Exploding Dots

4.2 MATH CIRCLES for Students and for Teachers

A Math Circle is a sustained extracurricular mathematical experience that offers participants, be they students, educators and/or general mathematics enthusiasts, the opportunity to engage in joyous, playful, and deep mathematical thinking and exploration. Math Circles come in a variety of styles and approaches. Some are based on practicing the art of problem-solving in explicit ways, others explore unsolved research challenges in a loose give-and-take of organic, but directed, discussion, and others adopt approaches a combination of the two.


One can learn about Math Circles and their philosophies (and find a circle near you) through:


SIGMAA MCST: A special interest group of Mathematical Association of America:  www.sigma.maa.org/mcst.


THE MATH TEACHERS’ CIRCLE NETWORK: www.mathteacherscircle.org.



Exploding Dots in a Math Circle


I have used EXPLODING DOTS many times as topic of discussion for a Math Circle session.  This course, with the order of the content and the words I use in the videos, really does match the approach I take with a live audience. Of course, I’ve necessarily pushed all the ideas forward in this online course in a linear way. But with a group I would pause, or step back, often and let participants discover the next steps, ask interesting and new questions, and direct the nature of the exploration.


I start by presenting the ideas of:



and then simply ask: What are these machines doing? The content of



naturally comes forth in the discussion.


Now I have a choice:


If it is appropriate to offer an “interactive lecture,” especially if it is one that is meant to be tied to classroom thinking, I lead the discussion through the ideas of LESSONS 1.4 – 1.7.


However, if this is a session for pure discovery and exploration, then right after 1.1 and 1.2  I introduce the machine of


and ask some of the questions presented in LESSON 3.1. (I usually present LESSON 1.3 in the first style of circle too!)




Of course, Math Circle groups can come together multiple times and explore the mathematics of “wild” base machines of all types. Each of the questions in:


LESSON 3.2, 3,3 and 3.4


can occupy a group familiar with base-machines for many hours. Be careful with the need to offer the group some chance of attaining  accomplishments (be it small) when exploring a research problem that is open and has not been solved at all!

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