Students understand their subjects and how to teach them

For over four years at Colgate, I have been surrounded by math. As a math major, I saw some of the proofs and theories behind math in courses such as Real Analysis and Abstract Algebra, and I have seen applications for mathematics in Differential Equations and Probability. Next semester I will be taking courses that further explore applications of mathematics, such as Mathematical Modeling in the Social Sciences. These college courses have given me an appreciation for the intricacies and complexities of math, but I have also had extensive experience with math at the high school level. I have been an after school math tutor at both Madison Central School and Hamilton Central School during my time at Colgate, and have been a math SAT tutor, and later a site director, for the Let's Get Ready program. Before I started student teaching I also observed math classes at Sherburne-Earlville High School once a week for a year, learning some of the strengths and weaknesses students have when learning math.

Overall, I have had enough experience thinking mathematically that I am confident that I can understand any topic I have to teach. Some of the Algebra 2 and Intro to Calculus topics I had to teach I had either not seen for five or more years, or perhaps were not taught when I was in high school, but with a little bit of practice and some help from my cooperating teacher I quickly understood these topics. Understanding my subject allowed me to focus on pedagogy, rather than curriculum. I worried about how to teach the math, rather than worrying whether I knew the math myself. This also helped me answer questions from students. I could follow their work and look for errors as well as appreciate unique ways to solve problems. I don't know that I had ample Algebra 2 knowledge before I started student teaching, but I had so much practice thinking logically and mathematically, that I could pick up on unfamiliar topics very quickly.

My experience with higher level math also allowed me to make connections that make math more interesting. For example, in Intro to Calculus we were solving a fairly simple problem, asking students to find the average rate of change between two points on a parabola. This comes down to simply finding the slope between the two points, something the students have been doing since middle school. However, I was able to show students that finding the average rate of change is directly connected to calculus, which they will be doing later in the school year. If the two points are arbitrarily close together, then we can say that the average rate of change is essentially the slope of the parabola at a point. This is the definition of a derivative - a calculus topic that may seem intimidating, but is actually a direct result of the middle school slope formula. Derivative then let us measure, for example, velocity and acceleration of an object given a graph of its position. Not only did students find this connection interesting, but it makes calculus much less intimidating and much more accessible. The original problem should have taken at most two minutes, but I was able to use my mathematical knowledge to expand it in to something much more meaningful.

Picture courtesy of math.hmc.edu