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What makes a good maths lesson?

In this episode we are joined by 2 experts in maths education in Australia. The host for the podcast is Allan Dougan, Allan is CEO of the Australian Association of Maths Teachers (AAMT). The AAMT are partnering with Texthelp to further maths education in Australia, so Allan is in a great position to host this session. 

Allan started his career as a Secondary Mathematics teacher in Scotland. As a celebrated teacher, he held various leadership positions in schools and was Scottish Teacher of the Year Runner Up. After moving to Australia, Allan quickly proved himself as a teacher of merit holding senior leadership positions in NSW schools.

Before joining AAMT in December 2020, Allan was Global Head of Education for a large EdTech company - a role that saw Allan present professional learning across the globe and develop a strong knowledge of current and emerging educational practices and pedagogies worldwide. Allan is passionate about relational and relevant education which helps young people become lifelong learners.

Check out more resources below:

8 Tips for a great maths lesson

We thought it would be helpful to explore this topic a little more. We wanted to pull together some practical suggestions for your classroom.

Teaching maths online

Visit to our guide to the best tools and tips for teaching and creating engaging maths online. There's videos, blogs and downloadable guides to help you along.

Transcript:

Allan Dougan (00:16):

Hello, and welcome to the Texthelp Talks Podcast, where we gather experts to talk about range of topics from education right through to the workplace.

Allan Dougan (00:25):

Before we get started, make sure you've subscribed to the Texthelp Talks Podcast through your preferred streaming service so you never miss an episode. And remember, if you want to get involved in the conversation today use the #Texthelp Talks.

Allan Dougan (00:41):

I'm Allan Dougan. I'm the CEO of the Australian Association of Maths Teachers. I'm not your usual host in this podcast, but I'm really excited to, as a result of the partnership that AAMT and Texthelp have, where we've joined together to further maths education in Australia. I'm really excited to host this podcast today.

Allan Dougan (01:03):

I'm really excited as well to be joined by our guest, Dr. John West. And today we are going to have a bit of a conversation about what makes a good maths lesson.

Allan Dougan (01:13):

As I said, we've got John West with us today. John is an independent maths education consultant and author. He's taught mathematics at primary, secondary and tertiary levels. He has worked as a casual lecturer in the University of WA and teaches in the WA Mathematics Problem Solving Program.

Allan Dougan (01:32):

John currently works as the WA Project Officer for the University of Adelaide Math in Schools Project, and is a fly-in/fly-out numeracy consultant for Coober Pedy Area Schools in South Australia.

Allan Dougan (01:44):

John has worked as a numeracy consultant for Australian Curriculum, Assessment and Reporting Authority (ACARA), the Association of Independent Schools WA and Catholic Education WA. John is currently president of the Maths Association of WA and former editor of the Australian Primary Mathematics Classroom Journal published by AAMT.

Allan Dougan (02:03):

As you see John is very well credentialed to talk to us today about what makes a good maths lesson. And in the next 40 minutes or so I'm going to try and get as much gold out of John as I possibly can. But before we talk about what makes a good maths lesson, John, I want to ask you a question around what is a good maths lesson?

Dr. John West (02:27):

That's a great question, Allan. And I think that while all teachers know when they've had a good lesson, it can be quite a hard thing to put into words. Remember also that every student takes something different away from a lesson. So what works really well for some students may not work as well for others.

Dr. John West (02:44):

I don't think you can underestimate the importance of having a really good task. In the US certainly, professor Jo Boaler talks about the low floor, high ceiling tasks, which are accessible to all students, but which can be readily extended to provide a greater degree of challenge. And of course, in Australia, professor Peter Sullivan, who was the chief author of the Australian Curriculum Mathematics talks about the idea of enabling and extending prompts, which is sort of that same idea of allowing a task to be differentiated to meet a range of student needs.

Dr. John West (03:16):

And in all of my work with beginner teachers and pre-service teachers, I guess probably the most challenging thing or the most challenging aspect they talk about in their work is that differentiating for their various student needs in their classroom. And of course, I think all teachers know that when they've got a great task, when they've got one that immediately engages all of the students and which the students have to be dragged away from at the end of the lesson because they see the value in what they're doing.

Allan Dougan (03:43):

I think that's a great point, John. And one of my hobby horses at the moment is the conversation around engagement and enjoyment, because I actually think that we have got to a point where we think that engagement and enjoyment is synonymous, or maybe we've not got to that point, but we run the risks sometimes of thinking that in order to be engaged, students have to be enjoying it. And that's not always the case, right?

Dr. John West (04:10):

Oh, you're absolutely right. I mean, I think in the age of iPads and these other technologies, the instant feedback that students are able to get, we kind of think that we need to get that immediate positive, that we have to have fun otherwise it's not working for us.

