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Ep 63. Teaching math so students learn with Craig Barton

This transcript was created with speech-to-text software.  It was reviewed before posting but may contain errors. Credit to Canadian Podcasting Productions.


In this episode, Anna is joined by Craig Barton, former secondary maths teacher, host of the Mr. Barton Maths Podcast, author of How I Wish I’d Taught Maths, and a new Tips for Teachers book series. Craig reflects on how his teaching evolved after engaging deeply with learning science and educational research.

Together, they explore what effective explicit instruction looks like in math from atomisation and worked examples to checking for understanding, purposeful practice, and problem solving across the I do, we do, you do phases. This practical, research-informed episode is essential for educators looking to improve math outcomes by aligning instruction with how learning works.

This episode is also available in video at www.youtube.com/@chalktalk-stokke

Order How I Wish I’d Taught Maths here: https://shorturl.at/rGTin 

Craig Barton’s Tips for Teachers books & websites: https://mrbartonmaths.com/ 

TIMESTAMPS

[00:00:22] Introduction

[00:05:09] What changed Craig’s mind about how to teach math

[00:09:07] When struggle isn’t productive

[00:12:29] Essential components of a well-structured, explicit instruction lesson

[00:14:54] An introduction to atomisation

[00:16:16] Purposeful practice

[00:21:08] More on atomisation

[00:23:58] Examples of atomisation

[00:27:03] Summary of atomisation

[00:27:49] How to deal with wide skill ranges in the math classroom

[00:31:36] Engelman & Carnine’s Theory of Instruction

[00:32:30] Tips for the ‘I do’ stage

[00:38:24] Importance of checking for listening [00:44:17] Tips for the ‘We do’ stage

[00:45:51]  A ‘we do’ fraction example

[00:49:23] Atomisation helps with struggle [00:52:24] Tips for the ‘You do’ stage

[00:54:13] How to use purposeful practice [00:55:24] How to set students up to solve non-routine problems

[01:03:56] How to effectively teach problem solving

[01:09:05] The importance of structure

[01:09:53] Can explicit instruction in math be interactive?

[01:12:01] Where to find Craig

[01:13:08] Final thoughts


[00:00:00] Anna Stokke: Welcome to Chalk & Talk, a podcast about education and math. I'm Anna Stokke, a math professor and your host. Welcome back to another episode of Chalk & Talk.

 

In this episode, I'm joined by Craig Barton, a former secondary math teacher, host of the Mr. Barton Maths podcast, and author of several books, including How I Wish I Taught Maths. In this wide-ranging conversation, Craig reflects on his own journey and how his teaching changed after talking to experts in learning science. We take a deep dive into what effective explicit instruction looks like in math, from atomisation and worked examples, to checking for listening, purposeful practice, and meaningful problem solving, including how to think about instruction at each of the ‘I do’, ‘we do’, and ‘you do’ phases.

 

This episode is packed with concrete, practical ideas for educators at all levels, but it also speaks to a bigger question. How can we give more students math success by aligning instruction with how learning really works? I hope you like it. Now, before we get started, I have an exciting announcement.

 

I will be co-delivering a four-session short course on evidence-based math teaching through Australia's La Trobe University School of Education starting April 2026. It can be taken by any teacher or pre-service teacher anywhere in the world. I'll include a link in the show notes for registration, and I hope to see you there.

 

Also, Chalk & Talk recently surpassed 300,000 downloads. Thank you so much for listening and supporting the show. If you find the show helpful, please do me a favor and take a moment to leave a five-star rating on your podcast app.

 

It really helps others discover the show. And if you know someone who could benefit from the conversations we have here, whether it's an educator, a parent, a policymaker, or anyone passionate about education, please share it with them. Your support helps bring these important discussions to even more people.

 

This episode is available in both audio and video. I'll put a link to my YouTube channel in the show notes so you can check out the video and please do give the show a follow-on YouTube. Now, without further ado, let's get started.

 

It is a pleasure to have Craig Barton with me here today. He's joining me from England, and he hosts the popular podcast called the Mr. Barton Maths Podcast. That's maths with an S. And he taught math in secondary schools for 15 years.

 

He was the TES Maths Advisor for 10 years. That's the largest professional network of teachers in the world. He's the author of three bestselling books, including How I Wish I'd Taught Maths.

 

And you should definitely check that one out. It's a great book. And a recent series called Tips for Teachers Guides.

 

And I think there's 16 of them in total, and I think I have six of them. So, for example, how to use mini whiteboards, checking for understanding, that sort of thing. And he's the creator of numerous websites.

 

For example, variationtheory.com, and these are used by teachers around the world. In fact, I've used them in presentations, so I've recommended them to people. In 2020, he was appointed as a visiting fellow at the Mathematics Education Center at the University of Loughborough.

 

 

And he has worked with teachers all over the world and students all over the world. And I'm very excited to have him here to talk to me about math today, which is, of course, my favourite subject. I think it's his too.

 

So welcome, Craig. Welcome to the podcast.

 

[00:04:07] Craig Barton: Thanks. And a very kind introduction. That'll be a big anti-climax now, but that was very kind. Thank you.

 

[00:04:12] Anna Stokke: In your book, How I Wish I'd Taught Maths, you write about starting your career as a secondary math teacher, and you were using inquiry-based activities and rich tasks, as we hear them called here, and what you thought were highly engaging lessons. And you were praised. So, you talk about being praised for it and being rewarded for it.

 

But over time, you started reading about cognitive science and how students learn. And as you talk to experts on your podcast, you started to question whether that approach really supported your students' long-term learning. And I guess that led you to rethink your teaching and shift towards maybe what we call explicit instruction.

 

So, things like retrieval practice and worked examples, etc. And today you're actually a leading voice on effective math teaching. So first off, maybe you can take us back. What were some of the main reasons you changed your mind about how to teach math?

 

[00:05:09] Craig Barton: This is a good question. I think having spoken to people for many years on my podcast, I think a lot of teachers, math teachers in the UK from kind of my generation, so I'm old these days, so I'm 43 now. So, I started training in the early 2000s.

 

That was a time when the trendy way to teach mathematics was all through open-ended tasks, inquiries. And it was certainly the feeling I got is if you were telling students how to do something, you were failing as a teacher. If kids are asking you questions or they were stuck, that was a good sign because they were struggling and struggle was going to lead to more learning and leading voices at this time were Joe Bowler.

 

And that was the only way I knew how to do things. And also, that appealed to me as someone who was good at math. I love struggle and this has really hit home for me over the last 20 years or so.

 

I think if you associate struggle with success at the end of that struggle, struggle's quite fun. So, it's a little hiccup in the road, but you enjoy the challenge because you know what in the end you'll get there and it'll all make sense and it'll, you'll enjoy it more because you've struggled to get there. But what I started realizing as I was teaching, it took me about 12 years to realize this, is for many kids, they associate struggle with failure and that struggle doesn't lead to enlightenment at the end.

 

It just leads to them thinking, Oh God, here's another thing that I don't understand. I think eventually I would have come to the realization myself because I would have taught enough students who weren't enjoying the subject and weren't becoming as successful as they wanted to, as they could have done. I think my methods that I used to do would have been fine for top set kids, but I think you can pretty much teach those kids any way you want and they'll be absolutely fine.

 

But for most students, what I was doing wasn't working. But what fast tracked me to that was really starting my podcast and it was, I think probably the two or three most kind of seminal episodes were when I had Dylan William on pretty early on and he made a point to me like, why would you ever ask one student to answer a question when you could get responses from 30 kids? And if that kid who you've asked the question is put their hand up and is a high achieving, high confident volunteer, the chance of their response being representative of the rest of the class is pretty low. That made me completely rethink my ways I checked for understanding and led me to use things like diagnostic questions.

 

I had Greg Ashman on early, who your listeners may be familiar with his work on cognitive load theory. He did his PhD with John Sweller. What Greg opened my eyes to was the limits of working memory and how the way I was teaching with these open-ended problems and just wasn't suited to many of my students whose foundational knowledge wasn't there, and they couldn't learn mathematics through these problems.

 

They need to learn the maths first and then enjoy the problems later. So that was a big changer for me. And I think the third one was when I had the Bjorks on, Rob and Elizabeth Bjork, and that really opened my eyes to the power of retrieval, interleaving, spacing, and what the Bjorks called the desirable difficulties.

 

And that made me realize that it wasn't enough for kids to just get things in the moment and then move on to the next topic. If we didn't systematically revisit those concepts, then all our efforts were going to be in vain because we're just going to forget them. And it's the most frustrating thing as a teacher, both for you and the kids when the kids got something on one day and then forget it the next day.

