Trapezoid gardens, Harvesting bark, and more. In this episode, Andy, Robin and Adam are joined by Darien Allan to chat about certain mathematical activities and topics that can help ignite young minds. How can we encourage kids to see the world mathematically? What topics have Darien’s school explored? Plus, does maths need a whole marketing overhaul and a big makeover?
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Hi, I'm Andy Psarianos.
Hi, I'm Robin Potter.
Hi, I'm Adam Gifford.
This is the School of School podcast.
Welcome to the School of School podcast.
We're back for another episode of the School of School podcast. We've got Robin and Andy with us as per usual. How are you doing team?
We also have a fantastic guest that I can't wait to talk to. Darien Allan. Darien, you are all the way over... Well, I say all the way over. This is like I'm looking at you and you're right here. But in Canada, Collingwood School, can you tell us a little bit about yourself, please?
Thank you. Thank you for having me. I am obviously a math teacher. I teach secondary, but I've taught everywhere from grade eight through 12th, through pre-service teachers, both elementary and secondary, taught in public and private systems. I am on a learning journey myself, just constantly trying to get better at what I do, which has helped kids learn mathematics.
Awesome. Picking up on that theme, I think that part of what you've looked at and I'm sure developed in not only the school that you're in now, but I'm sure over your career, is looking at creating tools and activities that can support that learning in maths. I know I'm fascinated by that. I know Robin and Andy will be too, and I'm sure the listeners will be, too. Would you mind taking us through some of those ideas, some of those activities and tools that we can use and put into our classrooms to help our learners?
Oh, yes. I'll just give you a bit of background. In British Columbia, we have, I think it came in 2016, what for us is a new curriculum, which is much more focused on many more competencies in mathematics versus sort of what had been a previous focus or at least valued, which was more of the content understanding and solving. Now we've really started to bring in communicating, making connections, reflecting on the mathematics we use and reasoning and analysing involving modelling, really just not so much just applying the mathematics, but getting a much bigger picture of it. I think that's so valuable for students just in terms of not necessarily answering the question of "Why do I need to know this," but "Where is mathematics?" It's not just within the classroom and it's not just abstract and it's not just this thing you have to get through that it has all of these applications in the world or it's just beautiful in itself.
Yeah, I could talk about the aesthetic there for a while, but I won't. I'll circle back to more of this competency base and building this conceptual understanding. In a previous podcast, I think Andy mentioned just in the elementary system, they have concepts that are often a little more concrete. You can use concrete tools and that's not as accessible in the senior mathematics classroom because it is more abstract. That's one of the things that I struggle with and I strive to do, which is bring in more of these, not necessarily real world, but more of these activities that stretch student thinking, get them to apply things and really have to determine for themselves, "How would I approach this," rather than "Here, use the quadratic formula to model the shape of your soccer ball when you kick it in the air."
That is one typical task, but what can we do to actually stretch student thinking? Maybe we have them collect some data and have them decide, "Well, what is the pattern here?" Having them select that for themselves because that's a much more authentic experience. That's what you're going to see in the real world. I'll keep going because I'll talk about a task I came across recently, and these ideas come from everywhere. I was sent something by a friend and it was basically an assessment or an exam, but it mentioned this algorithm that I'd never seen before or at least never seen formally. My colleague and I built a task out of it for students. I'll call it an investigation because it actually is something they're going to investigate, whether it's we're going to get them to do a little bit of research, so just "Pull out your phone, Google this, what do you know?"
But it has to do with credit cards, which the age of students we have are 15, 16 year olds. If they don't have one or haven't seen one, then it's a little bit of experience for them. That ties into a little bit of the finance, too, because kids should know so much more about finance then they have exposure to, but it has to do with how do you determine whether or not a credit card number is valid? This is just one thing, but it's this interesting algorithm for how it's done and it weights different values and they have to multiply them and add things. This isn't so much of a content like, "Oh, okay, you're going to use the quadratic formula," or "You're going to use linear functions," but it's applying a model. That's something that students will have to do regardless of which... Or they'll have to interpret somebody else's model, which is something they have to do all the time.
A message came out the other day about how two journalists misinterpreted this housing price and how long people would have to save for to be able to put a down payment on. I think that was just a deadline. It doesn't really speak to the mathematical competency of the journalists at all, but it's having these opportunities to engage with and have to try and think critically about information, data they're given, graphs. It's incumbent upon us to give students those opportunities. That's what we try and do. Also, if you can, it gives them some autonomy. I am sure you've had your credit card statements and you've looked at it and they X out a few of the numbers. Well, how secure is that? How many do you have to X out for it to be secure?
