Learning is Moving in New Ways - A Conversation with Professor Dor Abrahamson

Welcome to the Future Learning Design Podcast.

In order to enable children to engage their bodies, their intuitions, their heuristics, their common sense, in order that mathematics will make sense to them, it really is about a shift in the classroom norms. And the teacher is of course the model of what we're allowed to do and what we're not allowed to do when we do mathematics.

Hi everyone.

Thanks for tuning in for another episode of the podcast. My name's Tim Logan, and the podcast is brought to you in partnership with Notosh. My guest this week is Dor Abrahamson, who is the professor of secondary mathematics education in the area of cognition and development at University of California, Berkeley, and a member of faculty at UC Berkeley's Graduate School of Education. Dor has been pioneering a design-based approach to research called embodied design, and it was fascinating to talk to him about his work in embodied cognition. In his Embodied Design Research Lab, Daw and his team look at cognition and learning of mathematics through the body using different innovative technological tools. As one of Daw's fascinating articles from 2016 is entitled, "Learning is moving in new ways." And this conversation follows on from a recent episode with Keith Davids, where we talked about an ecological dynamics approach to skill acquisition and motor learning in sports.

Hello.

Hey, how are you doing?

Doing very well.

All right.

Well, thank you very much, Dor. It's really, really a great pleasure to meet you first. And we've had a couple of auspicious mathematicians on the podcast before, Joe Bowler and Dylan William. But I think you are perhaps the first virtuoso cellist mathematician.

Okay. We wear many hats. Thanks so much, Tim, for inviting me. I'm honored to be in the the virtual or audio presence of those other famous people you've been chatting with. Yeah, so.

No, no, it's great. And Keith David's put me onto your work and he's been on the podcast recently as well. And I was so interested to find out more about the way that you've been working with and thinking about maths firstly, but there's something much bigger than just maths going on, I think. And I think hopefully that's gonna come out in the conversation.— but if we start with the question about maths, I love the way you put it in one of your talks you were giving, that what in the world is maths? Like, what in the world is this thing? And like, there's something very situated in the world about the way that you seem to work with maths. So, but how does it kind of connect us in the world?

