about these numbers. Then even in the next steps of when we go to subtract, we say it as “17 – 9, 4 – 1.”
There is no place value understanding at all in the traditional algorithm unless you, as the teacher, are bringing it in. I know many of you are, but the hard part is that we know that kids don’t always hear everything that we say. Instead they’re seeing what’s on the paper or on the screen, and this is what they see. They don’t see any place value understanding whatsoever with the traditional algorithm. That is concerning because place value is a huge, huge piece in 1st, 2nd, 3rd grade.
We need to be building their place value understandings. And as the standard says, strategies based on place value. I do not believe the traditional algorithm focuses on place value. So some people will say that it’s based on place value so we can still do it in 2nd grade. I don’t think so!
A quick way to check your students’ place value understanding is a task you can see in this Marilyn Burn’s video.
So at the beginning of the clip it’s showing this girl, Cena, showing that she’s got this wonderful place value understanding. And then Marilyn sits down with her later, and one of the tasks that she has is they’ve counted out 18 of these little cubes and the girl wrote down the number 18 on the piece of paper. Then Marilyn asks, “so these were 18 cubes. You told me that we had 18 here. You counted them out. Can you show me with the cubes, this number right here, the 8? Show me the 8 with your cubes”. She counts out the 8, right? But then she asks Cena to show the value of the 1…and Siena put 1 of her little cubes up there.
When Marilyn presses her and says, “but I see you’ve got the 8 here and 1 here, but when we counted you told me all of these cubes make 18. So where do these other ones belong?” She just pushes them to the side. She’s like, “I don’t know. It’s one and eight.”
So that’s a quick little check to see do your kids actually understand the value of those digits? If they don’t, we’ve really got to focus on building that place value understanding way before we ever introduced that traditional algorithm because that traditional algorithm is so easy and so quick and it does not require any thinking hardly at all, especially place value thinking.
The algorithm doesn’t show the decompositions or compositions of the numbers.
This is what it looks like when we set up 57-19 and show the algorithm steps.
Kids who kind of grasp that number sense of place value, and they have that already, they understand what these mean, but for our struggling kids, especially when we introduce it too early, all that they’re seeing on this is now they see 417 -19 and somehow that magically becomes 38.
So overall, in general, there are two complaints we hear from parents, from teachers, & administrators around mathematics: Kids struggle with place value and they have no sense of numbers. They can’t even tell if an answer is reasonable or not. Well look at that. It’s 417 – 19, and somehow it becomes 38!
Just the symbols there make it very hard for kids to actually tell reasonableness. So instead, an alternative way in the early grades before we actually introduce the algorithm is to set it up similar to this.
Some people will call this expanded form. Some people call it partial sums. I don’t care what you call it, I just want you to understand it’s a great way to show the kids what’s actually happening in that algorithm before we ever introduce the algorithm.
We can break it apart and show that 57 is 50 and a 7. That 19 is a 10 and a 9. And
We’ve got to take away a 10 and a 9 from the 50 and 7 somehow. When we start with the ones and we say “7 – 9,” and some kids will say, “we can’t do that.” So we want to help them understand that we can, but we don’t have enough. So let’s come over and we’re going to “borrow” from the 50. But as many people mentioned in the comments, we’re not borrowing because we never really give it back. So the term of regrouping came about.
The reason we call it regrouping is because we are taking the 57 and we’re just grouping it in a different way. Who says that it has to be a 50 and a 7? I can regroup 57 into anything I want. The algorithm regroups it in a specific way, but you can really regroup it any way you want.
So with the algorithm we take 10 from the 50, and then we see that we’re left with 40. And that 10, we regroup over into the ones, and now we have 17 and 40. But the key here is not just the steps and procedures that we’re sharing with kids. It’s to help them see that we’ve decomposed 57. We broke it up into chunks that were easier to subtract from, but that it still is 57. We still have 57 there. We’ve just regrouped it. We decomposed it, but I can compose it back together, and I still have 57, the original amount I started with.
