Crocodiles, TikToks, and more. In this episode, Andy, Emily, and Adam are joined again by special guest, Dr. Mario Trono, to discuss maths visualisation. What memories does everyone have of early maths visualisation? What word did Mario struggle with? Plus, Andy discusses a frustrating maths exercise that'll lead to misconceptions.
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Hi, I'm Andy Psarianos.
Hello, I'm Emily Guille-Marrett.
Hi, I'm Adam Gifford.
This is The School of School podcast. Welcome to The School of School podcast.
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Welcome back to another episode of The School of School podcast. And today I'm delighted to be in conversation, not just with Emily and Andy, but also Mario Trono from Mount Royal University. Welcome Mario. I want to just throw something out there, kick something off. I have spoken to lots and lots of teachers people, all sorts of situations about visualization. Right. I keep hearing it's a pretty powerful thing. I understand that you've got an opinion on it or at least I hope you have Mario, and I'm keen to hear it. Yeah. Good. You've come to the right place. So I'm going to be quiet. I'm just thinking... For me often I'll talk about visualization in mathematics, but I've just untraced that just visualization generally if... Well, yeah, you've got an opinion, so let's hear it.
bWell, the thing is, it's very popular all the time whenever anybody wants to argue the importance of anything, try to hook it somehow to evolution, the biological reason that we're even here at all. And then of course that will imbue whatever you're saying with some tremendous significance, but that's fair enough sometimes when it comes to things like knowledge and visualization. Because what are we if we're not these cognitive machines, right. Were these bodies walking around, we're hearing, we're smelling, we're sensing, we're seeing.
I mean, ideas in your head can only function as abstractions for so long, a mathematical formula, a complicated series of words in some kind of sentence or linguistic formulation before you just want to bring it back to real space and time, whatever it is you're learning in real time, in real space. What is what I'm learning here have to do with what we so frequently call the real world? Well, that's your brain trying to get back to it.
Just to be evolutionary for a second more, think about oral cultures, cultures that never had writing and never had numeracy. Human beings have been in oral cultures for most of the time since we started grunting and communicating longer than we've been in literate cultures with writing and math and the differences in minds between people in oral cultures and literate cultures are extraordinary. And this is why visualization is important because you always want to get back to the real world.
I can give you a perfect example of this. When in the 19th and 20th century, when people from England went all around the world and studied oral cultures, if you walked up to a person from oral culture, and you said, "Where there's snow on the ground, the bears are white. Where there's no snow on the ground, the bears are brown. We are standing where there's snow on the ground. What color are the bears?" The person from the oral culture would look at you and say, "Are you a complete idiot? Are we hunting today or not?" Because that little line that we came up with is just a formula.
I mean, you could take out bears and snow and put in things that you don't even know what they are. And you'd still be able to get that question, right. Because it's just a little magic formula that we came up with in language, right. But that's your brain. But we still have oral culture behind the brains, we want to get our math or whatever it is we're learning back to real space and time. So that's satisfying click when you can finally get out of that mathematical sentence, even if it's just a pile of beans versus the size of another bottle of beans, it's thrilling. You see the application theory gears into practice and visualizations one of the best ways I think to do it.
So what about the other senses? See, this is fascinating to me because I've thought a lot about this, and you talked about evolution. I think evolution plays a huge role in this. Why the visual cortex? Why are we so fixated on seeing? We even say it, people say, "You can't believe what you hear." Right. But seeing is...
Believing. Now, why is that? Right.
Oh, that keeps going from there. I mean, it's everywhere in the language, in all languages, seeing is a metaphor for knowledge. Right. I see what you mean. Where do you see yourself in five years? I was blind but now I see.
It's everywhere. And you don't want to diminish the importance of the other senses, but arguably the visualization is the key piece. It's the one and that's one of the reasons I moved into cinema studies, and away a little bit from English literature, because we live in a world now more than ever that the fates of the world are being controlled by audio, visual soundbites flying around online. You know what I mean? And we've talked about... Educators talked before about how people are reading less and less, it's because of the relentless technological press of images. Right. So, I mean it's the new lingua franca. We're going to have to do everything through images pretty soon. I mean, people are communicating exclusively through Gifs, right. It's more and more so if we don't learn how to gear more abstract ideas into visualization, we may be doomed.
