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Dor Abrahamson discusses how Emerging technologies bring sensorimotor interaction into math learning. Ecological dynamics explain how students develop schemes and use symbolic artifacts for mathematical reasoning.

Image courtesy of interviewee. December 21, 2023

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This video video discusses the inherent capacities of humans as sophisticated mammals, shaped by evolution to adaptively interact with their environment. It emphasizes that mathematical thinking is a natural extension of our innate abilities, not an alien or separate skill. The speaker explains that mathematics has evolved as a useful tool for various cultural functions and advocate for understanding mathematical concepts through active engagement and interaction.

A key focus is on how children learn mathematics through active, embodied experiences, using examples like the "mathematics imagery trainer," a technological tool designed to assist in this learning process. This video video instrument allows children to perceive and interact with mathematical concepts, fostering new ways of thinking about and grasping these ideas.

The importance of physical movement and perception is highlighted as necessary for understanding mathematical relationships, asserting that our cognitive processes are fundamentally connected to our physical actions. The speaker draws on concepts from cognitive science, particularly the idea that perception serves action, noting that our senses are geared toward facilitating movement and interaction with our surroundings.

The discussion includes the significance of creating pedagogical conditions in educational environments that allow children to develop their mathematical understanding much like they would learn physical skills, suggesting that learning should be socially and culturally situated in a way that mirrors the work of historical mathematicians. The overarching goal is to foster a deeper engagement with mathematics through exploration and active problem-solving, ultimately leading to a more intuitive and grounded understanding of mathematical concepts.

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X-TIMESTAMP-MAP:LOCAL=00:00:00.000,MPEGTS=900000

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You and me and everyone around us we're,

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pretty sophisticated mammals.

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Over millions of years, evolution has shaped us

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to have these innate capacities

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for interacting adaptively within our ecology.

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We have these perceptual faculties,

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these sensory motor skills

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and cultural capacities to socialize

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and take part in, in civilization.

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Now, mathematics is not some bizarre outcropping there.

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We don't have a special math lobe.

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We don't have some, you know, extraterrestrial talent.

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It's just another thing that we do.

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And in certain, certain cultures,

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under certain circumstances, occasions occurred that,

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made the, the thinking mathematically, thinking,

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quantitatively noticing,

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certain relations within the environment,

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and writing them down, documenting them,

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creating a whole discourse around them.

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It, it served a certain purpose,

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whether it was administrative, logistical,

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litigious and et cetera.

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So this thing evolved,

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but it's not in any way as strange from, or like bizarre

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or transcendent from all the other stuff we do,

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like grasping a glass or drinking or, or what have you.

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It's just another way of perceiving a situation in order

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to increase our adaptive, capacity to interact with,

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with the, with the environment.

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When, when we speak about things like mathematical objects,

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we're not being too metaphorical.

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Well, obviously, you know, what is a, a,

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linear function, et cetera.

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What is an asymptote? What is a proportion?

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It's difficult to touch them.

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They're not necessarily tangible.

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And yet the very fact that we speak about them as objects,

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it's really, demonstrating for us

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that our brain can only behave that way.

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We can only think about things so much so that,

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a colleague in, in, in, in, around London,

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speaks about thinking like, it's not thinking, thinking

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that's a lamb forest.

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So when we, when we think mathematically,

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we are imagining things.

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And so they are not, like everyone talks about abstract

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abstractness, and then, and yet we call

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that the, the a word.

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You, you're not allowed to say that word in the lab.

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You have to put a penny in the pizza box, right?

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We, it is true that colloquially, we all use these words,

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but actually phenomenologically evolutionarily and,

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and nearly, we're always seeking to grasp things.

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And in order to grasp them, we need to figure out

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what they are in order to act upon them.

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So, so the entire work that we do in the lab,

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in the Embodied Design Research laboratory is founded on

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the premise that

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Mathematics is just another way of thinking.

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It's just another way of engaging with,

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with the world in collaboration with, my,

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my colleagues in, Urich University in, in Holland, and,

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and in, in other places, working with my colleagues here

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and students, we are developing, let's call this,

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technological architectures.

