I’ve Got 99 Problems… And Subtraction Is One

This is my first ever blog post so be gentle…

Introduction

As a full time maths teacher working with pupils from 11 – 18 years old, one consistent issue seems to arise no matter whether I am working with the new little year 7s trying to get their head around secondary school maths or the year 13s wanting to get into university – subtraction problems.

If I am working with a Year 11 class trying to get a grade C, I constantly see work of the following nature:

4 – 7 = 3

Whether it even be an older pupil simplifying some algebra I see work such as

4x – 7x = 3x

It has made me think very carefully about how I teach subtraction and where the issues may have arisen from. Now I do not have any evidence to support this blog piece and am merely speaking anecdotally from how I was taught, how my parents were taught, what I have seen teachers saying in classrooms, what I have heard myself saying at times and what I have tried out lately.

Please take a look at this video I made to elaborate on some of my ideas – I find that easier to do rather than writing it in a word document format.

Please watch video before reading on…

For me, I think one of the issues with subtraction problems that give negative answers comes from when pupils are introduced to column subtraction for the first time. For example.

751

-234

I admit to having said, admit to having heard and admit to having been taught that you “can’t do 1 – 4” so we “borrow a ten from the 5 and put it in the ones column. The 5 then becomes a 4”. So the method proceeds and there are various issues with writing it neatly with crossing out and “borrowing” all over the place.

To be fair, on the whole I find that the pupils can indeed get the correct answer in spite of the mess on the page and for me a lack of understanding of the place values of the digits involved.

The issue of “borrowing” (Example 1,2,3 from video above)

Personally I have an issue with the term “borrow” in the first place. If you are going to use that analogy of borrowing I don’t think it is consistent as in life when you “borrow” something you “pay it back” which does not happen in the method.

If we are in agreement that the “borrowing” method is best then we should probably use the term “exchange” such as “exchange one ten for ten ones” – I think it is more consistent.

The issue of several “borrows” (Example 2 from video above)

I think the layout becomes very messy and the place value notions get all confused.

The issue of a jump in the “borrowing”(Example 3 from video above)

Again I think the layout becomes very messy and the place value notions get all confused.

The issue of “can’t do 1 subtract 4” (Example 1,2,3 from video above)

Clearly this use of wording “can’t do” cannot be correct. I am not saying everyone does it but seeing what kids do when faced with 6 – 9 by writing 3 it makes me think as a profession we must be saying it often. I know I was taught that way, I asked my parents just now to talk me through a similar problem and they used the phrase “can’t do” and I know I have slipped a few times earlier in my career saying that.

I really think that this might be a reason why pupils are so against subtracting a bigger number from a smaller number and getting a negative. I would be very interested in other people’s anecdotal evidence on this point. Does anybody know of any research conducted into this?

Even if we agree the “exchanging” (not “borrowing”) method is best as we want to avoid any entry into the world of negative numbers which personally I think is a great shame, we should be saying something like “to make it a subtraction we are more comfortable with we are going to exchange one ten for ten ones” or something to that effect that does not propagate the error that you can’t do that subtraction.

Lastly, using my “new” method as opposed to borrowing allows pupils to get to grips with negative numbers sooner and not make them out to be more difficult that they need to be. I’m sure most maths teachers would agree with me that the problems they see coming from directed numbers are quite astonishing and wide spread. So perhaps it is better to not avoid the issue (and certainly not make the issue worse with the use of “can’t do”) and face it head on earlier. After all all it comes down to is moving along a number line, something pupils are familar with from very early ages.

What I do with my classes:

(1) From an earlier point I always refer to numbers as being positive and negative and make them use and write the appropriate symbols e.g. 8 is actually +8 (said “positive 8”) and notice the smaller + sign slightly raised. Similarly -8 (said “negative 8”). Notice here I do not call it “minus 8” as minus is an operation (it is subtraction) and I think this confuses pupils quite badly. I diverge here and could (and will try) to write a whole other blog post on that as that too is really interesting and I think a source of a lot of problems pupils have with directed numbers.

