S2 E2 - The Power of Puzzles with Catriona Agg

Episode 2 of ‘Digging For The Why’ explores the idea of enquiry-based learning in mathematics education.  

This week, we talk to Catriona Agg, a specialist in continuity between years 5 – 8, about the benefits of incorporating puzzles into your lessons. 

We also look at independent learning across educational levels and explore strategies for engaging students in mathematics and reflecting on education in the post-pandemic world.

Meet our Guest:
We are joined by Catriona Agg, group lead and experienced maths teacher. Catriona works with her local maths hub as a Work Group Lead developing A level pedagogy. She has also been a Work Group Lead for Year 5 to 8 Continuity, which, as listeners of season one will know, is very close to our hearts! Catriona is also well known in the maths-twitter world for her awesome geometry puzzles.

@Cshearer41

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You can find us on X
Andy - @desire2undrstnd/Bluesky
Alison - @AlisonHopperMEI

A talented young creative iingledew produced our intro/outro – check out his Soundcloud.

Transcript

Andy: 0:17

Hello and welcome to this episode of season two of Digging for the Why, the podcast for maths teachers where we explore thoughts behind asking why. I'm Andy, one of your hosts and with me is my podcast far better half, Alison. How are you today?

Alison: 0:30

I'm very well. Thank you for that introduction, Andy. I'm sure I'm not, but Are you?

Andy: 0:35

Absolutely. And I was thinking as well, like, you know, we've recorded one or two of these episodes so far. We're not quite sure what order we're going to put them out in, but we say it's a podcast for maths teachers, but it does feel a bit more wide ranging than that now. And it's generally getting into that idea of asking why, not just in maths, in teaching and in learning, but in all other areas of life as well. So it's quite exciting. For this episode today, we are joined by Katrina Agne, she is a maths teacher and well known in the maths Twitter world for her awesome geometry puzzles. She also works with a local maths hub and is a work group lead looking at developing A level pedagogy. She's also been a work group lead for 528 Continuity, which is listeners to season one will know is very close to our hearts here at Digging for the Y. Katrina, thank you so much for joining us. How are you today?

Catriona: 1:20

I'm so excited to be here. Thank you for having me on.

Andy: 1:24

That's okay. Now this is a free lesson and a lunchtime, so we're very conscious of stealing your precious time at school. So thank you so much for, for jumping on. Now we like to start this season with a pretty standard question and see what people come up with. So basically in the last two weeks, what has happened to make you ask why?

Catriona: 1:43

Right. I'm going to say something that happened yesterday. You can tell I'm really well prepared in my year seven lesson, and I haven't really thought it through yet, so maybe you've got some ideas. We were, we're doing division but we're trying to do division in slightly more interesting ways than just going through worksheets of division. So yesterday's lesson was, is it always the case that if you take a palindrome, so a number that reads the same forwards and backwards. And divide it by 11, you get, you get an integer as the quotient. So we started with four digit palindromes, like 2, 7, 7, 2, and they did always work and we were able to show that they would always work in a couple of ways, and then we moved on to other numbers of digits and one of my year seven said something, which was that it works whenever there's an even number of digits. But not when there's an odd number of digits. Is that because 11 has two digits the same? And I have no idea what the answer to that question is. It was a bit outside the scope of what I was going to do, which was just make them do loads of divisions.

Andy: 3:06

How good is that though? Like Alison's just quickly scribbled that down. She's like, Oh, resources around that. That sounds like a good little challenge. Yeah. I saw her start

Catriona: 3:13

writing. It's quite cool, isn't it?

Andy: 3:20

I love that. That's very

Alison: 3:21

cool. Yeah. Yeah. Start, but how cool that, that your year sevens are searching for reason and searching for meaning, and searching for the, why does that work? Why would, why doesn't it work then why might that, why might that be? I, it was just, that's a really brilliant, yeah. Mm-Hmm, . Yeah. That's,

Andy: 3:38

and that's what, that's what we want, right? Maths is about pattern spotting and, and things like that and going, oh. Well, I think it might be this and hey, you know what, I might be right. I might be wrong. And if it's wrong, that doesn't matter. That's going to inform my next question that I'm going to ask about what's happening. So that's cool. So what's the plan, Katrina? What's the plan? How are you going to respond to that?

Catriona: 3:57

Well, I mean, I don't really know. I've sort of vaguely been thinking about it but not in enough depth. Cause there's a question, isn't there? Like how, when he said it, my immediate response was, I don't think so, but I haven't thought about it long enough to be able to come up with an answer. Because I haven't really thought about it that way. What we were doing is, after we'd done all the divisions, actually it was a really good lesson and I, better than I was expecting. You know, sometimes you're like, oh, is this going to work? It actually worked. And I think I'm really lucky to work in a school where that's the kind of thing that appears on our department shared resources, is get them to practice divisions by looking at Is every four digit palindrome divisible by 11? And so we, we did that and then we, we've been trying to work on kind of working systematically and being a bit sensible in how we go about things. We're trying to kind of build these investigative skills. So, We tried, like laying it out in a table. That was quite nice because it turns out there's only 90 possible four digit pallet drones. And actually once we started trying to write 'em out systematically, we could come up with the fact there were only 90 because there are 10 that start with 1000 and there are 10 in the two thousands and so on. And then we were able to show from that, that if we could make the first column, so all the ones. that are in the 1000s. If we could show it for all 10 of those and luckily, because most of them had started trying to be systematic with a smallish one, they had covered most of those, so as a class we were able to tick off all of those. Everything as you move across a column sorry, yeah, as you move to the next column across, so as you move from the 1000s to the 2000s so 1771 to 2771 you're always adding 1001 but we know 1001 is divisible by 11. Because that was the very top left hand corner that we checked. So we're adding on something we already know is divisible. So we were able to kind of reason that actually all 90 were going to be divisible. So that was great. Then we started trying to do place value. You know 1771 is one lot of a thousand plus seven lots of a hundred plus seven lots of ten. Plus one lot of one, and if we kind of simplify that down a bit, we've been doing the distributive property, so we can call that A thousand and one lots of one, plus a hundred and ten lots of, what did I say, seven. And a thousand and one, we already know, is divisible by eleven, and so it's a hundred and ten, so this is good. And then, then we kind of moved on, they were able to do two digits very quickly, because two digits, they'd actually already written down their 11 times table before they started doing division, so that was okay, that was already there. There was three digits they quickly found sometimes worked and sometimes didn't. But then trying to explain why that didn't work and why five digits was, was similar, sometimes works, sometimes doesn't, was a bit tricky. And I'd kind of prepared for how we were going to prove that six digits always did work, because I think I was in a bit of an always, sometimes, never kind of frame of mind, you know, that question structure. So in my head, I was like, well, we can quickly prove that three digits falls into sometimes, sometimes. And so what we should focus on is, you know, how I'm going to explain that some of them fall into all ways, because that's hard. And we don't want to write out all the possibilities for six digits. So we're going to have to use the place value approach. But how far did