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We call it, the mathematics imagery trainer.

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And now this technological architecture interactive,

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it's been deployed in all kinds of media as long

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as there's some kind of way of interacting with, with the,

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the com digital computers, be it remote sensing or tablets.

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But here's the deal. We wanna create conditions for children

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to learn how to move in new ways.

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We actually, our, our lab's motto is learning is

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moving in new ways.

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And, what we mean by that is that in order

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to understand a mathematical concept, you need to grasp it.

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You need to come to perceive a new object.

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So here's a little demonstration of what we mean.

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And even though maybe the connection

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to mathematical won't be entirely clear,

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I think it does bring through what we mean by perception,

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evolving through action for action.

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So we have children working in some interactive medium,

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some digital medium, with an environment

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where there are two, what they call cursors.

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There are two little cursors on the screen where one

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of them can move up and down.

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You can move it that way.

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The other one, you can move that way.

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And these can be children who are

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before the age where they know about Cartesian,

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coordinate systems, graphs or Y axis.

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You're just moving this guy there and this guy there.

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And when you start out, the screen is red,

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and we just say, do what it takes to make this,

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you know, make the screen green.

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So the child tries moving this one, it's red,

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tries moving this one, it's red, tries moving both it,

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but, and suddenly it's green.

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Now, the reason it is green is

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because we have set things up on a console so

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that in this particular study,

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I mean, it could have been anything.

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Let's, let's just imagine this point here

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as my, like the origin.

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If the one cursor, it's height above here

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and the other cursors, it's just if these two distances,

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okay, that's like the, the y intercept

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and the x, if they are related by a certain ratio,

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let's just say one per two, to make things easier,

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if we have them this way, say, you know,

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a centimeter and two centimeters, 10 centimeters

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and 20, et cetera, the screen will be green,

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otherwise it'll be red.

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So the child sort of stumbles upon it and it's green,

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and it's like, now what?

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No, keep moving and try and keep it green.

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And they're like, I need to go.

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Now, to make a long story short, at some point

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we start noticing that they're moving much better,

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much better than before.

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Now at the same time.

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And as they are now being able to move in green

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and notice that my hands are all the time keeping, you know,

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so 10 per 20, you know, 20 per 40, et cetera.

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At the same time, we are using non-invasive eye tracking,

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goggles to see where they are looking on the screen.

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And we see something fascinating.

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If beforehand, they're trying to move this hand here

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and looking at this hand here and looking back

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and looking between the hand,

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suddenly they're looking here on the screen

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where there's nothing, there's not contour, no stimulus,

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nothing at all, just a blank screen.

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They're looking here suddenly.

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Now, why are you looking there? And we ask it,

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how are you moving it in green?

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And these are kids, they might be like on a vocational

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track, they might be very young immigrants.

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And it, it's not like they have some special mathematical,

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just like you and me, just any other person.

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They, so some of the kids tell us, and,

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and it varies what they say,

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we were looking at the line, the line.

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It's like, wait, what, what line?

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We're looking at this line. And later,

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later we see this in the eye tracking.

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They're looking at the what line.

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They say, we are looking at this diagonal line.

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Diagonal line. Where is that?

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Oh, so they, once they got green, these clever kids,

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you know, from the mouths of babes, they are,

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imagining this line.

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It's a diagonal imaginary line.

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Now, try, imagine with me, right?

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This diagonal line as I'm moving my fingers,

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please keep your eyes again on the line sub tended

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by my indexes, indices, index fingers.

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And, and, and you'll notice that it is

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as if I'm moving the line,

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so it's no longer moving the left hand

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and moving the right hand.

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And my gosh, it's like one of these, things, right?

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But rather, now I've invented a new object, this Dagon line.

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Now you don't think I'm crazy.

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I don't think you're like, we can talk about things

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that aren't quite there, but we can imagine them.

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So there's this line that I'm moving.

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I'm not losing, moving my left hand and my right,

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and my experience is now that I'm seeing,

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imagining this vivid imaginary line

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and moving it, I'm moving it across.