Thanks, that's an excellent question, Tim, because I mean, this question of what in the world, or like, what in the world, like, where is math? What is math? Now, you know, if you were speaking to, say, a science education person, we could speak about, say, magnetism. So I'd probably bring out a magnet, not that people at home could see it. But like, you know, if you want to talk about magnetism, you probably want kids to have a magnet there so they can feel what it is. If you want to speak about, I don't know, botany, you probably want to bring some flowers and, you know, seeds. Here's the thing. Or, you know, animals, zoology. But then you come to math and you say, okay, I want to bring the Whoops, wait, what do I bring? What? And in our exchanges, you mentioned Jean-Jacques Rousseau. He was, you know, speaking for the pedagogue of the eponymous Émile, the child in his charge. And he says, no, you know, you need to show the child the thing itself. Because if you just show symbols, if you begin from symbols, if you begin from algebra and say, all right, if you got x on the left, just move it on to the right, it will be minus. And you can teach people rules, but they'll never know what these things are about, what they are referring to. Unlike the case of, say, you know, here's an apple, I drop it, it falls. All right, let's speak about gravity. But in the case of numbers, what are the things, and how can we enable children to understand what all these symbols, all these, you know, semiotic resources, you know, tables, graphs, and equations, what are they referring to? So, well, you might say, oh yeah, but math is this special kinds of, you know, human knowledge and practice, it transcends the world, and only us privileged kind. No, no, no, no, no, I don't go for that because I come at all of this from a very evolutionary perspective. And so to begin with, when I want to know if how people do things and how better to teach them to do things and understand stuff, I will always, always begin from, okay, so who are we as, you know, cognitive biological organisms embodied with our— most of us have hands and legs and the, the sensory organs that most of us have. And so who are we such that we might have invented this thing called mathematics. And I always come back to saying, well, yeah, basically, and here I'm wording the famous Francisco Varela, and you mentioned other people like Evan Thompson and Eleanor Rosch. So, so let's talk about two tenets from what they call enactivism. That's one of the E words from the 4 E's. It used to be one and then two and three, and now it's inflating. One thing to understand is whenever we speak about Perception. People say, oh yeah, perception, seeing things here. Well, yeah, kind of, but no. So when we talk about perception, let's think about when we ancient, ancient millions of years ago time, we were these single-cell, two-cell, tiny microbe kind of, you know, mollusks or what have you. And it really helped while we were moving about to know if what we're moving towards or what we've bumped into is, you know, a friend or a foe. Should we, you know, have some kind of engagement and dissipate our energy with this other fellow here, or should we get the heck out of there? So it really, really helped if we had some kind of sensory capacity to inform our movement. So the point from enactivism is, if you want to understand what perception is, you have to go back and say, what, you know, how do we operate as organisms? So the moment we have some kind of sensory capacity, it informs our movement. And that could be just, you know, feeling. If it's tactile, like the feeling that we have when we brush our fingers over a surface. If it's haptic, to feeling when we maybe close out the grip of our hand on some object, can we hear it, etc. So all of this, all these sensory capacities are serving movement and ultimately action. So the basic tenet here: perception is always for action. If you want to talk about perception, ask yourself what action is it serving? And perception keeps changing according to what we might want to do with the thing. I might look at a water bottle, and if I'm thirsty, I probably want I see the grippable aspect, or what Gibson called affordances, the grippable affordance. But if it's a windy day and my papers are flying all over the place, I'll look for the, you know, a heavy thing to put on the paperable kind of aspect of this water bottle. So it always depends. If you want to know how I'm about to act, you need to know what I'm trying to do, and therefore you'll know how I'm perceiving. So one tenet is Perception is always for action. And so when we engage with the world, it's, you might say, we couple with the world through these perception-action loops. But, but how can we do that? So yeah, so the sensory motor, that's kind of below the surface of consciousness, luckily. I mean, just imagine you had to think as you walk, okay, move this foot here and I'll raise that. That, as people like Nikolai Bernstein said, 100 years ago, Soviet kinesiologists started the whole field. There's just no way we can deal with all of those, all those degrees of freedom of our body. Luckily, and this is something that's being championed by my colleague and collaborator Franz Metzner, luckily perception is what is leading our actions. And so how the heck does this get to math? Whoa. So, so if we start by saying, okay, all thinking is necessarily inactive, and we come to ask ourselves, so how does, you know, math belong to that? We have to shift away from thinking about math as this abstract stuff. In fact, in my lab, you're not allowed to say abstract. You pay a penalty. It's the, it's the A word, right? Following my own mentor Uri Wilensky from Northwestern University, I, he has this brilliant chapter from 1991 called something like Abstract Meditations on the Concrete with Concrete Implications for Education, and he makes this amazing proposal there that abstract is not an inherent quality to the extent that anything is inherent. But you know, you take a ball, you say it is a sphere, it is round, it is heavy, it is blue. But abstract, he says, it's not a property of an idea like this is abstract, it's just an is, but rather it's about a person's subjective relationship to that. That which is, you know, concrete to you, Tim Logan, oh yes, the binomial, you know, I might look at it and gape at it and gawk and say, whoa, that is abstract to me. But through learning, it will become tangible. So our capacity to do mathematics, I think of it from an evolutionary perspective as a co-opting, or what Gould called the acceptation of our sensory motor and perceptual action capacity. So if you think of birds' wings, all right, I think of this, you know, a heron, a black heron, it uses its wings to fly. And yet the heron, and as do many other birds like penguins, they do amazing, a whole sort of other things with their wings. What the heron does, for example, is called canopy fishing. It stands in water and it creates like an umbrella with its wings over the water directly in front of it or under its head so that the fish and the water will be in shade and can better fish. So wait, what came first, the heron or the egg, the flying or the— so you take something like wings for flight and then you use them for something like fishing. I believe It's the same thing with mathematics. We have this innate capacity, our neural architecture, the enactivists will say perception is always for action and cognitive structures will emerge from the recurring sensory motor patterns that enable action to be perceptually guided. Okay, I know that's lots of words.

It's getting a little, so I hesitate to use the word abstract, cause then I have to buy wine. But one of the things, it would be really interesting if we can kind of go into some examples of, in your lab about the way that you do that. But just, I mean, cause I think that will help make it really real. It certainly did for me, understanding the way that you're working and way that people are coming to understandings of maths through their bodies. Right. And I mean, your phrase, which I loved, learning is moving in new ways.

Yeah.

And I think there's something, as you said, kind of coupled to the environment, like you are moving in the environment and you are learning not only how to move and be in the envir— environment physically, but that's helping you conceptually or cognitively to develop different concepts perhaps. Mm-hmm. Like for example, around proportion. So I'd love to talk about that example if we can, but it, it, it also makes me think of the, the map is not the territory. Saying, right? Like you, that kind of idea that we create these maps, but that's not reality. And I think, I, I feel like education has been stuck in the map for a long, long time.

Yeah.

In the sense of we begin to slightly confuse the map as reality. We teach young people about the map. We don't really often always give them the chance to kind of start wandering through the territory and understanding and, or You know, allowing their navigation and their understanding of the terrain to develop and, you know, perhaps using a map to help them with that, but having this kind of relational thing going on between the map and the experience of the territory. So yeah, as I said at the beginning, this, I think this, this isn't just about maths and how to learn maths. This is, there's something really fundamental going on here about learning in general, which I'd love to just dwell on a little bit. Yeah. If we can.