 

So, I think those three things opened my eyes. And then as I was interviewing guests, they'd recommend papers and I kept reading more and more and more papers and realizing lots of the things I was doing in the past was wrong. And that then formed a bit of a narrative in my mind.

 

I thought, you know what? There's a story here. Somebody who thought they knew what they were doing, realized they were wrong and what they did as a result of that. That led to my first book, The rest is History, as they say.

 

[00:08:49] Anna Stokke: A couple of things you mentioned there that I'd just like to follow up on. The first thing I heard you kind of say that this sort of teaching open-ended problems, productive struggle, this was very popular in the 2000s or trendy. Is it not anymore? Because it's very popular here.

 

[00:09:07] Craig Barton: Really interesting. Right. So up until recently, I was visiting three or four different schools in the UK.

 

I went on a bit of a mission. I taught full-time, as you said, for 15 years, and then I kind of co-founded a company and I had a load of other things going on. So, I kind of stepped away from the classroom for a couple of years and then there was COVID and all this kind of stuff.

 

And then over the last few years, I thought, right, I want to get back into classrooms all around the country and get a real handle on what math education's like at the moment. And what struck me was there aren't that many maths teachers now who are teaching in the way that I used to teach. I think certainly in the UK, the message has got through that actually probably the most effective way to get kids successful at maths is some form of explicit instruction, some form of ‘I do’, ‘we do’, and then some independent practice to set the kids up for problem-solving later and that retrieval practice is important and all the things, and that formative assessment's important.

 

But what I did notice over the last three years, and as I said, I was watching probably a hundred lessons a month, was that whilst everybody was trying to do a similar thing, the different levels of effectiveness varied wildly. And that was why I wrote the series of books that you mentioned there, The Tips A Teacher's Guide To, to try and pick out what are the active ingredients that make a good ‘I do’? What are the active ingredients that make a good do now or whatever it is? Because it's all well and good everyone's saying they're doing the same thing, but if you can't pick apart what makes one effective versus what makes it ineffective, then you're not going to have the results that your kids deserve. So, I think teaching has shifted, certainly in the UK, to a more, in theory, explicit form of instruction.

 

There are still pockets of teachers, of course there are, who think I talk a load of rubbish, but my aim over the last few years has been, okay, if you want to go for this way of teaching, how can we make it as effective as possible?

 

[00:10:56] Anna Stokke: You've seen this sort of shift in the UK, and I think we're really waiting for that to happen in Canada and the US. I wouldn't say we're there yet. And I bet you with your podcast and books have helped a lot of teachers, to be honest.

 

You may have been partly responsible for that shift, right? You've done a lot of great work. And also, you mentioned Dylan William and I had him on. I also think he's great.

 

And he kind of shifted my thinking on that too. Like I'm doing a lot less, who knows the answer to this question, and a lot more things like plural response or finger voting in my own classes, which are actually quite large because I teach at the university level. And of course, I'm a huge fan of Greg Ashman.

 

He was one of my first guests. We should talk about teaching math because you've thought about this a lot.

 

You've talked to a lot of people; you've written lots of books. Let's go big picture to start, and then we'll zoom in. And this is a big question.

 

What do you see as the essential components of a well-structured, explicit instruction lesson? We'll say in secondary math, because I don't often talk about secondary math on the podcast. I'm often talking about K-8 math because I believe strongly in foundations. And I think that at least here in Canada, that's where we see a lot of the issues start, but maybe you can talk about secondary math and what you see as the essential components.

 

[00:12:29] Craig Barton: The benefit of listeners, my experience has been teaching students over here from year seven to year 13. So that's 11-year-olds up to 18-year-old. So that's the context where my teaching experiences come from.

 

Over the last few years, I've been very fortunate to visit lots of primary schools and particularly looking at the older age groups there. So, kind of age seven to 11, but I wouldn't profess to have any expertise. Well, I do have a six-year-old and a three-year-old now, so I'm getting very interested in those foundational levels of mathematics.

 

That's my context. I sound like a really awkward guest here, but where you kind of pick apart a question, but I wouldn't talk in terms of lessons. I think that's the single biggest mistake I made when I was thinking about my planning.

 

I think planning, thinking of the unit of planning is the lesson I think is a big error. So, I think in terms of what I would call a learning episode, which I define as the amount of time it takes for students to really understand a concept. Practically this might be like a block of time on a scheme of work or a curriculum document.

 

So, let's say for example, you've got a week or like five hours to teach students fraction operations or basic angle facts or whatever it is. I'd think, okay, I've got this kind of amount of time. What are now? What are the essential components I can fit in? And if some go over a lesson that doesn't matter, I can just start the previous lesson when the last one left off.

 

I'll talk through what I would consider to be my kind of key components of this learning episode and model. I would always start with some kind of purpose. I think people who would say I took a load of rubbish and also people who think direct instructions are a load of rubbish, I think they miss this bit out.

 

There's no good teacher just arrives in a lesson and starts teaching an algorithm to kids or a procedure to kids. What they do is they try and sell the kids on the dream of why this is important. It only takes two or three minutes and there are lots of different ways you can do this.

 

There's some, it might be a little story that you might tell about it, some historical context or something. I'm a big fan of Dan Mayer's headache aspirin approach where you try to say to students, try and show them the reason why this procedure is important because it helps them solve something that they can't. Now that's very different for me from productive struggle.

 

I'm not letting these kids struggle for 30 minutes on something and then show them the answer. It's a little 30 second struggle, which then allows them to say, all right, okay, now he's going to teach us something that's going to help with that. A little short section there on just the purpose of it, two to three minutes.

 

Then we go into atomisation and this for me has probably been the single biggest change to my thinking about maths education in the last five years. I taught a little bit about the concept without using the word atomisation in How I Wish I Taught Maths, but then in my two other books and then in this big series that I've written, it's at the heart of it because I've realized that if you don't atomise, in other words, break a complex procedure down into its smallest meaningful components, the chance of kids understanding it is minimal and we can dive into more what I mean by atomisation a little later. I'll just give you a top, an overview.

 

So, we've got purpose. We then go into atomisation where we break everything down. Then we go into the ‘I do’ where we put everything together.

 

So, the ‘I do’ for me is a teacher led ‘I do’. The only, I'm not asking any check for understanding questions. The only questions I'm asking are checking for listening questions because all that matters to me during the ‘I do’ is I've got my kid's attention.

 

And again, we can dig into any of this that you like later on. After the ‘‘I do’’ comes the ‘we do’, so the ‘we do’ checks for understanding. Has the ‘‘I do’’ made any sense? Big thing for me during the ‘we do’, I need mass participation.

 

I'm not asking for volunteers. I'm not cold calling. The kids are doing it all on mini whiteboards.

 

I need to see every step of their work. Following the ‘we do’, if that's gone well, comes the independent practice. While a lot of teachers will call the ‘you do’ phase of a lesson.

 

Then following the independent practice, and of course we could be two hours into a learning episode here. We might be two and a half lessons in. It could be at any phase here.

 

After that independent practice comes the problem solving and I have two types of problem solving that I like to use. There's one I call purposeful practice, which is where kids are still practicing a procedure, but there's problem solving elements in there. And then more traditional problem solving where kids are dealing with non-routine challenges and so on.

 

I would have all the components that I used to have in my teaching, but they'd just be shoved towards the end when the kids are in the best position to deal with them and be successful. And the final component of the learning episode for me, doesn't happen in the learning episode. It happens later and that's the retrieval opportunities.

 

So, I would schedule in for every learning episode, multiple opportunities for students to revisit the concepts that have been taught, whether that's in do nows, whether that's in low stakes quizzes, mixed topic homeworks. And so, we're constantly revisiting those things that we've covered. So that's a top-level overview.

 

I'll just say those things one more time. We have purpose, we have atomisation, we have the ‘I do’, we have the ‘we do’, we have the ‘you do’, purposeful practice, problem solving, delay, retrieval opportunities.

 

[00:17:26] Anna Stokke: Let's start a little bit with purpose. This is really important.

 

And you said something there that's so important and nobody starts teaching without purpose. I actually want to be very clear about this one thing though, is that purpose doesn't necessarily mean a real life application.

 

Because I think people often think that that's the really important thing. Certainly, you can motivate a math topic. And why this is useful is just to solve a math problem or to make something more efficient.

 

I always say economy of thought. You know, mathematicians we’re always thinking of nice ways to solve problems, right? Economy of thought, do things efficiently. You gave a great example of it in one of your books.

 

It may have been the big one, the how I wish I'd taught math. I think you had a quadratic equation. So, you have a quadratic polynomial equal to zero and you ask the kids, how would you figure this out? You could start guessing that's a really inefficient way to solve that problem.

 

And wouldn't it be great if there were this nice way you could use the quadratic formula or you could factor if you can. That's an example of providing purpose. Do you have any more to say about that?

