Statement of Teaching Philosophy

For a pdf version of my Statement of Teaching Philosophy click here.

I came to UCSD with a fellowship from Conacyt-UCMexus that would have allowed me, if I wanted, to work on my Ph.D. without having to do any teaching at all. However, for me, teaching is an integral part of the academic life, and something I have always enjoyed. During my time at San Diego I have sized every chance I have had to work with students, either as a TA, volunteering at the Calculus Lab or as a Tutor. When I arrived to the department and expressed my interest in teaching, I was sent to the Center for Teaching Development (CTD), were I received lessons on effective communication skills at the classroom. This class opened my eyes to the challenges of teaching mathematics, and gave me important tools to become an effective instructor.

One of my goals when working with students is to develop an understanding of the subject that allows them to solve their exercises in a natural, comfortable way. As I frequently say to them it's hard to memorize and use formulas correctly if you look at them as mysterious black boxes. It's much easier if they make sense to you.

One obstacle I often face while trying to achieve this goal is that some students don't feel confident enough about their mathematical abilities, and feel afraid of asking "silly" questions in front of the class. To help students overcome this fear I try to set up a learning environment in the classroom where people feel comfortable of expressing their doubts. One question I frequently ask them before section starts is "How is this class going for you?" They will start talking about what are the points that are confusing them the most, and what are the problems that they find more challenging. As people keeps arriving they join the conversation and add their own concerns. This informal talking helps students articulate questions they may feel too intimidated to ask during section. I have noticed that after this little talk people ask more questions and are much more engaged than if I just do a "cold start".

One thing I take very seriously when I'm in a classroom is to show appreciation for every question students ask, and try to answer every single one thoroughly with patience and respect. This may look like an obvious point, but is not as simple as it seems. Every quarter I meet students from other courses that follow me in my Office Hours at the Calculus Lab because, they say, I'm the only one who is patient enough to answer all their questions properly.

As a mathematician I deeply appreciate the power and beauty of a good abstract definition, and the importance that understanding abstract concepts has in problem solving. Experience has shown me that a great deal of the mistakes students make, come from the fact that they don't really understand what they are doing. Being mainly concerned about their grades, they are often not very interested in doing the extra work required to understand the abstract concepts presented in class, specially when they feel that this abstract concepts have nothing to do with what they are doing at the moment.

It's because of this reasons that when I have to go over an abstract concept I always do it using an specific example that I have previously chosen. At the time of choosing this example I make sure that it's very similar to one of their homework problems, so that I can have their attention, it's as down to earth as possible, so that students can focus on its solution and not in some specific technical notation, it should be simple enough so that students can focus on learning the concepts and not in the algebraic manipulations needed to solve it, yet it has to be interesting.

After I set up the example on the board I let students know that this is going to be a cooperative effort and that I expect their participation. As we work carefully toward the solution of the example, I show them exactly how the abstract concepts I'm introducing apply to the case at hand. I stop every time a new idea is introduced and ask for questions, feedback and ideas. I believe that by looking carefully at your audience you can identify nonverbal clues that tell you when people is starting to get lost. I also believe that your nonverbal clues tell students that when you ask for questions and participation you actually mean it and you are not just asking rhetorical questions. Learning new material is always hard and it's important to give students enough time to assimilate the concepts and formulate the questions they have.

After we have worked out the solution, we review the problem and we start working in describing an algorithm to quickly solve this type of problems. Then, I modify the wording of the example slightly and some of the parameters and we discuss how our algorithm should be modified to work with the new example. Here I make the observation of how easy it would be to get confused by the new example if we just try to memorize the algorithm and ignore all the concepts behind it. I finish by going over a couple of similar examples to help students develop their intuition and expertise with the problem solving.

I feel that this approach has been very useful as it helps students take out the mysterious halo that mathematics has always had to them, and helps them feel more comfortable with the subject. I remember once when I was covering the Office Hours of a friend who was TAing a Linear Algebra course, and some students came with question about their homework. After going over a couple of problems I realized they were struggling with some of the concepts, and I told them I wanted to review them. I could see their faces of "but I just want you to help me with this problem" but after I told them I would start with one of their homework problems they agreed. After this digression, we came back to working on their homework, and they were surprised of how easy everything was now. Seeing this, now they wanted to discuss some of the concepts they have learnt in the last couple of weeks and that were still confusing them. After a fruitful discussion about change of basis and diagonalization, we went over some of the problems of their last homework and they were able to quickly solve them. At the end of the Office Hours they were surprised of all the material we had been able to cover.

Experiences like that are truly gratifying and reinforce my wish of keep helping students appreciate the beauty that math has to offer. I keep working on improving my teaching and I'm always open to new ideas and techniques. But above all, I feel enthusiastic by the perspective of working with students in the years to come.