Yeah. So the experimental psychologist Stefan Harnad, he poses this general question which he calls the symbol grounding problem. Like all these symbols, like what do they mean? He has this funny story about being stuck somewhere in the airport where all the signs are in some foreign language and you're given a dictionary and you look up the words, but turns out it's a dictionary that just translates back into the language. And so often, so you go in loops and loops and he says, well, The joke is on us because so much of what we do is that way, and the meaning of symbols is never in the symbols itself. It's always whatever it is that they evoke in the body, the whole sensations and the motoric reaction. So absolutely, you know, I think that the tragedy of education, of certainly math education, something I know a bit more about, is that at a certain age we give up on trying to understand and we just assume that math is about manipulating symbols moving from side to side. If you ask a person, you know, what is the meaning of to add, like in math addition, you know, the plus sign, and you know, you've somehow implied to them that you, you know, you say mathematically informed, but you don't speak the language. Could you just help me understand this thing, this plus sign? If you keep pressing people, eventually they'll kind of gesture with their hands, like bringing stuff together. Stacking stuff, or means some fingers in the left hand, some fingers in the right hand, and bringing them together in some kind of cumulative actions. And then you go, oh, so that's what addition means. Means that plus sign is grounded in some sense of we're bringing things together. And we see when kids learn math, they'll have these, you know, manipulatives and little chips, tokens, things that they play with to learn. And maybe also when they learn multiplication, it pushes the envelope a bit. They'll know, oh, 3 times 5, oh, you have 5, oh, here's another 5 and another 5. We have 5 3 times. And it just gets into fractions, but then fractions is more or less when the whole thing collapses because fractions are just ridiculous. And the larger the number is in the denominator, the smaller the total number, right? And kids sort of lose it. And I think a lot of teachers lose it, you know. Famously, when you divide a number by a fraction, you're told you need to kind of flip the fraction and instead of dividing you multiply. So many have heard that, you know, the Doggerel, "Ours is not to question why, just invert and multiply." And I think that a lot of teachers have given up. So we see the work of our lab to come in and get in kind of there, just beyond addition, multiplication, where things become nettled and unpleasant— fractions, rational numbers, proportionality, probability, algebra, etc., etc. And try to do the world a service of making all those— I'll call them advanced, of course, not abstract— those advanced mathematical ideas, render them for people just as, you know, addition is, so that people will have a way of gesturing, but not for the sake of gesturing, but that would be maybe our measure, that they've somehow grounded it, so they, they have understood it, so that Professor Harnad will say, oh yes, that they've actually grounded those symbols. They have meaning for them. They can enact the way that they act. And the flip side is that when they see it in the world, they would recognize it. And that kind of brings us back to your question about the map, or we sometimes speak about a model. I mean, so many different meanings for that word. But basically, if you're hiking and you see just a creek of water,, and you start thinking, oh, ah, the rate of water, how many, you know, liters per second, or et cetera. So, so that you bring forth some capacity. It's, you recognize that moment in the world, nature, culture, what have you, as that. And we've seen recently in COVID, we had a striking situation where people didn't quite understand what exponential growth is. They thought of people getting sick more like playing tag. You know, I have COVID, I give COVID to Tim, Tim gives COVID to Maggie, and as if it's growing linearly. And we're going, no, no, no, because Dor still has COVID and so does Tim. So it's 1, 2, 4, 8, plenty. And people like, wait, how do they, how do you even make exponential growth? Is so, it's such a weird thing.

Yeah.

Here's a, if I may, a quick exercise. I once asked, I teach sometimes pre-service teachers. Mathematics. And just kind of to warm up, I ask them, imagine you have a sheet of paper like your regular printing paper, only it's, it's just infinitely large. Okay, so you fold it once, so now it's double as thick. Okay, and now you fold it again, so once again it doubles its thickness. And then the question is, again, remember it's infinitely large, just let's assume if you keep folding this sheet of paper How many folds would you need to make so that your paper's thickness would take us all the way from here to the moon? And I just, you know, the teachers look at me, hey, wow. And they say, it's like that, you know, Austin Powers scene, a million, trillion, billion, trillion, billion. And then somebody is really bold. I think only 1 million. I think, well, turns out it's more like 42 times. Yeah, like, whoa! And then you start doing the math and you say, yeah, wow, how did that— it's like our mind is not equipped to do that kind of exponential stuff.

Yeah, absolutely. But there's an interesting bias that we have towards pushing people as quickly as possible to the— again, hesitate to use the word— abstract stuff, right? It's almost like it— you would be embarrassed in a 9th grade classroom, if you wanted to reach for a manipulative to help you to understand something or count on your fingers, or, you know, this kind of idea of embodying that understanding has this association with being remedial somehow. Right. And it's this, this really weird and interesting bias towards getting to the very conceptual as quickly as possible.

