Intro to Proof in Four Weeks Part IV: Getting the Course Started

Earlier posts: Part I, Part II, and Part III.

The much-anticipated first day of class arrived with little fanfare. It's been a long time since I've had a schedule where I only taught in the afternoon, so there was this awkward time where I'd finished preparing everything for class and my colleagues were teaching the morning classes, but I had nothing to do. Nothing to grade. No students to come to office hours. The first test was too far away to be writing. Little did I know, about 48 hours later I would be desperately wishing I could go back to that laid back, even bored, moment.

I'm always one to expect my students to come to class prepared, and with only 20 days with my students, I knew we needed to hit the ground running on day one. We started, as I've already discussed, in the lab with \(\LaTeX\). Students had already been told to have it installed on their laptops and given some screencasts to watch. That was where I gave the students their first group assignments and their first homework assignment. I also gave a demonstration of some of the key features of Box, since W&L has an enterprise subscription and I was requiring students to use it to submit homework. Once we could all compile a \(\LaTeX\) document, it was time to move to our usual classroom to start the mathematical part of the course.

 The main part of day one started with a discussion the first two chapters of Burger and Starbird's 5 Elements of Effective Thinking, which the students seemed to enjoy. We took 5–10 minutes in class every day the first week to talk about another chapter and then it just became a bit of a recurring them where I'd point out how one of the five elements was really coming into play that day. (Or, almost as often, one of the students would note it.) I've really come to enjoy using this text with math students and plan to use it again in Calculus I this fall. Not everyone loves it, but most come away with the feeling of having read a self-help book in a math class in a way that opens their eyes to aspects of thinking mathematically that are quite valuable as the course proceeds.

I then spent a few minutes summarizing the course policies and tasked the students with reading them before the next class so they could come with questions. Since my course policies are generally in the 7–10 page range, I find this method works reasonably well. One of the few aspects of the course policies I discussed in depth on the first day was tests. I wanted to make sure everyone had an idea of how the tests would work, since they would be timed but out of class. Part of what I emphasized right away was the role of definitions in mathematics and how I would be testing them on these definitions. I'd heard Carol Schumacher speak on more than one occasion about the difficulty students have with definitions in the transition to abstract mathematics, and so I was really aiming to get that part of things right. (The challenge is that they've not really had to memorize definitions before, but as they get started in math, it's essential that they get all the quantifiers in the right order and such, so memorization is almost critical.) I tried to make a very approachable point about definitions on the first day: Functions are something that these students had been working with for years, but if I asked them to try to give me a precise definition, none of them would be able to do it. It turns out that this was exactly the right point to make as a starting point, but I wouldn't fully realize it until near the end of the course, as I declared in this Twitter exchange:

We then jumped into a discussion of the "Thought Experiment" from the beginning of Chapter 1 in Chapter Zero. (Technically a slightly modified version that I got from Carol personally, but it's only a slight modification.) The thought experiment is a list of statements and the students are asked to decide if each is true or false and then categorize them based on how confident they were in their proof/disproof (or if they were just stumped). This was a great mathematical ice-breaker for the students (and me). It got the students to start reading (very simple) quantified statements to see what it would mean to talk about statements involving "for all" and "there exists". The list contains the Goldbach conjecture, so they can't get all of them with a high level of confidence. Many of the statements are things that the course circles back to at a later date. One of the statements on the modified version is "If \(-1=1\), then \(1=1\)." Since the students didn't yet know what a conditional statement was (save the one student who'd already taken Discrete Mathematics I) from me, this one generated a lot of discussion. But it was valuable discussion, as I could foreshadow what was to come in the next class.

After we debriefed the thought experiment as a class, we started to get into the usual pattern of the class. Chapter 1 of CZ is the logic chapter, and the text really encourages you to blow through it as quickly as possible. I stuck to that plan, and we spent the balance of the first day discussing statements, predicates, and quantifiers. The second day focused on mathematical statements, implications (and everyone seemed OK with "If \(-1=1\), then \(1=1\)." being considered true at this point), and truth tables. The third day was HUGE, as we did more with truth tables, negated statements (including messy quantified statements), and then quickly ran through the ideas of existence theorems, uniqueness theorems, examples and counterexamples, direct proof, proof by contraposition, and proof by contradiction. My colleague who was using a more tradition text spent entire days on each of these topics, but I really like CZ's approach here. If you belabor this stuff too much early on, the students get bogged down in things that are "obvious" or at least trying to decide what things they "know" from their mathematical background are legitimate to use at this stage. By moving quickly, we were able to get an overview of the key ideas, have a quick discussion, and then the text and I could later team up to say things like "This is an existence proof. What does chapter 1 say about how to do an existence proof?" The students would then dutifully turn back to chapter 1, refresh their memories, and proceed to do the existence proof required.

On the subject of what can we use that we know from before, I learned a valuable lesson when I assigned the following problem: "Suppose that \((a,b)\) and \((c,d)\) are two distinct points in \(\mathbb{R}^2\). Prove that there exists a uniqe line passing through the two points." Existence is easy. It wasn't so great getting the students to edit out all of their scratchwork of finding the equation of the line. (I can't count how many times I reiterated CZ's point that an existence proof does not generally explain how the clacking waggler was found, but that all you must do is prove to me that the thing you claim is a clacking waggler is, in fact, a clacking waggler.) However, I still don't know what background material should be legitimate for proving uniqueness here. The text calls for mimicking an example argument in the text, but I don't really see how that applies. (The example is showing that the real root of a particular cubic is unique.) Eventually I settled on having them prove that any line through the two points must have the same slope and \(y\)-intercept as the line they came up with, but that wasn't really satisfactory to me. (I think they were all satisfied in the moment, but I bet if I'd reminded them of that problem on the last day of class, they'd be looking back and feeling unsatisfied by their arguments as well.)

Day three also featured the first true student presentation. There's a nice direct proof problem in section 1.12 that asks the students to prove that if \(x+y\) is even and \(y+z\) is even, then \(x+z\) is even. I was pleased when the student who'd taken my Discrete Mathematics I class volunteered to present this problem, since I knew that the proof would be right (or else I'd go give him an F for Discrete Mathematics I) and would live up to my writing expectations. I felt a bit like I cheated or had a "plant" for this presentation, but in the end, I think it was productive. Everybody got to see a nice presentation and it set a good bar. The proof had a few places where the writing needed a tweak, so it wasn't like the proof was flawless and set the bar so high that students were scared to present, however.

Don't worry, I'm not going to go day-by-day through the rest of the course. I felt the need to fully set the stage for what happened in the first couple of days, however. The remaining posts will discuss some of the specific highlights and lowlights of the term, with the ultimate goal being a list of things that I'll definitely keep the same next time and things that I'll definitely change next time.