Dr. John West (04:27):

And the reality is that sometimes we just have to learn things, whether they're fun or not. We all have to learn how to read, and obviously we all need to be able to use mathematics in our everyday lives. And it's not always going to be that much fun. You could ask people when they're doing their taxes, for example, that it may not be fun, but it certainly is necessary.

Allan Dougan (04:45):

Yeah. And I think that as we look at how we engage our young people to use a term, I guess, a technical term, we want them to be in their zone of proximal development, right? That whole place where the Goldilocks principle, it's not too hot, but it's not too cold. That whole idea of them being able to experience success, understand what's been asked of them, and be able to add value to their own learning.

Allan Dougan (05:14):

And I think you touched a minute ago on the role of technology in that space, but there's definitely a value that technology brings to that because there's dynamic elements of instruction that technology brings to maths and STEM we can enhance and add value to what's going on as we look at it, right?

Dr. John West (05:32):

Absolutely. Look, one of the huge challenges I've had in working with students over the years is that many of them lack confidence. And if they get up against tasks or assessments and they're constantly being told that you can't do this, or it's too difficult, they actually really struggle with that.

Dr. John West (05:49):

So the ability of having an experienced teacher who has that ability to scaffold the task, to move them beyond into that zone of proximal development is crucially important. But they also have to ensure that the students experience success because once the student starts to experience success, they start to feel better about themselves. 

I know Dylan Wiliam talks about this point that mathematics can actually decimate a child's self-esteem. They can start to attach their self worth to their perceptions of their ability in mathematics.

Dr. John West (06:18):

And really we want to get away from that idea. We want mathematics to actually build a student's self-esteem because they're able to solve problems. And at the end of the day, that's what most people need mathematics for.

Allan Dougan (06:31):

Well, lots there for us to unpack, lots there for us to talk about. Let's start for a second with the teacher. We know that there's a whole lot of work out there that talks about the importance and the role of us as educators in the classroom. We know that John Hattie's research suggests that teacher efficacy has one of the highest impacts or one of the highest measures on improving student outcomes for young people. Do you think we overestimate the role and importance of teachers in this modern day and age?

Dr. John West (07:09):

Look, again, another great question Allan. Look, I've known John Hattie for a number of years. He was actually a professor at the University of Western Australia many years ago. And his research has been embraced around the world. There's a lot that you can take away from it. But one of the things that I've particularly taken away from it is of the influences that Hattie researched on student achievement. I think it was about 250 he looked at in a series of meta analyses. 

He's actually looked at data that has been collected on over hundred million students. So over a third of a billion students around the world over many, many years, and looking at the effect sizes of all of those influences on teaching. And over 90% of all of the different things that you can try actually have a positive effect size.

Dr. John West (07:56):

So one of the things that John Hattie does say is that it almost doesn't matter which of these educational interventions that you try. He's got a whole list of 250 of them. They're things like improving teacher efficacy, clarifying the learning intentions. And obviously there's a list of over 200, but he's found that almost all of them will have a positive effect on student learning. And so there's only very few things like student absences that actually have a negative impact on student learning.

Dr. John West (08:25):

So students do just by their very presence in the classroom do continue to learn and grow over the course of a year. So being with their peers, being in a supportive environment with a teacher who knows them, and is willing to work with them, then I think almost anything that we do is going to have a positive effect. And what's beautiful about that is no matter where you are working, whether it's in a metropolitan area or in a rural regional area, then you can actually choose the influences that you can use most effectively that suit your particular context.

Allan Dougan (09:02):

I'm right in thinking that John Hattie's work would suggest that teacher efficacy is one of the biggest effect size measures in terms of actually driving and bringing about improvement in student learning. What should we take away from that? And what can we learn from that as educators?

Dr. John West (09:20):

Look, what I've found is that student confidence does depend on teacher confidence as well. And so some of the really experienced teachers often they have this way of making the subject make sense and younger teachers can learn from that. I think part of it is a depth of content knowledge. So seeing how the ideas fit together, looking for explanations that make sense, looking for ways to explain something in a different way if a student says, "I'm not quite sure I understand." So multiple representations are really important.

Dr. John West (09:52):

So teacher efficacy is that sense that the teacher or the teaching team understand what they're doing. They're confident. They're not worried if a student has a question or says that they don't understand because the teacher can regroup and come up with a different explanation. 

And I think that's really important because the student then has a sense that they're in a safe pair of hands. They know that there's a safety net. They know that the teacher isn't going to yell at them for being stupid, that the teacher understands the importance of developing a growth mindset.

Dr. John West (10:21):

And so Boaler's work following on from professor Carol Dweck, going back on mindset and applying it to mathematics tells us that students have to actually be positive about the idea of making mistakes in mathematics and learning from their mistakes. And that's part of the learning process.