Yeah. So that is absolutely fascinating. You know, when you, when you listen to the top mathematicians, of the world, like, like Terence Tao at UCLA, considered by many the top mathematician. I mean, he first, I think, was discovered, if you will, by Paul Erdős in Australia and brought to the States. And he was full professor at the age of 24, which I take as a personal affront. You can read online, he's so generous with his time and thoughts, and he shares how he thinks mathematically. And he's very open about it. He says, I have these crazy images of like a jet ski or fried eggs frying on oil. Or he says, on one time, he was trying to understand partial differential equations. He was literally rolling on the floor. And then he kind of adds sheepishly, well, of course I didn't put that in the book, but you know, because my colleagues and this and so on. And I'm thinking, no, Terence, you should have put this in the book because, so we have testimonies going back to 1945, the book by Jacques Hadamard interviewing people like Einstein and Poincaré, et cetera. People are thinking with their bodies, these top mathematicians. And I might say, you know, with all due respect, that when most people are teaching mathematics, they themselves are not mathematicians. And that's not in any way an evaluation, it's just that's the way the market works. Okay, so often they themselves have not had the true deep amazing experiences of thinking through mathematical problems. Now I want to talk about this child who wants to reach to the manipulatives and say that if we want people to be able, like Professor Tal, to roll on the floor in order to understand algebra or what have you, it's not so much about what the kids need, because to me it's clear that they need to be engaged in an embodied and active way to understand these ideas. I'd rather say it's all about, as you implied, it's about the norms of the classroom.

Yeah.

So there's a scholar, I think, in Texas called Professor Florian Voigt, and he speaks about what he calls the epistemic climate. So what the heck is that? Like climate, you know, in the metaphorical sense of like epistemic to do with knowledge. So when you're in a classroom, where does knowledge come from? Who owns it? What is the nature of knowledge? Who is allowed to author it? And what does it mean to engage in thinking? If I want to say, oh, when I think about probability, it makes me think of pasta going through a sieve. I had a student tell me that, and she looked at me, is that okay? And I said, oh yes, yes. Like, am I allowed to share these idiosyncratic metaphors that just pop up into my head when I'm thinking mathematically, or is that a big no-no? Would I be, you know, derided because, oh, mathematics is so serious? So in order to enable children to engage their bodies, their their intuitions, their heuristics, their common sense, and in order to make sense of mathematics, in order that mathematics will make sense to them, it really is about a shift in the classroom norms and the epistemic climate. And the teacher is, of course, the model of what we're allowed to do and what we're not allowed to do when we do mathematics.

As I see you gesturing with your hands, moving, moving them together, there is, yeah, this, this concept of tensional anchor and yeah, through, through your process of embodied design, like there's loads of amazing stuff there, but yeah, I think the example really helps. It certainly helped me to understand. So as you work through this process of working with teachers and in classrooms with students and, and, but using technology also to support the development of those understandings through movement in new ways, in different ways with the technology to support. So yeah, if you could describe that particular example. I think that's really useful to give a sense of what that is.