 

[00:18:40] Craig Barton: That's exactly right. That's a great, you've two things on that. So firstly, you're right. The kind of real-life angle is often counterproductive because you're having to shoehorn this real-life context.

 

The kids are like, what are you on about? And it's just an absolute waste of time. That example you picked up on that, that's a really good example of Dan Mayer's approach there. Let's create a mini headache.

 

So, for 30 seconds, try and find a value of X that makes this quadratic equal to zero. Anybody got one? Maybe somebody finds the positive one. All right.

 

But there's another one out there somewhere and they're trying, they can't find it. And he said, well, listen, I've got a method that make you find this every single time. And that 30 seconds of struggle is just enough for them to not get frustrated and demotivated, which can happen with the so-called productive struggle.

 

But it's enough time for them to really want that solution you're going to present. And the second thing I'll say, and again, we can talk more about this later on. I spend a lot of my time these days.

 

I have, I co-founded a company, Diagnostic Questions ED, and we're doing a lot of work with AI at the moment. I'm absolutely obsessed with AI.

 

I had been a massive skeptic when it first came on. A really great use case of AI that any of your listeners can use straight away is you can say something like, I am teaching, whatever, adding negative numbers tomorrow. Why do we need to add negative numbers? And you can stick around to chat GPT, Claude, Gemini, whatever you want.

 

And it'll spit out an interesting story, either historical context, a little problem where negative numbers have helped solve it. And straight away, you've got your way in. So, you'd have to think about these.

 

You'd have to think of these yourself. You can stick any topic in and it'll spit out. And then of course you can train the models to be a bit smarter.

 

You can give them three or four examples of really good ones and say, come up with a similar one, but just stick it in, just experiment. And you've got your purpose. Cut it two minutes, all you need, and then the kids are up and running.

 

[00:20:30] Anna Stokke: That was great advice. Use chat GPT for this.

 

[00:20:33] Craig Barton: Yeah, why not?

 

[00:20:34] Anna Stokke: We're going to find a purpose for what I'm about to teach today, if we're not sure, right? Because that's another issue.

 

You actually have to know the subject really well to be able to come up with those sorts of purpose exercises at the beginning, which should be quite short, but they do want to motivate people. You just have to be careful with AI that it tells you the truth because it does actually make mistakes. But I suppose it's not that important in this particular case because we're just talking about purpose.

 

I think that was really useful advice. Let's talk about atomisation. What is it? How do you do it?

 

[00:21:08] Craig Barton: I first heard about this, I was looking this up, I think it was probably 2017.

 

I had a guy on my podcast called Chris Bolton and listeners may have listened to my podcast. You'll have definitely heard him interviewed because I've interviewed him many, many times now. He was an early proponent of direct instruction and a big fan of Engelman's work.

 

So, he's one of the few people I know who's read Theory of Instruction. And again, I don't know if your listeners have ever tried to read it. It is one of the hardest books to read, Engelman and Carnine's.

 

but the actual ideas in it are, they're gold. They're absolute gold. And it's this idea of faultless communication.

 

Can you explain an idea so that if kids are paying attention, they cannot fail to understand it. They cannot fail to answer your questions correctly. And this was a whole new world to me because I'd been teaching kind of aiming for 80%.

 

If 80% of my kids are getting it, fine, I'm doing a decent job. Chris is like, no, no, no, 100%. If your kids are paying attention, a hundred percent of your students have to be getting what you're saying.

 

That is impossible. And over the last, what, seven or eight years, I've been fortunate enough to speak to Chris and also read a lot of the primary sources, experiment myself with my own students and also with teachers that I work with and coach. And I've started to see that actually, you know what, this can work.

 

This really, really can work. And at its heart, atomisation is really, really simple. And it goes like this.

 

You take a procedure that you're about to teach. Let's do something boring, right? Let's do add in and subtracting fractions. Cause it's a nice one that listeners of teaching any phase will probably have taught at some point.

 

The worst way to teach it, and I'm sure nobody would do this, but I've seen it done, is you just dive in and try and teach the procedure. You've done the purpose. And then you say, right, I'm going to teach you a new thing today.

 

And you dive in and teach the procedure. If you think what is involved in teaching adding and subtracting fractions, there's a hell of a lot going on there, right? Because you've got a load of prerequisite skills. So, if the denominators are different, you've got to be able to find the lowest common denominator or lowest common divisor.

 

Once you get those denominators the same, you've then got to know, all right, I have the numerators, but I don't have the denominators. Maybe then you've got to simplify the fraction. There's a load of stuff going on, right? So, the kids, if you think about a child, they're sat watching you.

 

They're trying to follow all these little steps, which at the moment we'll call atoms. And at the same time, they're trying to follow how these atoms fit together. And it's impossible. Kids will probably nod their way through it and you'll scaffold and you'll guide them to it.

 

But when it comes to them to try it themselves, there's absolutely no chance. And I see it time and time again. Kids sat there like dumbfounded whilst the teacher explained something.

 

The kids then have a go themselves. They get it wrong. The teacher assumes the kids haven't been paying attention.

 

And it's not that at all. It's because the teacher's tried to do too much. What atomisation is, is before you go anywhere near teaching a procedure or what Engelman would call a routine, you first need to break that routine down into atoms, the smallest meaningful components.

 

Let's do a concrete example. Let's take adding and subtracting fractions. Some of your atoms would be find the lowest common denominator, lowest common multiple.

 

Add fractions or subtract fractions when the denominators are the same. Make fractions equivalent. Simplify fractions.

 

All those things are your kind of prerequisite knowledge that kids would need. But there's another one in there as well, which is decide whether fractions are in a form ready to be added or subtracted yet. So, know that if the denominators are different, you need to do something before you add them.

 

Whereas if the denominators are the same, you can just crack on and do it. So, you get your list of atoms and that's the hardest part of this process. And it takes a while to kind of train yourself up to this, but you can do this with colleagues.

 

You can use a bit of AI, just say, break this down to its sub skills and then assess whether you think it's right or not. Once you've got your list of atoms, you're nearly there because then what you need to do is think, which of those atoms have my kids met before? Have I taught them lowest common multiple? Yeah. Have I taught them finding equivalent fractions? Yeah.

 

Any of those atoms that kids have met before, you check their understanding of. Not while she's doing the routine, but separate in perhaps a prerequisite knowledge check phase. You just have some questions on the board, find the lowest common multiple of five and 10, put it on your mini whiteboards.

 

Find the lowest common multiple of four and six, put it on your mini whiteboards. All right. Amazing.

 

We can do that skill. Tick. Let's move on.

 

Make these two fractions, make these two fractions equivalent. Put it on your mini whiteboards. So, you're doing just some checking for understanding of these atoms that kids have met before.

 

Any problems there, you've got to address it. There's no point you're moving on. You've got to address it.

 

But these are such kind of finite skills. And as long as you've been doing your spacing and retrieval, hopefully these are fresh in kids' minds, but this is a good way to kind of reactivate it. Then if any of those atoms are new, for example, that one I mentioned, check whether, decide whether fractions are in a form ready to be added or subtracted.

 

You teach that separate as well. So, you assess atoms separate that kids have met before, and you teach atoms that are new separate. So, you show kids these two fractions, two thirds and a quarter, they're not ready to be added.

 

But if I change that one quarter to one third, now they are ready to be added. And if I change those thirds to fifths, now they are ready to be added. If I change one of those fifths to a sixth, they're not ready to be added.

 

And you can communicate this in about 20 seconds. And then say to the kids, right, let's see if you've got it on your mini whiteboards. If I show you these two fractions, are they ready to be added or not? Or hands up if you think they're ready to be added, but just something as simple as that.

 

And that means that when you then go into the ‘I do’, when you put all these atoms together, everything you do, everything you explain the kids have seen before, their attention can be on how familiar things fit together in this new way. As opposed to what happens in 90% of the math lessons I see, kids are trying to figure out how unfamiliar things fit together in a new way. And that's too much.

 

So, atomisation sets up the ‘I do’. It sets up everything else that follows by getting kids into the ‘I do’, where they're understanding their confidence is really secure. So that is your kind of whistle stop tour of atomisation.

 

Believe me, listeners, you can take the world's deepest dive into this. So, every month, Chris Bolton comes on my podcast and he updates me along with a teacher who's kind of trying these ideas out, what's working, what isn't working, because you can get into all these things about the optimal way to teach things. But if you take nothing more away from this, then it's a good idea before you explain something to break that procedure down into atoms, assess the ones kids have met before, teach the ones that they haven't, and then put them all back together.

 

You'll get 80% of the benefits and it's an absolute game changer.