All of them.
All of them would be secure. Yeah. We think, but maybe not, right? Not with quantum computing coming down.
There's another one. It's called Benford's Law. These are both actually, and it has to do with the prevalence of numbers and the leading digit in the world. That's where my brain lives a lot of the time is, "Oh, this is a really cool idea. How can I bring this in? Or where would this fit?"
It sounds to me one of the fundamental problems, it's almost like maths needs a marketing overhaul, because a lot of what you're saying is that this is not just artistry, it's the practical application. The tolerances in engineering, the perspectives in art that hang in some of the great galleries of the world, all of these things that are mathematical in nature. But what I hear, to this day, from a lot of educators is it's like English and art itself are the creative subjects. Then you've got the dull ones like maths where it's just a series of facts. I think it also goes back to what you were saying about that shift from teaching certain things in isolation and effectively telling the children and doing that.
"There is no connection between what I'm about to do this month, and what we're about to do next month. It's just another topic in maths that we have to cover, and this is what we have to do. Don't start looking for relationships. Don't start looking for patterns. Don't bother with all that. These are the things in isolation, and you'll learn them like that. It'll just be a series of things that you learn in isolation and then get rid of because it's irrelevant. It doesn't have any bearing on the next section of what we are learning."
The four of us know that that's fundamentally wrong. The idea that mathematicians can be working on something for a lifetime and never actually come to an answer. These things, the creativeness that needs to be established in that isn't. For most people's view of mathematics, that doesn't equate, it doesn't fit, it's not right. It strikes me that if people just, once you get that lens on, then all of a sudden it's like, "Wow, this is pretty special. Oh, and that makes sense. Oh, I wonder if," and that curiosity comes in and it just seems like it needs a big makeover, something like that. I don't know, something along those lines.
I think that that's recognised, Adam, in things like the new curriculum in British Columbia where there's a push moving away from content and saying, "Actually, of course you need to learn the content, but that's not really what it's all about. It's about these bigger ideas, being able to generalise, being able to communicate ideas." All those things are really the aims. Recognising that mathematics, although it's quite complex, is essentially a language that helps us describe what's going on in the world around us. I think if we can get that into people's understanding of what maths is at the beginning, and part of that is how you teach, but the world where a lot of the curriculums emerged from was very different than the world that we were in right now. It's certainly going to be even more different from the world that our children will face.
There was a time when you actually needed rooms full of people who would sit there with sharpened pencils and work stuff out all day long. We needed to create those people, so the education system was largely created to create those kinds of people. The oddball people that we called mathematicians, the professors in the universities, there were so few of them, that was an elitist kind of thing. But the school system wasn't developed really to create a bunch of those people. It was created to create people with sharp pencils in rooms who could add large columns of numbers and get them right consistently. We don't need those people anymore or we don't need a lot of them, anyway.
Well, in mathematics... Yeah, it is a combination of what happened from people having to do things in the financial, I'll call it financial, but it's really just financial sector. When people were trading or accounting things and the theoretical, and I'm going to pivot here and go back. I have another example with our 12 and 13 year olds of just how mathematics and the realistic aspect of it. Our grade eights did a six-week project, over six weeks I should say. The ultimate goal was to estimate the number of trees in this area near us, in this Capilano Dam. The students went out and part of this was also to build in an indigenous perspective. They went out and they marked off a small area and they counted the number of trees. They did some measurements and then they had to estimate.
Different groups came up with different answers, and they're all like, "But which one's, right? Which one is the number of trees? Is this the right answer?" That was funny. But they also had to talk about, "Okay, well estimate what would be the volume, forestry." They had talked to someone who actually that's their job is they estimate the volume of wood in an area. They talked about surface area because with the indigenous, we have the birchbark, the cedar bark. The sustainability piece, you can't just pull off all of it, so how much can you harvest? There's all of these links, and that's one of my personal focuses is trying to build in more of that non-Eurocentric perspective in mathematics, because that is the singular understanding of mathematics that's been popularised, to just bring in those other perspectives, whether it's Asian, or Chinese, or Indian or indigenous.
