During the spring and fall of the 2004/2005 academic year, I took 4 classes in the Duke University Department of Mathematics. The courses that seemed to be the most beneficial were Geometry, the Pedagogy of Calculus, and Advanced Calculus.

Geometry

The Geometry course was an undergrad level course in which we proved the theorems of geometry starting from its basic axioms. It was the second axiomatic and proof centered course that I have taken, the first being abstract algebra, and it was quite valuable for me in understanding how the this field of mathematics was developed. It was especially valuable to see how dependent Euclidean geometry is on a single axiom, and how varied and extensive the notions of non-Euclidean geometry are if that axiom is not assumed to be true. The work itself was comprised of weekly homework assignments that involved completing proofs and other exercises from the textbook. There were three in-class exams during the semester. At the end of the semester, we presented on some sort of extension to the Euclidean or non-Euclidean geometries that we had studied. My own project dealt with projective geometry, and especially its foundations that are found within perspective drawing in art. The final product took the form of a PowerPoint slide show, and a final paper.

All in all, this class was valuable for a number of reasons. I had not taken a geometry course since middle school, so it was a great refresher. Moreover, I feel that I have a great background in how all that we teach in the field of geometry has been derived from a few simple axioms. This will be quite beneficial when I need to answer the “why” and “how” questions when teaching everything from pre-algebra to calculus. So many of the problems we use to teach math are related to geometric concepts, and I feel I have firmed the foundation of my own understanding.

The Pedagogy of Calculus

Also during the fall semester, I enrolled in a class called the “Pedagogy of Calculus”. This class was an independent study through which I taught a section of the MATH 25 calculus lab for undergraduates (mostly freshmen and sophomores) at Duke. In addition, I met weekly with a professor of mathematics education to discuss relevant issues, and I also wrote a survey of the history of calculus from ancient Greece up until the late 19th century. This three-part class was almost certainly the most relevant and applicable course that I took during the year. Teaching the lab was quite a rewarding experience. I taught an introductory calculus lab section that involved a problem-based method to help students master the ideas that were presented in lecture. Not only was this a great refresher, but it gave me great practice in explaining tougher math concepts. After teaching middle school math for two years, I was used to working hard to find a proper way to explain given concepts, but I was always sure that I fully understood the concepts myself when teaching pre-algebra. When it came to calculus, I found myself often having to clarify my own understanding of the ideas before trying to present them in a coherent manner to my students. I was also introduced to a well-constructed calculus curriculum, or at least reintroduced. While there was a investigative lab portion in Calc II when I took it during my undergrad years, it is quite different to examine the curriculum from a teacher's perspective. I had a much greater understanding of why students were asked to carry-out certain investigations and was much more attuned to what was included in the curriculum and what was not than when I took MATH 32 as an undergrad.

The “seminar” portion of the class was driven by my own interests, and I found myself researching a number of topics from the history and domain of the AP Calculus Tests to the relationship between the different branches of mathematics. While my undergraduate education in engineering was quite math “heavy”, my own knowledge of the different pieces of mathematics and calculus in general were not plentiful. To have the opportunity to discuss this and other ideas facing mathematics educators with a professor who was so knowledgeable and excited about mathematics education, not simply mathematics itself, was invaluable. The work consisted of independent research and developing my own questions to bring up during our discussions. I left with a better understanding of the history of calculus education, and mathematics education in general.

The research involved in writing the survey of the history of calculus was quite extensive, and answered some of the questions that were brought up in the lab teaching portion. The paper explains not only how calculus was developed by Newton, Leibniz, and those who came before them, but also on the questions that were later raised involving the foundations of calculus. This historical perspective will allow me to provide a more “human” perspective when teaching calculus. It will also allow me to ask questions similar to those asked by Newton and Leibniz’s critics to the students in my class, allowing for a bit of a debate on what can be pretty dry material.

Advanced Calculus

Finally during the spring semester I enrolled in an Analysis class, titled Advanced Calculus. The objective of the class was to explain how the methods of differential and integral calculus could be proved; in other words, we proved that the calculus actually does what it says it does. The class was presented not in a historical order, but rather in logical order, starting with set theory and the real number system and finally leading to differential and integral calculus. After writing a survey of this history of calculus, it was fascinating to actually follow the development of calculus from a different perspective. I gained a much deeper understanding of how and why calculus works. Also, having taken Abstract Algebra and Geometry and seeing how those fields could be built from a few axioms, I was curious how this could be done for calculus. While much of the material would not be accessible for high schoolers, there are parts that will definitely come up when I teach a calculus class, and I will feel much more comfortable when I need to prove certain concepts, like why the derivative of tangent is secant squared for instance. The work in the class was mostly proof based. We covered the ideas of sets, sequences, series, limits, continuity, and finally differentiation and integration. The work comprised of 11 homework assignments, 3 exams (with take home and in class portions), and a final exam. The problems largely dealt with organizing proofs and making use of definitions and theorems that were developed during class.