Andy: 7:37

you, how far did you plan for the lesson to go in your thinking? What were your

Catriona: 7:43

pre lesson

Andy: 7:45

plans?

Catriona: 7:45

I sort of half planned for, to do the next thing as well. I, you know, in my head, I thought maybe this is only going to be half, half a lesson, maybe they're going to get quite bored of dividing by 11 quite quickly. But they got really into it. I think because there's enough things to try that you're probably trying some different numbers to the person next to you. But there's not so many that it seems completely overwhelming. And actually, they kind of liked the fact that we, we filled our entire board with a sort of 10 by nine grid of all the possible palindromes for four digits. So they kind of liked the fact that they could sort of see that it was achievable and they could see the reasoning for four digits. And I think that made The rest seemed like it would be achievable. So they got really into it. So it took a whole hour. We, we got as far as explaining why six worked and we got as far as showing examples for the others, but that was sort of, I'd plan to end on, okay, six works, what about eight? What about 10 and kind of extend it that way. And then, so I wasn't really prepared for talking about the odd numbers. That's cool. I

Andy: 8:55

mean, my favorite, my favorite thing in that entire bit is where the kid said, you know, ah, will it work for all even ones and not work for odd ones? And your reply was. I don't know, but you know, we could look into it because it's so easy. Isn't it as a math teacher to go yes or no. And just kind of go, I've got 11 year olds, 12 year olds asking me a question. I don't know the answer to it. Can I admit, I don't know the answer to this. Like, and you've got to have that kind of that confidence and that humility to just go, but you know what? We're all in here doing this math together. So what I can do is use my experience as a teacher to figure out probably the best direction for us to. Start searching, but great question. That's go, you know, that ties in so much to what Alison and I have spoken about

Alison: 9:41

many, many

Andy: 9:41

times, that curiosity, that questioning, that, you know, not

Alison: 9:45

being to be the learner in, in your own classroom. I think that's that that's opened up so many interesting discussions with with primary age children who, and, and opening up what their whole perception of why they're there. And what, what I got from that is that those, those, those year sevens you're working with have are. You know, that they feel that they are part of, they're part of this. They're doing the work. And I think sometimes I've been talking to a couple of teachers recently and saying, you know, who's doing the most work in your lessons, if you're doing the most work, then we're not, we've not got something right here. You know, it should be the, it should be the pupils the students that are doing the bulk of the work in this lesson. We're a bit, we are a bit of the kind of like the blue touch paper and get that thinking going. And it's that's brilliant. And the other thing I love was all the links to different areas. Distributive law, there is the distributive law. And it's useful to them because it, it shortcuts things for them. They can apply that, that in their reasoning, which is really, really exciting. That sounds great.

Andy: 10:41

Yeah. I, I straight away, when you mentioned that went, I tutor a year nine girl and we've been working on bringing, you know, numerical, like the box method for multiplying out numbers by splitting them up and showing her how it's applicable to algebra and she sits there and goes, Oh, what? I can expand brackets using what I did for numbers. And I'm like, Oh yeah, like this is the whole point. You know, yeah. Algebra is numbers. Like, it's just the same thing. The same rules apply. We just have to think about it carefully. And, and it's just that, you know, when you can sow those seeds in year seven and you start talking about distributive laws and things like that, and just how we can break things up and how we can look at place value and go, well, what does it mean? Mm-Hmm., you know, let's, let's pull it apart and just, you know, I, I, I put on Twitter the other day, my, my little girl's six and them. We were talking about numbers and she, she's interested, you know, at school they're doing up to 20 in year one and I sit here like frustrated math teacher. No, no, don't, you know, don't do things she doesn't need to do. Like we need to do things in line with what they're doing. And she started coming home and talking about tens and ones and then I was like, Oh, okay, right. Place values freebie. Now we can go for it a little bit. So she saw a number in the hundreds and she was like, daddy, what's that number? Four nine one four. What, what, what's that? And I said, well, forget the four. What is it? She said 91. So she's got the two place values. I said, well, this column is just the hundreds. She was like. So 491, I was like, yeah. She said, well, what's after that? And I was like, Oh, well, well, Chloe, let's go crazy. So I drew like place value houses for the thousands house. And then just the hundreds bit and showed it. And then her mom was like. What's that? And there's no interest from that side of, for math at all. And so these are place value houses, you know, they allow us to see how it works and describe what's happening. And this is the thousands house. They're like, that's amazing. I was like, Oh, I know. Oh, I know. But I need to stop now before I ruin few exciting things you'll find in her future. But sorry, I get carried away with place value constantly in my life.

