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Criteria for any Math Circle class

Emily's lesson satisfies the properties of a solid Math Circle lesson:

The material was taught in an organized fashion
She built in individual work and collective participation into her lesson
The material was accessible to the audience; it required no more than high school algebra and a little bit of geometry
The material was appealing
She asked students to make mathematical observations and construct proofs and conjectures
She challenged students and made them confront the difference between cursory knowledge and true understanding

What makes the lesson on Platonic Solids exemplary?

Math Circle's philosophy is to immerse students in a participatory mathematical environment. Emily's lesson exemplifies this philosophy by the degree to which she engaged her students. She asked students to:

Think carefully about definitions
Prove, disprove, and make conjectures
Explain their work to each other
Answer questions raised by work of other students

Emily used the student's language whenever reasonable, and she asks students to explain their terminology to their peers. All students were active in her class because she communicated firmly that she wanted them to participate and built in a variety of ways to contribute: volunteering an answer in front of the class, individual work, group work, discussion. During the group work session, Emily asked the discoverer of the double-tetrahedral example draw this example on the board.

As a result, the students were doing some sort of mathematical exploration almost the entire lesson. This level of intellectual activity kept the students awake and engaged throughout the lesson.

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