Dr. John West (10:38):

So teachers have to be comfortable with the fact that occasionally the teacher will make a mistake and they need to model how to deal with that mistake. They need to model things like perseverance for the students, because it's very important that students if they can't solve a problem straight away, they don't give up. They actually sit down and have a little bit of a think ‘Is there a different strategy or a different approach that I could use?’. And that's how they begin to develop resilience, which is a crucial part of learning mathematics.

Allan Dougan (11:05):

So we're talking about what makes a good maths lesson and what that looks like in a classroom. And I think for those people who have heard me talk before around maths education in our classrooms, one of the things that I guess I hammer often is my view that maths in fact, education in general education is a social endeavour, it's not a scientific endeavour. That whole idea of the importance of relationships, the importance of creating an environment where young people can fail, where they can access the whole idea of ‘how do I fix this, or add value to what's going on?’. And I think what you're getting at there, John is that whole idea of, if we can create a safe, nurturing learning environment, then we are actually creating an environment where students can succeed and where students can add value to what's going on.

Allan Dougan (12:01):

Let me ask this question then. And I don't want to put words into your mouth, but the importance of creating a safe environment where learning can occur is what we've just been talking about and what you've just emphasised. How do we build that in ourselves? How do we feel confident and compelled to see your classrooms as a safe environment? For many of us maths is that daunting subject to teach. So how do I build safety and resilience into myself as the educator, so as to build it into my classroom?

Dr. John West (12:39):

Look, that's a really crucially important point as well. You'll know that many teachers burn out within their first few years of teaching because it is a very stressful job. And look, I think it's really important for teachers to look after their physical and mental health. And if anyone questions the importance of teachers, we've only got to look at the last couple of years. There have been people telling us for probably a decade now that computers are going to take over education and that one day it'll just be somebody on the other end of a virtual link up. And I think what most parents discovered over the last couple of years is that many students actually really struggle because they don't have the peer contact and the ability to actually see the demonstrations and use concrete objects and all of these, they can't see and touch the mathematics in the same way as they would in a classroom.

Dr. John West (13:28):

And so, while it's been great that we've got the technology to augment what we do in the classroom, we're not quite at the level yet that we can actually replace the teacher. And I hope that we are never really at that level, because I think that personal relationship is a crucial part of teaching. 

And I think that while the technology does augment what we are trying to do, trying to teach from the other end of a computer is quite difficult. And I've always thought of it as like trying to teach with one hand tied behind your back. You haven't got the same sort of level of access where you can use your facial expression and your tone in a way your proximity to students to actually help them with ideas and concepts. And many students are afraid to raise their hand just like they are at a university.

Dr. John West (14:11):

So working on your own self confidence, obviously the teachers look to us as role models. And they also really take into account what their peers and what their parents say. So Jo Boaler would again say, "We need to look at the language that we use in our classroom." So we actually have to establish positive classroom norms so that when students come into the classroom, they know it's going to be a safe place.

Dr. John West (14:34):

As you said before, if you want somebody to take risks, you want them to take risks in a safe environment. You don't want them to do that before they get their driving licence. So we actually have these programs where students actually practice before they're getting their driving licence by doing the theory and then all of these other programs that we do in schools. We are providing them with a safe environment so that if something does go wrong, they actually have, or they learn how they can deal with that situation. They learn that resilience.

Allan Dougan (15:04):

So one of the things that you said that really piqued my interest, John, was that there was a point in time where computers were deemed or ready to take over education and kind of replace the teacher. And I have a firm view that teaching is a social endeavour and teachers will never be replaced whatsoever, and I stand by that. But I guess I'm keen to hear what's your view, what's your thoughts on the role of technology? Is there a place for it? What is the place for, and what does that look like in a good mathematics lesson?

Dr. John West (15:35):

Look, that's a good point. The technology... Education definitely is a social endeavour. And one of the things that students learn in addition to the content that we teach them is that they learn to behave in a social environment. And so one of the things that the teacher does in addition to teach content is to model the way that they behave in adult society and how to ask questions and answer questions and be mutually respectful.

Dr. John West (16:01):

Using technology appropriately is also one of the things that we need to teach students how to do. If you look at the moment, there are huge issues with trying to get adults to stay off their phone when they're driving their car. That's an example of where technology use is not being done in a way that's safe and responsible. So where do students learn how to use technology in a safe and responsible manner? And I think if not in the classroom, where else?

Dr. John West (16:26):

And I think many schools have good access to technology. And I think one of the issues for teachers is that teachers often find it hard to upskill themselves about the new technologies as quickly as they would like. Because the technology changes quickly, the teachers need access to up to date research about how these technologies can be used in the classroom.

Dr. John West (16:48):

I remember when the first iPad came out, teachers were so enthusiastic to get their hands on them and to see what they could potentially be used for in the classroom. We still don't know what the limits are of these new devices. We're still finding those things out. And every day, teachers and researchers are finding new research and applications. Somebody like professor Catherine Attard is an excellent example of how the technology can be used in a mathematics classroom.

