## How to Teach Subtraction

If you are of a certain age, you’ll remember when you learned a rule about how to do subtraction. It was called “Borrow and Payback.” It was a neat little trick to get the right answer to those tricky take away sums. For example, if you were given a random sum (as you often were) 43 – 17, you would start by trying 3 – 7 and realising that this was impossible so you remembered the rule. Stick a one in front of the 3 to make 13 and add 1 to the 1 of 17 to make 2. (See below)

After that, you could get back to normal and find the answer. I assume, no one ever explained why this was the case, but it got you the right answer, which, at that time was more important that understanding how you got the right answer.

If you are younger, you may have come across the concept that is nowadays used throughout most of the English speaking world – the concept of renaming. For the same sum above, one builds upon their knowledge of renaming numbers to find out he answer. We know that 43 is a number containing 4 tens and 3 units. By renaming, we know that this is the same as 3 tens and 13 units. Therefore, we don’t need any tricks. Renaming 43 to “Thirty Thirteen”, we can do the subtraction easily enough.

Both methods have their logic. One can explain the “Borrow and Payback Method” mathematically. In this method you are adding 10 to both numbers in the sum. If I write it a little differently, it might be clearer:

- Start off with the sum: 43 – 17 = ?
- Add ten to both numbers: 43 + 10 is written as 4(13) and 17 + 10 is written as (1 + 1)7

In a way, we have sort of renamed both 43 and 17. However, since this is hard to explain, if one simply learns the rule off by heart, they have a fail proof method for subtracting numbers. The trouble of having to learn things off by heart is that if you rely on this, very soon in the world of maths, you’re going to have a lot of stuff to remember, rather than building on things that you already know.

I’m often challenged on this view and the two main arguments I get are as follows:

- “Weaker” students find the “Borrow and payback” method easier so for their sake, we should let them get on with it.
- That’s all well and good for two digit numbers, what about 1,000,000,000 take away 1?

Starting with the first argument, ignoring that I find it a little disrespectful, I cannot buy an argument that seems to be: “if something is easier to do, it’s better” I feel it’s a very short term view. As subtraction is a very basic concept, if we’re already finding shortcuts by learning a series of rules off by heart, in the long term, these pupils are going to get more and more confused. For example, when they get to long division, they’ll need to learn another rule off by heart, namely: divide, multiply, subtract, bring down. That’s two things to learn off with no obvious logic. Then let’s not forget formulae like area of shapes, calculating speed, distance and time and learning off ways to compute interest rates before they finish primary school. That’s a lot of stuff to learn by rote and it’s only one subject and we’re not even in secondary school yet!

The second argument puzzles me because if we are teaching subtraction properly, it shouldn’t be an issue. I suspect teachers are still using the term “borrow” in their teaching. I’m pretty sure teachers in Ireland still might teach our example above using the following narrative:

43-17. 3 take away 7, you cannot do so you

borrow a ten. That leaves 3 tens and 13 units…. and so on.

Using this method, if we try 100 – 8, the teacher might say:

0 take away 8, you cannot do so you borrow a ten… oh, there are no tens so we need to go to the hundreds and borrow there. That leaves zero hundreds and 10 tens and 8 units. Now let’s go again, we borrow a ten and that leaves 9 tens and 18 units.

No wonder the children get confused as the digits expand. Try doing a million take away one using this language! Why not simply rename the number. So:

100 – 8. Zero take away 8 we can’t do so we need to rename the number. 100 is the same as 9 tens and 10 units…

and so it goes on. Picking a bigger number, e.g. 2,002,010 – 123,456, we can rename easily enough if you take it in logical steps.

I think if we spent more time renaming numbers before tackling operations, we’d be doing ourselves a big favour. I also think we need to ban the word “borrow” from subtraction. Even the word “swap” is better. While the above example might seem difficult initially, if you follow simple Place Value rules as you go through the problem, it should be easy enough to deduce where you’re going. At least, there is logic to what’s going on!