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Uh, and that is enabling me to keep the screen in green.

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It turns out that our evolutionary neural architecture is

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such that when we are challenged by moving in a new way

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to operate some object in the world,

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and here it is an object in the world,

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even if it's not tangible, our inclination is

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to see the world to perceive new things there to witness

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and, and, bring forth new like geal, like a gestalt,

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like a con, a perceptual configuration.

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But it's a perce, it's a perception for action.

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So this thing popped up,

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and the remarkable thing about our brain as,

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Donald Merlin, calls it like, we not only move,

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we can also think about our moving, and we can describe it.

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Other children were saying, I'm looking at this point.

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And they're literally talking like that point there,

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you might call the Cartesian ordered pair, like, you know, X

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and Y and they say, it's like, I'm making the,

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I'm making this rectangle bigger.

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So, so they, they found it green,

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and now they're moving there.

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So they're looking not at diagonal.

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These other kids are looking here

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and kind of keeping what we would call the aspect ratio

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of this growing rectangle a constant.

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So they figure out some kind of new,

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we say like a new ontology, a new thing

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that come forth from the world, a new affordance

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that ecological psychologists would call it,

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that enables them to have a grasp on the world.

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They are gripping the world in a new way that enables them

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to operate this world.

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And the point is that we can create these kinds

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of interactions where we figure out how to

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instantiate a mathematical concept in a way that,

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that the perceptual solution, in this case, a line

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or so on, the perceptual solution

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to the motor control problem constitutes the, the, the, the,

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the embodied and act of grounding of the concept.

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So we work towards understanding, say proportionality.

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So we take this idea of proportionality as designers,

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and we instantiate it as the perceptual solution

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to a motor coordination problem.

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You know, in hindsight, we can say, yes, all

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of these ideas were there in the 19th

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and the 20th century, et cetera.

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But that's hindsight in what's the opposite

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of hind and foresight.

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Like when you're in the site itself, you don't, when it's,

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it's not necessarily in your site.

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So, but, but looking back, we see, the work of, say

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of Jean Pge,

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who's the famous Swiss cognitive developmental psychologist

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operated in Geneva, thought of as like the, the father

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by now the grandfather,

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or great-great-grandfather

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of cognitive developmental psychology.

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And his theory is called genetic epistemology, where he said

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that it's, it is not the case that we have representations

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of the world in our brains,

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or rather, we are born with the innate adaptive capacity

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to learn as much as we need about the world

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through interacting with the world.

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So that we're, when we are in situations that we perceive

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as similar enough, we already have ways of, of,

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grasping the situation so as to act upon it.

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So babies fumbling, try until they figured things out.

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Uh, the eye teach, the hand teaching the eye,

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and later the eye teaching the hand

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as we see similar situations.

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Uh, however, pge he had the,

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what he called the stage theory,

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and he thought that at a certain point we sort of

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graduate all these, you know, childish, infantile,

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sensory motor kinds of things.

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And now we have this kind of concrete form where we can sort

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of think through situations.

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We don't really need to interact with the world

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or with our body in a, like a, in a evidence, sort of,

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overt sensory mode away

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because we already got the ideas.

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But, famously he talked about things like conservation,

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like, you know, the,

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the famous PGE pouring from one cup to the, to the other.

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One is narrow. One is,

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why do we have more water, less water, milk?

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But we need to figure out these conservations.

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We believe that, that like Kennedy, he was right,

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about the sensory motor interaction,

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but that it's a lifelong sensory motor

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exploration and learning.

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And we know indeed, from, from testimonies of,

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of top mathematicians like Terrence Tao

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or historically from, Andre Pore

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and from scientists like Albert Einstein

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and others, that mathematical thinking is completely,

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it's this embodied enact of imaginary engagement

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with the world, the grasping for trying

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to get this new object that only later you sort

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of write it down and symbolize it.

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Terence Starr writes to us, he's very generous in

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documenting his experiences online.

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He says sometimes he, he, he, he, he literally has

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to lie on the floor

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and roll around in order to think,

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through some partial differential equation.

