The upside down subtraction bug

There are some issues with Maths that pop up across all agegroups, and the upside down subtraction bug is one of them. If you now have a mental picture of some sort of 6-legged brightly coloured ladybird creature hanging under a branch, then that’s not quite what I mean. I’m talking about the commonest error people make in traditional “Column Subtraction”.

The one where they say that 573 – 254 = 321.

It goes like this:     500 – 200 = 300, then 70 – 50 = 20, then lastly 3 – 4 = 1.

Most kids, when you point out their error, say “Oh yes”, and try again, doing the borrowing correctly and getting the correct answer of 319. But I just have a sinking feeling that the bug will reappear the very next time they subtract… nothing is solved. And I hate not solving a problem….

Working as a maths tutor in a primary school gives me a new perspective on how maths is learned and taught. The language that is used is pretty consistent – the teachers all sing pretty much from the same hymnsheet, and a lot of the kids make really good progress. And this is how this sum “should” be done….

3 – 4 YOU CAN’T so go next door. 7 becomes 6 and 3 becomes 13. 13-4 is…. 13, 12, 11, 10, 9. NINE. 6-5=1. 5-2=3. The answer is 319.

This makes me uncomfortable, it’s simply not true that 3 – 4 is impossible, and it set me wondering if kids’ later dislike of negative numbers partly stems from this rather strange approach to subtraction. What if the poor things really believe that you can’t do 3-4? I remember being 6 and if an adult told me I couldn’t do something, then it was TRUE.

So that’s one reason that traditional subtraction bothers me, but the other reason is that it bears no relationship with the mental method that is taught in primary, the whole concept of counting on. Using this method, 573-254 becomes:

+6                 +40              +100            +100          +70              +3

254 —–> 260 ——> 300 ——> 400 —–> 500 —-> 570 ——> 573

the answer is 100+100+70+40+6+3 = 219  and that is actually VERY difficult. So it’s only really suitable if the 2 numbers are a small number of steps apart.

I like written methods that flow seamlessly into mental methods, because kids feel SOO good if they can do maths mentally, but a lot of them do need to jot as a first step. So the other day, I tried out this with Year 6:

573-254 goes like this:

500 – 200 = 300

70 – 50 = 20

3 – 4 = -1

THEN

300 + 20 – 1 = 319

Several of them experienced genuine panic and distress – why were we working left to right? Surely this was wrong? 3-4 I REALLY can’t. They also believed that they needed to learn a whole load of new subtraction facts like 4-9=-5. It took a while for some of them to learn to cheat (ie do 9-4=5, so 4-9=-5) and yet cheating is pretty much the upside down subtraction bug, used properly…

What was surprising, after I had done this with 4 groups over the morning, was that the kids who had the biggest panics were the ones who ended up fastest and most confident with the method.

Like so many alternative methods, I don’t expect it will suit them all… but it opened their minds a bit, and empowered one or two of them to do some wonderful, fast, mental work.

Next time I see you, I won’t say hello.

I was introduced to another Maths teacher on the first evening of a conference and made an arrangement to meet him, 2 days later, to talk about some maths software I had written, and some he had written. Sure enough, at the arranged time, he was sitting in the appointed seat in the bar, and we booted up our laptops and discussed the programs. Then we had some more beer and the conversation became more general – I learned something about the way I come across to other people. I discovered that I was “aloof”.

He queried why I had ignored him every time we met over the weekend, and I had to confess I had not recognised him, and almost certainly never would. I am quite severely faceblind.

If a person has no distinguishing features within a particular group, then the chances of my recognising them seem to be almost nil. This very pleasant, medium height, medium build, middle aged Maths teacher had no scars, no crazy eyebrows, no beard, stoop, limp, nothing. He was intrigued. He hadn’t come across the problem before and clearly found it quite hard to believe.

The following morning in the carpark I noticed him loading up his car. I went over and greeted him, and he grinned and said “you see, you CAN recognise me!”. No, sorry, I have to confess I recognised your laptop case…..