Catriona: 12:43

Actually, you've reminded me that my favorite thing to teach this year to year seven is we have. taught them how to do exploding dots. Have you ever come across that? Like James Tanton's thing. It's so much fun. And it's, it's amazing. Cause it's just place value, but yeah, with dots and explosions. And so they love it, but it's actually incredibly useful and you can then use it to start explaining things that happen in different bases. And we used it today actually to model We're, we're moving on to dividing decimals by integers today and trying to explain why the decimal point always stays in the same position in the question and the answer in the kind of traditional bus stop method. So, yeah, they weren't great at kind of explaining where it's at. why it stays in exactly the same place. They could explain why the answer had to be that, that size, you know, the answer needs to be about four. So it needs to go here or it would be 10 times too big, but they couldn't explain why the right place would always be exactly above where it was before. Which is fair enough. Cause that is quite hard. Yeah.

Alison: 13:49

I was, reminds me of a, of a, of a session in in my PGCE, actually, and we had a, a maths tutor who tutored all the PGCE students, whether you were primary, secondary, or, or whatever, and we were looking at how you teach division. And so we had something like 748 divided by 4 or divided by 5, and and, and And it was like fives into seven. Oh no, hang on. We were the way around, sorry, four. So it was, we were dividing by seven, so it was like 496 divided by seven. So it was sevens into four won't go. And he said, but it's ridiculous'cause that falls really a 400. So there's loads of sevens in 400 said, but we can't, we don't write anything there. And I said, but that's 'cause it's like the hundreds column and there aren't a hundred sevens in four, four hundreds, so that you can't write anything there. We've gotta sort of take. And he said, oh. I've never thought about it like that. And so you can actually put your place value houses, I'm waving my arms around because it'll be okay. Because we can see each other. So you can, and I suppose that's the way I always look to see it. We are thinking how many, how many hundred lots of seven are there in four hundred? There aren't any. So So therefore we have to treat that, take that for and do something else with it and and and what have you. That, that was an eye opener. But the fact that it was an eye opener for quite an experienced ITT lecturer was, I think, the first moment where I thought, Oh, maths, that's interesting, yes, and I remember it distinctly as a, from a long time ago now. But yeah, really interesting, but yeah, that whole place value, where does the place value fit into those four, that formal method of division is a really tricky one. And it's, it's the, it's the why has it got to be there?

Catriona: 15:28

Yeah. It's so true that. Like an algorithm like that, that we've all been using for so long and you like, maybe I've no idea, like how I first learned that maybe I did really deeply understand what was going on when I first did it, but then it just became so background that then when you go back to think about it, you go, wait a minute. How, how am I going to explain what's going on here? I think even for secondary students, they've been doing lots of these algorithms for so long that, well, of course you start by lining up the digits. Why would you ever not do that? You know, they haven't necessarily thought through why that has to be, because it's become so, you know, that bit's been lost so far in the past. So that's what I just need to go back to.

Andy: 16:11

Absolutely. I've said before on this podcast that at the start of year seven, I always used to just put a long multiplication on the board and get the kids to go right. Just do it. They're like, how'd you want us to do it? However, however you want to do it. I don't mind. And you could see those kids who had been to schools where it was expected that they would, you know, play around with it and look into it and come up with different ideas. And they'd write two or three things down or they'd explode those numbers. They'd split them up to 20 and three rather than 23. And then you'd see the ones who just go 23 times 47 is put zero down and just start doing all this. And then you go to, and you go, so why have you done that? And they just look at you and go. So I get the answer. So why does that work? Cause it's right. Is it not right? Is it not right? I've got it wrong. It's like, no, no, you've got it right. I just interested in what you know about what you're doing and why you know about what you're doing. You know, it's evident that year nine girl, I tutor she, when we do, we were doing some work on fractions and we were talking about Multiplying multiplying. Yeah. Multiplying mixed numbers and also adding mixed numbers. And so she'd done these, I didn't make numbers and they were horrible. It was like, you know, seven and three 19th plus something else. So she's going through this process, right? So I've got these seven times 19 and then I've got to do this. I've got to work that out. And then I've got to add three on, I've got to do this. And then the other one was horrible. And I was like, okay. And she did it all. It took a good few minutes and she got it right. Brilliant. You know, great maths, really good written methods, all that kind of stuff. So I said, what about this? What if we just did like seven plus three? So we've got 10 whole ones and then we had three 19s plus whatever two 17s or something And so she was like, okay, so it wasn't it wasn't that complicated You have to do 19 times 17, but it was numbers that were awkward And so she did that really quick and she was like what how does that how does that work? And I said, well, let's take it back So, you know the long multiplication we've been doing or splitting these numbers up fractions are just numbers So sometimes you need to just step back Look at the question and think Oh, actually this way is going to be easier to do it, but you've got to have been encouraged from day one in primary school, especially to do that and to have that, you know, you, you, you gain your skill set, you gain your arsenal of things and you step back and go, right. What do I do? You know, and that's numeracy. That's making younger students numerate. So like the lessons that you started off this podcast talking about there with division, that's the kind of thing we need to get into our lessons, isn't it? To get the kids questioning and wondering and curious and putting the teacher on the spot and making them feel really awkward and all that kind of thing. Well, that's good, right? That's our classrooms. That's what we want them to be.

Catriona: 18:47

I do think there's a lot about. I don't know, what do children think maths is? Because from their perspective it like, it can turn into just doing loads and loads of calculations. I have a really distinct memory of, I think I was in year three and we'd been doing formal multiplication for the first time and just doing pages and pages and pages of it and then it got to the end of the year and we had to We got given our reports and we were supposed to write a little comment about it and my comment was I don't like maths because I have to do multiplying all the time And to me that was just what maths was like sit down and do loads and loads of multiplying but actually like that isn't what what doing maths should look like so We've got to find some way to show them what what maths actually is, like what do we mean by doing mathematics and, and part of it is being really fluent with things so that you don't get hung up on actually how do I divide this by 11. You can do it quickly and accurately and with enough brain power left over to think through, wait, why is this working all the time? How am I going to generalize this? You know, there is also that, hold on, what have I noticed here? How could I, How could I make this work? What are the limits of this? Like the conjecturing side of it and, yeah, students need to be exposed to that a bit because they're not magically going to come up with, yeah, that's what being a good mathematician looks like unless that's what we promote. So I think that's kind of what I try to go for with year seven. Not that I always get it right. Like they still end up quite a lot of the time doing really boring lessons where people are like, That I haven't really planned properly. And afterwards you go, yeah, that didn't work, but, but that's time, right? Yeah, exactly.