Dr. John West (17:14):

And I saw her at a conference in New South Wales last year where she was presenting at the conference and she bussed in two classes full of year nine students, and actually taught a class live to those students. So that's a teacher with a lot of confidence who is willing to try technology even in a situation where sometimes the technology lets us down. She was prepared to do that in front of a room of 200 teachers. And so that's the sort of confidence that is really inspiring. And the teachers walked away from that conference knowing that technology isn't something to be scared of. It is there to help us do our job, but it's also there to build on the skills of the teacher with the social interaction and to help us teach the content.

Dr. John West (18:00):

Nowadays, we are really fortunate that many companies, Texthelp amongst them, have made available tools that teachers can use in their class that are easy to use. And that is really, really helpful from a teacher's point of view.

Allan Dougan (18:14):

Totally agree. And I think one of the things I'm a big fan of is the work of Reuben Puentedura and the SAMR Model and that whole idea of how we actually consider technology in our classrooms. Because I think one of the challenges over the years, and I can see, as a classroom practitioner myself in my early twenties, I was absolutely an early adopter. I was absolutely “give me an iPad, give me an interactive whiteboard, give me something I'm going to use it”.

Allan Dougan (18:40):

But as an older, wiser and slightly balder person, I look back on that and I think I don't know if I added value to the quality of learning and teaching in my classroom with that technology. At the very bottom of the SAMR Model is that substitution. And I think, at best I substituted. And in fact, actually in the late nineties and early nineties it was death by PowerPoint. And the reality is that maths is not necessarily designed to be taught by PowerPoint clicks. Right?

Dr. John West (19:09):

Yes, absolutely.

Allan Dougan (19:09):

In fact, if I went back to my early 2000 lessons where I had PowerPoint I was getting RSI on my thumb with a number of clicks through PowerPoint. I would've been much better grabbing a piece of chalk and going back to my chalkboard. And so we need to think about how technology actually enhances the quality of learning and teaching. And I think there's things like interactive media. But more than that, it's about going back to that comment earlier on about zone of proximal development. It's about how we personalise learning opportunities for young people, making sure that if we think back to the whole host of work over the last couple years where we're all about adding value, how do we give students a year's worth of growth in a year and so on?

Allan Dougan (19:51):

And so technology really has to be about the personal learner. And that's not always about personalisation, but always has to be about the learner. It always has to be about adding value to the learning that's happening and actually doing something that is better or more. I don't want to use the word engaging because it sounds like entertaining. But it adds more value to the learning experience, adds to the understanding, adds to the depth, adds to the experience in a way. Would you agree with that?

Dr. John West (20:21):

Yes, absolutely. And I think as lifelong learners, teachers also have to be scaffolded about how to use the technology to do more than just substitute for what they were doing before.

Dr. John West (20:31):

Teaching is a fairly conservative profession. If you ask students, or if you ask teachers to predict what upper school mathematics subjects, for example, they should take, the teachers will often go for the safe bet because they don't want the students to fail. And in fact, what we really should be doing is to get students to have the confidence to challenge themselves, to do the mathematics which is going to open up as many doors as possible for them in the future.

Dr. John West (20:56):

I know a couple of years ago I had the opportunity to hear Professor Dylan Wiliam speak. And knowing that he'd worked as a mathematics teacher and teacher educator for many years, I asked him about the issue of the suitability or the supply of suitably qualified and experienced maths teachers.

Dr. John West (21:14):

And of course he said, "The problem is that producing great teachers takes time." And in his words, "Teaching is a career that it's impossible to master in just one lifetime." And as depressing as that may sound, what he was trying to say is he said, he knows of many people who have dedicated their entire career, an aspirational career of continual improvement to try and add to the skills and add to the value that they offer to their students. And they continue to do that until the day that they retire. And I know many teachers who've retired that continue to go back into schools and continue to mentor younger teachers. And I actually asked him what makes science and mathematics teachers or prospective science and mathematics teachers different from his other teachers that he's supervised over the years. And he said he has never found a prospective science and mathematics teacher who wasn't passionate about the subject.

Dr. John West (22:04):

And I thought that completely agrees with my experience in teacher education as well. And it's a great thing, but I think even though that's not a bad thing, he did observe that by contrast teachers of other disciplines and primary school teachers who are not subject specialists in the same way as secondary teachers were more passionate about their students and about learning.

Dr. John West (22:25):

And so what you were saying about the social endeavour is really, really important. Being good at physics or being good at mathematics, having a passion for the subject isn't enough. And so there's an important lesson there. In Australia I think Eddie Woo is one of the best proponents of that approach. There's actually a short YouTube video of Eddie explaining that he feels that the key to his success as a mathematics educator is empathy towards students. 