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Then he adds, and by the way, I, you know,

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he says in my mathematical proofs, I don't add, I don't tell

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that that kind of rolling on the floor story, but, but,

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but that's actually the way I think through it.

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So, we look for more to more radical, theories such

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as the work of, Francisco Varela, Chilean,

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biologist, cognitive, scientist

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and his, collaborators and, and, and students.

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Their work is called activism or an act of cognition.

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And people like Dan Hutto writing about radical

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and activist cognition with, with whom I collaborate,

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who say, well, look it there.

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Historically, I say hindsight, similarities

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to pge, but radical.

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And all the way through our capacity to,

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to perceive the world came to us evolutionarily only

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because it served our action since we were, you know, single

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of a cell or these simple organisms.

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It really helped us to know what we're going towards, so

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that we know if to, you know, to engage or to fight

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or flight, et cetera.

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Um, and so our senses are always in, in the service

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of our motor activity.

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So anytime we, we look, we see,

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we perceive the motor is always inherent

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because the perception is for action.

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So whenever we talk about perception, in a sense,

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we're talking about perception for action.

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So even if you think about some mathematical concept

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that you're imagining and say to yourself,

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it's static now.

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Well, maybe it doesn't seem to like anything's moving there,

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but the very fact that you can perceive it is

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because your brain is preparing you

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to act upon it, to grasp it.

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Now, the second principle is that,

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what we call co cognitive structures, some kind

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of stable neural, networks that then are, you know,

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the stuff that concepts will be made of, such

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as mathematical concepts.

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These, cognitive structures,

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they emerge from the recurring patterns

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of sensory motor interaction that enables,

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perception to, to, to guide our, our actions.

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So if I'm a child, an infant, I'm learning, I'm trying

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to exploring how the heck to

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Care until something, something proves,

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something proves efficient,

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then this little grasping thing will be repeated, repeated,

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so that next time I see this glass, I'm already prepared

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to grasp it in certain way.

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And just as I'm learning to grasp a, a, a, a glass here,

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the same thing will, will obtain in the case

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of mathematical concepts such as a fraction

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or linear function, et cetera.

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So that is what we're trying to create here.

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We are trying to create the pedagogical con con,

15:18.575 --> 15:20.925
conditions so that children will learn

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to grasp new mathematical concepts in ways

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that emulate the historical cultural, work of, of,

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mathematicians perception is serving action

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with the most simplest objects.

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So it is doing so with, with

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what we might call mathematical objects,

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because, essentially in terms of our epistemology,

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in terms of our capacity, the, the whole neural,

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networks that we have, that we build in terms of our,

15:50.145 --> 15:53.125
as we hear from the testimonies, people's raw phenomenology

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of mathematical thinking, it'll always be about things.

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It'll always be about engaging with objects

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or quasi objects that are there in the world.

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We know also from, from, cognitive neuroscience

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that when people are engaged in, in mathematical reasoning,

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their, their motor or premotor cortex is firing.

16:15.665 --> 16:16.925
And it's not just a byproduct.

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If you sort of arrest gestures

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and motion, it affects detrimentally the capacity

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to think mathematically.

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So we are inspired by an activist of philosophy

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and cognitive science, to both create new,

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theories that give us a handle on what mathematical is,

16:37.385 --> 16:39.765
and through that, to inform the design

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of learning environments that will create the conditions

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for children to experience

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and to, as people in, in Urich say

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that they will experience a guided

16:49.285 --> 16:51.205
reinvention of mathematics.

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Like we don't, it's not like we, you know, this idea

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of discovery-based mathematics, it's not

16:56.005 --> 16:57.565
as if you put a kid in a room

16:57.565 --> 16:59.885
and say, okay, reinvent, calculus,

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I'm gonna grab a coffee, I'll be back in two hours.

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Let's see where you are. No, but,

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but rather we're, we're looking to create conditions

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where a child learns mathematics just like she would learn

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to, to ride a bicycle or

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to flip a pancake or anything of that.

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There are artifacts out there, there are resources,

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there are tasks, and there's a, a,

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a human who's guiding them to move and grasp

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and interact in, in, in new ways.

