# The Harmony Learning Community Blog

News, Updates, Program Recaps, and Homeschooling Information

### Invariants

(3/8/2018) During our course on Invariants, the eight-year-olds spent most of the time exploring the Euler Characteristic (click here for details). This report is essentially a list of other activities we did to start or finish our sessions. An invariant is something that never changes. Piagetian Conservation Tasks – We did every activity in this article: http://www.cog.brown.edu/courses/cg63/conservation.html. Conservation tasks basically…

### The Euler Characteristic for Eight-Year-Olds

(Jan. 25 – March 8, 2018) The five students (plus one occasional visitor) in our math circle spent six weeks doing like mathematicians do – savoring a math problem, learning it in depth before any attempts to solve it. I felt like Andrew Wiles in his decades-long work on Fermat’s Last Theorem. BEFORE THE COURSE: THINKING ABOUT IT I didn’t…

### The Skills We Bring to the Circle

Experience the joy of discovery.   • the sense of humor to say “I want to whisper the function in your ear because I’m too lazy to do the math.” • the honesty to say “I want to whisper the function in your ear because I need help with the math,” and quite plainly, “I have no idea.” • the…

### The Platonic Solids: Never Mind

Developing collaborative problem-solving through the joyful art of mathematics.   Since Euler codified the concept of functions, we started with a function machine. It was challenging to get past the design phase to play with the math, but after applying a chimney to a fish, we did. The first rule was quickly deduced: “It makes it half; it takes half…

### What We Did in Math Circle, and Why We Did It

It’s fine to not solve a problem.   “I’ve got it!” said X, J, M, N, A, G, and D.  Almost everyone came up to the board (excitedly and without invitation) to demonstrate a solution to the famous Konigsberg bridge problem.  “Uh, wait…  I just had it,” said X, J, M, N, A, G, and D.   Each returned to her/his…