I then work with pupils to notice the symmetry of the positions of +8 and -8 with reference to 0 and get pupils to understand that +8 and -8 have a unique special relationship. For the number +8 (positive 8), -8 (negative 8) is the only number in the world such that +8 + -8 = 0 and vice versa (i.e. they are the additive inverse of each other).

(2) Having established the link between a particular positive number and its associated negative number, I get pupils doing several simple subtractions +9 - +1 ; +8 - +5 etc and then pose the question well what is +1 - +9 and +5 - +8?

Using our number lines we can see the answers will be negative numbers and better still we can spot the symmetry of the answers to the equivalent positive answers. e.g. +8 - +5 = +3 and +5 - +8 = -3.

When we were talking about addition I get the kids to understand that that the operation of addition is “commutative” i.e. +2 + +7 = +7 + +2 and together we discuss how subtraction is not commutative but that it does have a lovely symmetry.

(3) Then we would need to do some work on adding and subtracting positive and negative numbers. This can be tricky but a year ago I was introduced to a fabulous method using “Magic Soup” and “hot fire cubes” and “cold ice cubes” by Craig Barton from www.mrbartonmaths.com and my partner on HegartyMaths – Brian Arnold. I have used this ever since and this it is fantastic. I have attached a lesson (PDF Version) I have used for this but this topic again would require an entire blog post dedicated to it so I move on. But the outcome I go for is to have pupils understanding let’s say +8 + -5 = +3 and that adding a negative has the same effect as subtracting a positive.

(4) Then we would be ready for a slightly altered version of column subtraction. I have included a brief video I made to introduce the main idea which you have watched above. This used the idea of negative numbers and adding positive and negative integers.

Analysis of this “New” method from my classroom:

Prior to introducing this I did the 3 steps I suggested above.

Also I asked my pupils to answer 30 questions on column subtraction using their way.

Every pupil out of the 30 pupils used the “borrowing” method and none did mental methods despite me phrasing the question in an horizontal line like 234 – 123 rather than placing it into the column already for them. Again another blog post is needed here I think on the fact that kids go straight to algorithms and don’t think and use “Number Sense”, but see my video for a tiny exploration of those ideas at the end.

To be fair most of the pupils got the answer correct – it was a 90% correct score for the class. There were some mistakes and these were mainly due to messy working with crossing out and “borrowing 1s” in the method. Mistakes occurred much more frequently when a “double borrow” was needed or when you had to borrow from a column that was not adjacent (i.e. borrow from the hundreds when in the ones column).

When I asked the pupils to explain their workings on the board I realised they really did not understand what they were doing. In particular, their vocabulary demonstrated they were not really thinking about the place value of the numbers and were just subtracting digits. They also struggled to explain the borrowing and what it meant and why they were doing it. I again heard a lot of “can’t do” phrases.

I showed the new method to the pupils and they instantly were excited to see something new and picked it up straightaway. I then gave them questions exactly like the initial ones I gave at the start and got them to work it out the new way. They scored as a class 88%.

Now yes this is slightly lower than the method they had used before but I feel this was due to just seeing the method and a few early confusions with negatives and a few early mistakes that they did not repeat. Once I had spoke to those who made some initial mistakes they flew through the rest. I then gave this as homework and I got 100% correct answers using the new method.

Furthermore I asked the pupils to put their hand up about which method they preferred. All but 2 out of 30 preferred the new way. The two who didn’t just said they needed some extra practice and they got 100% later in their homework.

Best yet, I got them to come to the board and explain their answers and the understanding was so much better. There vocabulary for place value was vastly improved and their use of negative and positive numbers was excellent and something that would be really useful for them in future maths. We chanted as a class there is no such thing as “can’t do” in reference to the old method and I went on to finish with a bit of Growth Mindset about how this applies to anything tricky in life.

Really interested to hear people’s thoughts. Anyone interested in trying for yourself and feeding back – let me know?

Thank you for taking the time to read this.