Securing Number Facts: Practice Makes Permanent

Hi, my name is John Brunskill and I'd like to talk to you today about securing number facts. So I'd like to start with a problem. And that problem is 17 times 24. Take a moment and see if you can calculate that in your head without any pens or paper. If you're still stuck with it at the moment, and good for you if you are, you might be feeling a little bit of mental discomfort right now. It's effortful to tackle a problem like this. It's taxing on your brain. It can get a little bit frustrating. You know how to solve it. You know that you need to do 4 times 7, which is 28. Hold the 28 in your head, then do 4 times the 1, which is really a 10, which gives you 40. Add to the 28 that you already had in your head, which gives you 68 as a total. Hold that new total in and do your two, which is actually 20 times seven, giving you 140. What was the number that you had in your head? 68. 140Add 68 is 208. And finally you need to do your 2, which is actually 20. Add your 1 times your 1, which is actually a 10, giving you a total of 200, and then add together the third number that you had held in your head, which was 208, leaving you a grand total of 408. This is the sort of problem that we will give to children in key stage two, perhaps. And it's the sort of problem that people find challenging. And the good news is people find it challenging because of the way that our brains are built. But that feeling of frustration, that mental discomfort, can leave people feeling a little downbeat. It might lead to a feeling that sometimes described as maths anxiety. So what we're going to talk about today is a little bit about how we can see if we can solve some of that maths anxiety by securing number facts. Daniel Kahneman, Nobel Prize winner, looked into the way that the brain's life work is looking into the way that the brain processes and thinks. And the reason that people find this sort of a problem difficult is because what he calls your system two type thinking is invoked. There's two ways that your brain thinks. System one, Thinking is automatic, it's instant, it's involuntary. It's the sort of thinking that goes, that takes place when you look at a sign that says toilet and you immediately read toilet. You can't help it, you just immediately read it. It's easy for you. It expends no energy. But we also have system two. System two is really lazy. System two is when you have more complex problems and system two is effortful, it requires real attention. You can't do system 2 type thinking whilst you were distracted. If I asked you to solve this problem 17 times 24, whilst also asking you to navigate a busy road, perhaps across a road, you wouldn't be able to do it. You'd have to stop one of those two tasks because in both places, in both cases, system two system is working. So how do we, how do we solve this sort of maths anxiety? How do we stop people feeling like they're getting frustrated with maths? What I'm going to suggest is that we need to try and move some system two thinking into system one thinking. We need to secure our number facts so that they're permanent and they're involuntarily come to mind as soon as we start doing some maths facts. Before we do that though, I need to introduce you to your mind a little bit because when I say that we're securing number facts, really what I mean is we're remembering number facts. When I say that we're remembering number facts, really what I mean is that we're putting our number facts into our long term memory. So this is a very simple model of the mind. It's of course a lot more complicated than this, but for the purposes of teaching it's a useful model of memory and the way that we think. This is from Dan Willingham's book why Don't Students Like School? Fantastic book. Cognitive Scientists Applying Cognitive Principles to the Classroom. And he speaks a lot about memory. You can roughly break your memory down into two parts, working memory and long term memory. And you can see here that working memory and long term memory interact. Working memory is the part of your mind that's dealing with whatever is in mind. Right now working memory is really limited. It can hold maybe four items. It's why you got frustrated during 17 times 24 because you are trying to hold too many pieces of information in your mind. Long term memory, on the other hand is effectively limitless. Long term memory are those facts that information that you can recall at any time and that you have remembered forever. And we can bring stuff from long term memory into working memory to help working memory out, to assist it to remove some of those items. And likewise you can see that the opposite works in terms of this relationship. We can get stuff from working memory information from working memory into our long term memory. And that's going to be the goal of today's. Session. That's the goal with securing number facts. Because what we want to do is want to encode some of the information from working memory into long term memory, almost like saving it for later. But there's a problem, there's a problem with this remembering. And the problem is that we forget our minds, forget information as soon as we learn it. This is a famous graph by a German psychologist. It was first drawn in 1885 and it was drawn by Hermann