It's such a great world view of using real world experiences to demonstrate to students that these things are used all the time, because I'm sure you've heard it, I hear it from my own kids. It's like, "Well, when will I ever use this again?" That's just the common theme. To be able to say, "Oh, someone actually is calculating volume," I would imagine it would hit home with the students a little bit more like, "Oh, okay, we still do need to use these things," and maybe not everyone's going to need to measure volume, but there's going to be in some way, shape or form value added to knowing these things. It's just not "I have to learn this for the test."
Right. It doesn't have to be real world application for there to be that value. That's the other piece that I wanted to mention is that it doesn't have to be, "Where am I going to use this?" It's actually more "You are going to use the competency of thinking this way," whether it's logical thinking or algorithmic thinking, that this is useful. You could use the analogy of lifting weights. Do you ever really need to know how to lift a weight? No, but you do it, because it develops parts of your muscles.
It's also an appreciation, too. We can intrinsically understand very simple mathematical structures just naturally, but they're not necessarily representative of how the world around us works or the universe. Especially as things increase in complexity in the world around us, and whether it's in nature or whether it's in just our everyday lives and experiences that we're going to have in the future, it's getting more complex all the time. You mentioned logarithms. The universe works with logarithms. Whether you like it or not, that's actually, it's not the most intuitive way of thinking of mathematics. We tend to think in additive structures, but the reality is most things in nature work using logarithms.
A good example, it's like if you add a teaspoon of sugar in your tea and you taste it, it'll taste sweet. But if you add two, it won't be twice as sweet. If you add three, it won't be three times as sweet, and it's not multiplicative in nature. It diminishes. The more sugar you put in, the less sweet, relatively, it's logarithmic in nature. That's how nature tends to work. Whether you're looking at light, or if you're looking at something like all our experiences. If you don't understand the principles of logarithms, you're not really going to make a lot of sense about it. But finances as well, and things like that, multiplicative structures, if you don't understand how they work, people get in a lot of financial trouble because they don't understand the nature of how those things come together. It's really about your intellectual competence more than it is about remembering the specifics of how to calculate the area of a trapezoid or whatever.
It's the idea that you can work it out. If you have a trapezoid shaped garden, which I actually do.
You want to buy, let's say, artificial turf for it, how much do you buy? Well, even if you don't know how to work out the area of a trapezoid, but you know how to work out the area of a rectangle and a triangle, you can work it out. It is just being able to come to those things because know what? If you buy way too much, it's a real pain because the stuff's heavy, and then you got to bring it back. If you buy too little, that's a pain, too. It's just nice to be able to solve simple problems like that in your life. But yeah, anyway. Yeah. I like your idea, Adam. We need to remarket mathematics. I like Rosie Ross's interpretation of mathematics is not a speedy thing. It's a creative thing. The more of that we can indoctrinate in children, and I think a lot of that work needs to happen in primary because if they can come with the right mindset from elementary school into high school, you got a much better chance. All too often though, it happens at the beginning.
It happens at the beginning. I was talking to some reception teachers just today, and I think the phrase that strikes me the most, that I think is the one that I hope is picked up on most when I talk to educators of very young children is we want to encourage them to see the word mathematically and respond mathematically.
They love it.
Little kids love it, by the way.
Totally. Then all of these other things that we discuss become irrelevant already, just the way that they see the world. It's not something like we can develop the skills. I know we're wrapping up, but I do think that's really important, but we've got to believe it as well. The adults have got a job to do. We've all got a job to do. I guess it goes back to Rosie Ross in time, but I think if we can do that, then yeah, it's just that we see the world in the same way we do with the other subjects, and I think that that's pretty powerful. Yeah, we'll do our best to get it right young, and hopefully it continues through, and we'll keep fighting in our corner.
There's a great Netflix documentary series by Marcus du Sautoy, who spoke at one of our conferences a few years back about mathematics, and I don't remember what it's called, but we'll have to dig it up and put it maybe in the show notes or something. It's absolutely fantastic because he talks about the beauty of mathematics in nature. Like, "Hey, why do bees make their beehives in hexagons?" It's a good question. There's a great, and what's amazing about it is there's a fascinating answer, which is, that's the most effective tessellation. It's the tessellation that uses the less wax possible. What a remarkable story, what a great way to introduce the idea of tessellations, for example, is to look at honeybees. I think we don't do enough of that sometimes is tell the mathematical stories behind the great discoveries that we've made. Darien, thanks so much for joining us and sparking all these wonderful ideas.
Thank you so much for having me. I really enjoyed it.
Thank you for joining us on the School of School podcast.