Ice Cream, citizenship tests and more. In this episode Emily and Adam discuss fluency and understanding. What is the definition of fluency? Is it more than simply being able to recall a fact? Plus, discussion of the danger of relying on memorisation.

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Andy was one of the first to bring maths mastery to the UK as the founder and CEO of the independent publisher: Maths — No Problem! Since then, he’s continued to create innovative education products as Chairman of Fig Leaf Group. He’s won more than a few awards, helped schools all over the world raise attainment levels, and continues to build an inclusive, supportive education community.

With nearly 20 years of education experience, Emily has a knack for creating wildly successful learning content. Her past work includes publishers like Oxford University Press, Pearson and Collins Education. Currently, you’ll find her dreaming and scheming in her role as Head of Publishing at Fig Leaf Group.

In a past life, Adam was a headteacher, and the first Primary Maths Specialist Leader in Education in the UK. He led the NW1 Maths Hub’s delivery of NCETM’s Professional Development Lead Support Programme before taking on his current role of Maths Subject Specialist at Maths — No Problem!

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Hi, I'm Andy Psarianos.

Hello, I'm Emily Guille-Marrett.

Hi, I'm Adam Gifford.

This is the school of school podcast. Welcome to the school of school podcast.

Hello, welcome to another podcast. This is Emily, I'm here with Adam today and we are going to look at a really interesting, well I find an interesting question that comes up time and again with teachers around fluency, I'm going to give a personal account of my experience as a parent with fluency with maths, and you can tell me about this, Adam, tell me what is going on. Help me here. So my children, they're practicing their times tables, it's the eight times tables, they've been given a week, they've got to learn, come on, we're going to do the eight times tables. I'm all over it, Adam. We're doing it from one, we're doing it up to 12, but let me tell you we also, we change it around. We can do games, there's even online stuff we can do to try to encourage them because sometimes they're like, "I'm not doing that anymore."

I try to come with as many different ways, by the end of the week they go, they do their times table tests, oh, brilliant teachers. So proud of them. Woo, woo. They've got 20 out of 20. I mean, even if it's 50 now, 20 in the right direction, but 20 out of 20 I'm feeling like, yeah they've nailed this. Blow me five weeks later, seven weeks later, something not that long, but a reasonable amount of time. They're doing some maths homework, there's a question, it simply needs to be recalling the eight times table like six times eight comes up as part of the question. Completely forgotten. What was going on there? We worked hard, Adam, we worked hard.

Well, I think we've all been there, right? We've all been there. Be it as a teacher, be it as a parent, and I'm sure we've heard this loads of times. So we probably need to dig into what's happening. The first thing is, I always think is what's the success criteria? So whenever we do something, what are we learning it for? Now, the standard response, a certain standard response... No, I'm going to keep with that. Standard response might be, what are we learning these for? We are learning them for the test on Friday. That's what we are learning them for. Is it to use two years down the track? No, I'm not thinking about that. I'm just thinking about the test on Friday. I also know that my mum, Emily, is utterly thrilled when I get them all right, because she can't kind of feels like she's played a part in that, she's got shares in this as well.

So Friday kind of reflects her success or failure also. If she's successful through me by proxy, then we get an ice cream, or we go out, or whatever. So everything hinges on Friday, eh? Monday, what's the reward in knowing it? Well there isn't one. Not that I can see, nothing tangible. So what about for the teachers? Well, there's nothing better than calling out your class's, you're calling out your names, Anna, 19 out of 20, Dave, 18 out of 20, Emily, 20 out of 20. Oh, this is great, this is really warming my heart because it's suggesting that everything from this day forward, I will never have to worry about Emily in the timestables, or Anna, or Dave. They are honestly all over this. Experience tells us though that this isn't the case. So I think the starting point is, is if we aim to succeed on Friday, if that's our expectation, done. We can't complain.