Ebbinghaus, sometimes called Ebbinghauser's forgetting curve. And it shows what happens with information and the retention of information when it's learnt and how quickly it's forgotten. So you can see that when you first learn something, of course you know 100% of it. You literally just learnt it. It's there, it's fresh in your mind. But immediately you start to forget that material. And by the end of just one day you will have forgotten about a fifth of what you've learnt. And if we revisit that the next day, then you can get back to 100% retention. You remember everything that you were learning again, but you start to forget again. But notice that this time the curve is a little less steep, we're forgetting it a little bit more slowly and if we retrieve again a few days later the curve again less steep still, meaning that that information is becoming encoded in long term memory. So eventually the curve gets to a place where we instantly recall it and it's very, very difficult to forget and it's likely to be in long term memory for a very, very long time. That's where we want to get to and that's what we're going to talk about today. There's another cognitive principle which I'd like to talk about. We're going to touch on it in some of our examples. It's also a principle which you might want to apply when you are looking at some of these techniques and when you approach securing lumber facts with your children. It's the principle of interleaving, really powerful cognitive principle. A few different ways of approaching maths or any other type of learning. And that's what we can call the block method, this first method here or the interleaving method, this method here. So the block method would take you, would lead you to take subject A, it might be addition problems, it might be adding numbers together and then move on to block B, which could be fractions, block C could be identifying shape, block D could be, I don't know, division. And you'd spend a whole bunch of time Learning all about your addition, then you would move on to the next topic. The interleaving approach does something slightly different. It breaks those chunks down into much smaller, those blocks down into much smaller chunks and it mixes them up. So with the interleaving method, you might do a little bit of addition problems, but then skip straight through to the division problems and then go back to your fractions and you're constantly skipping and revisiting. Now this is quite a counterintuitive finding because this doesn't feel particularly nice when you're doing it. Chopping and changing gives the illusion whilst you're learning that the material is really challenging. It's what the psychologist Robert Bjork calls a desirable difficulty. It creates a difficulty in the mind. It's desirable because it leads to longer term retention. And that's what we want to aim for with these number facts. We want to aim for them to be in the long, long term so that all of the teachers who teach your children after you can thank you that they've got those number facts, secure that edifice, that firm foundation for the rest of their math. Now there's some recent research suggesting just the power of interleaving. It was conducted by Daniel Roh here and they used. These researchers used interleaved practice for maths problems with some seventh graders and tested them a day and a month after that material was practiced using the block method, where the children had a load of one particular type of problem and then moved on to a load of another type of problem after a day they retained about 80% of the. These are the wrong way around. Excuse me, about 74%. These should be the other way around. About 74% of the material. These two are the wrong way around. Okay, have you spotted the deliberate mistake? So using the interleaving method after day one, if you just imagine a line drawn from this one to this one and this one to this one, after day one on the interleaving method, 80% of the information was retained. It's about what the kids scored on the test. Whereas using the interleaving method a month later it dropped about 16% down to 64%. The block method, on the other hand, 74% in the initial test, after a month it dropped to below half. So using the blocked method, the children in terms of long term gains had really begun to lost some of what they learned there. So we're going to think a little bit about how we can try and use interleaving when we're practicing. And that's something that you can Consider as well during that practicing. Okay, so let's have a look at some practical examples of how we might start to secure some of those number packs, start to get some of them into our long term memory. There's no magic tricks here. Practice makes permanent. I think it was Bobby Robson that first said this when he was talking about his footballers doing their drills. But it's the same sort of principle with these cognitive facts as well. Dan Willingham, who I mentioned earlier, gives one of the sums up quite nicely where he says it's virtually impossible to become proficient at any mental process without deliberate practice. And without prolonged practice, your kids are going to need lots and lots of practice and they're going to need, sometimes it's called drill and kill or drill and thrill, if you're a little