 

[00:27:49] Anna Stokke: In a way you're talking about looking at the prerequisite skills for whatever it is that you're teaching. Hopefully you're refreshing the student's memory on those skills, but not always, because you're going to have students that didn't actually master those skills.

 

This could be quite complicated, depending on the topic. There can be so many prerequisite skills. I mean, I don't know what it's like there, but probably one of the biggest comments I get when people write to me is about the wide range of skill levels within classes, that teachers have so many students that are really far behind and they can't keep up.

 

This should help with that, right?

 

[00:28:32] Craig Barton: Yeah, so there's a few things here. I would go so far as to say this is the only thing that helps with that. The other option, teachers say to me, I'll never work, never work with my kids because they're so far behind.

 

Have you tried it? This is the only way for those kids to catch up. Or they'll say things like, well, my smart kids don't need this. Well, the thing is there that, how do you know your smart kids don't need it? Maybe they can do one or two problems on their own independently, but how secure is their knowledge? No kid ever moans, this is the thing, right? Teachers say to me, oh God, the kid, this is too slow for their middle, feels like it's holding them back.

 

No kid ever moans if they're getting things right. You've got to keep smashing these questions to get them all right. And the thing with atomisation, very quickly you can do some quite complex things.

 

Let me take that example of deciding whether fractions are in a form ready to be added or subtracted. The bottom line there is, if the denominators are the same, they're ready to be added or subtracted. You could take a class of 10 year olds, really mixed range of ability in that class.

 

And within 30 seconds, you could show them two fractions where the denominators are quadratic expressions and every child in that class will be able to tell you whether they can be added or subtracted because they know exactly what they need to look for. Optimization closes the gap and it doesn't close the gap by holding back your top kids. It pushes everyone up and everyone gets it every single time.

 

 

The thing that makes it go wrong is you don't break it down enough and you're not kind of careful enough with your choices of questions. But even a rough and ready approach will do infinitely better than not doing it at all because what's your other option? Your only other option is to crush your fingers and hope some miracle's going to happen. And that whilst you're explaining all these things that kids have either forgotten or never knew in the first place, somehow, they're going to remember how to do them, learn how to do them and then integrate it into this new complex routine.

 

Absolute madness. When you first start doing this atomisation, of course it takes a little bit longer and of course the kids are a bit resistant because it's a bit new, but pretty quickly when they start seeing how successful it can make them and you start reaping back all that time that's lost re-explaining things, there's no going back.

 

[00:30:41] Anna Stokke: You mentioned at the beginning the book by Engelman and Carnine. What's the name of that book?

 

[00:30:49] Craig Barton: Theory of Instruction. Put it on your Christmas list is what I'd say to listeners. If you're struggling sleeping, it'd be a good one because it is hard.

 

But I've just said two alternatives to that. So one, I sound like a dodgy salesman here. I've just literally put my atomisation book has come out two days previous where I've tried to summarize everything in one hour read and Chris Bolton, his book's going to come out in March, which will be the absolute Bible for all of this.

 

If you want to go to the source by all means go for it, but there are more digestible alternatives available.

 

[00:31:17] Anna Stokke: That’s great. And just so listeners are aware, Engelman was the original capital D, capital I direct instruction guy.

 

He came up with that program that was evaluated in project follow-through that I've talked about on the podcast before. So, he talks about atomisation in that book then.

 

[00:31:36] Craig Barton: Yeah, it's all about it, right? And what's interesting is in theory, it's subject agnostic.

 

You should be able to atomise everything. It's just maths works really, really well for it. I'm also experimenting.

 

We've got, I mentioned before, I've got a three-year-old. I've bought, listeners may have heard either me or other people talk about this, the classic learn to read in a hundred lessons by Engelman. So that is your kind of atomised way of reading.

 

It's a similar process. It just lends itself really, really well to mathematics with the way you can break maths down into its little components. All kind of Chris's work, all my work is all based on Engelman.

 

And it's just trying to make it work for teachers who don't have a hundred hours to spend trying to decipher exactly what Engelman's all about.

 

[00:32:19] Anna Stokke: We’ve talked about atomisation and I agree. That's very important.

 

And let's go on to the ‘I do’ stage and you can tell us about some tips for that.

 

[00:32:30] Craig Barton: I have strong views on this Anna. And again, this kind of splits the crowd. I'll just chat for a few minutes and then feel free to challenge or kind of come back at me at this.

 

One of my favourite things to do when I watch lessons is as soon as the teacher starts an explanation or a model, I just hit record on my phone, the audio recorder on my phone. And again, as a tip for listeners, next time you're about to do an explanation, just do this yourself, just on your phone, just put your phone on your desk, just hit record, just play and then just do your explanation as normal. Then after the lesson, just play it back to yourself.

 

And all I would recommend is have some kind of stiff alcoholic drink in hand because you will not believe the nonsense that comes out of your mouth. And I've had this with teachers where in the coaching session afterwards, I've said, talk me through how you explained expanding brackets. And they give me a pretty good explanation.

 

And I say, right, do you want to hear what you actually said in the lesson? And I play it and I've had teachers that got their heads in the hand, that they just cannot believe the nonsense that's coming out of their heads. And if you kind of boil teaching down to its essence, it's about explaining something that you know that somebody else doesn't know. That's the art of teaching.

 

That's what it's all about. The majority of lessons I watch, it's quite a poorly delivered phase of the lesson. And I think there are a number of reasons for this and they're all fixable.

 

So, the first is, I don't know many teachers who actually have planned out what they're going to say when they step up and do that. ‘I do’. The first time the words of how to explain something have left their mouth is when they're in front of 30 kids.

 

And that's never a good idea. A very simple thing that you can do straight away is just either rehearse it first, just like say it once to yourself or say into a voice recorder or something or bullet point a few things that you're going to say. That's going to improve things massively.

 

Second thing is, if you're going into your ‘I do’ having not atomised, your ‘I do’ will be a failure because for the reasons we've just spoken about because kids are trying to follow a new process with atoms that they aren't secure on. That's never going to work. Atomise first and you're giving yourself the best chance or your kids the best chance of following this new idea that you're about to explain.

 

Then I think the single biggest mistake that teachers make when they're doing their ‘I do’ is they try and do a bit of an ‘I do’, ‘we do’ hybrid. So, it's a bit of the teacher explaining, but it's also a bit of checking for understanding. It's a bit of kind of guess what's in my head and can you guess what I'm going to do next and all that.

 

And some kids get it great, but some kids just look at, watch it, thinking what the hell's going on here and you've lost them already. So, I think the ‘I do’ needs a really clear purpose. And for me, the role of the ‘I do’ is to try and communicate a procedure as clearly as possible.

 

And that requires a clear, concise explanation that you've thought through ahead of time. No questions that you're asking kids that they're just having to guess what the answer is because what's the point, that's the thing you're about to teach them. So why are you asking them questions on that? And instead, well, two options really.

 

One, you could just get, deliver a clear, concise explanation, 20 seconds, and then check for understanding in the ‘we do’ to talk about in a second, nothing wrong with that at all. Some teachers will spice that up a bit by doing silent teacher. I wrote about that in my first book.

 

So, they'll model it first in silence and then explain it. There's interesting research to suggest that's a really good thing to do. But the thing that I think is really effective that I've started doing over the last two to three years is checking kids are listening.

 

So, picking three or four key parts of my ‘I do’ and then getting data back from the kids as to whether they've heard what I've said, not whether they've understood what I've said, that's going to come later, but are they paying attention? And you mentioned before, and you could do this with call and respond. So, you could say something like, okay, I'm going to do the three multiplied by the X. And that gives me three X squared. I do the three multiplied by the X. What does that give me? Three, two, one.

 

And they say three, three X squared. So, you've just got a bit of data. You've got two things.

 

You've got data as to whether kids are listening. And as soon as kids realize you're going to be doing this, more kids listen. So, it's a driver of attention, these check for listening questions.

 

And it doesn't interrupt the flow of your explanation. It adds a few seconds onto it. But again, the teacher said to me the other day, and I had to laugh at this.

 

He said, I don't like these check for listening questions. It ruins my flow. It ruins the flow of my explanation.

 

And my point to there was, who cares what kind of flow you're in if no one's listening to you? How do you know kids are listening? Where's your evidence that anyone's paying any attention to what you're saying? Whereas if you ask these check for listening questions, you get data back, the kids feel good because they're answering questions correctly. You're getting data back to suggest that the kids are with you. And that's going to set up the ‘we do’ that's going to come.

 

So, to bring all that together, I think for an effective ‘I do’, you have to atomise first. Otherwise, you're asking for trouble. You have to have thought through in advance what you're going to say.