We've done that job. But the problem is, is that with all of our, I just call them basic facts. So whether it's the times tables, whether it's knowing number bonds to 10, and two numbers that add up to make 10, all of these things that we just kind of have to know, the part that is often missing is understanding the why or the big ideas that underpin multiplication. So for example, I'm just going to give you a little, really quick maths lesson. Really, really quick one. If I said to you five times four, we could think about that as five, four times, or four groups of five. We could think of it that way. So we could either add those five things together. We could add five four times, five plus five plus five plus five. Yeah, we could do that,

And we could work out what that is. What's really important and amongst that is that we realize that we're, whenever we're multiplying something, we are effectively adding equal groups. We're adding the same amount multiple times. Now where this becomes really handy is that if I know, if I just know that, for example, five times two, two groups of five, is 10. That can help me be more efficient in understanding when I'm asked what's five times three? Because I immediately, if I understand the concept of multiplication, if I understand that I'm using something I already know, which is five times two is 10, I don't need to go through a counting exercise. I don't need to go 5, 10, 15. I already know that two groups of five is 10. So three groups of five is one more group of five. So 10 of five is 50.

Now we expect that these sorts of things become what I consider fluent, but fluency to me is not just the immediate recall of effect. Fluency to me is the immediate recall of effect plus understanding the idea that underpins it. There's quite a big distinction there and quite a big difference between those two ideas of immediate recall, and immediate recall with understanding the big idea, because what we want children to do when it comes to fluency is apply the same logic elsewhere. So when it comes time to learning your seven times tables, let's just think about this for a second, chances are by the time you hit your seven times tables, you already know your one times table, and you know your twos, you probably know your fives. You probably know your 10s. So now your seven times table looks really diminished.

We know that the nine times table is one group less than 10, and I already know that. We know that the four times tables is one group less than five and I'm already fluent in those. We know that six times table is one group greater than the five times table, and I've already got the five times tables down pat. So what is it that I need to know? Well, I need to know seven times seven. Anything else? Well, yes, threes. Well, that's double in another group. Anything else? We've already got all of these things that we can then work on, and we're far more efficient if we understand how it all works. I don't need to, when it gets to nine times seven, I don't need to get my fingers out and start dropping certain fingers in a pattern. That might be useful,

Hey, listen, we all probably, I still have to do the months of the year on my fingers whenever I sort of come back into a country or yeah, what month were you? What month did you leave? Ah there's three, January, February, March. My little fingers are going like the clappers to work that's how it is. I don't mind admitting that, there's no problem. And yes, we can use these other things, but someone just said to a child, "Right, what's seven times nine?" Well, I know it's 70 subtract seven, I know my number bonds to 10, three and seven make, so it's 63, just done, simple. And if we practice it enough, of course we can get to a point where it is immediate recall in terms of the answering, but does it also help us with our mathematics going forward? Of course, it does.

So if I had something like, I don't know, 149 times seven, 150 times seven is a whole lot easier, and all I need to do is subtract one group of seven from 700, 350, 1050, 1043, done. Now does anyone know their seven times tables up to 149? Probably not. Those children who only learn it through just like saturation and memorization, good luck with answering a question like that. Those will be the children who will grab their pencils and a piece of paper, and they'll do a really born out, long thing. Do we want our children to do that? No, of course we don't, not if they can do it mentally and with the same speed using the same logical ideas that they've used to develop that understanding in the first instance.

So yeah, if we celebrate a Friday afternoon, and a sticker in a notebook, and I'm not suggesting for a second we don't, everyone works hard, children work hard, parents work hard, teachers work out all those sorts of things. Yeah, there's short. Let's celebrate that immediate recall, but if we are doing it without developing the ideas that underpin it, we're missing a massive track, and the success will be Friday. It won't be the following week, or the month, or the year down the line or two years down the line.

Or for life-

Potentially. Yeah, sure, sure. Absolutely.

Because I certainly, that's some that you just did super fast. I definitely did a lot of the learn everything by fact, learn everything by fact, and in a previous podcast, I was like, I think we had, there was a discussion and I can't remember now the detail behind it, we got to the point where we talked about times tables and I was like, "I do think people should learn their times tables, that they should know them." And interestingly, there's been a big push now from the government, even though maths mastery is the way, there is this big push for learning them, and kind of what I think you are saying is, yeah, fine. You can, yes, know them, but not in isolation. That's not going to help you.

And also, we need to be very careful about success criteria we put on it. So if it's purely to get a mark out of 20, that opportunity to do that may come up once in a child's life. So that kind of suggests then, that if I just get tested on the times tables once, or if I was in year two, for example, I might get tested on the two fives and tens or whatever sort of age appropriate if you like, then there is the potential for that success criteria to be used once. What message does that send? That we just learn something for that point. I'll tell you where it came true for me, I did my citizenship test.

Oh yeah. I remember that.

Yes. Tough man, it was really tough.