That’s a really big thing to emphasize with the kids is that when we’re doing these strategies, we’re doing things where we’re breaking apart numbers, we’re decomposing, but we still have to know that we can compose it back together.
We haven’t changed the problem. It’s still 57 – 19. I’m just regrouping it into better, easier numbers to subtract a 10 and 9 from. There’s nothing to say you have to do the 40 and the 17. You could regroup it in lots of other ways, but this one leads us into the understanding of the algorithm. This is important to have kids do with all sizes of numbers.
The important part is talking about if we still have 245. I didn’t leave it on here, but I’ll admit it. When I was typing up this slide I had 100, and then 140, and then 15. Because I was trying skip a step and not show the 30, and I just moved the 30 into 130. But when I did that, I made an error. I left it as a 40 and put 140. It wasn’t until I went to put the circle around it that I realized that doesn’t add up to 245. It doesn’t compose back to the original amount!
So how do we actually help kids do all of this?
The biggest thing is to focus on place value. Now the hard part is, most of the activities that we have in our textbook, and even when you go Google place value activities, the emphasis is usually on the place.
What place is the 4? Here’s another worksheet full of just having the kids put hundreds, tens, or ones for whichever place the underlying digit is in. That is emphasizing the place not the value of the digits, and we need to spend more time emphasizing value.
Here’s one that’s on the right track, but I still have a little bit of issues with this.
So I love that we’re bringing in place value blocks. That helps emphasize the value of the digits. But the hard part about this activity is that you could possibly get a false sense of security that your students understand place value just because they could match these things up.
The card sticking out says three tens, five ones. I could just look at digits, not even thinking about value. It has a 3 and a 5 and the card with the 30 has a 3 and a 5…and there’s only one base 10 block card that has 3 and a 5.
I don’t even have to really think about the values. All I’m looking for are 3s and 5s. I’m not looking for a 30. I’m not looking for three tens. I’m looking for a 3 and a 5. That can give you a false sense of security thinking your kids actually do understand place value when all that they were looking at were those digits.
So instead of the traditional place value activities that just emphasize place and digits, I want to encourage you to emphasize value.
How do you help emphasize value? It’s by doing lots and lots of bundling and unbundling activities. I don’t care what you have kids bundle, but getting things where you’re counting a big set of stuff, and you need to bundle them into groups of 10 to keep them organized. Then when we get 10 groups of 10, we’re going to bundle it into 100. But it’s also not enough just to bundle. Kids need to unbundle things as well.
One of my favorite activities comes from Cathy Fosnot’s work, the Context for Learning. There’s a unit about the T-shirt Factory. It’s all about how they have to bundle the t-shirts into bundles of 10 so that it makes keeping their inventory easier. Then as they sell t-shirts, they unbundle the t-shirts. It’s a fabulous, fabulous story to help you have kids see a context that goes along with the bundling and unbundling.
Let’s come back to this one really quick.
That card that said 3 tens and 5 ones, I’d switch it up just slightly, and flip it to say 5 ones, 3 tens. Sometimes it’s just as easy as that because the kids will want to put that as 53. They’re so used to us, and our textbooks, always putting it in order. We always say the tens and then the ones, so all they have to do is grab the digits and then put them in order.
I’d also modify the place value blocks for that same one. It’s still 35, but now it’s 2 tens and 15 ones. That’s important because that’s a big part of that standard algorithm. That’s what we’re doing to help kids move into the algorithm. It’s not just seeing 35 as a 30 and a 5, or 3 tens and 5 ones. We need them to be able to see it as a 20 and a 15, or even a 10 and a 25.
We want kids to be able to decompose and compose numbers back together. We don’t want to give them easy decompositions all the time, because unfortunately they’ll just look at the digits instead of actually building their place value understanding.
Another activity is having kids collect box tops, back before they became all digital. This was something I made my own personal kids, when they had to turn in their box tops, we would organize them into groups of 10, and we’d put them into little trays. Then when they had those 10 groups of 10, we would put them into a baggie, and then each baggie was worth 100.