Yeah, thanks for reminding me. Scholars across the embodiment theories and literature, from phenomenological philosophers and through to cognitive anthropologists and others, speak about the criticality of entraining other people's perception. What does it mean to get another person to understand something It is getting them to see the world in a new way that enables new actions. And so we are working within, in my lab, the Embodied Design Research Laboratory, developing this paradigm we call embodied design, which is a whole compendium. It's an approach to educational research which includes a design framework and methodologies, and in order to create and evaluate new technologies that support in my case, mathematical learning. And so if we get back to the example of addition, I said, oh, it's those 3 fingers and the 2 fingers, one in each hand, and then they come together. Well, what would it mean for me to bring a student to the point where, just as they have that embodied and active meaning for addition, they'd have that also for, say, proportional relations or ratios? How can train them that they can understand it? How can that bring them to new perceptions for action? And so we created this technological architecture that we call the Mathematics Imagery Trainer. And the Mathematics Imagery Trainer is a technological interactive platform that we've implemented in all kinds of media like Wii Nintendo and Kinect and tablets and now in virtual reality, etc.. But the principle is always that in order to learn a mathematical concept, you have to first, as you quoted, you need to learn to move in a new way. If you cannot move in a new way, then your understanding of the mathematical concept will always be superficial. It will be about moving, uh, symbols, pushing symbols from side to side. You'll never quite understand the concept. So how do we get people to move in a new way? So we work with sports scientists and with people in Feldenkrais Method and martial arts, etc., to understand how we teach and learn to move in new ways. And again, the principle that first you will learn to move in a new way, just as you'd learn maybe a little, a new dance movement or a new karate kata or one or something of that sort, and only then we will symbolize it. So just like for addition, we bring the fingers together. For proportion, one way we can think about it is placing our two hands on the desk and then getting the children to raise their hands at different speeds and appreciating that the heights of the hands over the desk as the hands are going up, one hand going faster than the other, the respective heights are always in equal ratio to each other. For example, you may put both hands on the desk. I hope you're not doing this while driving. Put both hands on the desk and start raising them where, say, the right hand is moving double as fast as your left hand. And so you will find at any given moment, your right hand will be double as high over the desk as your left hand. Now you are moving in a way This is a movement form that is instantiating proportionality. And now it doesn't matter that at this point you don't even realize it's about math because indeed there are no symbols. This is not mathless and maybe this is some kind of a video game you're playing where you need to do this in order to open, you know, a door onto the other lost kingdom or something. So you're standing there trying to figure out how the heck to move in a new way. And I might say for you, okay, whenever you're moving the correct way, I know that some green light will go up So it's a kind of peculiar way of teaching a person to move where I'm not really telling you what to do. Rather, I'm saying, okay, you just move your hands. Whenever you're moving correctly, whenever the posture is correct, the light will be green. And so you're starting to move your hand. Typically, a kid will move only the left hand, but the light is not green. Say the light begins from red, right? The light is red. They move the right hand only. The light is red. They raise both hands up and down. It's red. So eventually they start flailing their hands. At certain points, the light is green and they freeze their hands. Now we know the light is green because the right hand is, say, double as high as the left hand, if that's the ratio in question, 1 per 2. It could have been something else, but that's what we put in the console. Now the kids say, am I done yet? And say, oh no, sorry. Now keeping the light green, move your hands again. Now, this is a fascinating moment because the kid is sitting there and thinking, well, there must be some method in this madness. This guy's asking me to move my hands now, but there's— there are no cues, there are no frames of reference. All there is is this empty screen in front of me, my hands, and maybe opposite each hand there's like a cursor on the screen. So there are these two objects on the screen and they're floating in space. I need to move both of them And yet I need to keep something. Something has to stay the same so that the feedback, the green sphere, will stay the same. And the kids look around and all they see is the gap between their hands or between the cursors, the spatial interval, that which was outside of their attention. It was just negative space. They had not been attending to it. Suddenly this pops up and you might people at home, or you might look at the gap between your hands and say to yourself, blimey, there is something there. And even though it's invisible and immaterial, we can talk about this thing. We can talk about its length. We can make it longer or shorter. We can bring it up or down, talk about its elevation. And so the kids start to attend to this thing. And what they do, of course, is they lift this thing by lifting the thing. Again, it's invisible, but we are referring to it as a thing. So now for us, it is a thing. It's an ontology in the world which we can refer. They lift both hands, keeping the distance between their hands the same. And of course, the thing goes red because they have violated the ratio. But the kids are undeterred. They are theoreticians, as Annette Karmiloff-Smith will say, that the child is a theoretician. So what they will do is they will say, oh yes, yes, yes, I messed up. And so they make the gap larger and the thing goes green. And I say, move your hands again. Once again, they lock the interval, raise it up, the thing goes red, they fix it to green. Over time though, the child goes, hmm, I keep fixing it by making it larger. I wonder, seems like a kind of a variant, what, you know, Jean Piaget will say, it's I can still kind of assimilate this new situation, the world, onto my operatory scheme, sensory motor scheme. I can still kind of do that. But with time, the kids go, no, no, no, this is something else. Because even as I'm raising my hands, I'm already increasing the difference between them, meaning this is a different scheme. It becomes differentiated. And the kids go, oh, the higher my hands go, the bigger the distance between them. Meaning they are now coordinating between changes of this new object, the distance between their hand. They're coordinating between the length and elevation. And what we have here— and for a moment, I'm not saying this is in any way the definitive gesture of proportion, but for us, it had a lot of practical and pedagogical advantages, such as both hands moving the same directions, and you can start measuring things, et cetera.. But for the children, this gesture or this movement form becomes the meaning of proportionality. Now, of course, as you said earlier, we need to get from the world into the map because mathematics is so inherently about all those symbolic forms, about graphs and equations and numbers and various notations. So at that point, once they have learned to move in a new way, begins the shift over into mathematics. And we keep the grounding of the movement. All we do is bring in some instruments. We will often bring like a grid of, say, of horizontal lines. And what we see is when we introduce those symbolic artifacts, that stuff, that mathematical cavalry into the activity space, the children recognize in those artifacts they recognize in them embedded utilities for enhancing the strategies, enhancing whatever it is they were doing before, whether enacting better, explaining, or evaluating, they'll start moving up their hand and suddenly they'll stop at one of the grid lines. Then they'll raise the other hand soon enough to say, wait a moment, for every 1 unit that I raised my left hand, I raised my right hand 2 units. Oh, it's like 1 per 2. And we look at them wide-eyed. If you measure the pupils of our eyes, it dilates. We actually have that tool in the lab now. And because what we see here, we see the child rediscovering the mathematics, we see the child reinventing, or what Hans Freudenthal and others in Utrecht would say, it's the guided reinvention of mathematics. But they have found the mathematical tools and used them, and in so doing, they're shifting their mathematical thinking, all the while keeping it grounded.

I love it.

Amazing.

There's a certain irony in the fact that we are only trying to describe this in words, because given how much embodiment is going on in what you've just described and how much the visual also helps to see it, right? So I'll put a link in the show notes to the video, but I just, I think it's such an interesting experiment because you were just, I mean, you're like, they are, to use the frame I was using earlier, you know, they're kind of exploring the territory themselves. Absolutely. Just trying to figure things out and try and understand there's this kind of thing happening. They don't quite know what it is. And then as you talked through, they kind of make these realizations. And I love the moment where they realize that this space between their hands is really important. And I think that's what you then call an attentional anchor, right? It's like— Thank you.

Yes.

They suddenly acknowledge that there's a thing there. Like you said, it's like a negative space that is there but not there. Like it's really important, but they hadn't quite realized because up to that point they were only moving their separate hands.

