This month’s topics:
Crowd calculus.
When two groups of people walk in opposite directions through a corridor, they tend to form lanes consisting of same-directional travelers. But when the two groups move perpendicular to each other, there’s no lane formation. This phenomenon has also been observed in other “active flows” involving colloids, plasmas and cellular automata.
A March 24 article in the Proceedings of the National Academy of Sciences investigates this feature of crowd behavior. The authors (led by Karol A. Bacik of the Massachusetts Institute of Technology) investigated how the precise angle between two groups’ paths affects lane formation. As they describe it, they used a combination of “theoretical analysis, numerical simulations, and stylized experiments.”
The team set up a rectangular 8m $\times$ 6m area in a gymnasium, with 5 gates set up along each of the short ends. A set of subjects was grouped at each gate, with individual instructions to cross the area heading for a specific gate at the opposite end. The authors report the evolution of five “scenarios.” In the first scenario, the subjects were instructed to walk straight across to the same-numbered gate on the other side; in subsequent scenarios they were assigned more and more oblique paths. Here are stills, taken at $t=$4, 8, 12 and 18 from a 22-second video of the juxtaposition of scenarios one and five.
The authors observe some laning when the two paths are at a slight angle to one another, but there is a critical angle beyond which it disappears.
Their theoretical analysis supported this observation. The authors assume that all the “agents” are moving with the same speed, and analyze $\rho_{\theta}$, the spatial density of those with preferred direction $\theta$. The quantity $\rho_{\theta}$ changes both with position and time. They derive a partial differential equation for $\rho_{\theta}$ and explore the stability of its solutions. They determine that a crucial parameter is the standard deviation $\gamma$ of the distribution of directions: The disordered state is stable if $\gamma$ is bigger than approximately $13^{\circ}$. If $\gamma$ is below this critical angle, the disordered state is unstable and collapses into an ordered state characterized by lanes.
Embodied and dis-embodied math.
The creative tension between concrete and abstract, examples and generalizations, is a familiar topic to mathematicians. Danyal Farsani (Norwegian University of Science and Technology, Trondheim) and Omid Khatin-Zadeh (University of Electronic Science and Technology of China, Chengdu) explore what it means for mathematics education in a February 19 article, “Embodied and dis-embodied affordances in mathematics education.” They distinguish between two approaches to mathematics education. The first is more traditional, and uses “abstract symbols and abstract representations of mathematical concepts and constructs.” They term this approach dis-embodied. It’s contrasted with the embodied approach, the growing trend to engage in a process “in which mathematical concepts are represented in terms of body movements or other physical materials in the world.”
Farsani and Khatin-Zadeh cite many references showing that the shift from dis-embodied to embodied has solved many problems in mathematics education. Students are presented tangible and easily understood concepts, making the subject “more digestible,” in the authors’ words. A nice example they give is the representation in calculus of the derivative as the slope of a curve. The derivative’s formal definition as a limit of ratios is dis-embodied, and harder to grasp.
But the authors also observe that some important aspects of mathematics are not accessible to the embodied method, and that an over-emphasis on concrete examples in education may actually weaken mathematical thought.
Their main example comes from algebra. The concept of a finite group can be imparted in an embodied way. No examples given, but they may have in mind illustrating the group of symmetries of a polyhedron by manipulating a material model, or manifesting a small finite group by writing out its complete multiplication table on the blackboard. On the other hand, the important concept of an isomorphism between two groups is more abstract. As they remark, a group of matrices can be isomorphic to the group of symmetries of a geometrical object. Each of them can be presented in an embodied way, but “there is no easily-observable similarity between them.” The isomorphism has no sensorimotor features, it is a purely mental construct, and yet is an essential part of understanding groups.
The authors go on to derive the pedagogical consequences of their argument: “Both modes need to be emphasized in parallel and in a well-balanced manner.” They give no specific details of how this can be achieved, but they remark that currently this good balance “is not observed in our mathematics education programs (at least not in Iran, UK, Chile, Brazil, Norway and China).”
Missing from this discussion is an acknowledgement of the way the modalities of learning vary from student to student. As mathematicians know, what is “embodied” for a topologist may seem very abstract to an analyst, and vice-versa. An analyst can look at an equation with a long sequence of terms and “see” which ones to group in order to prove a desired inequality; a topologist can look at two very different projections of a knot and “see” how one can be manipulated to match the other. These differences in mental perspective are likely not just the result of training, but may well have motivated the choice of sub-discipline in the first place. A body of research going back to Stanislas Dehaene and collaborators shows that different parts of the brain are involved in different aspects of calculation; my feeling is that it is entirely plausible that different children are mathematically enabled in different ways, and that an ideal mathematics education would exploit these differences to give each student the best opportunity to learn. A non-scholarly but very engaging book, Marilyn Burns’s “Math for Smarty Pants” (Little, Brown Books for Young Readers, 1982), expands the idea of multiple mathematical intelligences to include logical, spatial, and linguistic; although it is addressed to youngsters it can be read by instructors as suggesting a variety of methods to reach a variety of students.