Posts Tagged ‘Math 114’

Problem Solving for the Liberal Arts

March 7, 2014

I taught a “Math for Liberal Arts” course last semester based on Pólya-type problem solving. I want to change some things the next time I teach it, and I should write it down before I forget it.

Just to remind you (and also me, actually), I will list the major points about the course structure. I have two more-detailed posts here and here.

But here is the short version:

  1. I taught the students the problem solving process, including some carefully-chosen heuristics (solve an easier problem first, find an invariant, etc). We spent most every Monday and Friday working on two new problems for students to solve (Wednesdays were quizzes or review). I (mostly) carefully chose these problems so that they could be solved by applying the heuristics we had already discussed.
  2. If a student solved a problem, she could sign up to present the solution in class. If we all agreed it was correct, the problem was closed and no one else could get credit for it. If multiple people signed up to present the same problem on the same day, I would randomly select one person to present, while the other people handed in written solutions of the problem. Everyone with a correct solution got full credit for the problem.
  3. Once a problem was presented correctly, it was eligible to go on a quiz. So the quizzes consisted entirely of problems students have already seen solutions to. Once a student gave a correct solution on a quiz, he never had to answer that question again.
  4. Regardless of whether a student found a solution to a problem, the student could submit a Problem Report on that problem. The idea was to describe their problem solving process and mine out instances of good habits of mind to present as evidence for a higher grade (see this for more detail).
  5. The grading scheme is basically this: a student got a C for the semester if she did well on the habits of mind in the Problem Reports; a student got a B for the semester if she additionally could reproduce solutions she had already seen (i.e. “did well on the quizzes”); a student got an A for the semester if she additionally could create solutions to problems she had never seen before (i.e. “correctly presented many of the problems from the course”).

Here are a couple of examples of problems I gave the students:

  1. How many zeroes appear at the end of 100!, where 100! is the product all of the integers between 1 and 100 inclusive?
  2. A dragon has 100 heads. A knight can cut off exactly 15, 17, 20 or 5 heads with one blow of his sword. In each of these cases, respectively 24, 2, 14, or 17 new heads grow on its shoulders. If all heads are cut off, the dragon dies. Can the dragon ever die?
  3. What is the last digit in the following product? (2^1)(2^2)(2^3)(2^4)\ldots(2^{201})(2^{202})(2^{203})?
  4. An enormous 5 \times 5 checkerboard is painted on the floor and there is a student standing on each square. When the command is given each student moves to a square that is diagonally adjacent to their square. Then it is possible that some squares are empty and some squares have more than one student. Find the smallest number of empty squares.
  5. Suppose you are in a strange part of the world where everyone either always tells the truth (a Truthie) or always lies (a Liar). Two inhabitants, A and B, are sitting together. A says, “Either I am a Liar or else B is a Truthie.” What can you conclude?

The last type of “Truthie/Liar” problem is a standard one in logic, and I started including a lot of them at the end of the semester. This was both because students really enjoyed them and the students needed a lot of help getting the Perspectives habit of mind. Students had a very difficult time figuring out what this even means, and I need to do a better job helping them understand it in future semesters.

One consequence of including so many Truthie/Liar questions is that I would like to add a heuristic to the class list: “Break the problem into cases.”

One other thing that I would change about the course is the quiz structure. What I did was to pull problems that had been previously solved by members of the class. Instead, I would like to find 15 or 20 problems, present them myself to help teach/emphasize/remind students about heuristics and the problem solving process, and use these on the quizzes. This would solve a couple of problems:

  1. I had three sections, so I had to keep track of three sets of quiz questions. This way, I would only have one set.
  2. This would give students more time to digest all of the solutions. As I did it, students may have only had two weeks to learn a solution that was presented toward the end of the semester. If I control the quiz questions, I could pace them so that the last one is solved for them by mid-semester, giving them at least half of a semester to learn the solutions for the quiz problems.
  3. Similarly, I can raise the expectations for how many solutions they learn if they all have at least half of a semester to learn them. Depending on the problems I choose, I think that I could realistically expect a B-student to know all of the solutions.
  4. Perhaps most importantly, some solutions are more instructive and valuable than others. I would be able to show them solutions that can be modified to solve other problems.

I would also change one detail of the Problem Reports. I required at least three in each category to be eligible for a C, six for a B, and nine for an A. I think that three was too low, so I would probably change it to 5 for a C, 5 for a B, and 10 for an A.

Finally, I spent too much of the class letting the students freely try to solve problems. I need to figure out how to incorporate more instruction into these. For instance, I could charge each team trying out an assigned heuristic on a problem, let them work, and then have the teams report how they worked to apply the heuristic. This would regularly review the heuristics and help the students get in the habit of using them (I think that most students did not consciously use them).

Does anyone else have any ideas about any of this—particularly concerning the previous paragraph?

Update on Problem Reports

September 4, 2013

As discussed in my previous post, my “math for liberal arts” students need to demonstrate achievement (I don’t know if “demonstrating achievement” is the best term, but I am going to go with it for this post) in several learning goals by submitting “problem reports” and pointing out where exactly they are demonstrating said achievement.

We had our first problem already. The students submitted their first problem report on Monday. About three-quarters of them did not explicitly point out where they were demonstrating achievement in the problem reports; they tended to generally state something related to this on the cover page, but they did not link. or “tag,” it to the specific place where the demonstration occurs. This is a problem for two reasons:

  1. I want students to have to be very aware of what they are doing, and specificity is important to doing this. So it is important to “tag” each demonstration to make both the student and the teaching staff aware of what the student was thinking.
  2. It is really, really tough to grade these if the students do not “tag” the demonstrations.

I did provide a sample problem report that included this tagging, but the underlying problem here is that I did not support the students enough with this new/unusual way of grading. So here is my two-part solution:

  1. We did not grade Monday’s submissions; rather, the students can resubmit on Friday. I explicitly told them to “tag” their demonstrations for their resubmissions.
  2. I spent today’s class (Wednesday) writing up another sample problem report in front of them. So students saw exactly the process of how to create these.

I have a good group of students, so I am guessing that problem reports will be good on Friday. This is just a good reminder to me that you almost cannot communicate enough when you are deviating from what is typical in a mathematics class.

Yet another grading scheme

August 26, 2013

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?