Problem Solving in Maths Lessons

I have been a member of the Association of Teachers of Mathematics  for years now, and very much enjoyed the 2011 conference – a brilliant opportunity for a Freelance Maths Tutor to do some networking and catch up with CPD. So I was very pleasantly surprised recently to find myself a member of the ATM Facebook Group, where more networking but at less cost, suddenly becomes a possibility. On February the 23rd Mike Ollerton is doing a session in Leicester entitled

“How can we develop a problem solving classroom culture?”.

I’d love to come but can’t, so I have had to content myself with holding this question in my mind for the past fortnight as I teach my Primary School groups.

It is SO tempting just to TEACH Maths, and it can be quite effective with some pupils, especially in the short term. I have  bunch of tricks I can serve up and they are simple, powerful, they go down well. But I have taken a break from that recently and given the kids some actual problems to solve… The results have not been quite as I expected…

How much does one Strawberry Pencil weigh?

I picked up a packet of 12 strange-looking sweets on the way into school recently, and put this question on the board at the start of each of the group sessions. The extension group, who decided to run this as a boys-v-girls challenge, sailed into a process with ease, using the “obvious” method, which was to read from the packaging the total weight of the sweets and divide by 12 using a bus stop division. It was so quick that they had time to work with another pack of sweets as well….

The interesting outcome for this group was they found three different answers. The division was 75/12, and I was offered 6.3, 6.6 and 6.25. We’ve done a lot of work recently on team work and collaboration, and so I offered this back to the group and asked them to resolve the differences. Pretty soon 6.6 was withdrawn (Oh, I made a mistake), but two remained for consideration. I was pretty chuffed when one of the girls (Kate) admitted to the group that she couldn’t do the division, and she knew her answer (6.3) was wrong. This took a lot of courage and I think was a complement to the level of trust she now feels in the group’s process.

She wrote her workings on the board and, predictably, several hands shot up accompanied by “Oooh”s and much enthusiasm. It would have been easy to get one of them (or several) to teach Kate how to do division when a remainder is not acceptable as an answer. However, I was skeptical whether she would get it, or remember, so I said to the group that we were ALL hugely tempted just to teach Kate, but I was going to NOT teach her, she was going to work it out herself. The atmosphere crackled – the hands reached higher and dislocated shoulders threatened… this was not at all what the audience wanted… one wannabe teacher appeared to be on the verge of exploding, so I offered her the chance to go and briefly run around outside. To the huge amusement of the group, she kangarooed theatrically up and down outside the window then returned…

Back to Kate…

I began to gently engage her with finding her own solution. I got her to rework the bits of the sum that she was confident with and identify the “unstuck” moment (which was the arrival at “remainder three”). She put “point three” into the answer area.. but looked worried. More hands reaching up. This time the exploding pupil was a boy but was sent out with instructions to kangaroo OUT OF SIGHT so the focus could remain on Kate. I continued to ask Kate questions about where she might put the three, where did she normally put remainders, just opening her to her own solution-finding strengths, and then the magic moment, the penny dropped and was missed by several of the group whose desire to teach had become almost unbearable – but Kate’s face as she inserted the required noughts, put the remainder 3 in the correct place and finished the sum correctly, was a picture of triumph. 100% agreement now on 6.25 being the answer.

We finished the lesson with a quick bout of the Inverse Function Dance to release the required steam.

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I think sequels are important. The long term effects of an experimental approach matter and it is important not to make judgements too quickly. So…

Ten days later

Kate assures me that she is still 100% confident with divisions that push on into the decimal world in this way. The group’s cohesion, focus and mutual trust continues to grow and they have spent two one-hour lessons now exploring “What is the largest (area) shape you can draw with a perimeter of 30cm”, they now believe it’s a circle but are very keen to spend next lesson getting to grips with HOW to best find the area of a circle – they are becoming dissatisfied with counting squares, and have found the investigation a genuine struggle in places, but seem to be experiencing a delight in their learning which is motivating them to go deeper.

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 A Quote..(from ATM)

The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner

Trying to buy a calculator

… I go online at my favourite supermarket, wanting my pet brand of Scientific calculator, and punch in “calculator”….

The phone coughs up “65 hits”. Great. No idea this retailer sold so many different types. Hit 1, the one I’m after.  Hit 2 is Sellotape.

In which universe can Sellotape be used to assist arithmetic?