So this semester, I’m excited to be teaching UGS 303: From Numbers to Chaos, with Dr. Mark Daniels. This course will be taught using inquiry-based learning (IBL) methods, which means that the students will learn by working together through carefully chosen problems, with guidance from the professors and TAs. This is a Signature Course at UT Austin, which is designed with the following purpose:
“The Signature Course introduces first-year [and transfer] students to the university’s academic community through the exploration of new interests. The Signature Course is your opportunity to engage in college-level thinking and learning.”
In particular, my UGS course will tackle this goal from the perspective of the discipline of mathematics. This seems like a pretty natural thing to do, as it seems like most, if not all, of my students have not yet had experience with the critical thinking that comes with studying, creating, and writing rigorous mathematical proofs. Furthermore, this course is an opportunity to show students some of the creativity and elegance that mathematics has to offer.
However, there’s a balancing act here in what I would like to accomplish: a majority of my students are not mathematics majors. In fact, I would say that most would not benefit from a typical intro to proofs course, where they learn to prove the usual basic results about divisibility, or proofs by induction, or finding limits using epsilons and deltas.
Instead, the course will blend rigor with exploring interesting ideas. We do so by asking interesting mathematical questions, and then having the students organically develop the ideas and formalism to answer these questions. So, as a supplement to this class format, I would like to use the discussion sections to achieve the following three goals by the end of the semester:
(1) Teach students how to communicate their ideas clearly.
(2) Teach students to determine whether an argument is rigorous or not.
(3) Show students the creativity and elegance of mathematics.
To achieve these goals, I have a few strategies in mind. The first and foremost is to foster a supportive learning environment where the students are comfortable talking. One reason I’m excited about this course is because the discussion sections are capped at 15(!) students, as opposed to the usual 30-60 in traditional calculus discussion sections. This size will allow for a seminar-style format, where everyone will get to know each other, and where everyone has a voice in the discussion.
In fact, the goal is to get every student to be comfortable to voice their thought process in discussion section, and ultimately to the whole class. To do so, I think it’s important to emphasize to the students that when you’re trying to solve a problem, it’s okay and even necessary to make mistakes.
There’s a lot that needs to go into creating an environment where this is possible. One thing to do is talk about mistakes that I’ve personally made, as well as anecdotes about mistakes famous mathematicians have made. Another is to, as a section, come up with some rules for discourse. In particular, I’d like to institute two rules, in addition to the usual rules about civility:
- When someone comes up with a suggestion during a problem-solving session, I’d like to have a five second pause for everyone to have a chance to consider the idea before we discuss it.
- After the five seconds, I will ask, “Can we make this idea work?”
These rules serve a few purposes: It firstly removes any immediate negative reinforcement for suggesting an idea, even if it might not ultimately work. More importantly, however, is that it shifts the paradigm of ideas needing to be either right or wrong. After all, we don’t truly know if an approach will work until we try it. And a lot of the time, we mathematicians can usually come up with an idea that is mostly right, but sometimes requires some small details to be fixed.
Another key component of creating a supportive learning environment is to make the students comfortable with asking questions. This is something mathematicians do exceptionally well, and it’s a skill I’d like my students to learn as well. And importantly, while the best mathematicians are able to come up with insightful questions, they often ask extremely basic questions as well. With that in mind, some things I’d like to encourage about question-asking are the following:
- If you have a question and aren’t being seen, it’s 100% okay to speak up and ask your question.
- It’s probably the case that a lot of others in the audience share the same question, and it’s even possible that it’s something they missed entirely. So ask away!
- In this discussion section, it’s 100% okay to ask “could you repeat that?” for any reason.
Again, these guidelines for questions are intended to reinforce the idea that math is a dialogue. After all, mathematical proofs are about convincing others that the argument is correct, and questions keep mathematicians honest..
To achieve goals (2) and (3), throughout the semester, I plan to show my students a variety of different proofs and proof techniques that can be used to answer interesting questions. The catch is, not all the proofs I show them will be correct, and it’s up to them to figure out if the proof is flawed, and why! For example, I have a few proofs of 1=0 for them to discuss and find the flaws in. Sometimes it will be obvious that a proof is flawed (for example, it concludes something we know is false), and sometimes it will be more difficult.
I think this will be a fun, interactive way to achieve goal (2). The process of examining a proof to make sure it’s not flawed will help the students not take proofs for granted. I think it will also be helpful in teaching them how to formulate insightful questions, such as “If this were true, what consequences would this have?”
Furthermore, developing the skills to critically examine a proof will in turn help the students to become more mathematically sound, as they can then turn these skills inward and check their own arguments to see whether or not they meet the standards of rigor for a proof.
So these are my initial thoughts about this course, and the goals that I have for this semester. I plan to post weekly about the things I do each discussion section, including what I did and how it went. There’s also a lot more to talk about, as I haven’t even really talked about the course content, nor have I mentioned that this course carries a writing flag, which means that I’ll also be teaching students how to write essays about mathematics! Of course, that’s an entire topic for another time.
Anecdotal Math 