Intro to Proof in Four Weeks Part II: Design decisions
This is Part II of a series on my experience teaching a four week introduction to proofs course for prospective math majors and minors. I suggest starting with Part I for some context.
Historically, I have not started planning for an upcoming course months in advance. A month out is generally good, as that gives me either the wind-down period of summer or the finals/grading period of fall to prepare. However, I knew that this Spring Term course was going to be brutal even if I had carefully planned. After attending the 2014 Kenyon Inquiry-Based Learning Workshop, I was bitten by the IBL bug. (In all honesty, when I attended the workshop, I realized that I had been gradually moving toward inquiry-oriented activities in most of my classes for a few years. In fact, when teaching combinatorics using IBL this winter, I went back and grabbed an activity on recurrence relations from when I was still in graduate school and only modified it slightly before using it. I didn't even know the phrase "inquiry-based learning" at that point, but it was definitely an IBL activity. That said, I am incredibly grateful for the IBL community's support in the past year.) I knew that this course was going to be IBL with a lot of group work as well as student presentations. Those decisions were made months in advance, but by late February, I was already sitting down and trying to make some decisions about my course.
I must confess that I didn't formalize my learning outcomes as early in the process as backward course design calls for, but I had these ideas in my head throughout the process. The precise language was adapted from a number of things I had used in other courses and found online. (Dana Ernst deserves a shout-out, because I lifted some of these almost verbatim from him.)
Students will write readable, concise, and mathematically rigorous proofs using standard mathematical English.
Students will identify incorrect proofs and inadequately-supported claims in proofs.
Students will use written mathematical English to express their knowledge of the terminology, concepts, basic properties, and methodology of symbolic logic, set theory, relations and functions, mathematical induction, and other topics selected for the course.
Students will identify correct proof structures and criticize incorrect proof structures.
Students will distinguish between true and false statements about sets, relations, functions, and other topics selected for the course.
Students will discover and explain examples and counterexamples of statements about sets, relations, functions, and other topics selected for the course.
Writing these down was really a straightforward task in the end. There's only so much you can hope for in a first proofs course, and this is kind of the optimistic list of skills. Although I'd never taught this course (even on a 12- or 14-week calendar) before, I had taught a lot of the material in the past. For the moment, I think I can say I've taught mathematical induction more times in my career than I've taught the fundamental theorem of calculus. I'm pretty comfortable teaching that topic. At the University of Nebraska-Lincoln, I taught one of their bridge courses, which is also an introduction to ring theory. Thus, the students have a lot of context in which they are learning to write proofs. However, they don't have any background in sets, relations, equivalence relations, or functions. I had a rough day when I realized that asking my students to prove a function was an isomorphism was going to be very difficult, since I knew they didn't know how to prove a function was a bijection but also had nothing in the text to help me teach them how. I've also taught our discrete math course here twice, which teaches just the basics of proof writing. We want the students to learn induction, and so we also teach some basic tools like contraposition and contradiction. Most proofs involve even/odd or congruence modulo \(n\) type arguments.
Past experience teaching students how to write proofs has convinced me that a portfolio is a worthwhile endeavor because it gets the students to actually pay attention to your feedback and make revisions. (The initial inspiration for portfolios was picked up from Robert Talbert.) However, a portfolio also takes a long time for the students to build up, and that's why I eventually decided against one for this course. In my UNL ring theory course or my W&L discrete math courses, my students spend 8–10 weeks building a portfolio. That gives them more time to think and me more time to give feedback. With the quick turnarounds involved in Spring Term, a true portfolio was not feeling like it would be within reach. I finally latched onto an approach when chatting with Paul Humke about a week before classes started. Paul is on the faculty at St. Olaf College, but he has spent his winters visiting W&L for many years now. Some years he has taught full-length winter-term courses, but he has also taught MATH 301 many times here. His approach to homework is that students type up their homework in groups. With that revelation (and as someone who does group homework and projects in calculus all the time), I was off to the races. Having only five homework sets to grade each day instead of 14 would mean I could actually expect written up versions of as many problems as I felt were necessary. In fact, I'd have time to process revisions, thereby allowing a portfolio-esque system to develop.
Once I had a scheme for homework, I had to figure out a way to get the students to pay attention to my feedback and revise. Given the nature of the course and its students, I figured that a lot of students would engage pretty well in the revision process. I learned at UNL that if you give a numerical score to a portfolio draft, students get focused on "What can I do to raise my grade to \(x\) points?" instead of "What are the weaknesses in my argument and writing that I should resolve here and consider in future work?" It was about then that I recalled Matt Boelkins' talk at the JMM about points-free grading in his introduction to proofs course. I dug up his slides and was off to the races again. Each homework problem would receive a mark of Exemplary, Mastery, Progressing, Started, or Incomplete/Incomprehensible/Irrelevant. I carefully cautioned students not to map these onto A-B-C-D-F, and I also laid out a rule that a revision could not earn a mark of exemplary. That meant that feedback even on acceptable work should be heeded, since following my advice on later assignments would help toward getting an A grade for homework. (Letter grades for homework were determined by a rubric that factored in the blend of E-M-P-S-I marks students earned. This meant that even a single S would make a B unattainable as a homework grade. This instilled a lot of persistence at the weaker end of the class.) Homework would count as 30% of each student's course grade.
I'd been waffling on tests and quizzes for this course from the day I started thinking seriously about the design. I knew from hearing Carol Schumacher speak on a couple of occasions that definitions are something students struggle with in making the transition to abstract mathematics. I also felt that having some sort of individual assessment would be beneficial, given the homework would now be done in groups. Again, I decided to run with Paul Humke's advice in this area. Paul gives a test every Friday. However, I know that IBL is a rather time-intensive pedagogy, and so I really didn't want to give up one-fifth of my classtime for tests. Fortunately, W&L's honor system means that I can be quite confident in giving my students self-timed, individual, closed resources tests. Thus, I settled on a test after each of our four weeks. Each test would require precise recall of definitions. To get an A would require five definitions to be completely correct. A grade of B or C would require four. The students would also be tasked with replicating something I produced in \(\LaTeX\), and external resources would be allowed for this part. The other part of the test is short proofs that should not catch prepared students off guard. Students had choices, but had to do at least three proofs (often one each from three categories). Each test would be worth 10% of each student's course grade. Revisions of proofs would be allowed, but not of definitions or the \(\LaTeX\) portion.
The final 30% of the grade would come from "classwork", to borrow a word from Carol Schumacher. For me, this includes presentations in class, sharing productive failures, coming to class prepared, and participating with group members in class. Honestly, I would be happy to give every single student an A for this part of their grade. We often say there is no A for effort, but in an IBL course, I think we're well-justified to give an A for part of the grade that's almost entirely effort based. In a traditional lecture course, students aren't expected to do much to prepare for class each day. In an IBL course, however, students must do a lot of preliminary work (much of it involving failing, dusting themselves off, and trying again) in order to come to class ready to go.
I'll leave this post off here having given a reasonably complete discussion of the course learning outcomes and feedback/assessment system. Full details can be found in the course policies and expectations. Next up will be a discussion of my plans for teaching and learning activities. Later we'll see how much plans and reality aligned.