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So we're taking an activist, philosophy

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and extension, implementing it in the form

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of educational theory

17:32.065 --> 17:34.765
and practice creating both the artifact that

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and technologies to enable that,

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and the methodologies such as eye tracking

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and various, computational, analytic,

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methodologies in order to make sense and predict

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and, the evolution

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of mathematical insight even as it is happening.

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So we are, creating a whole variety of

17:54.565 --> 17:57.005
what we call the mathematics imagery trainer,

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which are these interactive environments

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where children learn to, mathematical concepts

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by first developing new perceptions that enable them to

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coordinate bi-manual action.

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So, I'm working with collaborators in various,

18:13.925 --> 18:16.205
universities such as Urich in Holland,

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but also, doing work on, accessibility for,

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for blind and visually impaired students in collaboration

18:23.955 --> 18:26.965
with the University of Colorado and, St.

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Louis University, labs over there.

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Um, and we are, looking to see

18:32.905 --> 18:36.165
and document the micro process of coming

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to grasp a situation, a new way.

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And we, use eye tracking to see what are the

18:42.365 --> 18:44.805
constructions that children are making from that,

18:44.905 --> 18:47.125
and interviews and action logs,

18:47.305 --> 18:49.845
and taking all of this multimodal information,

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analyzing it in order to, characterize across all

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of the students the microprocesses

18:57.265 --> 18:59.925
of going all the way from learning to move in a new way

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and see it in a new way to start talking about it.

19:03.085 --> 19:06.925
Because language kicks in that it is one thing

19:06.945 --> 19:10.125
to ground a mathematical concept in a new coordination.

19:10.595 --> 19:13.365
It's another thing to take that to the next sort

19:13.365 --> 19:17.205
of conceptual, linguistic, interaction,

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with other humans, documenting it,

19:19.635 --> 19:21.125
symbolizing, formalizing.

19:21.625 --> 19:24.845
And we do that by like supplementing

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or interpolating into this working environment,

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various mathematical instruments, such, such as in the case

19:30.685 --> 19:32.485
of moving that way, we might put a grid,

19:32.745 --> 19:34.965
and then kids start, I see.

19:35.045 --> 19:36.805
I go up on here and two, one

19:36.805 --> 19:39.485
and two for every one I go up, I go two across.

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And that language comes sort of out of nowhere

19:43.275 --> 19:45.965
because now that they have these various grids,

19:45.965 --> 19:47.885
these various instruments, they, they kind

19:47.885 --> 19:48.925
of grasp onto them

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as giving them better purchase on the situation.

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It's where, thinking becomes distributed over the

19:56.885 --> 19:59.605
environments, over these culturally historical artifacts

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like a grid, which help us think, with the environment

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and to augment our, our, our reasoning skills.

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And then those things become, measured and marked.

20:11.625 --> 20:12.885
So we move on to paper

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and to what might look like more normative mathematics.

20:16.425 --> 20:18.565
But once you are on paper, it's not

20:18.565 --> 20:20.085
as if suddenly the world is static.

20:20.515 --> 20:23.125
It's just like if, if, if, if we read a novel

20:23.665 --> 20:26.685
and we're in this imaginary world, it's kind of silly

20:26.705 --> 20:29.405
to say, yeah, but, but the letters, they're static that,

20:29.545 --> 20:30.685
you know, stories are static

20:30.685 --> 20:32.605
because no, that, that's just a way

20:32.605 --> 20:34.205
of encoding the narrative

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so we can reanimate it into something that is lived

20:38.105 --> 20:39.805
and, and sensorial.

20:40.265 --> 20:42.925
Uh, so that's what we're trying to do with mathematics, so

20:42.925 --> 20:44.685
that when you get to the point where you're doing

20:44.785 --> 20:47.965
or doing the math, as teachers might say, you are doing it

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with, with the meaning that is evoked by, the perception

20:51.885 --> 20:56.285
and for action that these, that, that these symbols,

20:56.385 --> 20:57.885
encode for you subjectively.

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