Talking Stick Math Circle

An invariant is a quantity whose value never changes no matter what you do to the operation under consideration. For example, when you shuffle a deck of cards, the number of cards in the deck remains unchanged. Mathematicians consider invariance one of the most important concepts children need to know as they go through their math educations. In this course, we’ll engage in problem solving, games, flowcharts, storytelling, and a hands-on exploration of the Euler Characteristic to search for and understand invariants.

Neuroscience has provided empirical evidence of what we intuitively knew all along: that counting on your fingers enhances learning. The discipline of embodied mathematics employs gesturing and physical interactions with the environment to develop conceptual understanding and to facilitate articulation of mathematical concepts. Year after year, young students come into Math Circle with the idea that mathematics is all about quick computation and nothing else. This course will open students’ minds to the reality that math is about more than numbers and can be explored with more than a computational approach. We’ll use our bodies and surroundings to examine symmetry, 2D and solid geometry, equivalence, measurement, spatial reasoning, and arithmetic computation.

What are algorithms and how do they drive our culture? We’ll examine the Google page-rank algorithm, Cathy O’Neill’s National-Book-Award-nominated Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy, whether random number generators are really random, the mathematics behind “fake news,” the Euclidian algorithm, and much more. These topics will provide context for a study of the algebra behind algorithms: sequences of instructions that usually involve variables and (algebraic) expressions. These expressions can be organized logically into matrices, programs, flowcharts, etc., to produce solutions to well-defined problems. Then, of course, we will debate the appropriateness of labeling problems “well-defined.” While we’re at it, we’ll delve into some statistics and number theory and then compose some algorithms of our own.

Students should be familiar and comfortable with variables, although the ability to manipulate them is not a prerequisite. While this is a students-only course, interested parents and guardians are invited to participate during the final session.

Beginning with non-mathematical functions, then function machines, and finally – probably – ending with algebraic expressions and graphing on the coordinate plane, we will have fun discovering what a function is and how to express it in various mathematical ways. I say “probably” because Math-Circle is student-directed and the students’ mathematical interests could take them elsewhere once I present the initial premises.

Using both unsolved problems in mathematics and the book Avoid Hard Work, we will explore general problem-solving strategies. The goal is for students to move toward an understanding that (1) the pursuit of mathematics is not the same as memorizing a bunch of math facts, and that (2) there are ways of thinking that help to tackle a problem. The students will create their own problems for others to solve.

*The schedule may shift by one week to accommodate spring break.

In this course, we will attempt to discern whether math is about absolute truth or relative truth, and along the way will discuss how math is a creative endeavor. Using the book Camp Logic and other sources, we will engage in logical reasoning, induction, and proof to explore ideas including invariants and isomorporhism.

Our math circle will explore the storied history of Fermat’s Last Theorem and some of the underlying mathematics, such as Pell’s and other Diophantine Equations, and Fermat Proofs for Specific Exponents. We will discuss specific work by mathematician Sophie Germain, as well as the drama involved in Andrew Wiles’ Fermat proof.

According to Wikipedia, “Fermat’s Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n strictly greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics Prior to its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem”, one of the reasons being that it has the largest number of unsuccessful proofs.”

Led by Rachel Steinig, students will make ropes dance via specified moves. Rational Tangles was invented by one of Rodi’s favorite mathematicians, John Horton Conway, a living mathematician whose life we’ll discuss in the course. Rational Tangles is rich in mathematical content, including algebraic thinking, transformations, symmetry, classification, geometric equivalence, the order of operations, and some of the more interesting arithmetic of fractions.