# 2010 Circle on the Road Workshop Videos and Slides

## A Live Math Circle: In Front of and Behind the Scenes

### Robert and Ellen Kaplan

The session we propose with the people at the workshop would be on the actual running of a Math Circle class. It would take the form of a class with them as the students. Our topic will be one that some but hardly all are likely to be familiar with – those who know the story we’ll ask to keep it to themselves, and observe how others deal with it. But surely anyone who is a mathematician, and who also wants to hold Math Circles, will have the imagination to feign not knowing: if you can’t put yourself in your student’s position, forget it. The conversation will move between work on the problem and stepping back from the work to look at its dynamics: what is an apt question to ask here; what do you do when inquiry has dried up; when one person is dominating; and so on.

## The Geometry of Math Circles

Math Circles without Math? Well, no–by definition. But one way to think about math circles is as filling in the ‘holes’ in mathematics curriculum: using informal situations to introduce mathematical activities and concepts which are not easily accessed in more formal situations. Sometimes these activities lead back to formal mathematics related to curriculum. Sometimes they lead to mathematics, but not the sort that is studied in school. And sometimes, they simply help build the cognitive structures that are necessary for learning of mathematics, structures that are sometimes easily built with content that is not formal mathematics. The point of the workshop is to explore ways to access a more general perspective on mathematics than is usually obtained through curriculum.

Slides from this talk are available at the following links
Slides Part 1

## Math Circles as a Problem Solving Playground

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### Julia Brodsky

Math circle can serve many goals – such as teaching students to appreciate the beauty of math, to collaborate and exchange ideas, to solve non-standard problems. In this presentation I would like to focus on teaching the problem solving skills for young students. Instead of feeding facts, even the most interesting facts, to the students, we try to help the children to learn about themselves, about their strengths and weaknesses in their problem solving approaches, about potential traps created both by our nature and our education. The presentation will address various problem solving approaches along with examples of the problems for each approach.

Slides

## Re-Thinking the Math in Math Circles

### Mark Saul

Math Circles without Math? Well, no–by definition. But one way to think about math circles is as filling in the ‘holes’ in mathematics curriculum: using informal situations to introduce mathematical activities and concepts which are not easily accessed in more formal situations. Sometimes these activities lead back to formal mathematics related to curriculum. Sometimes they lead to mathematics, but not the sort that is studied in school. And sometimes, they simply help build the cognitive structures that are necessary for learning of mathematics, structures that are sometimes easily built with content that is not formal mathematics. The point of the workshop is to explore ways to access a more general perspective on mathematics than is usually obtained through curriculum.

## Exploring Kleinian Groups: An Activity for Students

### David Wright

Since ancient times, people have been fascinated with visual symmetry, and that has often provided a gateway to mathematics. In modern times, Mandelbrot’s incisive observations on the fractal structure of both physical and mathematical phenomena (including Kleinian groups) has led to a vast renewed public interest in mathematics. The symmetry groups named after the great German mathematician Felix Klein are connected to many advanced mathematical fields and yet they can be explored in a significant way with only a knowledge of circles, elementary algebra and complex numbers. We will show some of the incredible images and patterns that come out of this subject and discuss how to engage in your own investigations of this fascinating subject. Some of these ideas are found in the book Indra’s Pearls by Mumford, Series, and the speaker, but we will also discuss the ideas of several people, both amateur and professional, on investigating these symmetry groups.

The main slides from this talk are available at

Main Wright Slides
The intro slides are available at