So exactly. Thank you for reminding me about that, bringing me back to this. And thanks for reminding me that I'm being videotaped and doing all these ridiculous gestures. Flying around like the Black Heron. So this idea of the attentional anchor is so fascinating because it is very difficult to raise the two hands at different speeds and monitor these two things. And our mind, our brain is neurally inclined to search for ways of distributing this, this unruly coordination over the world. When we discover this thing between our hands, and talk, oh, we just need to make it bigger while our hands go up, suddenly our hands become sort of cognitively invisible because we no longer need to think about what the hands are doing individually. Rather, we're raising this thing, this thing here, like magicians go, there's nothing. We're raising this thing here, but while making it bigger. So lending, borrowing from our collaborators, Dan Hutto and Raúl Sánchez García, We are calling this thing the attentional anchor. It's this aspect of the world that comes forth. It pops out, out of negative space as a new affordance, a new handle on the world. But in so doing, because we have these extra kind of cognitive capacities, not just to do but also to think about our doing, we have this capacity to certainly recognize what Piaget called this reflexive abstraction. We suddenly kind of step back and say, oh, there, here is the thing, here's the thing that we are thinking about and manipulating. Now we can talk about it. We have these wonderful cases where teachers are interacting with students, and because the teachers had already kind of read our paper, they already know what these things are that might pop up. And if we have time, I can tell you about another one Just the other thing I just wanted to reflect on, because the attentional anchor is—

I, I just love that moment. But then also that transition moment where they go from the territory into the map. And I think like there's something really magic happening there. Like you say, when you overlay a grid onto the screen and suddenly this red or green screen that they were playing in front of, suddenly they're playing in front of this grid. And it's like at that moment, they— the movement turns into this more conceptual, mathematical, measured— all of those words— kind of experience. And it's, it's a really fascinating kind of moment of realization. Suddenly you've come through your body to understand this concept of proportion.

Exactly. And of course, the, the irony is that the child learning the concept, it's not like they go, oh, finally I have learned this, because that this if it makes sense, did not exist for them such that they would know that. But what they— they shift into a different, like, like, semiotic register in the sense that now they're talking about their movements but using these consensual terminology and, and ontologies from mathematical discourse. So they're talking about moving, and whenever we move from continuous space to discrete space, we shift from qualitative to quantitative language. It's no longer how much, it has to become how many.

Yeah.

And so we need something to count. And that grid suddenly gives us— it's like a lifeboat. We literally see kids hang on to that and start talking about these particular locations because it's really frustrating moving your hands in air and trying to kind of lock yourself into particular locations. Suddenly the grid gives that to you.

Yeah.

So you start working, and which is one of the key ideas from the, you know, the 4E, letter E, like the idea of the extended cognition or distributed. Our thinking is, you know, whereas the brain in and of itself is an organ in the head, I hope my daughter has on her helmet a saying, it says, I love my brain. The mind as such, we think of it as kind of distributed over the environment. And again, ecological psychology. So And the environment is not just the, you know, your natural habitat, the birds and the bees. It's also our cultural habitat. Evolution is sensitive to the cultural niche we are in and privileges those capacities to engage with artifacts, going back to knapping stones. So we do shift into this cultural discourse that changes the register of our thinking.. But our hope and ambition is that all the while it's still grounded in the sense of the rising hands, just like the counting is, still remains in some kind of cumulative feeling.

Can I ask you, have you done any then more longitudinal study of those students who have learned proportion in that way? That's the question that comes to mind for me is like, do they have a kind of somehow deeper whatever that word means precisely in that context, but a deeper understanding of, through their bodies, of this concept of proportion. Does that manifest in any other way longer term, or—

Alas, not yet. Mia culpa. And, you know, I just yesterday I was kidding with my seminar. I was talking about what a cross-sectional study is, and I said how much cheaper it is than a longitudinal study, because you just go in and speak to kids of different ages rather than wait for a particular kid to grow from the age of, you know, 4 to 11. Who's gonna fund that? But I would absolutely welcome that kind of work. And but, you know, we do know from parallel work, if you will, from programs such as the Maria Montessori, we do have insights from, from those programs on the capacities of more striking. I recently read a book and apologies, I do not remember the name of the author who looks at the work of Friedrich Froebel and looks to the people who study— Froebel, the guy who invented kindergarten.

Yeah.

And And he was the first person who brought in all these things, manipulative, the whole pedagogical regimen of how to learn by interacting stuff. And you have people like Frank Lloyd Wright who studied that way, people like Buckminster Fuller who literally kept on building these things. So in this lovely red, it's a tall red book with lots and lots of examples of this kind. I have it, you know, as you walk into my lab, you see it.

He's, I mean, that his famous phrase is head, hand, heart, right? I mean, then he— and that idea of gifts, I think, is also— is that rebellion? Like the idea of absolutely gifts that are given for his children to explore and manipulate and understand the world through these objects.