 

When you're doing that explanation, you probably want some data back as to whether kids are paying attention or not. And if you do those three things, I think the effectiveness of that ‘I do’ is going to go through the roof. And I'll tell you the other thing it's going to do is going to save you a hell of a lot of time because that ‘I do’ is probably going to take 30 seconds versus these hybrids that I see when the teacher's asking all these questions and kids are getting things right and wrong.

 

Sometimes they take six minutes, seven minutes and that's where you're asking for trouble. So, a nice, clear, concise teacher led ‘I do’ and that'll set up the ‘we do’ that comes next.

 

[00:37:52] Anna Stokke: I agree with you about checking for listening.

 

I have to agree with that because even teaching adults, I teach 18, 19-year-olds. You'll be surprised. If you stop when you're teaching something and you could ask them a very straightforward question and get them to, like in my case, I'll do things like thumbs up or like hold up your cue card, yes or no.

 

You'll be surprised how many students might get it completely wrong, even though it's a very straightforward question. And that tells you they're not listening.

 

[00:38:24] Craig Barton: Well, I'll tell you a quick story.

 

I've had a very similar experience. So, I am up until recently, I've done a lot of kind of PD and stuff over the last kind of 20 years. I'm taking a little bit of a backseat from that.

 

When I've done CPD on check for listening, the same thing always happens. Teachers are like, no, I don't like this. I don't need this.

 

My kids are listening to me. All it takes to dispel that myth is to turn the tables on the teachers as you've talked about with your students there. So, what I often do, just during my PD, I'll just have a slide up and I'll just tell a bit of the story, blah, blah, blah.

 

And then I'll take that slide away and I'll ask the teachers three questions on what I've just spoke about on that previous slide. And I'll say, write on your mini whiteboard. And you can see straight away their faces drop because they've not been listening.

 

Of course they've not been listening. And the point I make there is all the evidence that I had in the moment suggested they were listening. No one was talking.

 

Everyone was quiet. Everyone's eyes were on me. They were like my dream students.

 

And yet not a single one of them can answer the three questions that I've just asked. And the teachers protesting like, I didn't know I was supposed to be listening to that, or you didn't explain it clearly, blah, blah, blah. And the point I make there is, how often does that happen in your lesson? How often are you explaining something that you really want your kids to pay attention to? They're giving off all these visible signals that they're listening, when in fact they've probably not heard a word that you've said.

 

Unless you get that data back from them, you're just crossing your fingers and hoping. And once teachers realize that they're not listening, then the penny drops that actually these kids who are quiet and looking at them may actually not be listening as well.

 

[00:39:58] Anna Stokke: I mean, it's human nature that your mind will drift.

 

It's also human nature to think that people are listening to you when you're speaking. Because we feel like we're doing a great job.

 

We've put all this work into planning this lesson and you can find out quite quickly that often people aren't listening. So yes, I agree with you there. We should move on to your next stage that you like to talk about.

 

You've got another little book on this. That's a ‘we do’ stage, right?

 

[00:40:25] Craig Barton: It's an interesting one, Les. And it's the shortest book in the series.

 

You can probably read it in about 15 minutes, but it's got a few really core ideas in it. I'll just run through a couple of them. The first is when you're doing a ‘we do’, first make sure the question is assessing the right thing.

 

Often, I see ‘we do's’ that are either too similar or too different to the ‘I do’. So, two similar ones are where you've literally just changed the number and the kid doesn't have to do any thinking whatsoever. And they can just kind of pattern spot and just figure it out.

 

So, they never work. But I also see ‘we do's’ that are too different where you introduce a bit of a twist. And the problem with that is if the kids get that question wrong, you don't know whether it's because they haven't understood the twist or they haven't understood the core procedure that you've just demonstrated.

 

We do selection's important, but the single most important thing about a ‘‘we do’’ is you need data from every single kid in a manageable way. And for me, the only way to do this is to do what I call step-by-step with mini whiteboards. Let's go back to our adding fractions example.

 

You've modeled on the ‘I do’ how to add these fractions after you've atomised. You then put a well-thought through ‘we do’ question up where you've changed some numbers, but it still requires some thinking from the students. And you say, write me down the first step to solve this problem on your mini whiteboards.

 

All I want to see is step one. Don't want to see anything else. Hover your boards, three, two, one, show me.

 

Now, the advantage of that is firstly, you're seeing data from every single kid. So straight away, you're beating cold call or volunteer because you've got 30 pieces of data versus one piece of data. But what you're also doing is your miles better than if you ask the kids to write all the steps that work in and put them on the mini whiteboard.

 

For two reasons. One, you can't take in that data. There's just too much information held up on those boards.

 

And secondly, if kids have gone wrong in step one, you're then having to try and play detective as to figure it out. You can't do that in the five seconds that the mini whiteboards are on. So, assess step one of the ‘we do’.

 

If all the kids have nailed it, confirm it on the board. By all means, you can ask a kid a question or whatever. And then say, okay, on your boards now, write me step two.

 

Now write me step three. Now, if kids go wrong, you know exactly where they've gone wrong in the process. And your diagnosis ability just goes through the roof.

 

And also, the kids get lots of opportunities to feel successful. They get four or five chances to hold up their board and four or five opportunities to think, you know what, I'm really, really getting this. So, the first ‘we do’, I always do step by step on mini whiteboards.

 

Break it down into four steps, five steps, however many steps it needs to be. And then I always do a second ‘we do’. I kind of do an ‘I do’, ‘we do’, ‘we do’.

 

And the second ‘we do’, again, no twists or anything, just a similar question, but this time the kids do it in their books from start to finish in silence. And then we mark it using Adam Boxer, who's a science teacher in the UK. He came on my podcast and told me about the tick trick that he uses.

 

So, if listeners aren't familiar with the tick trick, very, very simple. The kids do it in their books from start to finish. And then what you say is, right, I'm gonna go through it on the board.

 

If your first line looks exactly like my line, give yourself a tick. If your second line looks exactly like my line, give yourself a tick. And in fact, on that second line, if you put a three there, give yourself an extra tick.

 

Right, line three, if it looks like this, give yourself a tick there and a tick there. By the end of it, let's say you've allocated nine ticks, something like that. You then say to the kids, okay, swap with your partner, swap books with your partner, just check they've allocated their ticks correctly.

 

Okay, hands up, whose partner got nine out of nine ticks? Perfect, amazing. Who dropped a tick somewhere? Anna, where did you drop your tick? That piece of the work and so on. So, what the tick trick does more than any other technique I've ever seen is it forces kids to not just look at the final answer, but to look at the structure of how they're setting their work out.

 

Because kids could not care less about that structure. All they're bothered about, does my final answer match your final answer? Yes, it does. I'm not gonna listen to a word you've said about the show you're working and all this nonsense.

 

Tick trick forces the kids to look at their work now in a really efficient way. We've got three now, right? We've got atomisation, teacher-led ‘I do’ with checks for listening. ‘we do’ first with step-by-step with whiteboards, secondly, with tick-tricking books.

 

You have set your kids off or with a really solid foundation to go into the independent practice. And of course, if any kids are struggling at that point, you know exactly which those kids are and why they're struggling. So, when you set the rest of the kids off, you can go exactly to those kids or gather in a small group or whatever, and you can give that targeted intervention.

 

So that combo, I think is really, really powerful to set up the independent practice.

 

[00:44:59] Anna Stokke: You're talking about fractions, which traditionally is a place where a lot of people kind of fall off the math wagon. And actually, fractions just, they're really, really important, for algebra and any math you're going to do later on.

 

And you really need to be able to do fraction arithmetic really well. You mentioned that when you're starting off and you're doing sort of, I guess it's like the worked example effect that we're talking about, and you're using the, you're in the ‘we do’ stage, that you want your example to be not just the same with the numbers changed, but not too hard. You're going to start off, say with fractions that have the same denominator.

 

Your example is, I don't know, one fifth plus two fifths. What example do you give the students then?

 

[00:45:51] Craig Barton:  Let's do a bad example, right? So let me write this down. So, you've got one fifth plus two fifths.

 

That's your kind of, your ideas that you would do. Now, again, technically, this is where we could really get into the weeds. I wouldn't do an ‘I do’, ‘we do’, the one fifth plus two fifths.

 

So that would be what Engelman would call a transformation. He wouldn't call that a routine. For all the geeks out there, if you really want to get into this, ‘you do’ an ‘I do’, ‘we do’, if it's not immediately obvious from the input, how you get to the output.

 

Say you were doing one fifth plus three quarters. It's not obvious how that question immediately spits out whatever the answer is, however, many twentieths. Whereas one fifth plus two fifths equals three fifths.

 

Most kids are like, right, I see where that's come from. When it's obvious where the inputs led to the output, that's what's called a transformation. You can actually teach that really quickly.