Andy: 20:28

You just, we'd all love, and obviously Alison and I don't teach anymore, but we'd all love to have the time to create and think about those lessons ahead of time to go, you know, I know what I want to do and we can make a brilliant lesson here. You just don't have time because you've got year 8 class, year 9 class, year 10 class, year 12s, year 13s, like, you know. So it is that compromise about having to kind of find out the best way of doing something.

Catriona: 20:51

Yeah, and I think also some days I'm in a bad mood and so a kid might ask a great question and I'm like, Ugh I don't know. And kind of forget about it and move on. You know. Yeah, I don't want to paint a picture like I'm some magic maths teacher and you know, always like doing these amazing lessons and talk about all this. So yeah, no, it doesn't always happen, but sometimes it does. And I think it needs to happen enough that, that the kids kind of know what, what. being a mathematician should look like.

Alison: 21:21

Yeah, I think that's really interesting. I've, I've just been reading a couple of of things recently where John Mason was quoted quite a lot. And a couple of things that you've just said made me really resonated with some of the things. One of them was about talking about mathematical noticing and what you're, the lesson you described really shouted out. You, you knew, you knew what you wanted to do. The students to notice. And so the lesson was planned in order that they would notice that the place value could think, thinking could come into it, that the distributive law could come into it. And that made me think, gosh, when I'm working, so now I don't have a class of my own anymore, but I work with teachers and I do, I do collaborative planning sessions with teachers. And again, it's like, what do you want them to see? What do you want them to talk about in this lesson? It was really interesting. And the other one that made me think about your experience back in year three is that I think he talks a lot about. The difference between working on and working through. And if you're just working through a load of examples, then you get to the end of it and you think, well, I've done those examples, that's fantastic. But what what, what he advocates is this idea of working on the mathematics. And again, you could have given them whole pages of this. dividing numbers by 11. And but actually you didn't. You said, we want to have a think about this. And so by working on these examples, we can come to a deeper understanding and we can bring other things. And it was those two things that just, the way you described that lesson, just really shouted out. And I wonder if there's a sense that even when we're not, when we're, where our heads are full of everything else, that, that, Perhaps, even if, and the lesson maybe isn't the most inspiring one that we've ever sat in front of, if, if there's a way of just keeping those two, two little things like that, how are we know, what am I getting to notice? Okay, it's a bit, it's a bit of a, of an odd lesson today, but what do I really want them to notice?

Catriona: 23:09

Yeah. And how do I

Alison: 23:10

get that idea that we're working on this maths and not just working through it? It's just, yeah.

Catriona: 23:15

I'm thinking back now to a kind of contrast between a good lesson and a bad lesson, which were exact, I taught exactly the same lesson, period two and period three to two different year seven classes last week. And in, in the first one, it was going to be about division. So in the first one, I thought I'll just give them like some quick numeracy practice and I wrote up three random questions on the board. But then in the second one, I just changed the question so it used the number, so it was some big long number divided by two equals, and then, but then I used the answer to what I knew the answer was going to be, because I'd just done it the first period. This divided by three equals, and then I had them times their final answer by six, and so obviously it got back to the beginning. And, I mean, it was the same amount of practice, they still did two divisions and a multiplication. But then you can say, ah, what do you notice? Can you make another set that works? And suddenly there's like such a better task and it took exactly the same amount of time. It was, it was just so much better because I had that, that experience of doing it before. Basically, cause I put 20 seconds more thought into it rather than just writing up random numbers. Yeah, it just really reminded me of that, that there's such small things that you can do and I've been using lots of Don Stewart's resources where the answers make a pattern which are amazing for practicing arithmetic. It's just like, it's just such a small thing, but it makes such a big difference and they love it and they love the fact that there's always that kind of extra thing to look for. I love those.

Andy: 24:42

I love those as well. I love the And again, it goes back to that pattern spotting thing that if we can ingrain when they're doing something and it looks like, hang on, that's a bit weird. Miss, why, why is that doing that? You've won, like you've won on that 10 minute task that you're doing, because you just need one student sometimes in lessons to ask that question. And then it can just kind of filter around the room. And, you know, even though you're doing this such a small task, like you were saying, now, you know, you have, you had 20 seconds extra thought for that same task for the second class. But that goes right back to the start again, when you said, you know, this only happened yesterday and I've not really had time to reflect on it yet. And for all teachers listening, that is the most important thing. Find some time, whether it's to drive home, whether it's just having a cup of tea when you get back, you know, you've got your kids to bed and there's this moment of downtime, I've yet to find that book when you find it. Think about what happened in that day. Cause if you can give yourself a few minutes just to think about your lessons, it can be so powerful to feed into what. What happens next time you teach that lesson, you know, or similar lesson or that class or whatever it might be.

Catriona: 25:48

Yeah,

Andy: 25:50

I've been sat here and I love listening to you talk about this and what happened in school But all I can hear in my head is ask her about the problems, ask her about the problems, ask her about the problems. You know, I really really want to know about your, I don't, I don't want to call it obsession But you know, if you scroll through your Twitter, I think it's close. The geometry problems And puzzles and pictures that you set.