So that, that which we now call manipulative, he kind of invented this regimen, even though of course, you know, creativity probably came from some— but, but he called them the gifts. Absolutely. He was this broad philosopher, visionary, psychologist, activist. So he had a set in, in his box. I think there 15, 17, 19, some prime number of gifts. And the very first one is a yarn ball. And I just recently— we have actually in press in the Springer Handbook of Digital Resources for Mathematics, I have a chapter with Kimiko Ryokai and Justin Dymel where we look at the history of educational resources. And we begin with this yarn ball because Froebel gives a whole list of like 101 things to do with a dead yarn ball. You know, like you can roll it, you can put it in a box, you can do that, you can kind of yo-yo it, all these things to do with a yarn ball. And in this chapter, we make the claim that technology is only now beginning to catch up with our little squishy, cute little yarn ball because people are going VR, XR, but we've lost that loving feeling. We love that, that, that, that haptic tactile feeling.

Yeah. Yeah.

Yeah.

And I mean, I, I really do want to ask you about the, the technology and how, how it's advancing, cuz I think that's a really fascinating and potentially a little bit scary subject. But before I go there, I'd just love to ask you about how does then when young people are connecting with each other and collaborating, like how do you find that that adds to this example? Because I, there's a, an individual thing that you've been talking about with the movement of the hands and the attentional anchor, but then when you put them together in groups to explore, do you see anything differently happening there? Or it struck me that there's, that's kind of what a lot of what happens in more, let's say, progressive educational methods where young people are working together in groups, exploring project-based et cetera. What's that thing that's happening?

Well, we can go to the opening page of Ludwig Wittgenstein's 1953 Philosophical Investigations, where that's the turning point where he shifts from the early to the latter self and from the analytic philosophy to more continental. And he says, oh, actually, what is the meaning of words, even mathematical words? Where does it come from? Well, it's to do with its function in human interaction. If I say, hey Tim, would you please pass me that wood beam? And you pass it to me. Well, the meaning of beam, there's nothing deeper to it than the way that it enables us to engage in this joint collaborative action together. And so language is really important as an impetus. If we spoke earlier of, you know, you start raising your hands and suddenly you notice the thing and you start talking about it. The need to talk about how you move, the need to— we sometimes use, it's kind of funny, we use the word language as a verb. We say to language, the need to language those tacit aspects of your perception, that gap between the hands or all kinds of imaginary attentional anchors that come forth for you. The need to speak to other people is the cognitive impetus as Dan Sloven calls the thinking for speaking. It's the thinking for speaking that brings forth these ontologies as things that become conscious for you. So really, when you're talking to another person, it's self-serving in the sense that you become conscious of what it is that you were thinking. One of my doctoral co-advisors, Karen Fuson, she had this wonderful example, a way of thinking about what it means to think before you know what you're thinking about. And she says, imagine you're you're writing a paper to, I don't know, to a loved one or an article or what have you, you're writing away and suddenly you kind of, your fingers freeze and you look up looking for the word. Is it allege? No. Impute? No. Ascribe? Yeah. And you put that down, right? She says, well, clearly, when you found the word, it matched something that was there. It's not like you were a blank slate. You had that notion in you.. So people like Radford will call that the pre-symbolic notion. But you were just going along, but then you had to write it. You had to find the word. You had to objectify that pre-symbolic notion. So that's one function of this collaboration, working together. And the other one, if I may quickly, it's just that we often have different, like, duck-rabbit perspectives on things. And it's so important for us to see the world as another person does. That goes far beyond, you know, mathematics. It goes into, yeah, human interaction.

Absolutely.

And yeah, increasingly important, I would say, right now. But just before we finish, I'd love to ask you about the technology question, because I think, yes, from, from understanding all the different, or lots of the different pieces of research using technology you've done in classrooms, I mean, you've been working with, for example, with AI in classrooms for a long time, and, you know, obviously with advanced AI now Khanmigo and all of these bots coming into as AI tutors, et cetera. I mean, given the, and you mentioned the eye tracking, pupil dilation measuring, all these different kind of technological gizmos that you've been working with, how are you currently now feeling about the landscape around technology and education in terms of, obviously, mm-hmm, from a research perspective is your main lens, but I'm also more thinking perhaps from a pedagogical perspective and the way that teachers might engage with some of these things.

Yeah. So, so to begin with, you know, whenever I speak about this in seminars, immediately people just yesterday that were writing a text, yes, but what about privacy, surveillance, all these Foucauldian nightmares? So to begin with, like, you and I are talking now, and, you know, excuse me for saying this, but I'm tracking your eyes, right? I can see where you're looking, and probably to the extent your pupils are dilating, I'm noticing that, um, any facial expressions So that's just human interaction.

Communication.