 

It's kind of, it's very similar to ‘I do’, ‘we do’, but you essentially just do two quick examples. So, I would do, I'd write on the board in silence, one fifth plus two fifths equals three fifths. And then I'd probably change one of those things.

 

So, I'd probably rub out the two fives on the bottom and make them quarters. And I do one quarter plus two quarters equals three quarters. Take me 20 seconds, 10 seconds even.

 

Then I'd say to the kids, okay, you have a go. One seventh plus three sevenths on your mini whiteboards. And every kid gets it right, of course they do.

 

And you just mess around with that. Okay, what if I changed instead of one seventh plus three sevenths, one seventh plus five sevenths, what would that be on your mini whiteboard? So that's how you would teach a transformation. Two very quick examples without saying a word, and then just get loads of data back from the kids.

 

Whereas when ‘I do’ and ‘I do/we do’, when it's not immediately obvious where the, how the input leads to the output. So, what we'll choose, we'll do fractions where the denominators are different, right? So, let's do a one fifth plus two thirds. Let's imagine that's your ‘I do’, okay? A bad ‘we do’ would be probably to do one fifth plus one third because what you're doing there is you're not challenging the kids with enough aspects of that routine, probably.

 

Okay, it's not the world's worst ‘we do’. It's absolutely fine. But maybe you want to perhaps test their ability to find that lowest common multiple a little bit more.

 

So maybe you want to vary something different. So maybe you probably better varying the denominator more than you are the numerator. But again, that's kind of neither here nor there.

 

The bigger mistake to make will be you change it too much. So, all of a sudden you took a mixed number in there. So instead of it being one fifth plus two thirds, now all of a sudden it's one and three sevenths plus nine and two sixths or something mad like that.

 

So always err on the side of caution, but just ask yourself, can my kids get this thing right, this ‘we do’ right without having understood the ‘I do’? Can they pattern spot? And if they can pattern spot, it's probably a bad ‘we do’. So, there's, that's just kind of some general guidance for that.

 

[00:48:47] Anna Stokke: I just want to back up on that.

 

So, you were doing one fifth plus two thirds.

 

[00:48:52] Craig Barton: Yes, that's the one. Yeah, yeah, yeah.

 

[00:48:54] Anna Stokke: A bad example is if you kept the denominators exactly the same.

 

[00:48:58] Craig Barton: Not bad, just I don't think that's as optimal as changing one of the denominators.

 

[00:49:03] Anna Stokke: But another bad example might be doing something where one denominator was two and the other denominator was four.

 

So, the example you gave, the denominators were relatively prime, right? If that's the concept you're teaching, your example that you're giving the students, you probably want the denominators to be relatively prime.

 

[00:49:23] Craig Barton: This is interesting, right? So, you've opened a whole can of worms here on it because this is interesting. So, this goes back to atomisation.

 

If you have atomised and your kids are going into that ‘I do’, being experts at finding the lowest common multiple, then I think it doesn't matter whether your denominators are co-prime or have a shared factor. Because that phase of the routine, all it is to kids is find the lowest common multiple. And I know how to do that.

 

If you haven't atomised, I think that's where teachers fall into this trap of, right, let's do a whole load of fractions where you just multiply the denominators to find the lowest common denominator. Now let's do a whole load where they've got a shared factor. Now let's do a whole load where one's a multiple of the other.

 

And the kids get really confused there because now they're like, well, what types this, what types this, what types this? If you atomise and you isolate that atom and you do a load of practice on mini whiteboards with finding the lowest common multiple, then I think you are fine to vary those fractions because that's not a concern for the kids. All it is to them is, ah, just find the lowest common multiple. So that again showcases the power of atomisation.

 

[00:50:31] Anna Stokke: That was really helpful. And if you have a good resource, it should really lay this out for you.

 

[00:50:37] Craig Barton: I don't know what it's like in Canada, but this is madness.

 

It is absolute madness that teachers, novice teachers, or even teachers of 10 years are having to make up these sequences of questions themselves, right? You get some textbooks that are kind of close to it, but I've never seen anything perfect. So, I had to, again, back to Engelman. I don't know if, again, your listeners will know.

 

So, he has a series of textbooks called Connecting Maths Concepts. So, I thought, amazing. This has been what I've been looking for.

 

I bought a load of these, and I thought, right, let me look up the fractions chapter and let me see the optimal way to sequence teaching, adding fractions or whatever. Well, that was my first mistake. So, there's no fractions chapter because the whole point of this book is all interleaved.

 

So, every lesson covers about nine different concepts. So, fractions actually spreads itself across about 300 lessons or something like that. So, you don't get this kind of careful sequencing that I was after.

 

And it's probably the most effective way to teach, but you kind of need to start from square one with that. You can't just drop that in with your kids as a little experiment. Again, back to AI, I found the most effective way to do this is to essentially train.

 

The new version of Gemini that dropped last week, Gemini 3.0 is really good at this. So, if you describe exactly what you're after in terms of I want a really carefully varied sequence of examples, where one critical thing's changing, it still requires the kids to think, they can't pattern spot.

 

Here's the example of the kind of thing I'm thinking about. Now, can you have a go? The stuff it's coming out with, I would say is better than 95% of the paid resources that I often see over here in the UK. But you're right, it is madness that teachers should have to try and figure this stuff out for themselves.

 

[00:52:20] Anna Stokke: So, we did, ‘we do’, and now we're at ‘you do’.

 

[00:52:24] Craig Barton: Correct, yeah, and again, it's an interesting point here, Anna, right? The term, shared vocabulary, I'm obsessed with this. I think ‘I do’, ‘we do’, ‘you do’ means different things to different people.

 

Firstly, the ‘I do’, as I say, is more often than not an ‘I do/we do’ hybrid, where it's a bit of instruction and a bit of checking for understanding. Whereas for me, the ‘I do’ is very clearly a teacher-led worked example. The ‘we do’, its purpose is to check for understanding of the ‘I do’.

 

And we've spoke about how I want mass participation in that. A lot of people call the ‘you do’ the ‘we do’, which doesn't help. The ‘you do’ for me is consolidation practice.

 

Its purpose is to give students time to consolidate the procedure you've just shown them with the ‘I do’, that you've checked for understanding with the ‘we do’, but consolidate it at their own pace. So, kids need this, and it only needs to be, you know, three minutes, five minutes, something like that, depends on the sizes, on the length of the procedure. It also depends on how many ‘we do's’ you've done.

 

So, some people like to do five or six ‘we do's’. They actually do a lot of the consolidation practice on the mini whiteboard. So, in that case, you only need a shorter ‘‘we do’’.

 

But it's just got to be some well-chosen questions, no dodgy context, no problem solving, no twists in there. It's just an opportunity for students to just get that procedure consolidated in their mind so that you can then either cycle back to atomisation ‘I do’, ‘we do’, ‘you do’ for the related procedure, or you can go on to asking them to do something more challenging with that knowledge. In other words, to do some problem solving.

 

So, the purpose of that ‘you do’ is just to allow kids to consolidate what they've just seen at their own pace.

 

I've loads of things I could talk to about that, but that's the kind of top, top line of this.

 

[00:54:06] Anna Stokke: For the practice piece, how do you design that? You actually have a book on purposeful practice.

 

[00:54:13] Craig Barton: I do, yeah. Which is different from this kind of consolidation piece. So, I'll draw a distinction between them. Let's stick with fractions.

 

Let's stick with this adding and subtracting fractions with different denominators. So, we've chosen our ‘I do’, we've done a few ‘we do's’. Now it's time for the kids to consolidate.

 

It could be as simple as six or seven well-chosen, potentially quite boring looking fractions for kids to add and subtract from a really good textbook. Or I'm sure your listeners have all got their favourite sources. Certainly, over here in the UK, we've got things like Corbett Maths, Dr. Austin Maths.

 

We've got loads of teachers who freely share really high-quality resources that allow kids to get this much needed consolidation practice. Now this isn't death by worksheet. It's not doing a hundred questions where once you've done two of them, you know what you're doing for the rest.

 

But likewise, it's not, all right, just one question of normal practice and then twist after twist after twist. It's just a carefully selected group of questions from again, a well-established source. And that then sets the kids up to do the most sophisticated thing, which would be for me, purposeful practice followed by problem solving.

 

[00:55:20] Anna Stokke: How about problem solving? What do you do for that piece?

 

[00:55:24] Craig Barton: You've got to do purposeful practice first, right? So, this is the thing. I think the jump between consolidation and problem solving is too much often.

 

By problem solving, I mean, solve a non-routine problem. So, the best visualization I've ever heard about this is from a guy over here in the UK called Colin Foster. And he, I'm a visiting fellow at the University of Bluffborough Mathematics Education Center.

 

So, he's the head of it basically. So, he's one of the UK, if not the world's leading thinkers of maths education design. He has a free math curriculum called the Lumen Maths Curriculum.