Catriona: 26:16

Yeah. Where did

Andy: 26:17

that, where did it come from? You know, what, why did that start? And I'd look at them and think, how do you come up with these problems? Like so consistently, and there's so much fun and just intriguing. Like, where's that come from? What in your nature led you down that route?

Catriona: 26:34

Do you know, well, the enjoyment I get out of it, I think is what Alison was describing before about like mathematical noticing. So lots of it is. I just draw loads of pictures, like I've got notebooks full of scribbles and, and quite a lot of algebra that goes with the scribbles and it's kind of, oh, I wonder what happens if I kind of, you know, stack these shapes up in this way. Kind of what can, what can you work out? And you sort of hope that you can find something that's, that's nice. And often I find something that's nice and then you realize later that it's not all that surprising. And I don't have my notebook here, so I can't think off the top of my head, but I think I was doing one recently where it was just like a triangle. And I realized that this triangle, it had side lengths that were really nice, like maybe like one, two and five as the side lengths of the triangle. And then it contained a 45 degree angle and I was like, Oh, that's really cool. And then I did a bit more work on it and realized that that wasn't that surprising. It wasn't one, two or five. It was so, but it. basically by stacking two other triangles together. It was not that surprising that that it had this nice, nice angle in it. And so a lot of it is kind of near misses, but then I think how could I disguise this in a way that would lead someone else to notice this nice thing that I think is quite cool. So most of the puzzles actually come from Wait a minute, like, there's this cool fact, like, for example, an an angle drawn anywhere in a circle is going to be the same, the angle at the circumference is the same anywhere. How could I hide this, like, hide the circle, but in a way that allows you to still somehow deduce that it must be there, and therefore it's there? You know, two angles are going to be the same when they look like they might be different. Or rotations are quite nice. If you take a shape and then you rotate it around, it's still the same shape. How do you hide that in a way that doesn't make it obvious it is the same shape twice by rubbing out a few lines that can, but not so many lines that someone can't deduce it to go back in. And so, yeah, I think it's, I think it's disguise. I really enjoy that feeling of trying to. I've noticed something, so trying to design a puzzle that will make someone else notice the same thing. Now what normally happens Do you bring that into your

Andy: 28:50

lessons then?

Catriona: 28:51

Ooh, I don't know, like there is a sense of it in, in what I was just describing, that you, you have an idea of what I want them to notice here is that dividing by 2 then dividing by 3 can be undone by multiplying by 6. How can I set up a series of questions that will lead them to, you know, Notice that fact, so yeah, I guess there's the same kind of attitude to it. That's kind of what I enjoy. Like, I love that. It's going to sound really smug and horrible, isn't it? You know that feeling where you know a secret that no one else knows yet. And then you just sort of watch them as they, as they work it out. And you're like, I love that. I used to have

Andy: 29:30

that with with trig identities in A Level. And you'd be working through it on the board and you'd just giggle to yourself. And the kids would be like, why are you laughing? And it's like. I know what's going to happen. I can see it and you'll, you'll see it, you'll see it. And then it'll, you know, slowly go around and go, Oh, there it is. That's it. That's it. We've, you know, we've talked about that mathematical kick or that, whatever you want to call it, that students get. And, you know, I still get answering right angle problems. We were talking off air beforehand about right angle competition and how beautiful and difficult those questions are. And I still get a kick out of getting those right. Even though, you know, I taught maths for 20 years and stuff like that, but it's just that, that challenge and that, that kind of, I don't know, success that you can feel. And when you see it happening, it's like, Oh, I love it. I love that. I still love that.

Catriona: 30:16

Yeah. It's exactly that. And I get that bit on Twitter because. Thankfully, there's a really nice community of people who, if you post a nice looking picture, will try and solve it with like a full picture of all of their workings. And I think geometry is the absolute perfect thing to do this for because, well, A, it's pretty and I like coloring things in and I have like some really nice Packs of felt tip pens. One of my year 13 students last year, as she left, got me as a present a pack of 100 different colours. And it is the best present I've ever received from a child. It's so good. So yeah, I love colouring things in. And they look really pretty, but also, like, there's almost colour. About six different ways to solve the problem. So I'll have in my head, like, Oh yeah. And I can make them notice this and this and this, and then someone will do it in an absolutely completely different way. And you go, Oh yeah, that's really neat. How did I not spot that? I just, I really liked that fact. So you get that kind of, I get a bit of the discovery back because people see it in very different ways. And I think it's very open to that. Like I can imagine that if you were setting a load of algebra problems, There maybe wouldn't be so many different methods to do it. But when you see something and you're like, Oh, how did I miss it? That's kind of, there's a bit of a mathematical thrill there as well, isn't there? Like, Oh, I'm kicking myself now. How did I not see that that was a possible way to do this?

Andy: 31:40

Do you ever use them with like a, you know, good year 11 class? Throw them up there.

Catriona: 31:45

Oh, do you know what? I don't, but I do have a really good year 11 class at the moment. So maybe I should. Maybe that'll be a, I've used them a bit with a level classes, but generally with kind of tweaks so that it's more specifically, like there's something that I want them to discover. So there's, there's a really nice one that I've used teaching vectors where they get lots of practice of, of using vectors, but then. At the end of it, they find that all their vectors have one particular form at the end. And then that's a definite like, wait a minute, why did that happen? And then I show them the puzzle and I show them non vector ways of solving the puzzle that this whole thing's been based on. Which is quite fun. So, but yeah, they're not always the easiest thing to, to fit in.

Andy: 32:33

Yeah, yeah. Especially, you know, you lose so much of that. Fun geometry at A Level, even though you want that A Level thinking and kind of understanding and the willingness that they've chosen A Level maths, for example. But actually, there isn't that much fun geometry to be had, if you like, that's overly relevant in the curriculum. Whereas at GCSE you can have all sorts of fun with it.