bit more sympathetic towards it. So we are going to think of a few different ways that we can start to drill these and practice them in hopefully fun ways. Now, some children might seem to be better at this than others, but the principle that we have to remember is that all children can improve with practice. And I think very often when children seem to be really speedy with their number recall, what we're probably actually seeing there is a child who's done lots of practice at home, maybe they've got a tutor, maybe they're going along to a Saturday maths class and they're getting that advantage because they're having that practice. It's our job as teachers to level that playing field and make sure everybody gets that practice. So everybody has that secure foundation that gives them the speed, that gives them the confidence to tackle those more complex problems and to free up that working memory to prevent overload. So I first saw this, I'm going to call this number bond rockets. I first saw this from some Chinese teachers teaching in England looking at maths and they were a little bit fed up as they were teaching how children were constantly going to concrete aids to solve quite simple problems. Number bonds, for instance, the numbers that add up to 10. And they felt that that wasn't a scaffold for the children. That was a barrier that was becoming an obstacle because all of their working memory, all of their thinking was focused on manipulating these concretes when in fact they should have been focusing on a larger problem. And so they were getting stuck at that first level. And so one exercise that they use, which I quite like and going to talk a little bit about now, these number bond rockets, you can put any number at the top here I put number bonds to 10, but you could put Any number at the top here you could do your number bonds to seven. And to begin with we might start out working quite systematically so that the children can understand the relationship between this side getting one bigger and this side getting one smaller because the balance is shifting over and we're really going for speed here. So this is an exercise you can do maybe at the start of a lesson or any other time during transitions. And in fact we're going for speed so much that when I saw this being done by the Chinese teachers, they didn't even use the add and the equals symbol. They just wanted the children to say the numbers. They learnt those number pairs and they learnt those numbers together. They made this total. So they would do some choral stuff. But you could do this written down as well. You don't need to have these printed out. We're trying to make this as not trying to make this too resourcey with lots of bits of pieces of paper going around. A bit of a nightmare for teachers. This is something that children can sketch on their whiteboards very, very easily or in the back of their books very easily as well. So it's something that can instantly fill a minute. And the more regularly you do it, remember Ebbinghaus forgetting Curve, the more regularly you do it, you know that the more it's going to be encoded into that long term memory. So it'd look a little something like this, we go 0, 10, 10, 1, 9, 10, 2, 8, 10, 3, 7, 10, 4, 6, 10, 5, 5, 10 and so on. And we might start with these completed and we might slowly start to take them away. So when you're first doing this with children, then you might have all completed so that all the children can take part and if they need to they can have a look and read off and they're just learning that muscle memory of seeing those two numbers together and making the total. And then a nice way to allow the children to self differentiate and a good sort of low threat challenge that I like to add in, I might put them all up so that everybody has the opportunity to see all of these numbers. But then I'll say to them, if you're feeling really confident with this then just cover your eyes as you say them. And the children love showing you that they're covering their eyes and if they need to they can quickly just have a little peek. And it's a really nice sort of high challenge, low threat way of self differentiating. So we'll start to go through and those children will learn those number bonds and we can add different Numbers. But let's think a little bit about that interleaving that we were talking about earlier and think about how we can modify this example a little bit to see if we can make it a little bit more challenging, but hopefully a desirable difficulty that helps these number bonds go into long term memory. So we're still going to have our number ones to 10, the same rocket, the same format. And I'd encourage you when we're, when we're looking at securing number facts to really try not to make it too fancy. It's really tempting because it can seem a little bit boring or dry. It's really tempting to try and put lots of different images on there or maybe try and make it exciting with some other sort of gimmick. But that can be really distracting for children and it means that their attention, their mind is not focused on the maths, it's focused on all that other distraction. So really try to keep it as simple as possible and you can get that excitement and that engagement through your delivery with the children and through keeping it really pacey. So let's have a look at how we can modify this to add in some of that interleaving that we were talking about. So I've got the same number bonds here, I've still got all the number bonds to 10, but this time I've changed the order. So if they were relying on that systematic order and it's important for them to know that so that it can spot those patterns, but we also want them to instantly be able to recall any number fact. So if they see 14 add something equals 20, they know that that missing number is 6 because they know that 4 had 6. 