Feeling positive about negative numbers

Now sats have finished yet term stretches out before us – what shall we teach year 6 in maths?  The test and the demands of the curriculum behind it dominated the landscape for so long, now that’s all done and dusted, it can all feel a bit anchor-less – what now? What shall I teach them now I am free to teach them anything I want, without anyone checking up on me? It’s all a bit disconcerting.

We usually opt – as I am sure many do – for doing lots of problem solving. I bet the website analytics at nrich show a peak in June and July. And there is nothing wrong with that.  It was difficult to yield precious curriculum time to their more open ended problems when there was so much  stuff  to be got through. But now – now we have 7 whole weeks to fill and spending time on actually applying all that maths we’ve been learning seems like a great idea. Consolidation and application; isn’t that what mastery is all about?

And, to a degree, it is. But 7 weeks of nothing but…I’d go crazy – never mind the kids.  I know maths isn’t all about getting the right answer but its nice sometimes to be able to get the right answer fairly quickly – and be pretty sure that you’ve got it!  And while some activities are great – this is my all time favourite -others are just boring number crunching.  Also – you actually have to be pretty good at maths reasoning yourself to teach these well. To use these effectively, you have to know yourself what would be an effective solution so you can prompt the children appropriately in the right direction.  nrich and other maths problem solving sites are written by mathematicians who find such matters trivial. They don’t supply the correct answers – what would be the fun in that? In the example I dismissed above as boring number crunching – Im sure if I actually sat down and spend a few minutes I would see that  – of course – if I did such and such that would be better than something else. It’s not something I immediately know and frankly would prefer some pointers to save me time.  The same goes for the wonderful Don Steward. His website is aimed at secondary teachers who presumably solve the problems he poses in almost instantly. He doesn’t write for primary practitioners who mostly stopped learning maths aged 16. If only he would put the answers!

Also – all that maths we’ve taught year 6 – it’s amazing how quickly it begins to dribble out of their ears once you stop the highly focused daily practice that marked lessons before sats.  There’s a sort of ‘match-fitness’ to a lot of maths that rapidly declines once you stop.  Consider times-tables facts. Remember how rusty yours were when you first started teaching and how much better they are now? Once term ends, they will probably have 6 maths-free weeks. If we tack on another 7 weeks when they don’t really learn anything new or practice much old, then no wonder secondary school maths teachers sometimes think sats ‘levels’ (50p in the swear box) are inflated. They may have been a 4B/at the expected level/secondary ready in early May; by September they have probably slumped way below that. Spaced repetition doesn’t like spaces quite that big.

So, what to do?  Giving them a flying start to secondary school obviously. So if they don’t yet know their tables or understand place value or can’t do the four operations in their sleep – then you should do those. But I’m taking it as read that nearly all the children can. There is no point in accelerating them through ks3 content because  *mastery* and also because if you have ever looked at a ks3 maths textbook, part from the algebra it’s just just the yr6 curriculum again. Seriously – we bought some k3 textbooks for the level 6 children last year – when level 6 still existed and we had to jump to halfway through year 8 to find anything sufficiently challenging.  I am sure secondary school teachers are well aware of this and use such text books selectively.   They probably even have new ones for the new curriculum.  Although I am slightly haunted by the experience of my first son who spend the first half term (yes the whole half term) revising what happens when you multiply or divide by 10 and multiples thereof. But that was some time ago, I really must move on.

So the obvious contenders are those things we whizzed through prior to sats in the mad dash to cover the new curriculum and suspect the children do not really understand deep down. They just know a few tricks.  As I worried about here.  So possibly we could revisit fractions, especially multiplying and dividing which are so easy to teach as procedures and so difficult to understand conceptually. Hey, we could even do some nrich investigations on them. This is the only one I could find that focus on multiplication. Most focus on equivalence but I am assuming the class is pretty solid on that.  This one from Don Stewart is good – but see what I mean about answers being useful – took me a minute or so before I realised that drawing a bar model made these ridiculously easy.

