I am teaching our “mathematics for liberal arts” course for the first time. This is a topics course, so I can teach whatever I like; I have chosen to do a Pólya-type problem solving course.
In class, the procedure will likely go like this: students get new problems to work on on Mondays and Fridays. Students will work on old problems on Wednesdays.
The grading of the course has five components: problem reports, correct solutions to problems, quizzes, a final exam, and a project. Without going into too much detail, here is how the final grades will be determined:
- Students will get at least a C if they provide a modest amount of evidence that they have achieved the learning goals (see below) and get at least a CD on the final exam (“CD” is like a C- or D+).
- Students will get at least a B if they provide a good amount of evidence that they have achieved the learning goals (see below), do well on the quizzes, do a project, and get at least a BC on the final exam (“BC” is like a B- or C+).
- Students will get at an A if they provide a whole lot of evidence that they have achieved the learning goals (see below), do well on the quizzes, do a really good project, get at least a AB on the final exam (“AB” is like a A- or B+), and get correct solutions to many of the problems.
In short, C students are able to demonstrate good habits of mind, B students are also able to understand and replicate solutions, and A students are also able to generate solutions to problems.
The learning goals are this:
- You will improve your written and verbal communication skills.
- You will be in the habit of providing and demanding evidence for any assertion.
- You will be in the habit of employing supposition when you encounter new ideas (“What if the idea were tweaked to be slightly different. What would happen then?).
- You will be in the habit of employing different perspectives by determining how other people think
about a situation.
- You will be in the habit of making connections between new ideas and old.
- You will be in the habit of planning before acting.
- You will be in the habit of using heuristics (“rules of thumb”) to help you solve problems.
The way students demonstrate evidence that they are achieving the goals is this: each student specifically states which of the learning goals were “used” in the problem report. For example, the first problem in class is a simple variation of the game nim; I would expect many students might claim that had to employ different Perspectives in solving the game, since they will have to think about how the opponent will respond to each move. Additionally, the student might have a partial solution strategy; if the student provides a “proof” of why the strategy is guaranteed to work, the student can also claim that they displayed evidence of the Evidence goal.
So the students are responsible for realizing what they did (although I have a grader who is going to verify that the students did what they said they did). I like this because it encourages students to use these good habits (“I need a Connections, so I had better try to think about whether this problem is related to something I know”), it forces students to reflect on what they did, and this is how most of the “real world” works (When I apply for tenure, I am the one who needs to provide the evidence that I deserve tenure. Similarly if I were to ask for a raise.).
I help my students by telling them where they can often find opportunities to provide evidence of the learning goals. For instance, students can cite each problem report as Communication, although it must be well-written to get credit. I tell students that the Game Theory questions and the Knights/Knaves/Liars/Truth-tellers problems are good for Perspectives. I also tell students to pose new, but related, problems in each problem report (from The Art of Problem Posing); this is good for satisfying the Supposition goal.
Now tell me this: what could possibly go wrong?