6 equals 10 straight away. So we can give them a number bond rocket. But this time we have the first number bond added in and the children's job is to fill in the missing number. So this time when we're chanting, we will say 4, 6, 10. Same again. We're not going to do 4, add 6 equals 10. We want the instant pairing of those numbers to keep it really pacey and to make sure that they get those numbers are paired together to make the total. So this time we'll go, let's try this one together. This time we'll go four, six, ten, seven, three, ten, two, eight, ten, one, nine, ten, six, four, ten, two, eight. Excuse me. Eight, two, ten, one, nine, 10, 10, zero, 10, three, seven, 10, five, five, 10, zero, 10, 10. You can make it even more exciting with the simple addition of a stopwatch. Put some of these up here. Stopwatch makes everything exciting with children. As soon as you've got a stopwatch and say, let's time this, it's a competition. Let's see if we can be faster than last time. Children will love to try and beat that, beat their last time, and we'll immediately make it much more exciting and get everybody engaged. So they're really focusing on these numbers. And that excitement is going to mean that when it encodes in their memory, it's going to be more likely to stick. Fantastic. So let's think a little bit now about how we can use this same technique to derive some other number facts that are really important and that we're going to be really interested in because they help to form the foundation of so many other number facts. Of course, if you know 3 add 7 equals 10, then you know, 30 add 70 equals 100. These really are the bedrock. So let's just focus on this one. 3 add 7 equals 10. Using the same format is going to be really helpful for the children because they build on what they already know. And so they're going to feel comfortable and safe with this. This is something I know really, really well. So this next step is going to be less of a jump. There's going to be nobody thinking, oh, gosh, what have we started with here? I'm not sure on what this part is. So the jump is going to be too much because we're starting with something that's really secure. So we know 3, 7, 10. We know 3 add 7 equals 10. Let's see if we can derive some, some more number facts from that. If we know that 3 add 7 equals 10, then in fact, we're going to leave this because before we move on to this, before we move on to deriving facts, I'd like to bring in some. I'd like to bring in some concrete now and. And concrete manipulatives are a really good way of demonstrating the concepts that are underlying a lot of these principles. So far, I've been talking a lot about drill and repetition and choral responses, which is so important. But if we want those memories to be lasting, then matching up the conceptual understanding with the fluency and that sometimes one or the other goes out of fashion. If you've been in education for a long time, I'm sure that you've seen fluency come in fashion and then out of fashion and back into fashion. And there's a debate about should it be fluency or should it be conceptual understanding? And of course, they both need to go hand in hand. And they strengthen each other. You're going to be more likely to be able to be fluent in both your procedure and your recall, if you understand the concepts underlying it. And so a bar model is a useful way to demonstrate this. I like bar models for a few reasons. And you can introduce them right from reception. And I think introducing them as early as possible is going to be helpful. They remain consistent because you might be using them just for number bonds to 10 in perception year one, year two, when you get into year six, then you can use the same bar models to work out percentages of holes or solve 2, 3, 4 step problems involving some quite complex calculations. And so that consistency is going to help children. Let's start off with this. The other reason that I like bar models is that if you've got your unifix out, which you're using as a concrete aid, they marry up with the bar models really nicely and the children can see the link from the concrete to the pictorial representation really clearly. So if we're looking at this maths fact, 3, 7 equals 10, then we can use our bar model, we can use our concrete here to help demonstrate exactly what we're talking about here. Well, we've got 10 altogether, we can count up to 10. We've got 10 things altogether, 10 items all together. And I can break that 10 up into 7, 1, 1, 2, 3, 4, 5, 6, 7 and another 3, 1, 2, 3. And if I add those together, I've got 1, 2, 3, 4th, 5, 6, 7, 8, 9, 10 altogether, I've got seven, I've got three, got 10. Seven at three equals 10. And we can show this with our representation Here you can see