It's not the depth of his content knowledge, which is huge, or his enthusiasm and his passion, which is also huge. He's a lovely man. And these students really respect him because they know that he cares about them. He has been in a situation. He explains that he was the student that didn’t understand what was going on. And he needed a teacher to take the time to explain things in a different way. And since then, everyone knows that Eddie's been very, very successful.

Dr. John West (23:18):

A very similar situation with Salman Khan from Khan Academy in the US, he was supporting his cousins by creating little YouTube videos, saw no reason to keep them private. And now the world has an incredible resource that really shows and supports teachers and learners all around the world for free, which is just huge. I think it's really, really important that these are available to students.

Dr. John West (23:41):

And Jo Bowler's work tells us that a good teacher can continue to have a positive influence on students even years after they've left that teacher's class. So it really is down to the teacher, their passion, their enthusiasm, but also their empathy.

Allan Dougan (23:57):

So there's a hundred things I want to unpack on what you just said there. And the role of Eddie, of Salman, of Byju Raveendran as another example of someone who set out in exactly the same way to support someone shows the importance and role of technology.

Allan Dougan (24:15):

But I want to zoom back actually to something else that you spoke about and you kind of touched on this and moved on, and I'd love to hear a little bit more about it because we're trying to talk today about what makes a good maths lesson and what that looks like.

Allan Dougan (24:28):

And I think that one of the things you spoke about is that teachers never stop learning. A lifetime's not enough to become a good teacher. And one of the things, one of the dichotomies about being a teacher is that often I'm trapped for lack of a better word in my own classroom. And so teaching is not a team sport, nine times out of 10. Teaching is an individual sport. But the reality is that you talk about teachers who retire and then go back to mentor younger teachers. The importance of us sharing good practice, of us having feedback, of us seeing what other people in our schools in our local areas are doing. That's a really good way for us to understand what a good lesson is, and to begin to develop effective lessons, right?

Dr. John West (25:13):

Absolutely. Teachers need to see good lessons and they need to see the masters at work. And I think anything that we can do to kind of help facilitate that is really, really important. I know that a couple of years ago, when we were still allowed to travel, I was fortunate enough to go on the Australian Association of Maths Teachers Inaugural Singapore Tour. So that was back in April, 2019. [crosstalk 00:25:36].

Allan Dougan (25:36):

I'm just going to make a really shameless plug here. We are hoping that in a post COVID world, we can restart some of these study tours. Because I do know that people saw them as an invaluable way to have practice shared and reflected on. So watch out if you're listening to this and you don't know about those tours, check them out on their website and we are hoping to start them up in 2023 again. Sorry, John.

Dr. John West (25:58):

No, that's great. And look, one of the things that I will say is that it was one of the best professional learning experiences of my career to actually have. It's a real privilege to be offered the opportunity to sit and observe a teacher at work.

Dr. John West (26:10):

Some people feel that it's quite a private thing because a teacher is someone who really opens themselves up to their students and their class. So a lot of trust is involved to be able to do this. And the Singapore teachers did that without any reservations. And Singapore has for the last 20 years or so led the international rankings in terms of their maths and science achievements. So we were really privileged to be able to watch what they did. And to be honest there wasn't anything that we could necessarily put our finger on straight away that really was hugely different about Singapore classrooms. It was actually much more of a societal thing.

Dr. John West (26:46):

One of the things that was really great about the tour is that we had 20 teachers drawn from all around Australia and many of us have kept in contact. And so that's great to actually have a network of teachers who are all interested in sharing what they've learned across the state boundaries and with our international colleagues as well. And one of the things that was really noticeable though, is that teachers in Singapore obviously enjoy a very high level of status, but at the same time they also work incredibly hard and have held to a very high level of accountability. I'm not saying that teachers in Australia don't work incredibly hard, they do.

Dr. John West (27:20):

And one of the things that the tour taught us is that both Singapore and Australian teachers are incredibly good and incredibly effective in their own context. But the context in Singapore is a very different context from the context in Australian classrooms.

Dr. John West (27:34):

One of the things that I learned, because I like my statistics, is that Singapore, the country has an area of about 700 square kilometres and a population of around 5 million. So it's about the size in population-wise of Melbourne, crammed into this 700 square metre area.

Dr. John West (27:52):

Western Australia has an area of about 2.6 million square kilometres. So it's a problem on a different scale entirely. And the Singaporean teachers have trouble getting their head around those numbers, just like we have trouble getting around the idea that there are 5 million people and that whenever they need to have professional learning in Singapore, they have a centre, the Association for Singapore Teachers that provides professional learning for Singapore teachers every single day of the year, the school year.

