After years of frustration, reflection, experiments, reading, failure, and observations I finished a day of school ecstatic and even sent and email saying “I LOVE my job.”
If you read through the posts from my first three years teaching, you’ll see that while I had good moments, and thought that my job was important, I never expressed such excitement over it. I finished Friday full on energy and ready to attack life and do more lesson planning.
Some of my students said, “I can’t believe this is math class, it isn’t even boring.” Another said, “I can’t believe class went by so fast” when I looked up at the clock and realized that I hadn’t even had time to conclude the lesson and pass out the homework.
What happened? I was put in charge of designing a class on logic and proof and finally have the classroom management and interpersonal skills to put my curriculum dreams into action.
We have only had two days of class so far. We’ve solved a couple of great problems. The first is “Someone claims there are 204 squares in an 8 by 8 chessboard. How can you justify this?” The second involved finding the perimeter and area of the Koch Snowflake fractal. If you google it you’ll find that it has infinite perimeter and finite area and even a high schooler can understand why with a basic knowledge of infinite geometric series.
We also looked at instances of great proofs online such as the Banach Tarski theorem which says you can break a sphere into five pieces and rearrange those pieces into two spheres that are equivalent to the first. The pieces are collections of unconnected dots and not possible to construct in the real world but it checks out mathematically. I hear my first ever “what the f?!” in response to a math theorem. I was excited that the kids was so shocked at what type of conclusions one could draw with deductive logic. We connected logarithms(which they had just learned in Algebra 2) to the prime number theorem which says that the number of primes less than a given number x is approximately x/ln(x). And of course Euler’s formula popped up. I majored in math to figure out why this formula works and maybe someone else will too.
While trying to solve the Koch snowflake problems kids were jumping up to the board, moving around the room to share ideas, looking for patterns deductively and inductively and having a good on-task time. In the other class all of the students were able to answer the chessboard problem including the students I’d had before who hated Algebra.
Of course, math is more than a collection of problems. There is vocabulary to know and types of proof to be familiar with and geometric properties and theorems that will appear on the SAT. Next week we are reviewing first semester of Geometry in which students used inductive reasoning to arrive at a number of conjectures about triangles, quadrilaterals and parallel lines and starting our foray into Geometric proof. I worry about killing the enthusiasm of the class by trying to teach proof, a notoriously hated topic. I learned while applying to grad schools about the progression of geometric thought that most students pass through and that an alarming number of students can not complete a proof at the end of a course on geometry because they started the class without an informal grasp of ideas such as necessary and sufficient conditions.
I also worry that geometric proofs are often boring because the outcomes are obvious. The other teacher(who is awesome to work with) definitely wants to cover a fair number of geometric proofs and I wonder how I can make this accomplish the overall goal of the course which in my mind is to help my students love math and become better problem solvers and logical thinkers.
I can’t believe how much reading scholarly literature on mathematics education has inspired me and informed me and therefore improved my classroom. I’m finally seeing students who are curious about math and who leave class saying “don’t tell me the answer. I want to solve this!” It gives me hope that I’m not crazy to think that my intuitive philosophy of math education will resonate with students.