I've got three here, you can see I've got seven here. If we make this one bigger, if we make this four, then you can see that we go down to six. We're building on that relationship earlier learn. But this is helpful for another reason, because this can help us to derive three, three more number facts. Once we show this part, part whole model, we know that 7 add 3 equals 10. What's another equation that we know? Well, if 7 add 3 equals 10, 3 add 7 also equals 10, linking those number facts together means that you get a bit of a cheat. You get a four for one, you get four number facts for one. What are the other number facts that we know? Well, we know our whole. This is what we've been talking about. We've been talking about what, 10 altogether, we've got 10, our whole is 10. If I take away three from 10. 10 take away three, I've got seven easy. I know that this is seven. And if I start with my 10 and take away seven this time, how many have I got? I've got three. Lots of repetition of this. Seven add three equals 10. 10 take away three equals seven. Really linking those number facts together and helping children realize that once you've learned one number, one, which you've learnt one number fact, you can use that to derive other facts, which means that you're going to build your foundation of number facts to be even greater. So we return to the pictorial and the abstract represent. Sorry, to the abstract representation here, because using these concretes is really important, but we also want the children to be able to apply that to their abstract representations as well, because it's no good if they're just stuck on the concretes all of the time and then get lost as soon as they see written problems. Because they're going to have to deal with written problems. Yes, we want the concretes to give them that number sense. Yes, they're a useful scaffold. Yes, they're a useful way of demonstrating those conceptual underpinnings of what we're talking about. But ultimately, the goal is really speedy fluency with our abstract representations. So we know four equations right through from key stage one, we can derive four equations from any of these number facts. This is useful because it doesn't matter whether We've got the 3, 7, 10 or 30, 70, 100 in each case. We get this number family. So we get 3, add 7 equals 10. Can we fill in the rest of the number facts? 7, add 3 equals 10, and then our two subtraction equations. If you need to go back to demonstrate, then you can. Very often children will still want to put one of the parts in. You can show them what we're starting with. We're starting with the whole and then we're taking away three. Tell the story. We're starting with the whole. So we start with the whole and then we're going to take away one part. Let's take away three and we're left with seven. And finally, obviously, we can do 10, take away seven equals three. Lots and lots of practice, lots and lots of repetition. Interleaving as often as you can and switching between those concretes to be able to show the concepts underpinning it and also the abstracts so that they can get fluent at applying them, means that you're going to stand a good chance of those facts being remembered for the long term. Let's move on to times Tables, another key number fact, another key set of number facts. Times tables underpins so much of the rest of maths. Children who are really fluent with their times tables are going to be less likely to have that maths anxiety as they move through school and move on to more challenging maths problems because they've got that firm foundation. Whenever I speak to secondary maths teachers and ask them, so we're teaching these kids X, Y and Z in primary school, what would you like us to focus on? The response is almost always the same. They say, I don't really care about any of this complicated stuff that you're talking about. I don't care about word problems. What I would love is if they could just come up knowing their times tables. We'll do the rest. Just send them up knowing their times tables. It's going to be a times tables check coming check perhaps euphemism there check test is perhaps in year six in the coming years, which means that all children are going to have to be really secure in this. But it also really is the Rosetta stone to lots of the rest of the maths. So it's something we should be focusing on. So how can we start looking at our times tables? Well, something that I've done with my classes and that we do in my school is skip counting or rolling numbers. And this can be a really useful precursor to learning our times tables and helping the children understand how times tables work. Well, it looks like this, I'm going to play the teacher and the student here. So the teacher will start off by saying, team, team, good as gold. Let me see your fingers roll. And then whichever times table you want them to look at, we'll do the threes. Team, team, good as gold. Let me see your fingers roll the threes. And then the children with lots of excitement say, yeah, and they put their hands together like this. The hands are now ready to count. And this is really important because as they're counting, they're doing their times tables 1 times, 2 times, 3 times. But they just roll the numbers to begin with. So it looked like this. Team, team, good as gold. Let me see your fingers roll the threes. Yeah. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 and 33. 36. And then you can put in a little cheer for them to celebrate their success. Something like, this team's got it going on. Going on. Oh, yeah. Maybe let them choose their own cheer just to put a little bit more excitement and enjoyment into it. This is really useful because as they're rolling those numbers, they're seeing 3, 6, 9. Stop. Stop them there. 3 times 3 is 9. Great, let's keep going. 3, 6, 9, 12, 15, 18. Stop. How many fingers have you got up? Six. How many threes have you got? Six threes. What's six threes? 18. We just said it. Really useful way doing this. If you're doing this all the time really often, building it together as a class, it's something you can do as you're walking to playtime, rolling your numbers, it's something you can do whilst you're waiting in the line for your hot dinner. Something you can do. Do as you're putting your coat, as you're taking your coats off, coming in from playtime or at the start of the day. It's something that builds lots of energy and gets all of the children really excited. And it's also low threat because they're all doing this together. You can use the same technique that we talked about with number bonds in terms of sticking that number sequence on the board and then letting them cover their eyes if you want to. Or here, because they'd be using their hands closing there. So they can start to learn that sequence and really fix in their head. Practice makes permanent. Repeat it as often as possible. And then we might like to, once we've got that speed, that fluency, which is going to give that confidence, it's going to give. Do you know what threes are? Something that I feel okay with. Threes is something that makes sense to me. I already know all the different numbers in the three number sequence. We can start to build in some more conceptual stuff which hopefully is going to start to fix this. We're just going to look at arrays for now. So an array is a way of pictorially representing times tables. And let's start with one times three. In the times table, we're just going to draw three dots. And so we can see we've got one set of three. The next times table is two lots of three. And so the next line in the array will be another set of three. Now, we already know how to skip count in threes. So we can do 3, 6. And we don't just count 1, 2, 3, 4, 5, 6. We can count in threes and straight away get to two times three equals six. And the children can see that it's getting greater by an order of two. Here they can see that it's the same amount. Again, helpful for your doubling as well. If we add in another three for three times three, the array builds More and more. And we can see that we've always got three across the top. And we're getting another three each time. Three times three is nine. And then we can build in another three and we can go down. The children can draw these and they can make these recounters if they're finding this a little bit challenging to get their head around, to help them make sense of what they really mean when they're saying, hold on a sec, why are we suddenly going up from 15 to 18? Well, we're getting another three. That's what's going on there. That's why it's jumping from 15 to 18. That's why going from 15 to 20 or 15 to 17 would make no sense, because we know that 17 is two more. We know that each time we're adding three. This picture in their mind is going to be helpful for them to make sense of what's going on. There's another way that we can start to build conceptual understanding and fluency with our times table. And my thanks here go to Gill Manzerg, who I first saw this with a video from Gill, but this was a technique that was quite popular under the National Numistry Strategies, but something that I haven't seen going on too much in schools. And I love this technique because it just uses a counting stick, which is something that you can have by your desk and grab at any time. Not very resourcey. Really quick. So let's have a quick look at what we can do to help build children's understanding of what's going on with our times tables. So we have a counting step which is split into 10 divisions, and these are going to be our three times tables. Of course, the beauty of this is that we can use this for any of our times tables. We just need to change the post it notes again. This can also transfer. This could be 100% instead of 30, so it gives the children a consistency as they go through. And now we're going to start to give them a little bit of help in terms of the relationships between the numbers so they can start to see how times tables work, not just with the three times tables, but with all times tables, because the counting stick is going to be what's consistent here. So we start with zero. We know that we're starting with zero and we're doing the three times table. So that's going to be our first number. Our first number is going to be three. And we're going to leave the others blank for now because we're going to Build up to these so that the children can see why these little tricks that we're going to teach them work and gain those little tricks. So we're doing our three times table. 