Catriona: 32:55

It is a bit of a shame that There's so much less geometry, but I do think that's about the only reason that they work on Twitter is that it's one of those things that people can reason a lot about, but I don't know, you, you don't learn hundreds of shortcuts in lessons. It doesn't feel like school maths. And I think that's partly because it does drop out of the curriculum fairly early on, you know, it's not like. Solving a load of equations where you'd sort of feel like, Oh, this is like doing stuff we did at school. It's my feeling. I think that's part of the reason they're so popular is you don't need a high level of kind of education as it were to get into them, but you do need a really high level of reasoning and so that kind of makes them quite fun.

Andy: 33:42

Oh, Alison, you're, you are muted, by the way.

Alison: 33:46

They don't look off putting, I think, your puzzles as well. I come across them on Twitter and I think, ooh, ooh, it's one of, one of Katrina's puzzles. I like the look of that. And you've drawn it yourself and it's beautifully coloured and it looks really appealing. It looks like the sort of thing I want to get into. Sometimes people put up something that looks like it's come from an exam paper and immediately there's that bit of maths anxiety comes back. Oh, it's an exam question. I'm going to have to solve this. It's really interesting. I really liked your, the way you were saying about designing them because you, you know, you want to lead them to notice something

Catriona: 34:17

and I've

Alison: 34:17

been thinking quite a lot about this. And I had a, I had lots of discussions with different people about, about what, what sorts of, what the, what problem solving, being an efficient problem solver is. And And we were talking the other day about, about, I think one of, one of these other, in one of the other ones about number of times I've said to children, just have a think about it, just have a think about it, and then realize that I don't actually know that they've got the mathematics to think about it. And I really like this idea of designing, designing lessons and designing tasks for them to do where We want some, some maths to fall out of it, and we, we want them to notice that this is here. They want them to realize that they can apply this understanding and this understanding, and therefore I will get to, to an answer. And I think sometimes we sort of, we see possibly in primary problem solving as either being a word problem that wraps up the maths I've just taught you, or something really interesting. from over here, and, and, and random and very large that involves listing a lot of things. And that sometimes it's finding that middle ground where I've actually, I've actually wrapped up in a problem here some understanding of fractions and how they relate to ratio that I want you to spot that you can use those two understandings here. And I've, I'm not, and that's maybe something I need to think about is how do, how do we design those problems? that force that attempt, force them to notice or are designed to allow them to notice that. Yeah,

Catriona: 35:50

I've been rereading Colin Foster's things about mathematical etudes recently and trying to think about that. My A level pedagogy group have been kind of looking at this and I mean, like when you find a really good task like that, where they, get lots of practice, but in the course of their practice, they're kind of working towards a higher mathematical goal. Like, they're just gold, aren't they? But then when you sit down and try and design them, it's really hard. So it, I mean, it's such a skill to be able to write good tasks like that. But you're right. Like there's such a payoff because Like, yeah, like students need loads of practice. And I was a bit worried that my students spent an hour yesterday only dividing by 11, but actually I think they, they probably did get enough practice of using the division algorithm, given that they do know what they, they, well, they have seen it before. Some of them are pretending they've never seen it before. I know they've all seen it before. I've seen their SAT scores. They have all seen it before. But, but they also get that kind of, larger sense of we're, we're being mathematicians here and like there's, there's something to achieve out of it. So yeah, I do think that point there. Exactly. And that's the

Andy: 36:57

difference, right? Between the year three, Katrina, who's like, oh, maths is just multiplying. But actually those, hopefully those year sevens are leaving, like you get home and how was maths? Oh, it was great. Like, we were just looking at dividing by 11. Did you know that if you've got a four, a four digit pandrame and then all of a sudden there's this enthusiasm there. And then you know what next lesson they're going to look forward to maths again. Because it might be a, but it, it might be one of those lessons, which you described as, you know, more boring or whatever, but you still got that buy in. You've still got that kind of, yeah, but no maths is cool. I did this last week. It was brilliant. Like, you know, can't wait to do that kind of thing again. And as long as you sprinkle a little bit of that in as often as possible. Then it's golden, you know, that is the, what all we, all our lessons need to be like, or should be like, or hopefully could be like again, if we had time, but it's, that's, that's what you need. That's what you want from those students.

Catriona: 37:50

Yeah. And doing it once in a while is, is okay, isn't it? Like you don't have to be absolutely 100 percent all the time, like doing it a little bit is actually, it's fine. And yeah, we,

Andy: 38:02

I taught in a couple of private schools at the end of my career and it was, you know, it was. You had more freedom than you did in some of the states we were taught in, but also you were, you were expected, if you like, that the students would get good results. So you had to be a little bit too much, most of the time focused on, well, what are you going to, how are you going to get a grade nine in your GCSE? This is what you're going to have to do. But it also meant that we could kind of go, well, you know what, every half term, can we just have a a problem solving lesson for every Key Stage 3 class? And we'll just take that lesson out of the curriculum. And. We could do that because we could work a bit quicker through things that we needed to and things like that. And it was great. So even if it was only once every six weeks, you knew you were getting this stuff into them throughout case issue, which was great. You know, we tried to do it further up, but it just becomes more difficult, but it is, it, it doesn't have to be every lesson, just anything. But I think like Alison alluded to earlier, the key thing about in those lessons where that first year seven one you did where. You did the same lesson twice and it was a bit rubbish, but actually the fact that you're the immediacy, even if you haven't planned, that's one of the key strengths, right, of good teaching is reacting to what's happening, reacting to what the students are doing, the questions they might have, and we can hopefully then adapt the lesson on the fly a little bit to kind of follow them and follow those questions and follow that curiosity. I was really, this is not part of any kind of questions that, that we think about ahead of time, but it's really got me thinking about post pandemic stuff. Now I taught a little bit after the pandemic and I felt like my key, my year 11s that I'd had in year 10, and then I kept them on in year 11, they were much, much better at learning on their own after the pandemic. And it had kind of worked isn't, you know, the right word, but it had worked for them in the sense of. They'd got the chance to test themselves and think about it. I also really noticed that my year sevens that came into school, they were my form, were not like, in inverted commas, normal year sevens. They were, they hadn't finished primary school. What's, what's the kind of knock on effect now for how students learn? So those year, what would they be now? Year nines?