The other candidate is negative numbers which we whizzed through at the start of the year and then realised some children were confused about when we did algebra. There wasn’t time at that point to go back and address those and actually, the algebra questions in the sats paper were far easier than those we had been practising. However, what  gift to the child and their future teachers to have a rock solid understanding of negative numbers on starting secondary school. Including, of course, a firm grasp of why when you subtract a negative number, you end up adding. Algebra gets really tricky if you are not secure in your understanding of positive and negative numbers – so let’s give our leavers something that will really set them up well for the coming year.

Now when I started writing this post, I fully intended to explain at length about using integer counters to teach negative numbers.   If you don’t know what integer counters are, watch this. (There’s the added bonus the teacher sound a bit like Officer Dibble from Top Cat – the original series obviously).  But then I did a bit of pre-post googling, to se what else was out there and stumbled across this from Tess India which is simply brilliant and uses integer counters as well as various other good ideas.  I particularly liked the bench number line  where you use PE benches to make a number line with children describing how they are going to move from say -4 to +3, and the ‘feeling positive’ idea where you ask students to think about things that make them feel positive and things that make them feel negative.  If you add yet another negative thing you become even more negative whereas if someone takes away a negative thought you become more positive. Simple – but brilliant! So there is no point in me explaining much more about negative numbers – read the lesson plans in the link.

Indeed the whole Tess India resource is a treasure trove of wonderful ideas – well the bits I’ve managed to look at so far are anyway. And it is humbling to read the plans and realise they are aimed at a context where a class size of 60 is not uncommon and interactive whiteboards don’t exist.  Alongside the maths there are handy tips about how to make flashcards from old cardboard…rather puts things into perspective. There are English and science resources alongside the maths and it covers primary and secondary. It’s really well worth a look.

But back to integer counters. These don’t seem to be well known about in the UK. Indeed, we use the wonderful Primary Advantage maths scheme in key stage 2 and while they bang on about CPA everywhere else – for negative numbers they state that no concrete materials are possible and go straight to number lines.  Now number lines are all fine and dandy but some students get so confused using them and it all seems a bit arbitrary why you are moving forwards or backwards.  I  love  integer counters because I love being able to see why the maths works.  When I found out you could even model  why -4 x-2=+8, I beamed for days. I kept on showing people my newest party trick. (I didn’t get invited to many more parties after that.)  And here’s a great link showing how to divide negative numbers.  And no I am not suggesting you teach year 6 how to multiply and divide negative numbers. I’m just banking that if you’ve got this far, this sort of thing brings you great joy.

Using integral counters is the final idea in the Tess India resources.  To be able to understand them, children need to understand the concept of a zero pair, made from one positive counter and one negative counter.   These can then be added to any equation without changing its value. I could try and do some badly drawn graphics to explain it properly – but others have done so with so much greater flair I suggest you look at these instead. Unfortunately Officer Dibble’s video on this seems to be missing. However, this from Learn Zillion  is perfectly serviceable although this is a bit more fun, if a bit more complicated. Can’t quite place the accent.

With a bit of practice, children soon learn to just draw themselves + and – signs if they want to check a calculation, rather than need to counters.

One of the problems children face with this topic is that we never make it clear that all numbers have polarity –  that they are either positive or negative – and that strictly speaking we should write 3 as +3 etc. It’s a shame that the polarity signs are the same as the operator signs – I’m sure it would be a lot easier if they weren’t.  When we write 5-3 do we mean

+5 + -3 or +5 – +3?   I’m sure if we did a lot more work with counters showing that they give the same answer but actually represent something different, that might help. No wonder children get muddled and think -3-5 equals 2  or possibly 8 (because you have got 2 minuses and they half know something about two minuses making a plus. How much better to act it out with counters and see the maths before you very eyes. Works for me.

Then we can return to where this post started and have children investigating negative numbers.  Back to nrich.  And here’s a great reasoning activity from maths pad. If they can articulate why certain statements are or are not true then that’s job done. Secondary schools- here we come!




Feeling positive about negative numbers