So have kids collect anything, like pop tabs off cans, have kids bring those in and ask if your class can be the ones who count them for the whole school. Collect anything where you get the opportunity to have kids bundle and then even unbundle those together.
Another one of my favorites are these Base 10 riddles.
These come from John Van de Walle and friends in the book “Teaching Student Centered Mathematics.” It’s all about giving kids these base 10 problems but mix up a little, like this first one, “I have two tens and 13 ones.” The place value activities they tend to get, it will just say, “I have 3 tens and 3 ones. What am I?” All the kids have to do is just pull the digits out.
But when we put it in these mixed up ways, they really have to think about that bundling and unbundling to figure out the amounts.
Place Value Check-in
So how many tens are in 53?
Now, how many tens are in 243?
We are so used to just asking kids what digit is in the tens place. So in the first one, kids will naturally say 5. But then when we give them the second one, they want to say 4.
They want to say that there’s 4 tens in 243 because they’re so used to looking at what digit is in that place value. Now, if I asked what digit is in the tens place, the answer would be 4, but I’m asking how many tens are in 243. That’s a completely different question, and it trips up even adults.
It’s really 24. We can put it into all kinds of contexts, but let’s put it this way. If I went to the bank, and I wanted to get cash out, and I asked for $243 in tens and ones, how many tens would I expect them to give me?
I don’t want just 4. I need 24 tens to make the 240, right?
If I have 243 cupcakes, and I could put 10 cupcakes in a box, how many boxes am I going to need to hold my cupcakes?
I’m basically asking how many tens are in 243. We have these scenarios all the time in our lives, but we don’t ever think about it in this way. Those kinds of scenarios are really asking, how many tens are in 243?
Here’s a little bit tougher one. How many tens are in 1,037? Most people will say 3, right? If I went to the bank and asked for $1,037 in just tens and ones, how many tens should I get from them? If I have cupcakes, if I baked 1,037 cupcakes, and I could put them into boxes of 10, how many boxes am I gonna need? If I think about the t-shirt factory, Grandma Masloppy has 1,037 t-shirts, and they’re bundling them up to put them into inventory. For every 10, they bundle them up with a rubber band. How many rubber bands are they going to need? If it’s stickers on sheets of paper, and you can put 10 stickers on a sheet of paper, how many sheets of paper am I going to use?
I don’t know what your scenario is. Think about a scenario that would resonate with your kids, but that’s a big piece that we want kids to be able to understand. It’s super important because of problems like this.
They gave this problem to 6th graders, and only 9% of sixth graders answered that correctly. The majority of those 6th graders in that study tried to do the operation. They tried to do long division, and you know how horrible people are with the long division.
All that problem is asking is how many hundreds are in 48,638?
We want kids to be able to do operations, whether it’s addition, subtraction, multiplication, or division, but we also need to build their place value along the way.
But our algorithms are devoid of place value.
They don’t help kids decompose and compose numbers.
It is just a bunch of steps that kids aren’t really connecting to and once the algorithm has been introduced, it’s very difficult to get them away from it because it is so easy. It’s so easy and it doesn’t take any thinking. It takes no number sense. It doesn’t take any place value understanding. It takes none of that.
We want kids to be able to use place value, use their understanding of numbers to help them do those operations. No matter how hard you try to avoid the term ‘borrowing’ or not introduce the algorithm too soon, you’re going to have students who have a different teacher, a parent, a sibling, etc that introduces that terminology or the algorithm to them. So as much as we can, we want to connect what they’ve seen with that algorithm to place value, build their understanding of place value, show the connections between them so that kids aren’t just seeing it as another separate thing. We want them to use their understanding of numbers of how you can decompose numbers, compose numbers, and of course their place value understanding.
So as always, I hope that this helps you build your math mind so you can build the math minds of your kiddos.
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