Dr. John West (28:19):

And they have anywhere between 2500 and 3000 teachers come in every single day for professional learning. And so instead of having to find professional learning opportunities in your nearest capital city, teachers can come from all around Singapore because that is possible. And they can all meet in the same place at the same time for their professional learning.

Dr. John West (28:41):

So they have different challenges than we have here in Australia, but they also have different affordances that we could only imagine having here in Australia. Trying to roll out professional learning, for example, or IT all around a country, as big as Australia is a logistical nightmare. And I think we do pretty well.

Allan Dougan (29:01):

Okay. I agree. And I think I don't disagree with anything you've said there, but I think there is an element of professional learning and professional reflection that starts even before that. I think some of the... You said that the trip to Singapore was one of the most impacting professional learning journeys that you've been on. I think for me, some of the most impacting professional learning experiences I have had have been in school, on the job like that on the job professional learning. And I think sometimes we talk to our young people about the importance of a learning or a teaching opportunity, fail fast, fail often and fix it. That's something I would say in my classroom to young people often. Fail fast, fail often and fix it. And we can do that together.

Allan Dougan (29:48):

And I think sometimes, not for me, maybe for others listening, I would forget about that with my teaching. So I would walk out of a lesson and you said early on in this podcast that what's a good lesson. It's really hard to define, but you know when you've had one. The converse is true as well. What's a bad lesson?

Dr. John West (30:04):

Yes.

Allan Dougan (30:04):

I know when I've had one. And I walk out of my classroom and there's one of two emotions for me personally, when I had a bad lesson. And unfortunately there were more of them than I'd like to admit, but you walk out of a classroom and you either go, oh my goodness, that's awful. And I'm in the wrong career. And I've done a disservice to these kids or number two. And sometimes maybe, maybe I'm admitting more than I should.

Allan Dougan (30:25):

Number two, thank goodness. That's over. Oh my goodness. And I think that I would often forget... Because it's painful to do, I'd often forget to reflect. I'd often forget to go, "Okay, what made it bad? What can I do differently? How can I improve that? If I had the chance to do it all over again, what would I do?" And for me, I think back to my first year teaching and I had the most wonderful year eight class in my first year teaching. And they were a class of 32 and I can remember them, I can see them. That's how impacting they were. And I remember teaching them to divide fractions and it was to turn the second one upside down and multiply. That's what I taught them to do.

Allan Dougan (31:12):

And I remember there was a kid in my class, his name was Bobby. Bobby, if you're listening, I want to hear from you. But there was a kid in my class, his name was Bobby. And he said, "Sir, why did we turn it upside down and multiply"? And that was the first time in my career where I was put on the spot and embarrassingly, and I'm prepared to admit it here. Embarrassingly, I didn't know why. Right, I've been taught back at school that to divide the fraction, you turn it upside down and you multiply it. And it worked for me. And it got me through a uni degree and it got me through a post-grad and it got me through all sorts of things. But I couldn't actually explain to Bobby and the rest of that year eight class why that worked.

Allan Dougan (31:44):

And I remember my response to Bobby was, and I'm embarrassed to say this now. Was, "You don't need to know why, you just need to be able to do it. Turn it upside down and multiply. Now, there's some examples, get on with them." And I walked away from that lesson. And my first thought when that class left on that Thursday afternoon, and you can tell how much of a traumatic experience for me it was because I can remember that it was a Thursday afternoon back in the early 2000s or early 90s.

Allan Dougan (32:10):

And the first thing I thought when they walked out of the room was thank goodness that's over. I can't believe I was put in the spot. And I remember driving home that night and the curiosity started to go, well, why is it? And I could explain that, I could algebraically explain it and so on. But I began to think about how would I explain it to the class. And by the time I had them, the next day that Friday morning I had decided I was going to redo that lesson and I was going to redo it and actually explain to them why and help to develop some understanding.

Allan Dougan (32:37):

And I say that to highlight the fact that I went in and I retaught and I apologised and I said, "I'm really sorry, Bobby. I shouldn't have done to you what I did yesterday. The answer is, I didn't know how to explain it to you. I've gone away, here it is and so on."

Allan Dougan (32:52):

And two things I learned those couple of days. Number one was the importance of reflection and advice and interacting with my peers. Because I went and asked others, how would you have done it? What would you do and so on. So working as part of a team, whether that be a team that's in your school or feeling a rural remote place, connecting with other peers, using digital technologies, that was the first thing I learned.

Allan Dougan (33:15):

But the second thing I learned was that a safe learning environment, going way back to earlier on in our conversation, a safe learning environment doesn't mean that I know everything. It means that I'm confident enough to go, "Do you know what? I'm not quite sure. Can I come back to you on that one tomorrow? I'm not quite sure. Can we investigate this together? I'm not quite sure. Maybe we can look at it. You don't have to know everything. 