10 times three, we know 10 times three is 30. That's an easy one. So we can start with 30 as well. Okay, so we've got zero and then we've got 33. So the next thing that we can look at is what will this number here be? We've got three here. Well, it's just another three because each one of these sections counts for is equal to three. So another three, we can double three and then we're going to get six. Linking our doubling knowledge here and we're going to use our doubling knowledge. That's another number factor. I'm not talking about it today with doubling and halving key secure key number fact foundation to make sure the children are secure. And you can see that we're going to use it here. So we've got double three. Giving us two times three is six. The next one we're going to go to is we're going to go to this one here. 4 times 3. So we know that we've got 2 times 3, which equals 6. If we double this again, if we double 6 again, we've got another 6 here. What's double 6? And double 6 is 12. So we can add in 12, which is 4 times 3. Now, this is useful because it works for any times table. Four times is often a times table that children find tricky, especially when they start to get towards their sevens, eights and nines. And it helps them to realize that we can double and then double again to get to four, four times. And of course we can do that again because if we know that this is 12, we can use our doubling again and go to 8 times. So the same amount here, this is 12, 4 times. So another 4 times is another 12. 12 add 12 we can add in our 24. So we've already built 1, 2, 3, 4, 5, 6 of our times tables we can go to 5 times 3. Now, few different ways to get 5 times 3, but most children will go to 10 times 3 and then halve it. They know that it's going to end in a 0 or a 5. Another quick little trick that they can use, 5 times 3 gives them 15, 6 times. Always a little bit tricky. But we can see that if we've got five times three and we know that each one of these represents three, we can add three to 15, three to 15. If we know Our number bonds is nice and easy. It gives us 18. Three times three. We've got a few different ways of doing this. We doubled to get to six, so we can just add another three on. We can take away from four. It's a little bit tricky because we have to cross 10. So most children will probably want to add three on to the six, giving them three times seven times. A little bit of a trick with three times table. Often with times tables, you'll find that there's one bogey, one that children find really tricky. So we can emphasize this by saying, this is our favorite three times table. This is our favorite times table. So here I'm going to say my favorite times table is 7 times 3, and it's 21. We can add on or take away. But my favorite times table is seven times three. It's 21. What's my favorite times table? Seven times three, which makes 21. And then nine times three. We know what 10 times three is. That's easy. That's 30. Let's take away three and we get to 27. Now, we built our times tables up really carefully, looking at what the relationships are. We've got that fluency in terms of skip counting. We can now start to talk a little bit more about the relationships, which will allow the children to extrapolate other times tables. It will make other times tables, especially, like I said, the sevens eights, which are sometimes a little bit difficult for children to learn, makes them a little bit less daunting because they immediately know how to get the first few times tables. They know that they can double to get two times and they can double again to get four times, double again to get eight times. They know to get to nine times, all we need to do is go up to the 10 times and take it away. We can then set the children a little challenge, which is see if we can roll the threes. Taking away the numbers one by one, you can say, well, we don't need the 30 or the 15 or the 0. They're really easy. Can we roll it without those? And especially when you're learning a new times table, children love the challenge of just incrementally taking one away each time. We knew that three was easy because that's the times table that we're doing. It would work with the seven times tables, eight times tables. We know it's a seven times table, so we would take that one away. And then how do we get this one? We double it. We're reminding that fact that we double it. So we don't need that because we know we're just doubling our times tables. We're just doubling three and we know that that's six, so we don't need that one. Which other one did we know about doubling? We knew this one. So we can take away this one each time, roll it each time, see if you can, and then see if eventually he can get to a completely blank counting stick. Really huge sense of satisfaction for the children if very quickly they can learn a new times table or count all of their times tables with a blank counting stick. So I'd like to finish. We've spoken a lot about different ways that we can in the classroom using quite traditional techniques in terms of course responses, repetitions, tests, some different conceptual techniques. The counting stick arrays show how we can start to build those number facts and hopefully encode them in long term memory so the children remember them for the long term and are able to use them to alleviate some of that overload