I don't think I'd pass it by the way, but anyway.

No, but I think this is a case in point where I was learning things about... Some of it that I'm genuinely interested in, like the Magna Carta or various things, but there were some facts there that I was learning purely, I knew as an adult, I would forget them immediately after, because the only criteria for success was that day, and that test. The relevance of learning it was zero other than that. So if I feel that about those, what's stopping anyone from feeling that there's no more relevance beyond Friday for the seven times tables?

Everyone's placing such massive importance on this one day to get it all right. After that, does anyone care? Do we keep practicing? Or do we move on to something else? So I just think we need to be mindful that the messages like does the effort stop once Friday stops? Does the message we receive from, I don't know, parents, teachers, our peers in our classroom, does that change once we've got our mark out of 20? Maybe, but, I mean, listen, as long as we keep referring to those things and we learn the big ideas that underpin them, then we're okay. But if we rely on memorization, then we're in trouble.

So this links, interestingly, to a question that the Maths — No Problem!! customer services team had, which I thought was a really interesting one, because it was from a teacher who highlighted that in their school, they, some of the team, highlighted that one of the key principles of maths mastery, one of the kind of blocks or pillars if you like, was fluency.

Yeah. Yeah, yeah.

And so they were asking for some help to kind of prove that rote learning, which is what I'm calling it, that might not be appropriate, but just learning and regurgitating it.

Yeah.

That it might appear to be fluency. What would you say to that teacher to help them to go into school and say, "Here's the toolbox for your governor's meeting." Or whatever meeting it is, this is what you need.

Yeah, I just think when we talk about mastery, because it's not at odds with fluency, or it's not at odds with... It depends on what people see it as. If it's a very simplistic view of just memorization of basic facts, then that's going to have limited usefulness to accessing new learning. Mastery is about gaining the tools so we don't have to think about it anymore, like driving a car, when you get to a point where you can change the gears and you're not even conscious, you're changing the gears in the car, we would say that you are fluent in changing the gears. You're also fluent with the idea that you know when to change the gears, otherwise the car's not efficient. It's revving too high, or it's not revving enough, or whatever. So all of that plays into, if someone says, well, as a driver is Emily fluent with changing the gears?

I'm not just thinking about the mechanical act of you being able to change the gears, I'm thinking, does she know when to do it? Does she use it to help her braking? Does she...? All of these things, when you reach a point where you are no longer thinking about changing those gears in any situation, then you are ready to learn something new, and I would argue that you are fluent at changing gears because in any context, you know what to do, you know what that looks like. So if it's raining, or if you're on a gravel road, or if you're on a, I don't know, a tarmac road or whatever. In all of those situations, you know about changing gears. So if we just memorize one thing, "Oh, I must go from one to two to three to four through my gears." Then there's going to be situations where that's not going to work.

So if the context change, if the roads change, or the conditions change, or you're going uphill or downhill, then if you're just thinking, "No, I still have to plow through one to four." Then the outcome will be, as a driver, you won't be very good. You'll always find it difficult in certain situations. It's exactly the same in mathematics. So what I'd say is, is that a mastery approach is about developing ideas that allow people to access new ideas. So I think I'm pretty sure it's a Japanese model where they start by saying, learn the one times tables, then the twos, then the tens, then the fives. And there's a structure to it because everything relates to something else. If I know the 10 times table, and I understand that it's just like five groups of 10 and that's 50, then I can understand the five times tables because it's half of the tens.

I can then learn the nines because it's one less group than the tens, and I already know the tens. So by the time we get further as said before, by the time we get to those like sevens and eights, we've already got the majority of those multiplication tables boxed off because we understand the relationship between the groups and what happens in multiplication. That's a mastery approach. So we're taking things and we are able to access the new learning of the seven times tables, but we're doing it in a way that's efficient and brings into it the big ideas that we've already established from the very first time we started to learn about multiplication. So I think we need to use those things, not ignore them. We need to understand how they work. And of course, we strive to reach a level of fluency in everything that we do so we can do something better or we can build on it. That's true for everything, not just mathematics.

Adam, that was so helpful. I feel that I've got a sense of fluency as a parent. I think that addresses the question as to what do we mean by fluency from a mastery perspective, and I hope that there's some top tips there so that, that maths teacher can go in and prove the understanding of fluency in terms of mastery. So thanks ever so much.

No worries.

Thank you for joining us on the school of school podcast.

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