Okay. Here's a crazy idea. Would you say that the kind of communication... I got a teenage and young adult children, right. And they communicate in a way that's completely different. Right. They send visuals back and forth. Is that a new language? Is Snapchat a new vocabulary?
Well, yeah. There's lots of research on that. I mean, it's turning into a kind of a grammar, isn't it? And the thing that's most interesting about it is, it's so hardwired to commercial platforms, and at the same time it's hardwired to in and out groups and social class, it's extraordinary. I mean, we all know TikTok comes out of China and we know what's politically going on in the world with China right now. But yeah, you bet it's a new kind of language.
But the thing is though, as educators, we have to be a little stodgy. We have to sit back and say, "Well, as fun as cinema is, or this kind of visualization, or using Twitter to kind of get conversations going around the world about something you're studying in school, it's thrilling and everything like that, But we got to dance with who brung us to this extraordinary civilization we have." And that's the English language sentence and the mathematical formula or equation.
Those are things that can carry so much informational power. That's extraordinary. It's always going to have to come back to that. So I would say that the visualization has to be key, but oh my gosh, we have to go around and tell some anecdotes. See if anybody has an example about how visualization saved your life, because it did mine. You know what I blocked on in primary school? And this is humiliating to say the concept of greater than, and less than. I didn't know what that meant to say that eight was greater than seven. Because I got blocked on the word greater. This is what happens in math with word problems. The language can sometimes mess you up.
Yeah. Try writing a math textbook. Try writing-
Yeah. No, thank you. No, thank you.
Wrestling math with words, but anyway, sorry to interrupt Mario. Go ahead.
Not at all. So I thought the word greater meant awesome. I didn't know it meant a higher number. So when somebody would say to me, eight is greater than seven, I'd be like, "Well, isn't that just a matter of your personal opinion? I happen to think four is pretty great myself."
I thought Muhammad Ali was the greatest.
Exactly. How could one not be the greatest? Exactly. Number one's got to be the greatest of all the numbers. So I had huge math anxiety. I remember begging my mom to stay home because people were getting it right in class, and when it went around to Mario, he just got it wrong and it seems such a simple complex problem. So I remember I was playing cards with my mom. As a little kid we played cards all the time and she... Brilliant woman. She said at one point... Because I just wondered. She said, you see Mario, that number seven helped you win there because it's greater than six. And I had all these playing cards out in front of me and I was in the lived life context of just having won a game. And there all these cards in front of me, it just recast visually the problem. And I didn't say this literally at the time, but I felt like giving the world a smack and saying, don't say greater and lesser say higher and lower.
It's just a higher number. It's not necessarily better. It could be terrible. Even more of something, it could be bad. We say a higher case count of COVID cases. We say a lower case count. So I should start a movement online, get rid of the words greater and lesser.
So look, this was a wrestling match when we were writing the first edition of the Maths — No Problem!! textbooks, because it was like, okay, is eight greater than six? Is it more than six? Is it larger than six? Is it higher than six? It's all those things depending on the context. Sometimes times it's after six, sometimes it's less. Right. You came in eighth place is not greater than coming in sixth place. Is it? And here's another question right. Is one greater than negative 500 billion.
Like this value judgments all over the place here. You need the drama of the visualization and it'll click it instantly.
There's no more problem.
Yeah. So exactly. You're absolutely right. Unless you can pin it to something, is abstract nonsense. Right. There's no reason why one is greater than negative 500 billion. Negative 500 billion is a great number. Look how big it is. Right. Can you imagine owing that much money?