If one insists on defending the Borrow and Payback method, at the very least, the teacher must explain what is happening. Using our big number above 2,002,010 and taking 123,456 away from it, the Borrow and Payback can be done quickly but I find it hard to explain why I keep adding one to the bottom digit each time. The first time I’m adding 10 to both numbers, the second time I’m adding 100 to both numbers, the third time 1,000 and so on. Yes, it works but even with an explanation of what I’m doing, it feels a little odd.

Subtraction seems to bring out heavy emotions in the teaching world. It seems to be the first hurdle where students falter in primary school maths. I have witnessed teachers shouting at each other defending their own points of view. There’s suggestions that we should teach both methods to children – a sort of compromise between the old and the new but I’d argue that’s like a doctor today sprinkling vinegar over a sick patient as well as giving him modern drug treatment because back in the day, sprinkling vinegar often worked though they had no idea why! I guess my final point about this is to ask the “borrowers” to teach me their method. I promise I’ll ask “why” throughout and you’d better have a decent answer!

**Last Update: ** August 9, 2017

Pam4th July 2012 at 9:11 amI’ll preface this comment with the caveat that I’m not an expert

in teaching Maths to children so forgive me if I’m making incorrect

assumptions. I do however have 2 children who have learned subtraction in the

‘new’ way and I learned in the ‘old’ way so I do have some experience.

I would argue that there is no right way in this – I think that

both have their place and in fact I wouldn’t see that they are as different as

the above suggests. I think you’ve hit the nail on the head in one key point

though and that is how the subject is taught. The assumption that the old way

was not explained at all, but was simply a rule is not true, certainly in my

case. The assumption that the new way is not sometimes taught as a rule is

probably also not true. What it comes down to in both cases is the manner in

which it is taught. Learning to recognise the rule and applying it, without

having some understanding of what that rule is achieving and how, is not good

regardless of what the situation is.

I think one key factor to remember is that people see things in

different ways, particularly in Maths, and so, approaching problems from a few

vantage points would in fact be a positive thing and not a compromise. I am often surprised when covering what I

would regard as basics topics, how many people don’t get it when I try a

particular approach, and often trying a slightly different approach can help

more to see what I’m getting at.

I would see the two ways as complimentary and would argue that

regrouping or renaming and borrow and payback both have a place in Maths

education in primary school. In both

methods you are moving along one of the numbers taking place value into

account. I think there will be as many

children who actually get what you are doing with the new method, as there are

who got the old method, and there will be as many who go through the motions

without really thinking much about it, for both methods.

I’m not an advocate of either method particularly but if pushed

would probably come down on the borrow and payback method, but maybe that’s

down to the way I learned it and the fact that I’ve done well by it 🙂

Pam

Pam Moran4th July 2012 at 11:29 amWhen my son was a 1st year (freshman)

student at the University of Virginia- a top two public university in the USA,

he happened to room with a S. Korean student who planned to major in chemical

engineering. (Just for today’s U.S. history lesson Thomas Jefferson,

author of the U.S. Declaration of Independence, also founded The University of

Virginia and considered it one of his great achievements. Today is the Fourth

of July, anniversary of America’s Declaration of Independence so it’s a big

deal in my hometown which also was home to Jefferson’s Monticello and the

University of Virginia- both World Heritage sites.) Talking with this young man

from Korea over a Thanksgiving turkey meal was an “aha” moment for me in

regards to where the U.S. arithmetic program falls short in creating a

math-competent, math fun-loving citizenry.

Within days, this international student

was tutoring several groups of kids who were products of the U.S. mathematical

“textbook” curricula. This young Korean explained to me that

our “top of their class” high school graduates whom he tutored in

college engineering and math courses could not visualize solutions to math

problems. They had no “strategy” to solve problems in multiple ways.

