As I reflect on this past school year (which is quickly heading towards its end - yeah!), one of my favourite things about teaching P3 has been how number talks have developed my own understanding of how the different operations work. It's been humbling to realise, at times, that my 7 and 8 year old pupils sometimes seem to have a better intuitive grasp of these operations than I do. I have learned so much from them!

​I wanted to share another one of those moments - I hope that these stories might encourage you to give number talks a try (and persevere with them!) in your own classroom.

​Division is a tricky operation. When you google Number Talks on youtube, the division number talks are few and far between - perhaps we teachers aren't quite sure enough of our own grasp of this operation to want to allow others to observe us when we are exploring it with our classes.

​But if you are willing to take the chance, your class will rise to the challenge (even if you yourself have a few stubbles along the way - that's OK and part of the learning process).

​This past year, I had a student teacher who was doing a division number talk. She chose the equation given above. The class found this pretty straightforward, and the first kid to defend the answer '4', gave a standard 'think multiplication' strategy: they knew that 4 x 10 was 40, so 40 divided by 10 was 4. Fair enough. 

​Then another child offered to give a different strategy. He said that he knew that 40 divided by 5 was 8, because 8 groups of 5 are 40. BUT - if you double the number of groups you have (from 5 to 10), then you must half the number of things in each group (from 8 to 4) - so because he actually needed to divide by 10 (rather than 5), he knew he needed to half 8 to get 4.

​Did you follow that? Because my student teacher struggled to. I knew intuitively when the child offered this explanation that he was right (and that his strategy would always work), but I would have struggled to articulate it at the time. So we congratulated the pupil on his thinking, agreed to come back to it later, and moved on.

​My student and I then sat down with manipulatives after school to figure out exactly what this pupil had known instinctively! If you found it difficult to follow the strategy as I explained it, get out manipulatives as well, and you'll soon see what he did (this need to use manipulatives as a teacher, in order to really understand some of the different strategies, has been a very helpful reminder for me that our children MUST have many repeated concrete experiences with mathematical concepts in order to internalise and understand them before they move to working with the abstract equations alone).

​Once you've followed what this pupil did (and if you didn't need to use manipulatives, I'm very impressed!) how does this strategy help our classes with division? My student's initial reaction (once we'd figured out what he did) was that this pupil took an easy problem and solved it in a more difficult way.

BUT - if we understand this strategy, we can use it whenever we are asked to divide by 5. And it makes division by 5 a very simple mental operation - no need for paper ever again. You can see this in the picture below:​

Why do we double our answer? Because if we are halving the number of items in a group (from 10 to 5) then we need to double the number of groups (from 32 to 64, in the first example and from 43 to 86 in the second example).

​Not every child in your class will follow every strategy that other children are using. But with repeated exposure to different ways to manipulate equations, as well as repeated hands-on experience of solving equations using manipulatives so that they can see how different strategies work, your class will learn to think and reason mathematically. They won't always need to wait for you to explain how to do something - they will have the confidence to approach a problem and figure out how to solve it for themselves.

​Give it a try (but maybe don't start with division!).

As we come to the end of the year, I've been thinking about how I've used number talks in my class this year. As I moved from Primary 1 (kindergarten) last year to Primary 3 (2nd grade) this year, it has been a bit of a challenge at times. Why? Because my understanding of mathematics is largely procedural. I can look at a problem and follow the algorithm I was taught years ago and get the right answer. BUT - I often have only a vague understanding of why the algorithm works. And I definitely struggle to 'think outside the box' and solve problems in ways I wasn't taught.

This year, I've found that I've needed to work hard (and pull out concrete materials to aid my own understanding of why a given strategy works!) to really understand what is happening when I use the multiplication algorithm, for example. Or how I can split numbers apart and combine them in different ways to mentally solve problems that the 2nd grade me would have definitely used paper, pencil and algorithm for.

​I think this is the main reason why number talks can be daunting for teachers at first - we are asking our kids to reason about mathematics that we might find just a little bit tricky ourselves (luckily, I've only been teaching Primary 3!). And what if the kids come up with a strategy that we can't follow, or that we can't make clear to the rest of the class? I benefited, I think, from starting the Number Talk process in kindergarten, so the different strategies I was getting my head around were simple.

​But when you teach this way, you really open up an exciting mathematical world for your pupils. As the year has progressed, I've often found myself thinking that some of my class have a much better intuitive grasp of mathematical fundamentals than I have (oops). They can reason with numbers in a way that often just never occur to me.

​Take as an example, the 2 fraction equations given above. I put the first equation to my class yesterday morning, and many of the kids were able to tell me that the answer was '6'. To help, I put up 24 different magnetic cubes, and we split them up into our 4 groups, so we could all see the 6 cubes in each group.

​Then we moved to the second equation. Again, a reasonable number of children could tell me that the answer was 18 (although this concept is definitely trickier for a lot of them!). But when I asked one little boy to defend his answer, this is what he told me:

​I knew that 1/4 of 24 was 6, and I had 3/4 left, so I took 1/4 (which is 6 cubes) away from 24 to get 18.

​Oh, My. Goodness. Of course - how simple. But it never in a million years would have occurred to me.

​Implementing number talks can be tricky. And you will definitely stretch your own understanding of mathematical processes as you implement them. But the rewards for your pupils are amazing, so I would really encourage you to give it a try. Start simple - even if you are teaching an older class, if they aren't familiar with number talks and having to explain their mathematical thinking, they will thank you (and you will thank yourself!) for starting slowly. But give it a try - you won't regret it.