And that's when a picture is worth a thousand words. Right. Like I hate that phrase because sometimes a picture is just useless, and then you need the words to say, "Well, that's a coniferous tree, and it's sitting in a geothermal zone." The picture can't communicate any of that. Right. But what you were just talking about there, that situation of context, the picture, the visualization can give you that effortlessly.
When you were talking about visualization and greater and less than I thought, he's had the same experience as me with maths. Because-
Yeah, but you didn't. It was a different point, but it might be related. When we were taught we were told, think of crocodiles, think of crocodiles. So the crocodile mouth it's like this. So if it's eight is greater than six, the crocodile's mouth is open and I'm a very visual and imaginative person. So like this for me is now already undistracted with the task in hand.
There's lots of crocodiles. There's a whole world. There's a jungle going on. We've got the water, the crocodiles and eight there now needs to be eaten. There it is. The mouth's open. It's therefore greater than the six. But I was like, yeah, but what if the crocodile has snapped the eight, therefore the six is greater at the end. His mouth is closed because he ate the eight. Yeah. I think the six might be greater than the eight and there's all sorts of crazy stuff's going around.
And I think whoever started that off with me caused me... It used to take me ages to get through the greater than, less than questions because I had so many crocodiles and things being eaten and other creatures coming in and greater and less than. So, sometimes people trying to be helpful with maths when they give you crocodiles. Just...
This pisses me off to no end, right. Sorry teachers, if you're doing that, you are cheating. What you're doing is you're trying to give the kids a meaningless trick to get a correct answer and they'll have to wrestle with that idea later on and unpick it just like what Emily is describing. Shame on you for doing that. I appreciate you may not have thought about it, but shame on you. That is just rubbish. That's rubbish teacher. I'm sorry. I know that's controversial, but stop doing it. For God's sake stop doing it.
Speak your truth.
You're messing everybody up. Oh, Adam. I know you don't agree with me or maybe you do either.
No, No. It's a true story. It's a true story. But I think maths and maybe other subjects, but I'm not sure. I'm just going back to my own childhood. We had a slight change and a slight difference to what you got told. I don't know what it was, but it was just the hungriest. Right. So, the sign was hungry. So, they always wanted to eat the most. So yeah, the sign's looking around furiously.
No, I know. You're attaching the concept of greediness, right. To mathematics. Right. Which is just so stupid. I flipped out and I overreacted the other day when I was reading this nonsense document called ready-to-progress. That's been written by the NCETM. Look, why is it nonsense? This is my own opinion. It's not necessary. Right. It's well intended, but it goes off on tangents that it shouldn't go off on. I blew my lid because as I was reading this, I say, why, why, why, why, why? There's so many clever people that worked on this, and everything in here makes sense, but you're not helping teachers. You're sending them in all kinds of crazy directions. Right. You're being too didactic.
Anyway, look here I'm on my podium now, I'm freaking out. Right. So this is the lens that I'm now reading. If I'm halfway through this document, I'm reading it with this lens on, and I'm kind of gritting my teeth as I'm doing it. And then I see that in the early years or year one or somewhere, they're talking about fractions and they throw in the word fairness. You can equate it to the concept of fairness. Right. Equality. They're trying to say equality is the same as fair. My goodness. Is that ever a stupid idea to try to put in kids because you'll be on picking that forever.
Because if you're sharing chocolate chip cookies, anyone who's ever had kids, right. You're sharing chocolate chip cookies. The amount of cookies is not the only factor that matters, right, when it comes to fairness. Because some cookies have more chocolate chips than the others, right. Now in mathematics, that's not relevant. Because the unit is a cookie. All cookies are equal in mathematics when you're counting cookies. Right. So this concept of fairness is a construct that's equally valid, right. Fairness is a valid construct, but it does not equate to equality.
Right. And you talk about anyone about human rights or anything and ask them if fairness and equality are the same thing, or a philosopher, and they'll talk all day about that. Right.
