Academic journal article The Mathematics Enthusiast

Clockface Polygons and the Collective Joy of Making Mathematics Together

Academic journal article The Mathematics Enthusiast

Clockface Polygons and the Collective Joy of Making Mathematics Together

Article excerpt

(ProQuest: ... denotes formulae omitted.)

Introduction

Mathematics is often treated as an "individual sport." A strong image persists in our cultural imagination of the solitary mathematical genius, engaging with abstract, disembodied entities in Platonic realms. This image persists in spite of strong empirical evidence that professional mathematical scientists work in teams and invoke body-based resources when tackling difficult problems (Gonzales, Jacoby, & Ochs, 1994; Ochs, Gonzales, & Jacoby 1996); that the basic conceptual infrastructure of mathematical thinking, like all thinking, implicates the body and lived social experience (Frejd & Bergsten, 2016; Lakoff & Nuñez 2000), and that learners and teachers think through communicating socially (cf., Sfard, 2007) and through embodied acts that provoke, accompany, or even precede learning (Goldin-Meadow, 2003; Cook, Mitchell, & Goldin-Meadow, 2010).

Historians and philosophers of mathematics have also made the compelling case that disciplinary and representation-embedded thinking is social, at least in the sense that mathematical thinkers must generate a "dance" of disciplinary agency (Pickering, 1995) in their work, simulating the presence and response of the field and of the subject matter itself. In this way, mathematical actions and conjectures have features in common with fundamental aspects of other collective "forms of life" (Wittgenstein, 1953). In particular, mathematical actions are human actions and inherently social, sharing the "dialogic" nature of all "utterances" (cf, Bakhtin, 1982) in that they are posed in anticipation of a response - in this case, a disciplinary response. Analysis of the historical record substantiates this conception of mathematical activity as a speculative, dialogic exploration of the potential of models and representation systems (MacLeod & Nersessian, 2018; Gooding, 1990), and theories of learners' activity increasingly recognize this pattern of "coaction" (Moreno-Armella & Hegedus, 2009; Moreno-Armella & Brady, 2018), particularly where technology can support and illuminate processes of learning in interaction.

Moreover, collective cognition and socially distributed inquiry has begun to affect the working practices of professional mathematicians in potentially transformative ways. Fields Medal winner Timothy Gowers posed the question of the feasibility of what he described as "Massively Collaborative Mathematics" in the journal, Nature (Gowers & Nielsen, 2009), and in his widelyread blog. The question was ambitious: could a collective mathematical entity prove new theorems, where individuals had failed in the past? Since then, thirteen "Polymath" projects have run, involving medium and large groups, sometimes including non-professional mathematicians and hobbyist participants, have sprung up in direct connection to Gowers's blog alone (Cranshaw & Kittur, 2011). Each of these projects has followed a pattern where a number of topics are proposed, and discussed in an open forum on Gowers's blog. When a critical mass of speculative comments gathers, a date is set, and a concentrated collective effort begins (Cranshaw & Kittur, 2011). This approach has yielded published papers and proved theorems. (e.g., Polymath, 2014). Polymath is thus not just a speculative experiment: it is an emerging new way of doing mathematics. Moreover, the connections between Polymath activities and the recent advances on the chromatic number of the plane by amateur mathematician Aubrey de Grey (de Grey, 2018) suggest that collective modalities may also open the way to broader participation in mathematical work.

Given the success of the Polymath Projects and the strong and increasing emphasis on the social dimensions of mathematics thinking and learning in education research, then, we are led to ask whether there are ways to bring these approaches into classroom-based learning. As participants in Polymath have learned, there are dispositions, skills and expectations that favor productive, creative collaboration; and students in our classrooms may benefit greatly from developing those skills and that experience. …

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