He described how he had learned math as very different from our

procedure-driven math and said that U.S. grads wanted to just go to a textbook

and find the formula. He said, “to really be able to do higher level math,

you have to be able to just look at the problem and see many ways to the end of

it. It’s not about a procedure, it’s about understanding the ideas of mathematics.”

( See Singapore math http://www.keysschool.com/Documents/Bisk.pdf and Korean philosophical approaches to

math as writing and visualization http://www.prontolessons.com/how-koreans-teach-basic-math-philosophy.html

Unlike teachers in some other countries,

U.S. elementary teachers usually take little math in college and what they do

take often has emphasized reliance on textbook procedures and doing math as a

recipe/formula. This isn’t true everywhere in the world- for example –

Finland where both the overall ethos of learning and teacher preparation is

very different than in U.S. classrooms http://www.cimt.plymouth.ac.uk/journal/malaty.pdf

.

Many of our primary teachers can’t

explain basic concepts underlying NCTM’s mathematical understanding expected of

children in elementary or secondary schools as discussed by @irasocol in

http://speedchange.blogspot.com/2012/01/changing-gears-2012-maths-are-creative.html

http://speedchange.blogspot.com/2012/01/algebra-without-numbers.html

http://speedchange.blogspot.com/2011/12/stop-asking-questions-if-you-know.html

Following arithmetic procedures and

understanding and applying mathematics represent two different outcomes for

learning. One might help you buy eggs at the grocery store – as long as you

don’t need more than a couple of dozen. The other helps you see the world

mathematically and solve problems through multiple strategies because you

understand the language of mathematics, not just a few words to get by. If what

you want for children is to understand mathematics, not just do simple

computation, then do your research so you choose approaches designed to help

children conceptually understand maths really well. You won’t get that as a result

of children who are taught to memorize U.S. math recipes such as “carrying and

borrowing.”

Currently, I’m interested in Matt

Peterson’s ST math work to look at a math visually through technology-driven

problems http://www.youtube.com/watch?v=2VLje8QRrwg

We are exploring this in schools where I work through a “seed”

project and it seems to have a place in the continuum of strategies we use to

get at math learning differently than through procedural texts. We know

the teacher’s disposition and competency is key and the focus of math curricula

on conceptual understanding is essential. We see tech learning tools as adding

value to the math work and play of learners.

Lastly, doing well on standardized tests

isn’t the end game of learning math in school.

Instead, it’s about learners becoming confident, competent,

entrepreneurial thinkers so they can generate multiple solutions that work when

faced with math problems that on the surface they don’t know how to solve.

Ventry Weather4th July 2012 at 10:04 amDebate about the methods is one thing, it’s the unintended consequence that drew my initial attention. My son Age 9 had always done well at sums but suddenly became unstuck at subtraction this year. Then I saw the mess on his copybook and understood right away what the problem was. Diagram 3 above explains it perfectly. That was done on a computer; now try this yourself with a pencil and paper. Messy, right? There was no way a teacher could discern whether the method was being followed correctly based on my son’s work. So I went and asked what was going on. That was when I first learned about the changeover from the Austrian system by the DoE. The teacher confirmed my worst fear: they only checked if the answer was correct, not if the method was followed correctly. When I asked why, I was told that the US method was illegible. So in other words they had no idea that the class was struggling with the new method; just that homework was messier and that math performance dropped. The teacher was very grateful that I had picked up on the nuance that some students may be struggling with the new method and that the teacher was unaware the messy method was the cause. They redoubled their efforts with some solid teaching and sorted it out. However as pointed out here http://bit.ly/hardsums it really is a problem. Ask an American to do the sum in Diagram 3 without resorting to paper. Make sense? US/Austrian methods aside, my own was of the kitchen table variety. I put 5 apples on the table; I asked my son to remove 2 apples. That’s subtraction.

simonmlewis6th July 2012 at 10:17 pmFrom what I can gather from your comment, I think the problem is less about the method, but more in how it’s taught!