And I think often when we're teaching maths and often when we think about a good maths lesson, we think good maths lessons are born out of areas of expertise of mine. They're born out of places where I'm really "Good" in inverted commas at the maths. And yes, we have to have competency in the maths we're teaching. Absolutely. Yes, we have to have a degree of understanding. Absolutely. But that does not need necessarily determine a good or a bad lesson. The importance of creating that environment must never be underestimated. Right?

Dr. John West (34:10):

Absolutely. And if you go into a classroom and you set yourself up as the content expert, you're always going to run across somebody who is able to throw something at you that's going to cause you to scratch your head. But I think the importance of what you said there about the emotion, you can either deal with those negative emotions or you can avoid them. And it is much more pleasant to kind of avoid them and say, "I don't want to feel like that again, because I was very uncomfortable."

Dr. John West (34:34):

Now, if you turn that around and you think about that student who was in that situation, feeling really uncomfortable about mathematics, unless the teacher shows them how to deal with those negative emotions in a positive, constructive way, they are going to develop a fear or a resentment perhaps towards mathematics.

Dr. John West (34:49):

And we know that many students do leave school unfortunately, feeling that mathematics is a series of hoops to be jumped through rather than the problem solving tool that it should be. If you see mathematics as something that is designed to hurt you rather than help you, then you've got the wrong end of the stick. And unfortunately in many cases, it probably comes back to some experience that maybe a teacher was having a tough day, or maybe they hadn't thought. I think some of my beginning teachers this year have used the word, "A teacher's job is really to articulate concepts." And that's a really good word, I think for sort of trying to think about how do you make these concepts make sense to students who haven't had the time or experience that you have as an adult to kind of work out or maybe they don't have the life experience.

Dr. John West (35:36):

I think the role of the teacher is... Charles Lovett would talk about being a lesson collector. A collection of these ideas or ways to teach concepts that make sense to people. And once you start thinking about it in those ways, the teacher, I think can start to relax and think it's okay for me to say to the students, "I don't know everything. I don't know the answer to that, I can go and research, or I can go and figure out the answer because I have the skills I need to teach myself this concept. Or I know someone that I can ask who can help me about this concept." And that's a much more sustainable approach to teaching is realising that you are part of a community with other learners and we're on a shared journey.

Dr. John West (36:14):

There's nothing worse than seeing a student feel that all of maths has been discovered. It's all been written down in the textbook. There's nothing left to discover because they're robbed of the sense of enjoyment and discovery, which is a hugely important and attractive part of mathematics.

Allan Dougan (36:29):

Completely agree with you. But let me be the other side of that coin. The importance of the discovery is about the beauty of maths. But there is absolutely an importance to building automaticity of those basic skills and basic understandings.

Allan Dougan (36:50):

If we are going to facilitate future success, if you like, if we're going to give young people the space to actually discover something new, to have the confidence, we need to build in space to build automaticity and I guess, multiplication facts and the ability to work with numbers and some algebraic understandings and so on. It's not an either or conversation here, is it?

Dr. John West (37:16):

No. Many years ago, I came across a bit of research in educational psychology that pointed out that students who have a really strong command of basic facts, for example, times tables use as much as 80% less conceptual effort when learning some higher level mathematics.

Dr. John West (37:32):

So if you were doing, for example, a long division, I know that we don't really do those anymore, but knowing the times table facts to hand means that you are actually able to proceed with the more complex calculation without having to stop every couple of minutes and actually do a secondary calculation.

Dr. John West (37:49):

And what that means is that for a student who's actually going through a series of these problems in a lesson, a student who has that automaticity walks out of that class feeling quite fresh because they've solved a bunch of problems. One of the other students who’ve been having to work hard this entire time and multitask between the various problems and the basic facts can walk out of that classroom exhausted. And of course, when you're feeling exhausted, you do get prone to those negative sort of emotions and frustration and tiredness, and that can alter your perception of the subject.

Dr. John West (38:18):

And we do have a lot of students that certainly would find it easier to master those more advanced concepts if they just had a little bit more command of those important facts. And so, yes, there's absolutely a place for that. There is absolutely a place for teaching concepts clearly and explicitly. But it's really important also just not to rob students of that opportunity for discovery.

Dr. John West (38:46):

It is equally bad to just say to students, just investigate something without any structure or scaffolding, because many students actually don't have the investigative skills or the reporting skills that they need to be able to do that investigation successfully.

Dr. John West (39:00):

And so that scaffolding that we were talking about, that zone proximal development is crucially important, no matter what level you are working at.

Allan Dougan (39:09):

And so I guess if I try to bring a little bit of a summary to some of what we've spoken about today. I think for me, it comes back to that whole idea of something that we spoke about earlier, which is the role and the importance of a teacher.

