TEACHING STATEMENT

TERENCE TAO

At UCLA, I have primarily taught upper-division undergraduate and graduate mathematics courses. At these levels, they have already completed their basic training in mathematics and are ready to move beyond a purely memorisation-based or formulaic approach to the subject. I therefore try to challenge my students to start thinking about mathematics on an intellectual level, as well as on an intuitive level. I attempt to do this in a number of ways.

Firstly (and perhaps counter-intuitively), I do not try to hurry in my lectures to the most advanced portions of a subject, but instead dwell leisurely on the basic material, which the students are already familiar with, but delve into it more deeply, showing how even the simplest topic can be used to illustrate important concepts (such as the importance of a rigorous proof, or the power of abstraction, or to showcase some useful mathematical method in the most simplified context available). My philosophy here is that all too often in advanced classes, students are struggling with conceptual difficulties (because of the flood of new material) simultaneously with more fundamental issues such as learning how to prove things rigorously, or how to write coherently, or how to harness one’s innate intuition on a subject. I find it better to tackle these difficulties separately, spending the first half of a quarter with basics, and then moving on in the second quarter with advanced topics once the students are secure and comfortable in their basic knowledge. For instance, in my honours Real Analysis class I spent the first two weeks on the foundations of the number systems; a typical homework question was to prove that (a + b) + c = a + (b + c) using the Peano axioms. My honours students complained (somewhat wryly) that while conceptually their homework was much easier than the homework of their non-honours counterparts, the homework was in fact much harder due to the emphasis on rigorous proof. However, they largely rose to the challenge, perhaps because the questions seemed so obvious that they were much less willing to concede defeat.

In the second half of the course, when we moved rapidly to more advanced topics, they were able to focus on the new concepts being introduced, whereas their non-honours counterparts were now having to deal with the concepts and with the need for rigorous proof, which they had not been seriously exposed to in the first part of the course. In fact, despite the leisurely initial pace, the honours class eventually overtook the non-honours class in terms of topics covered.

Another way I try to force students out of the memorisation habit is to provide lecture notes in advance, and to make them freely available on-line, especially since students for one reason or another are not always able to attend every class. Before I did this, I noticed that my students would spend most of a lecture simply copying down verbatim what I wrote on the board, and did not have the time to actually think or digest the material. With the notes in hand, the students had little choice but to try to listen to the lecture instead. (Sometimes it took a while to adjust; in some classes, the students spent the first few weeks highlighting my notes instead of taking notes, but eventually they learnt that they could get more out of the course by paying attention to the lecture directly.) Actually, I try to use the printed medium and the lecture medium to complement each other; once the students have the careful details in their notes, I am freed up to talk instead about the ideas and motivation underlying the material, and the connections with other topics.

I find that the homework is also a fundamentally important component of a course, as it is the one component of the course which the students complete on their own schedule, using their own methods - in short, it resembles how actual problem-solving and research is done far more than, say, sitting in a classroom or studying for (or taking) an exam. It is also the primary motivator to get the student to review, digest, and practice the material he or she has (usually passively) absorbed in class. By the time the student gets to the upper division or graduate level, I feel it is no longer necessary to give many formulaic homework questions - the sort which are minor variations of an existing problem worked out in class or in the textbook. Instead, I try to find questions which illustrate some point in as simple and quintessential a way as possible - for instance, a problem which is simply stated, but will be frustrating to solve until you see how a certain method can resolve the problem easily - in the words of one of my students, my homework questions were incredibly time consuming but then he always “kicked oneself once I saw the answer... and then it was burned into my brain.”. Homework is too often either too trivial to be of educational value, or a laborious process which yields no insight even after the problem is completed, or so challenging that it simply discourages the student. A good problem should be inviting enough to naturally draw the student in to attempt it, challenging enough that the student recognises that something nontrivial has to be done, and clean enough that the insight provided by the solution is not obscured by a mass of time-consuming calculations. It is here that a student can finally get a taste of how one’s mathematical power increases when a tool is understood and used correctly, and how one can use the current class material to revisit earlier subjects and clarify them even further.

In constrast, I do not rank exams or quizzes as highly as a pedagogical tool, especially ones with an artificial time limit such as one or three hours. A midterm exam, for instance, serves the useful purpose of allowing the students to gauge their own progress and understanding at an intermediate stage of the course, but is otherwise a rather artificial and unnecessarily stressful experience, and so I try to make the midterms as “clear” as possible, with the format telegraphed in advance (with practice exams provided), and the each question simple and designed solely to test the student’s knowledge of exactly one key point. (More complex questions should instead be assigned as homework.) I tend to avoid quizzes entirely, or use them solely to motivate the students to do the homework (by announcing in advance that the quiz question will be drawn from the homework). Instead, I sometimes offer online equivalents of quizzes (such as self-test multiple choice questions) which the students can perform in their own time to assess their grasp of the material. (One particularly popular such online quiz was my logic quiz, which tended to bring home to the students the need to develop clear and logical thinking skills.)

I believe that students at the upper division level and higher are (in general) mature and motivated enough that they can learn at their own pace and in their own style. Thus, I tend to eschew artificial constraints on learning such as mandatory attendance, or prohibitions on collaboration in homework, and instead design my courses for flexibility; for instance, as mentioned earlier I tend to put lecture notes online so that lectures are not strictly necessary, and also prefer take-home examinations to in-class ones. It is true that insufficiently motivated students may thus not take full advantage of the course, but this is their own loss, and I feel that at this stage of their education, students should be responsible for managing their time effectively. The lecturer is obliged to provide every opportunity for the student to learn (through class, office hours, homework, lecture notes, online material, or alternative examination students), but it is up to the student to grasp these opportunities.

I have experimented at times with more non-standard teaching techniques. For instance, I have written some Java applets designed for the students to experiment with mathematical concepts, or test their own knowledge, at their own leisure; for one course (linear algebra) I even hired a programmer to develop these applets. For the honours real analysis class mentioned above, I offered a “bonus point” system in which the students were invited to find errors or corrections in the notes, with each error scoring a point which added a nominal amount (0.1%) to the final grade. This turned out to be very successful, with both the average and good students reading the notes more carefully, and amongst the top students it even became a sort of game (I posted ”high scores”, using student-selected nicknames, on the web page), with one student ending up detecting over 50 errors. (I capped the grade benefit of the bonus points at 20 points or 2%, but the best students continued to enjoy finding the errors.) As a side benefit I obtained a substantial amount of proofreading and other feedback from the lecture notes, which eventually became a textbook.

In conclusion, I tend to demand more out of my students, in that they need to devote more of their intellectual faculties to my courses than just rote memorisation or pattern and keyword matching, but they tend to get more out in return. For some, even the insight that learning is in fact an intellectual experience is a revelation (and an absolutely necessary one when transitioning from the undergraduate level to the graduate and post-graduate levels).

Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA.

  E-mail address: tao@math.ucla.edu