on their working memory to help them feel more confident and tackle more complex problems. But we also have some. I want to draw attention to a few fantastic online resources which are becoming more common in schools and have the advantage that the children can continue to work on these at home. And we know the more practice, the better they're going to get at those and the more likely they're going to stick in their long term memory. And these resources have the advantage of adding in a little bit of gamification. So it almost feels like they're playing a bit of a video game. And I've yet to find a child who isn't really excited by these and doesn't want to beat their score the next time. So the first one is Times Table Rockstars, which has both a paper and an online version, allows children to have a league table and build an avatar, which they really like to do. As they complete more times tables, they can use the points accrued to make their avatar even better, maybe buy them a new guitar or whatever. Times Table Rock Stars is the first one that I draw your attention to. Mathletics, another common online resource to start to build up some of those key number facts. And if you get children using this a lot at home, then really a lot of your hard work is done for you. You can continue doing all of this in your class, but you'll find these children are already coming with super speedy number facts, which just makes teaching so much easier because it means that when you're talking about those more difficult problems, thinking back to our original one, when you say, well, let's start with 7 times 4. They go 28 straight away and you think, right, fantastic. We're not going to get stuck on that first obstacle. Mathletics allows you to monitor what the children are doing as well in terms of looking at the amount of time that spending and the different problems they're solving. However, both of these are subscription based. There's a cost associated with them. And so I'm going to finish by talking about, I think my favorite, which is called Hit the Button. This is completely free. I love the simplicity of it and I love the subtle gamification which children find really, really exciting. So Hit the button uses all the key number facts that we're interested in. Doubling, halving number bonds and number ones going right up to a thousand, but you can go within 10. There are different levels available. Also our times tables, our division facts. It incorporates some interleaving that we were talking about because there are also mixed approaches where you can have different kinds of calculations, different kinds of equations going in there. The children get the equation at the top seven AB1 and they need to find the answer and hit it as quickly as possible. They just get a minute each time. So it's really snappy. Get a nice little time filler if you have access to computers or tablets within school and they need to see how many they can get. And children love trying to beat their personal best. Better still, stick it up on the whiteboard, you have a go and then challenge the children to see if they can beat your score. And the goal is that you'll have somebody in your class that that's quicker than you by the end of the year. So to recap what we've talked about today, we talked about how instant recall of those key number facts is critical to preventing cognitive overload, which is going to allow children to solve more complex problems, but also diminish that sense of frustration, that lack of motivation that maths anxiety that sometimes see in children and boost their confidence. We know that practice makes permanent. There really are no shortcuts. The more practice you do, the quicker you get. So give your students regular opportunity to record these facts. Make as exciting as possible. I would urge you to go for short, snappy sessions so that they don't get too tired and boring. Just go really quick, short, snappy stop. Then that allows you to give, that gives you that space practice as well because you're doing it in increasing spaces. Consider trying to use some interleaving in your practice. So interleave practice for more long term games. Maybe when you're first introducing a topic number bonds, for example, you might just want to stick with additional. But as soon as possible, see if you can start to mix up those different topics. It's going to seem like it's more difficult. It might seem like the children in the short term and whilst it's happening, it might seem like, oh gosh, they're finding this really tough. Perhaps this is the wrong call. It's counterintuitive, remember, and there's good research suggests that actually that's a desirable difficulty and results in longer term gain. And finally, still use those conceptual models, still use those concrete aids, still use those pictorial representations alongside that fluency work to help reinforce the fluency work that you're doing. Explain what's going on when you're doing the fluency work and hopefully the fluency work will also filter into their conceptual understanding so that children have that little bit of a leg up, that trampoline onto the conceptual stuff that's going on. Thanks very much for listening and good luck with securing some number facts into a long term memory with the children in your class.