Absolutely. And so if a machine is picking up on that, it's not like it's invasive in some sense that it's looking into your mind and into aspects of your being that you don't wish to expose. So just to begin from that, you know, from that, the terror of the surveillance. And actually, it would be useful for me if I'm reading a text, I'll step back. When I, when I, if you're teaching me how to tie my shoelaces, it's really useful for me that you know where I'm looking.. Yeah, so that you can say, oh no, Dory, you shouldn't look at there, you should look here. So that's useful for me. Now AI is coming in despite all the anxieties and all the Luddists, etc. It's what can we do that anything which, which is, you know, today's science fiction, tomorrow it's going to be here. And we've been around enough to know that the craziest things that we could hardly conceive suddenly they become a product on the shelf. So it's going to come in. Artificial intelligence is going to be everywhere. And if now we're going, oh my gosh, the software completed my sentence on Google Doc. Oh my God. So that's going to be, you know, tell your grandchild that and you say, what? You know, Grandpa, that's all it did back then. It completed your sentence. Get out of here. Just like we're looking at people 10, 20, 50 years ago who didn't know what an iPhone is. Right. So it's, it's going to go wild and rampant and it will be ubiquitous and people will greatly benefit from this outreach dissemination, the ubiquity of AI supporting learning. The trick about it is how we design it, what we want the AI to pick up on, what we call MMLA, multimodal learning analytics, picking up all of these aspects of behavior and EEG and, and skin connectivity. What do we want them to pick? Like the Fitbits that people wear when they're jogging, like what do we want them to pick up and how do they process and, and respond to that?

Yeah, it's really fascinating because I, I mean, I completely agree with you and clearly, I mean, it comes back to your exponential point earlier, how quickly it's changing. So we feel that like in our bodies almost, like we feel the exponential there. But the thing that I find interesting is the, the way that it's intersecting with a more progressive call for changing education towards certain things like competencies, 21st century competencies, whatever you want to call them. That's where I find it's quite interestingly worrying is, you know, your Foucauldian nightmare perhaps. But if, if we are interested in where a child is looking in terms of developing the skill of tying their shoelace.

Mm-hmm.

Or learning proportion, that's one thing. If we're trying to use biometric data, blood, et cetera, et cetera, to understand how empathetic they are or how much compassion they're demonstrating towards another peer, because we've, we begin to want to shift away from the knowledge paradigm towards this kind of fuller, more human flourishing competency framework. That's an interesting intersection of that kind of technology with those kinds of ideas, because then you start into this this interesting space of engineering the— could say the inner parts of human beings in terms of their emotional life, their spiritual life, all of these. That's where I find it interestingly scary. Not, not so much learning calculus or supporting them to learn a new language. I don't know what your reflections.

Well, firstly, I, I believe that both, uh, research and technology are about to take us to new echelons of understanding the, the human complex. And that which we've ushered aside to say, oh, there's a cognition, but oh yes, then there's all this emotion stuff, the affect. No, I don't study that. Go down the hall or the next building, other floor in psychology, the wishy-washy people. We more and more since Damasio, we— that this feelings and sensations are inherent to decision-making. It's only retroactively we say, oh, I decided. Really? Your stomach decided and you felt it before you could verbalize it. In our labs, we can predict when people are about to have an insight before they know it. So, and just to, just as one example of where technology can work with aspects of psychology, and perhaps not necessarily, you know, this 21st century skills, I never quite understood what people actually mean by that. I think that it, you know, should have been 20th. But, but so I have a colleague in Canada called Alyssa Antle, and she was working with Nepalese orphans of the big earthquake that was there, and she created for them biofeedback tablets. Where they just by trying— there was like, I think, like a hot air balloon which they were trying to sort of raise on the screen. The software was connected to their body, feeling their, I think, their heart rate. And they didn't know what they were doing, but they just tried biofeedback. They tried to get the hot air balloon to rise, and they figured out how to do that. And in so doing, they could calm down so that they could begin to study. So, you know, what would you categorize that under? That's technology that's working with— but it could— and I know other people who are, who have various— so for children, for example, with neuro differences, maybe children on the autistic spectrum, etc., there are all of these technological provisions that support their garnering the types of information that perhaps they are not attending to Right, or if in, in various circumstances it could really help to have some visualizations of aspects of human interactions that are not clear to you. For example, I, I have a bit of prosopagnosia. If I met you in the street, I probably wouldn't recognize you. It would really help me if like on Zoom your little name would pop up there.

No, it's interesting because there's a kind of an instructional intention of using the technology, and then there's a kind of an evaluative intention of using the technology. And I think there is a difference in those intentions as well. So I think my worry is more from the harvesting of the evaluative data.

Well, about the evaluative, you know, I like saying that the assessment tail is wagging the pedagogy dog, meaning teachers are teaching for tests and for examinations. And exactly, you know, I cheer them to do that if it's my kid who needs to pass the exam, but then what do kids actually understand? Because it's coming back to our beginning. It's not grounded.

Exactly.

No, 100%. And, um, thank you, Dore. This is, I mean, this is deeply fascinating work. And as I said at the beginning, I think we've got into the, the weeds there of some of, you know, your research and the specifics. But actually, if we kind of zoom out slightly, there is something really important happening, which is a shift towards this more embodied, enacted, embedded, extended, all of those E's that you were talking about at the beginning. Way of viewing learning, and hopefully the pedagogy and the instruction can kind of take heed of all of that and begin to shift away from this kind of neck-up, conceptual, abstract, map-only, no territory experience. So thank you, João. I so appreciate the chance to chat to you.

Thank you very much. Really enjoyed this conversation.

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