Mathematics is a wonderful subject. Of all the classes a student might take during a college career, I believe mathematics offers some of the greatest potential for overall intellectual development. On the one hand, it is able to describe and explain a wide variety of specific real-world phenomena, while on the other, it requires the use of more general intellectual processes such as abstraction, conceptualization, and logical reasoning that form the building blocks of a mature, thoughtful worldview. Unfortunately, many incoming students neither appreciate nor value the riches math can offer them. This may be due to several factors, including poor experiences in past math classes, a tendency to take math only because it is required, and a more general feeling in society that math is simply “hard” and anyway unnecessary to really learn well. But whatever the cause, I believe my role as a teacher of mathematics is to respond to this situation by (a) motivating students to see mathematical concepts as being worthy of their effort, (b) communicating those concepts during classtime in a clear and engaging manner, and (c) facilitating the retainment of those concepts through classroom organization and appropriate technological tools.
To motivate students that mathematics is worth their effort, I emphasize as often as possible the power of modeling and interpretation. First, I try to frequently use everyday examples to illustrate and practice mathematical concepts. For example, calculus students could calculate the amount of force and degree of accuracy required to throw a 50-yard touchdown pass, while differential equations students could apply the theory of first-order equations to learn lifelong lessons about the details of smart investments. Second, beyond just using accessible examples to illustrate a day’s point, I require students to formulate their own models, whether for simple story problems on tests and homework, or for larger, more open-ended questions in class projects. This modeling-centered motivational approach serves two purposes — first, it makes abstract mathematical concepts come alive, and second, it gives students an appreciation for the power mathematics offers in understanding the world.
To maximize the effectiveness of actual classtime, I strive to keep my lectures well-organized, focused on communicating essential concepts, and full of student participation. In my own experience, having a general conceptual framework in place makes it vastly easier to learn individual facts within that framework. Therefore, I always teach classes in an outline format, summarizing each day in words what will be done, and why, before actually performing any calculations. A second strategy I employ is to stress the comprehension of foundational concepts over the memorization of specific formulas. For example, when it comes to various integral formulations, my favorite high school teacher constantly repeated the mantra “slice, approximate, integrate,” to remind us of the basic geometric concept behind integration; he consistently achieved a 100% pass rate on the AP calculus exam. Finally, I try to create an environment where students participate as much as possible. Asking students to summarize a general solution strategy, supply key steps in an on-the-board calculation, and even perform pre-assigned calculations in front of the class are all ways to give them a sense of ownership in the learning process.
To facilitate students’ understanding and retention of classroom material, I try to employ two observations from studies of learning — that multiple exposures to a concept enhance its retention, and that students can mutually benefit by exchanging perspectives on a given concept. Thus, for homework and class projects, I try to organize students into groups whenever possible, ideally based on the results of a brief quiz on prerequisite material. Additionally, I require students to read each day’s material ahead of time, and post their responses the night before class in a group-visible online forum (idea from Chad Topaz). These strategies serve several purposes. A group format encourages students to help each other, which aids retention for both the helper and the recipient (my own experiences in group-format classes have been very positive). Second, requiring students to read classroom material in advance encourages multiple exposures, and additionally allow me to tailor each day’s instruction based on the previous night’s comments. Last, requiring individual reading and group discussion promotes one of the greater goals of a university education — the ability to acquire knowledge oneself.
I realize that most of my students will not go on to be mathematicians, and that many will enter class on the first day of the semester with a sense of dread. The tasks of motivation and communication, then, demand the ability to communicate mathematical principles effectively with non-math majors; something that my past experiences as a tutor and member of interdisciplinary research efforts have prepared me for. I feel my experiences as a peer tutor at SMU and member of an interdisciplinary program like Northwestern’s have prepared me well to do just this. I therefore hope to instill in each of my students at least an appreciation for the skills mathematics has to offer. Besides encouraging general intellectual tools such as abstraction and logical thinking, it offers students the ability to fluently speak the language that describes so much the world around them. Whether they use these skills to breeze through the quantitative details of later studies in science and engineering, or simply to make wise decisions at the grocery store, I hope they will at least look back and think that mathematics was worth learning.