### by Ari Nieh, Mathematics

#### Teaching Effectiveness Award Essay, 2006

Linear Algebra and Differential Equations, Math 54, is among the earliest math courses in which students are expected to produce clear proofs as well as calculations. Unfortunately, most students entering this course have only the vaguest understanding of what constitutes a proof. They are likely to have encountered a handful of specific types of arguments: two-column proofs in high school geometry, or epsilon-delta proofs in calculus. With few exceptions, however, they have not developed much facility with general mathematical reasoning.

I noticed this issue early in the semester when I taught Math 54. The homework problems that generated the most confusion among my students were not particularly long, complicated, or computationally arduous; rather, the difficult problems were the ones which involved formulating a rigorous argument. Faced with any problem that used the word “prove” or “show,” the class was unsure how to get started.

It was clear to me that my students needed several skills to become competent with proofs. First, they needed the confidence and initiative to get their feet wet. They needed to grasp the connection between everyday reasoning and formal mathematical logic. Most of all, they needed a strong familiarity with the structure and flow of a proof so that they could construct one on demand.

I devoted a sizable chunk of section time to proof techniques. I emphasized the procedure of logical deduction by requiring students to take as many steps as possible without my help. At every stage of the argument, I would prompt them: “What is this problem asking?” “Is this even true? Why do we believe it?” “What should the first line of our proof be?” “Knowing this much, where can we go?” “What is the final statement which we’re trying to reach?” When we got stuck partway through a proof, I would encourage them to analyze their options carefully: “What would we like to be able to say next?” “Do we have any tools that relate to this concept?” Throughout this process, whenever a student made a suggestion, I would first repeat their idea as clearly as possible in precise mathematical language (and again in less formal terms, when necessary). I deliberately did not refute approaches that led nowhere or were mathematically flawed. If nobody in the class noticed the mistake, we would continue until we hit a dead end, then retrace our steps and figure out what had gone wrong.

This procedure had several beneficial effects on the section. First, because the class was fueled largely by student participation, the students were kept highly engaged with the material. They could not sit back, tune out, and rely on the GSI to show them the answer. They were also required to think critically about arguments, and they grew accustomed to evaluating and correcting mathematical ideas. They became familiar not only with the structure of a proof, but also with the procedure of constructing one, along with the toolbox of methods used to connect the logical pieces.

I observed a marked improvement in my students’ ability to assimilate new mathematics as the course continued. They approached proofs with increasing confidence and skill. In the final weeks of the semester, they were able to understand the subtle conceptual relationship between solving linear systems and solving differential equations. I believe that the emphasis on proofs was a major factor in their learning; armed with the correct analytical tools and perspective, they achieved a new level of mathematical maturity.