Psychology and Leadership in Mathematics Education

Students coming to MathPath have a lifetime of learning mathematics ahead of them. More importantly, they are likely to assume leadership positions in the field, starting math circles, developing and teaching courses, managing projects, authoring books, organizing programs and departments, designing software and facilitating online communities. This course provides introductory topics in neuroscience, cognitive and social psychology, which participants will find immediately helpful in their current mathematical adventures, and useful as they advance in the field. The experiments we will conduct and analyze are exciting because they help us see mathematics from yet more points of view: human aesthetics, information processing and communication.

The course can be visualized as a progressive jpeg, where short individual experiences contribute to increasingly detailed understanding of overarching themes, visible from the start. This structure supports flexibility, allowing to spend more or less time on topics, or add and remove topics based on the interest of participants. Here are some of the topics I consider including. Note that there are more of them than fit in a week, to have some choice as the course progresses. Each activity takes from three to fifteen minutes, depending on its nature and the activities of participants. I worked out some of the activities to provide a sampler of their types: modeling, critiquing and comparing, self-experimentation, discussions in the course of designing something.

Note this is the grown-up description of the course. I make it a point to expose children, even young, to the deep and fundamental topics, but I realize it needs to happen in appropriate forms and levels. My past teaching experiences, and reviews from course participants, show that I can make it possible for students of many levels and backgrounds to participate in rather advanced work, contributing on their own terms.

Our relationships with quantities

  • Short-term memory capacity: history of algorithms and devices as "assisting technologies." What changed in the last decade and is expected for the next with software and social media development? Activities:
    • Measure your memory capacity
    • Discuss historical algorithms and tools associated with them (from Mayan abacus to Wolfram|Alpha)
    • What do you predict for the near future, and what do you want to do to make it happen? Videos on the topic (Wolfram, Robertson).

  • Subitizing, counting and visual literacy. Activities:
    • Find your subitizing range (statistics).
    • Can toddlers understand exponents? Experimental visual number systems of the last two decades. Critique and compare several. Model your own (paper or computer-based).
    • Is multiplication repeated addition, and why is this question causing "math wars" (Keith Devlin). The splitting conjecture (Confrey) and the counting scheme (Steffe). What is your side in this debate?
  • What color is five? Surprisingly many mathematicians experience strong synesthesia, simultaneous and automatic simulation of sensory pathways while doing math. Some use it in their work. Activities:
    • Explore your synesthesias, while discussing theories explaining it. Do you use your synesthesia? Can you acquire one and should you try?
    • Brainstorm: how can we use synesthesia? Examples of prominent mathematicians and known savants.

Data analysis and visual literacy

  • Information theory's take on the human sense of beauty as data compression. Symmetry, fractals and other patterns as beautiful, and mathematical reasons why.
  • Computer model and visualization tools, and mathematics involved in using them well. Social implications of data mining. Free and open source tools (using some if we have the computer lab).
  • Self-tracking, citizen science and data aggregation. How can you participate, and whether you should (privacy, change of behavior concerns).
  • Mathematical writing and mathematical visualizations.

Everything I need to know I learned from computer games

  • World of Mathcraft: the human need to see progress. DIY U, computer tools and "the big unbundling" (of curricula) as the theme of the coming decade. Data structures and computer models that play prominent roles in this social process.
  • Math is not linear. What is your path? How can you learn differential equations before calculus (short demonstration) or algebra before arithmetic, and should you try?
  • Math game mechanics (I co-lead an interest group on this). Serious games, designing your own, using game mechanics in math learning and math communities.
  • Why do 98.7% of math games suck? The temptation and the corruption of extrinsic game mechanics - and extrinsic design principles behind many modern math curricula.

Where mathematics comes from

  • Where mathematics come from (Lakoff, Nunez) - metaphors and the foundations of mathematics. The use of metaphors in theory-building, problem-solving, software and game design.
  • What is epistemology and why do we care? Comparing and contrasting mathematics with other human endeavors (using particular problems students work out - e.g. an artist, a programmer, a scientist and a mathematician constructing a square). Discovery vs. invention.
  • Meaning and significance of mathematics (using a particular problem). Personal relevance, community authenticity, applications, pure mathematics.
  • Ethnomathematics, and renewed interest in multiculturalism from social networks perspective. Online projects and communities supporting plural notations and approaches (e.g. Intergeo, ATM).

Multiple representations

  • Traditional mathematical representations by area: grids, formulas, graphs, words. Representations and learning/thinking styles, analyzing your style (e.g. spatiovisual vs. sequential). Representation traditions of different cultures and times.
  • Use of multiple representations in problem-solving and proofs. Which match your thinking the best?
  • Computer modeling, dynamic links among representations, CASs (computer algebra systems).
  • Manipulatives for grown-up, 3d printing, touch and gesture computing, multipoint touchscreens and the "maker culture": what mathematics is in high demand and why.