Catriona: 40:14

Yeah, I

Andy: 40:15

think. What are they like as a year group compared to say the year sevens now that Come back to some normality with primary school.

Catriona: 40:23

That's such a good question. Is there a difference

Andy: 40:24

in how they learn?

Catriona: 40:26

It's because it's something I haven't really thought about for a couple of years. Like it's, it was such a focus for, for that first year, obviously, when we came back and everything was different and you had your tape on the floor that you were never allowed to cross that line and, you know, like break times at ridiculous times so that different year groups never met each other. And then it's sort of. it's scaled right back so that all of the kind of external signs of of COVID are now gone from the school other than occasional kids missing for it. And so I just haven't thought about it. So the honest answer is, I don't really know. Like I could tell you how our year nine cohort is generally different to the year seven cohort, but I've never really stopped to think about how much of that is affected by the fact that, as you say, they never finished primary school.

Andy: 41:16

Yeah. Yeah, and I'm sorry, you know, sorry to throw that on you. I don't know if, Alison, I don't know if anything's cropped up in the primary work that you do that has looked into it at all and what effect it's had on some of those younger learners.

Alison: 41:29

I mean, well, prioritisation curriculum, looking at how do we cope with these children with gaps? I think there, you know, there are still teachers I work with who say, oh gosh, they do still have gaps. I think one of the key age groups, is the very young, our youngest children, so children going, starting early years now. Who, or maybe the ones in, in, in Key Stage 1, certainly, who missed out on some really vital socialize, socializing experiences. And I was talking to a, in an infant school, and they were saying that their reception classes are really difficult this year, because they, they feel, even now, they, you know, they're sort of, You know, we're a couple of years out of it, but they, they missed all the early stuff. They didn't go to the toddler groups. They didn't go through all the snatching things off each other. At a point and being, you know, being sort of that, that sort of stuff was missing. And so there's still a lot of, of, of backfilling of that going, going on. So there are still gaps. I think, I think you know, they. relatively resilient. I'm thinking about mine who missed out on quite a lot of upper end of key stage three before I went back and did, and did was the first year of actually taking GCSEs. And to be honest, I'm not sure, I've nothing to compare it to, but he seems to have made a fairly decent transition on into A level study and doesn't seem too, too traumatized by it. I would say the one thing is that they've found and developed other ways of communicating.

Andy: 42:59

I

Alison: 43:00

think older, older, older students, I think, you know, he spends his, all his waking hours at home communicating with friends online. And so that level of communication has, has changed dramatically, but but yeah, there is still work going on looking at, at key stage two. And I think particularly at younger, how it's impacted on our youngest. learners in, in school with that the fact that they're learning, not only having to learn a curriculum, but that whole, the social personal development as well has been quite severely impacted for some of them.

Andy: 43:32

I saw it in I went to our sports day for reception last year, and I was talking to the guy that goes in and does PE there. And I was like, you know, my daughter's fairly sporty and, you know, just never stops. So she, she can run, she can do gymnastics, all that kind of stuff. And I was saying to him, you know, out of the 20 odd kids in the class, a good half of them, I don't really understand how to run, like, and I was saying to him, is that, is it normal? Like, you know, this is obviously the first sports I've ever been to with five year olds. Is this a kind of a normal thing? And he was like, just don't forget, they've not done sport for a year and a half, unless, unless their family have taken them to do stuff, they've not done any. So those, that coordination, that agility, that balance. Isn't there for a whole load of them? And I was like, God, I hadn't thought about that in my head. I started thinking about the math that's been missed. And that's a natural thing to think about, but you just don't think of the other side of it. And, and like you say there, you know, those kids who have a three year old and he's going through snatching everything phase and that's fine. Cause he's at nursery and that's what it's all about. But of course you'll have five year olds who didn't go to the nursery in that phase and didn't go to those, you know, preschool things and stuff like that. Yeah, it's, it's just. Wide range, you know, and you know, that's not on our normal questions, but it just kind of struck me as a you know with katrina being in school and Seeing it day to day. I just wondered what what effect it might be having. Maybe that's a whole let me that's season three Maybe we look at you know, three years on from the pandemic season three of digging for the why What's happening in our classrooms? I am Conscious of time as I, as I always say around this point of the recording and I don't want to take up all of your free period and your lunch time so there's a couple of questions that i'm interested to just kind of finish with and i'm going to ask him in a slightly reversed order to what we sometimes do this was I think it was a A quote kind of or something that allison heard at something and somebody said people don't become Truly independent learners until they're postgraduates Now that struck us as Really? Because, you know what, we've seen primary school kids and three year olds who are completely independent learners, and that's the whole thing. I just wonder what your thoughts are on that, Katrina, in a secondary school environment, you know, you must have kids that are independent learners. If they are, what do they look like? You know, what, what makes an independent learner as a teenager?