So it takes Them right out of the lesson and into more of a social studies piece that is inherently way more dramatic, and potentially interesting. And it's going to take you away from the lesson. That's interesting Andy, what you've all been saying is. I hadn't thought about that because I fall in love so much with visualizations and metaphors and ways of lighting the lesson up like a Christmas tree. Because you love it when you see a class explode into excitement and people are laughing and debating because the visualization worked. But this has given me a lot of pause. At what point in time does it just yank them out of what you're trying to teach them? Because you've introduced either a blood bath, or a socialist versus capitalist fraction system. And then they start thinking about their brothers and sisters, the time they took those darn cookies.
But one of the reasons that mathematics is so hard for young children to learn, is that they have to un-pin these other concepts, because they develop misconceptions. Mathematics is brutal, right. There's no gray areas in... Well, there are areas, of course there are. Especially when you get in advanced mathematics, but the point with mathematics, especially when you're talking about something like cardinality, like how many cookies do you have? Right. You don't count the chocolate chips. Just like you don't count the chairs or the people or the plates or whatever. Right. That's the whole concept of mathematics, and it's kind of a really difficult abstract concept for very young children to understand. And when the minute you put the word fairness into there, you've all of a sudden introduced a misconception that they may never walk away from.
This is anecdotal, and this is what I see in schools. And I think this is another thing that... I'm just going to talk about maths, but it may well be true for other subjects too. Right. As a teacher, I go in and I think this is true for a lot of countries. As a primary school teacher, the maths shouldn't be challenging for me to find the answer. Right. Okay. So let's just take that as a given, that I've been through primary school, I can do the maths. Does that mean I can teach the maths? No. Is there a massive misconception that just because I know how to do it I can teach it? No. Teaching it is infinitely more difficult than just getting an answer to primary school maths, especially as an adult.
So if that's a starting place for some people that, "I know maths." So there's no problem. English, but tougher, because we are trying to create pictures and these sorts of things and we're allowed to sort of go... I'm not so sure about this. So what I see happening a lot, is I'm teaching fractions or I used to see it a lot with subtraction. So I get to a point where a child's stuck, instead of me learning about the maths, because I can do primary school maths. So I don't have anything else to learn. What I'll do is I'll couch it in terms that this child will understand, except my understanding's not there. So I'm going to go with what I believe is the right thing.
In subtraction, borrowing sugar off the next door neighbor, go down and see if you can borrow some sugar. What I'm talking about is renaming or regrouping and just sort of a very clear process that you would do. If you had a single bar of chocolate, it might be a better analogy. You have to break it into pieces, but to borrow something off someone else, it's a nonsense, because you're not giving it back.
But the choice that the teacher makes is, either learn more about the subject, right. In which case then I can present it a way that the visualization can be correct, or I can say to myself, "I don't need to learn more about the subject, because I get it. I can do primary school maths." So what I'll do is, I'll keep going around the houses with whatever story I need to tell, until the child nods their head. And oh, they're nodding their head because they're borrowing sugar off the next door neighbor. Brilliant. That works. I'll use that forever in a day.
And it goes back to what we were saying earlier, is how you measure success in mathematics teaching. Often it's about producing the correct answer, right. If I can give you a shortcut by telling you a story to produce a correct answer, right. Then I've done my job. But that's wrong. That's nonsense.
That's absolutely wrong. Yeah.
Yeah. There's two issues there. One is probably professional development. Look, I get passionate about stuff. And sometimes I say inappropriate things like what I just said earlier, but they are my opinions. There's a reality, right. At the end of the day, the teachers being judged at how many of their kids gave correct answers. Right. That's their measure. So systemic, right. Systemically we've created this culture of box ticking and it's okay to take shortcuts in order to win. It's kind of like cheating. I think I called it cheating earlier. Right. It's like it's okay to cheat as long as you win. Right. It's well intended. Right. Of course that's nonsense. We got to switch that mindset around. Right. Sometimes concepts are difficult to learn and they take time and you need to spend the time, and you need to struggle with it because that's the only way you're ever going to learn it. Right. Is when you wrestle with a concept. Thanks for listening everyone.
Thank you for joining us on The School of School podcast.