I had an odd extra day in one of my precalc classes last week and picked up my college combinatorics and graph theory textbook for some math to explore. We spent the period playing with graph theory. This fun problem from Play With Your Math was an awesome place to start:
We played with this for a bit, and then we looked at colorings of graphs. How many colors do you need to color a graph of the United States? What about Canada? (Hint: they’re different!) Then we played with the Bridges of Königsberg. The map below is of the Prussian city of Königsberg, with bridges highlighted in green. Leonhard Euler wanted to go for a walk crossing each bridge exactly once, without repeating any bridges. This led to explorations of Euler and Hamiltonian paths.
It was a pretty relaxed class. I’m not sure how much students learned, but it was a fun one-day excursion into some neat math. The best thing about the class was that it relied on so little prerequisite knowledge; students didn’t need to know about fractions or factoring or functions, but could play with new and challenging math on their own terms. I’ve taught Algebra II and Precalculus this year, and I hear from so many disaffected students that geometry is the only math class they ever enjoyed because they didn’t feel behind from the beginning and the math made sense to them. But geometry is an outlier. Most of our math curriculum is designed sequentially in the race to calculus, building on ever more complicated layers of algebraic manipulation.
Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. Which topics would we eliminate? Which topics would we add? Which topics would we teach differently? My current teaching load feels like it is designed to do the opposite; there are so many places a student is likely to feel confused because of something they missed or forgot from a year or several years before. How does that make students feel? Which students who aren’t invited into the math conversation now might be if we approached math in a new way?