An important part of a mathematics education is mastering the process of understanding a handful of general concepts and then being able to apply them to a specific situation.  With many Penn students focused on their final grade, they find the “just tell me how to do it” strategy initially more convenient. I take it as a challenge to convince students that learning how to think mathematically will in fact pay off, not only for their comprehension of the material, but in their final grade. My teaching philosophy revolves around exactly this point, that emphasizing a firm foundation in mathematical problem solving skills, the “yoga,” over rote memorization of facts will both help students learn the material and foster a sense of personal confidence when dealing with future mathematical problems. To make this philosophy succeed in practice I find it very important to provide extra support in office hours and private meetings, to carefully illustrate how I think my way through problems, have a commanding presence in the classroom, to solicit continual in-class student participation, to be flexible with my lesson plans and in-class examples to fit students’ needs, and to exude excitement for the subject matter and methods at hand. 

Often I witness students who know “how” to solve a particular type of problem, but are frozen in front of a slightly modified problem. Surely they could learn how to solve that particular modified problem as well, but where does it stop? It’s as if I individually memorized 1 × 101 = 101, 2 × 101 = 202, 3 × 101 = 303, up to 13 × 101 = 1313, but can’t say what 14 × 101 is, in short, I never learned a general principle behind multiplying by 101. Helping students to learn general foundational tools and strategies appropriate to their situations will enable them to robustly tackle a wide range of problems. For example, setting up area integrals for surfaces of revolution in Math 104, I emphasize drawing a picture and using the basic  principle of “slicing into small cylinders” each time I work an in-class example, rather than memorizing the formulae on a case-by-case basis. Though it seems like more work up-front, students tell me that relying on their geometric intuition and a picture rather than memory often suits them better on the final exam, when they’re faced with a slightly nonstandard version of the problem. In Math 371, when computing a basis for a free submodule given explicit generators, I emphasize the general idea of converting the question into one about the image of a homomorphism, so that matrix elimination techniques can be applied, instead of giving them a step-by-step algorithm for finding the answer. In this way, the problem is broken into smaller, structured, cohesive, and more manageable parts, rather than becoming a long list of steps.  Stressing these general techniques, and their applications to examples and challenging problems for which they are necessary, fosters real mathematical problem solving skills, increased confidence, and a respect of deeper learning, especially in the upper level courses. 

The best way to convey the importance of general problem solving strategies is through their active use in examples. When I’m asked to solve a particular problem, I’ll first suggest how to place it in a larger context with similar problems, then with the help of the class, we’ll “create” the solution live at the blackboard. Starting with only the general tools available, and we’ll highlight exactly where the problem breaks into pieces, and how to handle each piece.  Of course, I’ll have prepared notes, with some examples worked out and some discussion subjects in case the students are at a loss for questions, but when I’m actually in front of the class solving a problem, I’m actively thinking my way through the problem without notes. This live transformation of the classroom into a space of mathematical creation achieves two things.  First, it’s more conducive to student input and the flexibility to change the direction of the solution based on their suggestions and ideas. Second, it reinforces the fact that mathematics is an experience, something actually done by humans, and it’s fun, not a scripted set of instructions to be memorized and regurgitated. 

As an example, while writing out group multiplication tables in Math 370 for the cyclic groups Z/2Z, Z/3Z, and Z/4Z, of orders 2,3, and 4, respectively, a student asked if the emerging pattern of the multiplications persisted, or whether one could “do other things with the elements” to fill out the table. The situation flashed through my mind. Though I had planned to introduce only cyclic groups, by anticipating the existence of noncyclic groups, this student had brought into focus the contrast between the two. In that moment, I replied “Yes, these patterns do persist, and there are also other ways of filling out these tables,” as I drew the table for the noncyclic group of order 4, Z/2Z × Z/2Z. In response to this inquisitive student’s question, I restructured the lesson to focus on the two different isomorphism classes of groups of order 4 and the concept of group isomorphism in general. I discovered that the sudden change in topic was just what the class needed, and in fact, I found myself continually referring to ”that
one day we saw that there are two isomorphism classes of groups of order 4.” The lesson for me was that being flexible and responding to students’ questions can dramatically change the course of the lesson for the better! 

An essential part of fostering students’ confidence in their abilities is putting them at ease and creating a space where they feel comfortable asking questions. This takes place in the classroom as well as in my office hours, usually full of students asking additional questions and bouncing ideas off of each other. I think it also helps that I’m genuinely excited by the subject matter, and clearly feel joy and satisfaction at the blackboard on successfully tackling a problem in a mathematically sophisticated way. Taking my lead, students realize that it’s not only the final answer that counts, especially not to me, but the process of getting there, that is perhaps the most interesting. They also know that’s what I’m looking for while I’m grading their papers.