Seomra Ranga4th July 2012 at 8:01 pmThis is an interesting debate which surfaced last night and really could go on forever with those for and against a particular method. I take Pam O’ Brien’s point about the benefits of knowing different methods of approaching a sum. I also understand Ventry Weather’s point about subtraction sums sometimes being illegible, especially working with larger numbers.

However, I still think that pupils should understand why they are using a particular method to approach a sum, and, if explained properly, the “re-grouping” method is the most easily understood and can be applied to sums with larger methods. Yes, it can get messy, but if an answer is incorrect, I always try to trace back to find at what point a pupil made a mistke in the calculation.

We will probably never all agree on this subject, however, I find myself in the same school of thought as Simon on this one – just because a particular method makes it easier to arrive at the correct answer, does not mean that it is educationally or mathematically preferable to another method.

Pam4th July 2012 at 10:29 pmI agree that students should understand why they are using a particular method but I would be worried about the one size fits all strategy. Both methods are mathematically sound and are different sides of the same coin in my view. I am often surprised by the strategies my own children have for adding numbers together – e.g. to add 8 – add 10 and take off 2. This seems strange to me in some ways because adding 8 is like breathing, I don’t have to think about it but for my child this is a very sound way of adding 8. I suppose what I am trying to get to is that the important thing is that the child understands what they are doing and why and I think if alternative methods help children to do this then that can only be good.

Pam

m5th July 2012 at 6:20 amI find parents are convinced to use borrow and pay back method and use this at home, I teach the modern method however in all honesty if I was doing my own calculation for something I would do the older method e.g. working out electricity bill. I do conclude if children haven’t got renaming by end of 2nd class another method may be used (why keep teach something a certain way they can’t get) and then in senior classes a variation often used ‘just cross off digits’ for speed.

simonmlewis6th July 2012 at 10:15 pmI feel if they haven’t got renaming in 2nd class, they haven’t got Place Value so I’d teach that.

IT Support9th July 2012 at 10:32 pmI feel like this method is a little bit complicated since I was taught subtraction the old way. Maybe they can start teaching this method to children at a young age, so they adapt easily.

ireene13th November 2012 at 11:40 amif u already understand this new method its easy, but the teaching needs to be less complainted, iv had a very good collage education and its to much detail all at once. when i was in school the books we had explained how the sums were done in the back of the books, so parents could go over it, now their is nothing for us to learn the new methods. i thought my 8 yr old how to subtract, the borrow and pay back method during the summer and now the teacher is teaching the new method and hes getting confused. if it was in the book i would not have done that so the books and the system are letting the parents down and putting to much pressure on kids which in turn knocks their confidence etc

simonmlewis13th November 2012 at 11:32 pmHi Ireene

Thanks for your comments. I’m afraid I’m going to have to disagree with you regarding your thoughts on the system letting parents down. When we were in school, the maths book seemed to be the tool that taught us maths. In today’s classrooms, the book is only a resource and there are lots of other ways that a teacher now teaches mathematical concepts. Many schools ask parents in to show them new methods for operations like subtraction. If you’re interested, I can write an article on how to learn the “new” way of doing subtraction.

ireene14th November 2012 at 12:00 pmhello simon, yes i would be very greatfull if u would do that and with examples. i have asked in his school if they can show me how and i got a very quick run down b4 the class started, which is not what i actually ment but i felt greatfull that she done that much for me. i understood it leaving the room but by the time i got home id forgotten some parts and i could not put it all together myself . thank you and looking forward to hearing from u

thank you and god bless

ireene14th November 2012 at 10:15 pmhello simon

thank u for ur reply, i would be really greatfull if u could explain it and with examples. i did ask in the school so she just went throught it quickly b4 the class started. thats not quite what i ment but was just greatful to get that much. thank you and looking forward to hearing from u. god bless

ireene

admin17th November 2012 at 11:32 pmNo worries Ireene. Your wish is my command. There’ll be an article or two on this over the coming weeks.