Hello everyone, I’m really excited to talk about our research on “constructing embodied
algebra by sketching” I am Saquib, and my co-authors are Rubaiat,
Li-Yi, Gloria and Deb.
The work was done at the MIT Media Lab, in collaboration with Adobe Research and UC Irvine.
I want to start by talking about the typical trajectory of learning and using mathematics.
In our childhood, we usually start math by activities with tactile shapes, drawing, and
manipulating them using intuitive interactions and gestures.
Then there’s picture based math: for example, working with small collections of objects.
Then the jump to abstract math is almost arbitrary, we now deal with symbol manipulation with
seemingly arbitrary rules.
Plots and visualizations are important to make sense of this paradigm, but they usually
do not directly connect back to embodied math paradigm
And of course we move on to more complex symbol manipulation later on
If you notice carefully, you will see that in our current culture of mathematics, doing
abstract math and data analysis is often about moving between these different representations.
The bigger question is: Because the embodied math activities and interactions are intuitive
to most humans, can we bridge abstract math and embodied math,
by using these fundamental interactions and concepts?
So how do we do that?
In this paper, we take a first step to create a design framework for what we call “embodied
algebra”.
We have also implemented the framework as a sketch interface where users can draw iconic
elements and functions to define algebraic relations using a visually descriptive medium
like sketch.
We chose sketching as the primary medium because this medium is flexible to convey complex
concepts we can iteratively work on any sketched structure
and they are also widely used in the sciences, engineering, and math to communicate concepts
On the other side of the spectrum, there is the composition of abstract algebraic expressions.
We would want to be able to construct and deconstruct abstract formulations in the sketch
interface, which is something that we currently do on paper or using computer algebra systems.
and ideally we should be able to observe the effects of construction and deconstruction
in some intuitive manner
the act of sketching creates iconic elements however, we somehow have to be able to define
variables and functional relationships by sketching too.
Together, the iconic elements and functions create an embodied math representation
And by manipulating the iconic elements and functions, we generate algebraic expressions.
So, how do we do this?
In a nutshell, we have designed a few brushes that let us draw objects, collections, and
functions.
The iconic element brush can be used to create different categories of objects, like circles,
rectangles, and stick figures.
Once we have the iconic elements, a list brush lets me draw lists, which can count the number
of elements inside it.
Since these are brushes, any collection or function can be drawn in any shape or form.
In this case, I am basically drawing mathematical sets in the form of houses.
Mathematical functions can be drawn using a function brush.
Here, I am drawing a function container, in the form of a street that contains these houses
and adds their outputs.
All elements are manipulable.
So, an algebraic expression can be formed by bringing the lists inside the function.
An important characteristic of each of these compositions is that they can be explored
in three layers of mathematical abstractions.
In this case, we can see that the function can be displayed as the iconic representation
or the symbolic formula representation.
We also demonstrated that using the brushes shown so far, we can cover a good portion
of the American Common Core school-level mathematics.
What that means is, a lot of our abstract, symbol manipulation exercises could be converted
into drawing-based math exercises.
In this example, I demonstrate how we can calculate pi using Buffon’s needle formula.
The formula relies on how many times a matchstick has crossed some lines, and derives pi from
a probability calculation.
Every time it crosses, I draw an object inside a list representing the line.
And instead of calculating the probability using a symbolic formula, here we have a visual
drawing activity, which is also the algebraic formula itself.
This is an example from physics, an example of a transformer coil and how we calculate
voltage as we add more turns in a coil.
This example shows that books and research papers can be made more interactive and personalizable,
based on how users draw on top of them.
We can create linked documents and books, in which sketching one entity updates all
other linked sketches and formulas.
This is an example produced by a neuroscientist.
It is a simulation of neuron action potential mechanism.
The visual formula drawn using the brushes are quite visually descriptive of the process
itself.
I invite you to read the paper to explore the design framework.
So send me any question or comment at this email address if you are interested.