 

L-U-M-E-N. If you just Google Lumen Maths Curriculum, you get all the resources and so on on this. So, when Colin talks about problem solving, he describes two tunnels.

 

So, the tunnel on the left, as you enter it in your car, you can see the way out. It's a straight tunnel. Now it might be a really long tunnel.

 

You might be in that tunnel for a while, but you know exactly where you're going. That's not problem solving. That's consolidation practice.

 

If it's a short procedure, like adding fractions, you'll be in that tunnel for 20 seconds. If it's a long procedure, like solving a pair of simultaneous equations, you'll be in that tunnel for two minutes, but you know where you're going. Tunnel on the right, you're entering the unknown.

 

You can't see where you're going. You've just got to feel your way through the tunnel. And again, you might be in it for 20 seconds.

 

You might be in it for an hour, but that's your non-routine problem. It's not immediately obvious when you look at it, how to solve it. Now that might be because it's wrapped up in a context.

 

It might be because it's a wordy problem. It might be because it weaves together two different areas of mathematics, but that would be problem solving. And that for me is the holy grail of maths teaching.

 

You want your kids to be able to be given any kind of problem that's on the curriculum, whether it's in an exam or whatever, and then not to be phased by it, for them to understand what the problem's asking them to do, and then go ahead and do it. For me, the way you get them to that position is first you do the atomisation, ‘I do’, ‘we do’, ‘you do’, because that gives them the skills. Because what you don't want is the kid to say, ah, I know what this problem is asking me to do, but then they don't know how to actually do it because they haven't got the skills.

 

But also, you want them to have that resilience and also toolkit to be able to piece apart what these problems are asking. For me, there's a few ways of doing this. So, one is to sandwich in a bit of purposeful practice in between consolidation and problem solving.

 

And purposeful practice are activities where the procedure's still at the heart of it, but kids have an opportunity to think mathematically. So classic purposeful practice activity structures would be Venn diagrams. So one of my many websites called Maths Venns, and it's a Venn diagram website where kids, the labels on the circles of the Venn diagrams would be things like, let's say you're doing straight line graphs, it would be something like, one circle would be has a positive gradient, one circle would be has a positive Y intercept, and one circle would be passes through the point two comma three or something like that.

 

And you have to think of examples that fit in each of the eight regions. Now, what you're doing there, you're still practicing the core procedure, but you're having an opportunity to generalize, to conjecture, and so it eases you into this way of thinking. Every learning episode, I try and provide an opportunity for kids to do this purposeful practice, and then we dive into the problem solving.

 

[00:58:42] Anna Stokke: Do you want to talk more about the purposeful practice?

 

[00:58:46] Craig Barton: I'm a big fan of activity structures. In the past, I had like a favourite activity for Pythagoras, a favourite activity for decimals, a favourite activity. Problem with that is, when activity structure is new to kids, they spend a lot of time and attention trying to get their head around the structure, which takes away time and attention to think about the mathematics. So, what I do now is I'm really tight.

 

I only have three purposeful practice structures and three problem solving structures. And the logic being there is, the more I reuse the same structure, the better I get at delivering it. And the more familiar my kids get with the structure, the more time and attention they can dedicate to the mathematics.

 

So, my three for purposeful practice, Venn diagrams, just spoke about that, completion tables. So, if listeners aren't familiar what a completion table is.

So, a classic for this would be, if you had, so negative numbers is a good one here. So, this would test people's spatial, visual kind of awareness here. So, if you imagine a table, and let's say that the columns across the top of the table are labeled A plus B, that's the first one, A minus B, the second one, A times B, A divided by B, and then going down, the rows are labeled with values of A and B. So, the first row might be, A is negative two and B is three.

 

And then the next one might be, A is negative five and B seven or whatever. And the point of a completion table is, you can remove any of those values of those rows and columns, and depending on the information the kids are given, they either have to work forwards or backwards to try and fill out each of the cells. Now, what's interesting there is again, all kids' attention is on the procedure, but they're having to think mathematically because they're having to think, right, I've got this information here, so which cell can I work out next? What do I have to work out before I can work out that one and so on? They have to think, there's a lot more depth of thought involved than just simply being given a worksheet.

 

So, I'm a massive fan of completion tables. And in my purposeful practice and problem solving book, I put links to all these sources. You can get these freely available everywhere.

 

So, you've got Venn diagrams, you've got completion tables. And my other favourite purposeful practice structure from the US is the open middle problems. I'm a big, big fan of those.

 

So again, if listeners aren't aware of those, that's where you get given a problem with constraints. So, it's often a fill in the blanks one. So, it'll be something like, it'll be an equation.

 

It'd be like something X minus something equals something. And then below it, something X plus something equals something. And you have to fill in the somethings with numbers between one and nine to solve certain constraints to get the biggest value of X or the biggest difference or something like that.

 

And what I like about those is you have to keep practicing the procedure. It's the only way to solve them. But again, you have that opportunity to think a bit deeper.

 

You have that opportunity to conjecture, to reason, to hypothesize, to compare answers with your partner and so on. So those three structures, Venn diagrams, open middle problems, completion problems, they're for me a nice sandwich, a nice bridge in between just pure consolidation practice and pure problem solving.

 

[01:01:52] Anna Stokke: Are you familiar with the instructional hierarchy? Do you ever talk about that?

 

[01:01:56] Craig Barton: Describe it to me and I'll see if we have like a different phrase or something.

 

[01:01:59] Anna Stokke” And so, the way it works is students go through four stages when they're learning anything. So, you start in the acquisition stage where you don't know how to do things at all and you're just kind of learning it.

 

Then you move to fluency where you get faster. The goal is to get faster. And then generalization is now where you're kind of having to think a little harder.

 

 

So, the examples that you talked about there reminded me of that. It's the difference between if I say, minus three times five is what? And the student says minus 15 versus minus three times blank equals minus 15. The second problem is more in the generalization stage that you'd use more in the generalization stage when students have become fairly fluent because they've got to think harder about what the answer to the problem is that if you just give it to them.

 

[01:02:52] Craig Barton: I think so. I think that's a really useful framework. goes back to what we said before as a teacher and I'm pretty experienced, I don't want to be having to think every single topic.

 

How can I get my kids to think more harder about this? I don't want to have to keep coming up with these problems. Whereas what I've found with these three structures, completion tables, Venn diagrams and open middles, I've yet to come across a topic where I can't use one of those as this means to get kids thinking harder about a certain procedure. That's been the biggest change I think in the way I've approached this over the last few years.

 

In the past, I would have had a load of individual problems for different topics. I don't do that anymore. I constrain myself to these three structures and I've found that that just means, I can just say to the kids, we're doing an open middle problem today and they know it.

 

They know the setup. We're doing a Venn diagram today. They know the setup and I know how best to deliver it.

 

So, I think there's a lot of virtue in kind of constraining yourself in the types of activities that you do with kids.

 

[01:03:50] Anna Stokke: That's really great. We have not too long left. So, is there anything in particular that you think we should cover?

 

[01:03:56] Craig Barton: Let's just talk problem solving for a little bit because that's the kind of the holy grail to get to the end of it. So, I kind of tease the problem solving is by its nature, unpredictable.

 

 

So, it's very difficult. You can't kind of train kids up in the sense that you can show them every type of problem they could ever get. But again, I think there are three activity structures, problem solving activity structures you can do with kids that give them the best shot at being able to cope with the unpredictable.

 

So, what is my SSDD problems that I wrote about in my first book? And I have a website ssddproblems.com that stands for same surface, different depth. So, I got this idea from the Bjorks and it's this idea that if you give kids problems that on the surface look quite similar. So maybe they've all got an isosceles triangle on them, or they've all got the context of a ratio in them or they've all got something to do with perimeter.

 

And yet each of those problems is actually about a different area of mathematics. Then it's really good training for kids in terms of building up what the research calls their discriminative contrast. So, their ability to be able to say, all right, this problem's asking me to do this.

 

Whereas this problem's asking me to do this because they have to sift beneath that tempting surface feature. So SSDD problems are a really strong structure for problems that look similar but actually have different depth. So that'd be one that I use.

 

A final two, no number problems. I'm a huge fan of these.

 

So I don't know if you get this with kids or if listeners can relate to this, but often when kids get a wordy problem in an exam, they just kind of go to pieces and they just grab the first two numbers they see and just do a random operation with them and kind of cross the fingers and hope for the best. No number problems is a good way to avoid this and train kids to get better. So, no number problems, you take that exact same wordy problem, but you just scrub out all the numbers and you give it to kids with no numbers.