Allan Dougan (39:26):

And one of the standards, or one of the conversations we talked about is the importance of teachers knowing their students and how they learn. And if we understand our students and how they learn, then what we understand in all of this is what they need in order to have a good and successful lesson. And in some respects, we should be empowered to be very explicit in our instruction. If we know that's what they need at that particular point in time, or be very scaffolded with the problem solving discovery phase that we might want to engage them in.

Allan Dougan (40:01):

And it's not a dichotomous conversation. It's actually complementary. It's about sometimes there's an importance to be quite intentional about our instruction and very structured about the instruction with a need to build in some skills development, some automaticity and so on and so forth. Other times it's about scaffolding what they do with that knowledge, what they do with that acquired knowledge and how they then apply it.

Allan Dougan (40:25):

So I guess what we're seeing in our conversation today, John, and I'm going to challenge you to sum up better than I have. But what we're seeing in some respects is that a good lesson comes down to the teachers knowing their students and how they learn, and then being incredibly confident to pull the right pedagogical tool out of their toolbox at the right time, and be prepared to ask for help, advice, input from their colleagues and their peers as, and when they need it.

Dr. John West (40:52):

Yes, absolutely. The importance of having that toolbox is really important. And we know from looking at the Australian Professional Standards for Teachers that knowing the content and how to teach it comes after knowing the students and how they learn. 

And so obviously we need to know that those students, we have to have that social and emotional intelligence and we have to remember that emotions, be they positive or negative, have a profound influence on learning. And so pushing ahead through the content if students are having a negative experience is not really going to have much success. And so we have to really understand that these are people that we are working with, and we have to actually think about their motivations, their own experiences. And we have to think about the balance between what the different students are taking away from our lessons.

Dr. John West (41:40):

And even if you have what you think is not a great lesson, there may be students who come back to you afterwards and actually say, "Look, I really enjoyed the explanation or that part of that lesson." What I've learned from my experience and from my heads of departments over the years is that students are actually incredibly forgiving. And they come back the next lesson ready to learn from you. And if you have to reteach a concept they are quite happy for a different explanation. And I think they're very supportive and very understanding. I think any other teachers also would understand that if you came to them for advice, I think most teachers would be quite honoured when a colleague would come to them and ask their opinion on how would you teach this?

Dr. John West (42:19):

Because it is a way of showing respect and deference to people who've been in your shoes. And we do learn as a profession. And I think being able to share that amongst the profession is one of the things that we have to do.

Dr. John West (42:33):

As you touched on earlier, Allan, teaching can be quite a lonely profession if you stay in your own classroom. And it is a wonderful profession to be part of. People are incredibly supportive of each other. Anyone who knows what it's like to be in a classroom and not to be able to articulate something will understand what that feels like. And so they will be wanting to try and support you through that process. And so I'm proud to be part of the teaching fraternity. I've never run across any of my former students who've had a bad word to say about me as much to my surprise as anyone’s, but it is absolutely lovely to be part of that community. And people can come up to you and say, "Hi, Dr. West, remember when you taught this lesson on whatever the topic was," and that can happen 10, 15, 20 years down the track.

Allan Dougan (43:21):

That's awesome. Well, John, it's been great to chat and I feel like we could chat for another hour and still not have got anywhere near covering the entirety of this topic. If people wanted to connect or follow up more with some of this stuff that you have said, what's the best way for them to find you?

Dr. John West (43:40):

Well, look as President of the Maths Association over here in WA I get the auspicious email address of president@mawa, M-A-WA .edu.au. So I'm always happy for people to send me an email. I do have my own website, Drjohnwest.com.au. But that tells a little bit about some of the professional learning that I've been doing in schools. And a little bit about my fly-in/fly-out work at Coober Pedy, which is a whole other discussion. But I feel incredibly privileged to be able to be working as a mentor for a couple of beginner teachers who are in their second and their third year out. And I'm able to sort of reveal a whole world of resources and techniques and things like that to them freely, because I've been doing this job a little bit longer than they have. And it's a wonderful community to be welcomed in as part of that school community and to be welcomed by all the students.

Dr. John West (44:33):

So, Allan, thank you for all the work that you do with AAMT, the Maths Association in WA being a long way away from the rest of Australia really understands how important it is to be connected as part of the community of Mathematics Educators in Australia.

Allan Dougan (44:47):

Excellent. Well, thank you. And I want to thank Dr. John West for joining us today. And I want to thank all of you for listening. I hope that you have found this podcast useful, and there's some nuggets of gold that you've managed to get out of it. If you want to connect with us, as in AAMT, you can find us at www.aamt.edu.eu. You can tweet us at AAMT Inc, or you can tweet myself at Allan_Dougan, or you can use the #Texthelp Talks to join the conversation for today. And John and I will both look out for those conversations as they continue.

Allan Dougan (45:24):

Please, don't forget to subscribe to the podcast to make sure that you catch each new episode. Until next time thanks and goodbye.