Catriona: 45:55

I mean, I find that so I find it very interesting, and I suspect that whoever said it presumably has quite a different idea of what being an independent learner means. It was, it was

Alison: 46:07

in a, it was in a university context. And they were talking about the fact that, that as they saw their students coming through, they didn't become truly independent learners until they got engaged in postgraduate study. And I happened to be in this discussion. We were, I can't remember quite what we were doing. And I said they're quite independent in early years. And there was a sort of tumbleweed moment. And so that heads out questions, what do we do? Why do we, where does it go from year

Catriona: 46:33

one onwards? I do think potentially in early years that they get more independence in the sense of the only thing I can think in which this kind of comment, the context of it must be in terms of directing their own curriculum, you know, all the way through. School, like certainly secondary school and undergrad study, the curriculum's directed for you. You know, somebody tells you what it is that you've got to learn and what you're going to be examined on, and that kind of becomes your focus. And I can see that postgraduate study is much more about building your own curriculum, but I think you're right, possibly, right? Early years, there's a certain amount of that as well. There's not a sense of, right, you're gonna learn this and then this and then this, and here's the next small step. There is a bit more freedom to kind of. I guess play,

Alison: 47:20

but

Catriona: 47:20

I think that means that there is no independent learning up to that point. I think I'd go for a broader definition of independent learning. I think there's room for kind of mathematical play and things earlier on as well. Just maybe they're not assessed or looked for because we have this idea that we're going to assess everybody against a really set curriculum. I always think we can create situations.

Andy: 47:48

Yeah, yeah, yeah. I always think we can create situations in class and you've talked about it a little bit already on this, where you are creating that independence because you're giving them that opportunity to explore. And I've, I've often talked about the bowling alley with the bumpers on. You know where you want them to get to at the end of the lesson, you're the bumpers, so you're just making them go down the right path, but there's enough space for them to bump around and explore and things like that. And I think that you are absolutely getting independent learners, aren't you? If you, if you're getting that kind of questioning and thinking and your year seven lesson with dividing by 11, well, does it work for this? I mean, how is that not an independent learner? Kids are asking those questions.

Alison: 48:31

Yeah, I think that, I think what you said about, How do we define independent learning? I think that's really, that's really interesting because maybe we can't give them the freedom that they have in early years where, yes, there is a curriculum and we are aiming to get them to a certain point by the end of it, but there's this wonderful balance that we try to achieve of self, of child initiated as opposed to adult initiated. And I think you're right, we, the adult initiated becomes the whole thing or can seem to become the whole thing and but yeah, lessons like the one you described gives back some control to the, to the pupils and they feel, you know, kind of where they, where they're going, but they've got that. They're definitely the bowling ball traveling down and having the freedom to move. Yeah. Yeah, and

Catriona: 49:17

teenagers, teenagers learn all kinds of things like that, don't they? Just not necessarily the things in lessons. And so I guess it kind of depends what you decide counts as acceptable learning and whether it's, you know, it's not always what's in school. Teenagers are very independent learners. About some things that they can't learn in school, you know, they, they go and teach themselves. So, yeah.

Andy: 49:39

Absolutely. Well, there are a whole bunch of teenagers moving behind Katrina at the moment, which must mean it's lunchtime. So we'll, we'll wrap this up. Thank you so much. What, what would be. You know, if you're a math teacher listening to this, and we always like to think of what, what's the takeaway for them, you know, like we've done a good lesson plan here, we've got an idea, we want to take something away. What would be your takeaway that you'd, you'd hope people would get from listening to this Katrina?

Catriona: 50:06

I know what mine would be but I'm going to put you on the spot. I really should have thought this through so that I can do it in a nice pithy way. It

Andy: 50:15

wouldn't be digging for the why in that stance.

Catriona: 50:18

Oh, I could give you a soundbite, couldn't I? I think it's that, if you can, like, The more thought you put into a lesson, the more the students are likely to get out of it. And ideally what they get out of it is in addition to whatever kind of focused learning objective you have that you want them to practice. They also, maybe not every lesson, but over the course of, you know, a year, they get a sense of what maths is. And so building in something that allows them to kind of be a mathematician for a bit. And see what that looks like in the course of doing what you do anyway, which I think you can achieve with, you know, like some small tweaks to little tasks, like making a pattern in the answers actually can have a big effect like that. Sorry, that wasn't pithy at all, was it? That was about as far from a soundbite as I could have got. That was great. I

Andy: 51:16

thought that, I thought that was wonderful. Alison, anything else from you? Before we finish, anything you want to ask?

Alison: 51:21

No, I'd just like to come and be one of your year sevens, Katrina, at some point. That would be really lovely. I'd love to, I'd love to take part in a lesson. A little MEI trip to your school to come and look at

Andy: 51:32

maths education.

Alison: 51:33

No, no, really good. Thank you.

Andy: 51:35

I will put links to, to everything that's been mentioned some of those activities and the, and the people. And I'll link to people's Twitters in here so you can go see Katrina's puzzles if you've not seen them. And if you want to shout out or reach out to us, then please do. Thank you so much again, free periods are not great to give up, so we really, really appreciate you doing that for us. You can go and enjoy your lunchtime hopefully before now. And

Catriona: 51:58

before I go, can I ask you what your takeaway was?

Andy: 52:02

It was. The idea of kind of what you said about your puzzles, and I think if people can plan something which Kind of, even though it's a simple thing, but you're undoing something, you're peeling back the layers, if you like, to reach this kind of thing. I think that's something you could get into lessons. And I think it comes from the same thing you said about giving yourself time to plan something, even if it's just the starter. You know, the thing that you want to get going in that lesson. Divided by two, then divided by three, all of a sudden you're multiplying by six. What does that mean? And it's the same, you know, you're not spending any more time on some, a small task, but you're getting more out of it because the student gets more out of it. And I think for me, that is, that's, that's the thing that reflection, that thinking about next time I do this, this would be really good if I did this, you know, because like we've said throughout this podcast, you can't have. Outstanding lessons every single time. It's impossible. You know, you can, you can spend hours planning lessons. This happened to me loads of times thinking it's perfect. And then the kids walk in and they're tired and they're upset. They're wet. It's windy. Right. Well, that's gone out the window. You know, we just survived the lesson and start again next time. And that's teaching, right? It's the beauty of it and the frustration of it all in one thing. Okay. Thank you so much again. Thank you for listening at home. I hope you got as much out of that as we did here. And we'll be back. For another episode of digging for the why at some point soon, we look forward to hearing from you. so much for listening See you again soon