Dissertation Title:  Embodied Mathematics by Interactive Sketching

Zoom link: https://mit.zoom.us/j/99803767730 Password: 644883

Abstract: 

The language and formalization of mathematics historically evolved as an interplay between abstractions and their grounding in real world objects and events. The embodied mathematics philosophy posits that our mathematical capabilities are centered around our embodied experiences. Certain elements underlying mathematical thinking are intuitive to most humans without formal training. For example, counting and putting together small object collections, doing simple part-whole analysis, moving along a path etc. They also form the basis for arithmetic and higher abstractions by acting as helpful metaphors. Unfortunately, while journeying into mathematics, many of us lose the natural, intuitive underpinnings of the abstractions and resort to memorizing the rules of manipulation (“symbol pushing”), which are cognitively arbitrary and are severed from their grounding. Wittgenstein pointed out that the rules themselves form mechanics and metaphors of their own, and are often too many layers of abstractions up to establish a connection back to their intuitive grounding and explanations. I propose that one way to reduce this gap between embodied math and abstract math is to come up with unified representations and interaction designs that are useful for both ends of the spectrum.

In this dissertation, I explore  a computational representation in which some of our innate mathematical capabilities and embodied interactions---while being useful educationally---could also provide an interface for abstract symbolic mathematics. Specifically, I develop a design framework, and subsequently, an interactive sketch interface to combine computer algebra algorithms with layers of sketched, visually interpretable compositions. The framework re-imagines some abstract mathematical activities to sketching manipulable structures, enabling the user to form embodied mathematical representations that are personalized and meaningful to them. The challenge in designing such a system is the mapping between sketching, iconic objects, symbol algebra, and functions. The design represents some models of computer and symbol algebra down to three kinds of sketched primitives and two key interactions. These are inspired by formative studies on a survey of scientific diagrams and the embodied cognition literature.

To evaluate the framework and interface, I present examples from the Common Core curriculum, demonstrating how the three primitives and the interactions cover a wide range of exercises from Kindergarten to 8th Grade. Two playtesting studies with children are then presented. One with children in Bangladesh in schools that have a range of math curriculum (a British curriculum school, a remote village school, and an Islamic school or madrasah), and another with children in the US who trained under the Common Core curriculum. The qualitative findings are compared to point out the strengths and weaknesses in the proposed design and its learning curve. An online drawing experiment is also presented that tests whether people prefer drawing descriptive visual forms for mathematical compositions.

Next, I describe collaborations with scientists and mathematicians (in astrophysics, neuroscience, physiology, epidemiology etc.) that explore how the proposed construction methods can reimagine some advanced symbolic math, such as numerical integration and differentiation. The ability to do symbol algebra on iconic representations opens up opportunities for doing mathematics with a wide range of objects and media. To demonstrate such affordances, I present symbol algebra and simulation on some sample objects, such as guitar, subway maps etc. The current scope of the work is limited to arithmetic and standard symbolic algebra. I finish the dissertation by discussing the implications of my exploration for broader mathematics outside of these domains, and what future research needs to happen in the math, scientific computing, and HCI communities for the landscape of abstract mathematics to adapt embodied mathematics.

Committee members: 

Deb Roy (Professor of Media Arts and Sciences, MIT)
Gloria Mark (Professor of Informatics, UC Irvine)
Dr. Rubaiat Habib (Senior Research Scientist, Adobe Research)