 

And you say to kids, what's the first step to solve this problem? And what that forces kids to do is to read the problem and engage with the structure. The distraction of the numbers goes. So now all of a sudden, instead of kids doing three times five randomly, they're doing the number of buses multiplied by the number of students.

 

They don't know how many buses there are or how many students there are, but they know that's the first step multiplied by the cost divided by the number of miles or whatever it is. So, by removing the numbers, you force kids to engage with the structure of the problem. And the beauty of those no number problems is you just grab any exam, scrub out, pick a wordy question, scrub out the numbers and give it to kids.

 

And if you do this regularly enough, it forces kids to get much better piecing apart those worded problems. And then the final thing to say is that my third structure, it's a bit cheating in a way. It's kind of everything else.

 

It's kind of like a surprise, a non-routine problem where it's something that kids have never seen before in their life. And what I do there is I train kids up in a way to deal with the complete unexpected. So, I have a real clear structure to this.

 

Let's say we've got the class, 30 kids in the class. I say to them, first, I kind of set them up for it. I say, right, I'm going to show you something on the board.

 

It's going to scare a lot of you. A lot of you are going to look at it and think, what the hell is that? But it's okay, don't worry about it. We're all in this together.

 

So, here's what we're going to do. I'm going to give you that. And for 30 seconds, no one is allowed to speak and no one is allowed to write.

 

Now that's quite interesting because no speaking is obvious. No speaking stops the kids shouting out or saying, I don't get it. Or like, it just creates a calm environment.

 

But asking the kids not to write forces the kids to just not jump in with their first instinct, but to actually override that and just to start to study. So, I have 30 seconds of no speaking and no writing. Then 30 seconds of silent writing on whiteboards.

 

And then two minutes, physically put your whiteboard in between you and your partner and compare your method, compare your solution. And then let's open it out to the class. Whose partner came up with something interesting there? Who disagrees with their partner? And we can start to do that.

 

But that approach of silent, no writing, no talking, silent writing, and then a structured paired discussion by physically putting those whiteboards in between you and your partner. I found that this really helps students get to grips with those unfamiliar problems. And it stops the kids saying, I don't get it.

 

I don't get it. Because they can't, they're not allowed to. Silent thinking and writing.

 

So, I found that helps quite a lot.

 

[01:08:16] Anna Stokke: That was really helpful. We started with atomisation.

 'I do’, ‘we do’, ‘you do’, purposeful practice, and then the problem solving.

 

[01:08:28] Craig Barton: And the final cherry on the cake, of course, you blow everything if you don't provide those retrieval opportunities.

 

Next week, let's include a question in a low stakes quiz. The week after, let's put it in a mixed topic homework. The week after, let's put it in a do now.

 

Let's keep that knowledge bubbling up so it doesn't slip down that forgetting curve.

 

[01:08:46] Anna Stokke: Yeah, for sure. And making sure all those pieces are in place so that students are set up for problem solving is really important.

 

And even the way you talked about it, actually your problem solving is very structured as well.

 

[01:09:00] Craig Barton: I'm boring. Everything's got a structure.

 

Everything's got a structure.

 

[01:09:03] Anna Stokke: Everybody likes structure.

 

[01:09:05] Craig Barton: And the kids like it, right? They feel safe.

 

They like the consistency. And again, if we accept that kids have got limited working memory, which I think is pretty well established, what do you want them thinking about? Do you want them thinking about the structure or do you want them thinking about the maths? Willingham says kids are going to remember what they think about. So, let's get them thinking about the maths.

 

Let's keep that structure similar and get that hard thinking about the mathematics.

 

[01:09:26] Anna Stokke: We've seen these videos on X or various social media platforms of people like Adam Boxer or Pritesh.


These are science teachers, right? And they've got these great videos. I'd call it like an overhead projector or something. It seems kind of old fashioned, but there's like a lot of interaction and they're calling on the kids and there's call response and there's cold call, et cetera.

 

So, is this possible in math?

 

[01:09:53] Craig Barton: I think it's easier in maths than anything. Again, I'll be honest, I don't think we have any excuse as mathematicians.

 

And the main reason is the mini whiteboard, right? Like I don't know a single question in maths where we can't use mini whiteboards to get responses from every single kid. And unfortunately, that's just not true in every other subject. It's much harder in humanities.

 

It's much harder in some aspects of science. It's much harder in modern foreign languages. So that constantly engaged teaching.

 

And in fact, I have a newsletter, my tips for teachers newsletter. And every week I pick out a research paper and I review it. And the one that's kind of towards the end of November is about the engagement from direct instruction. The engagement, how it's driven up by constantly getting data from the kids. And we can do this in maths with mini whiteboards.

 

We can do it in call and respond. We can do it with checks for listening. We can do it with cold call.

 

But the more ‘we do’ this, two things happen. One, we get data from our kids in terms of their levels of effort and understanding so we can respond. But more than that, because the kids know we're doing it, they pay more attention.

 

And that's the underrated bit that I always draw the same analogy, right? I say to teachers, have you ever drifted off during some departmental training or some whole school CPD or whatever it is. And everyone's like, yeah, of course I have. And I said, well, let's imagine now that the person giving that training every 20 seconds pick one of you at random and ask you a question or ask you all to do a call and respond or ask you all to do your mini whiteboard.

 

How would that change how you approach that? And of course, everyone sits up, everyone would pay attention more because you know that's coming. And as soon as you know that's coming, you start paying attention more. So, the checks for listening aren't the important thing.

 

It's the fact that kids know the checks for listening are happening. Almost the checks for understanding. Sure, they're important to get you data back, but they're also important because the kids know they're coming.

 

Creating that culture where you're constantly getting data, the data's useful, but the fact the kids know the checks are coming is even more powerful.

 

[01:11:57] Anna Stokke: Okay, so is there anything else you'd like to add that we missed today?

 

[00:12:01] Craig Barton: I'll tell you one thing. So obviously I've got the books and check those out if you want, but the thing that first started it for me, I had a website, MrBartonMaths.com. And I started like sharing PowerPoints that I'd written when I was like 23 and looking back and some were terrible and so on.

 

I've not updated that website for about five years, but this is my next project now. So, I'm working on a massive update and it's my project for 2026. So, it'll always be free and I've got some really exciting free maths resources are going to start appearing on there.

 

So that's MrBartonMaths.com. Don't go on there now. It's an absolute mess. The coders just fall apart, but sometime in December, maybe January, brand new website and your loads of free resources.

 

That's perhaps that was something your listeners can check out.

 

[01:12:44] Anna Stokke: I think they will for sure. And I would suggest getting the little books, like pick the ones you're interested in or preferably if you can get them all. Like that's the blueprint for effective direct instruction in math, which I think a lot of people are looking for resources, particularly at the secondary level, but they apply to all levels.

 

[01:13:08] Craig Barton: Yeah. So just on that, there's two reasons I wrote them. One is that I'm not going to be visiting as many schools anymore.

 

My job's changed. My company ED and stuff, we've got a lot going on there. I'm not going to be in school.

 

So, it felt like kind of a bit of an end of an era thing, having visited three or four schools a week for three, four years, having seen like I was in the thousands of lessons that I've seen. So, I thought it'd be a nice way to draw a line, but also it was a mission. I wanted to travel the country and find what are the techniques that don't just work in one school.

 

My criteria for inclusion in any of the books is I had to have seen them work in three different scenarios. So, every idea there has been tried and tested and it's got a decent chance of working. And I've tried to make them the most practical books out there.

 

So, every single one, I've tried to summarize the idea as quick as I can and then just get you to try it out the next day in your classroom. So yeah, if that sounds appealing to listeners, then yeah, they're all available.

 

[01:14:02] Anna Stokke: Thank you so much.

It's been a real honour to meet you, and I've followed your work and several people of course have recommended that I have you on the podcast. I was glad that we could make that work and it was just wonderful to talk to you today.

 

[01:14:18] Craig Barton: Thank you. I really enjoyed the invites and yeah, I really enjoyed the conversation.

 

[01:14:21] Anna Stokke: Thank you. Thank you so much for listening.

 

If you enjoy this podcast, please consider showing your support by leaving a five-star rating on Spotify or Apple Podcasts. Don't forget to subscribe on your favourite podcast app or on YouTube so you never miss an episode. You can stay connected with me on Instagram, Facebook, TikTok, X, Blue Sky, or LinkedIn.

 

All links are in the show notes and check out my website, annastokke.com for more information. This podcast is funded by a grant from La Trobe University and from the Trottier Family Foundation through a grant to the University of Winnipeg to fund the Chalk & Talk podcast.

Anna Stokke

a.stokke@uwinnipeg.ca

Department of Mathematics & Statistics

The University of Winnipeg

515 Portage Avenue, Winnipeg, Manitoba

Canada R3B 2E9

204-786-9059

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