Problem Solving for the Liberal Arts

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?

14 Responses to “Problem Solving for the Liberal Arts”

1. TJ Says:

Hi Bret. Do the students work on the problems out of class or in class, or both?

Anyway, I really like the idea of assigning heuristics for the first few weeks. Once they have seen them, you can more reasonably expect them to work independently from you (maybe still in teams).

And be sure to tell them that you are giving them the tools they need:

“This problem solving strategy is called X. It is often useful, and you should use it often. Eventually I will expect you to do it, so that if you want to ask a question, I might ask you if you have tried this first.”

In my experience with this type of audience, they don’t necessarily pick up this kind of thing unless you have a parade.

• bretbenesh Says:

Hi TJ,

The students started working on each problem in class, but were left to finish the problem at home. Do you think it would be worth revisiting problems again in class if no one solves them?

I did a good job giving them a parade; I did not do a good job giving them multiple parades for each heuristic, which is what I think I need to do. My standard reply when someone asked for help was, “Hmmm. . .is there a heuristic you could try?” They got to the point where they started solving easier problems, but didn’t do so well with the other heuristics.
Bret

• TJ Says:

My gut feeling is that it might be good to group the problems for each strategy early in the term. And let it be ok if a problem doesn’t get resolved exactly. Slowly build a list of engaging problems that have a variety of methods tried on them. It might even be good to have a solved problem reappear, but ask for a different attack!

Anyway, at some point in the term, you will have covered all the main ideas, and then you ask questions which are more obviously open ended. Encourage discussion from students comparing effectiveness of different approaches, etc etc.

But in my head, that all fits with a lightly modified Moore method structure. Some group work at the beginning. Lots of presentations. Eventually, group work disappears into presentations and whole class discussions led by students. This presupposes lots of out of class thinking, which I know I don’t always get from my LibArts students.

Problems near the end of the term would have a variety of flavors, so students would have an opportunity to develop insight, while you gently fade away…

Um. How big is a section of this class?

• bretbenesh Says:

Hi TJ,

Each section is roughly 25 students.

I like where you are heading here. I have some questions.

“My gut feeling is that it might be good to group the problems for each strategy early in the term.”

Do you imagine that I tell the students which heuristic(s) would be helpful for each problem? Or would this list only be for me?

“Encourage discussion from students comparing effectiveness of different approaches, etc etc”

Like.

“But in my head, that all fits with a lightly modified Moore method structure. Some group work at the beginning. Lots of presentations. Eventually, group work disappears into presentations and whole class discussions led by students. This presupposes lots of out of class thinking, which I know I don’t always get from my LibArts students.”

I definitely did mostly group work at the beginning, and it morphed into “a small amount of presentations.” I think that the main problem is that the students really didn’t put in a ton of time outside of class. But I think that this is probably because I didn’t support them enough (this conversation should help with that).

“Problems near the end of the term would have a variety of flavors, so students would have an opportunity to develop insight, while you gently fade away…” Said a different way, I think that part of my problem was that I faded away too soon.

Here is my take-away: I solved roughly 10 problems for the students at the beginning of the semester, and I ended up giving them 40 problems for them to solve on their own throughout the course of the semester. Perhaps it would have been better for me to give them fewer problems but spend more time on each problem—maybe 25 problems instead of 40. I would need to reconcile this with my Problem Reports structure, but it is definitely doable.

I like this whole idea; I am going to think of it as I am driving to Grandpa and Grandma’s. Bret

On Fri, Mar 7, 2014 at 10:13 PM, Solvable by Radicals wrote:

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• TJ Says:

Well, 25 students stretches a Moore method class.

And while we are largely on the same page, I would ask many more questions! Maybe 100. Several of them would likely be easier things near the beginning, to build confidence. Maybe some hlaf-measure questions: “this problem is intimidating, but we are using the SOLVE A SMALLER PROBLEM strategy. Find two different ways to make a smaller, easier (if still challenging) problem to work on.”

• bretbenesh Says:

“And while we are largely on the same page, I would ask many more questions! Maybe 100.”

How do you envision meting out these problems? Give them all of the problems on the first day of class? Giving the students 2 problems per class period for the entire semester?

You seem to have a vision for this class. Your vision is hazy to me, but what I can see of it seems very promising.

2. suevanhattum Says:

I don’t have any suggestions, but I am loving some of the problems you posted. I get 7 empty squares on the chessboard, but I don’t have a proof. The dragon problem made me laugh. We did a great dragon problem at the Math Circle Institute last summer.

• bretbenesh Says:

Hi Sue,

I am about to email you offline—I don’t want solutions/hints on my blog. Some of these solutions are easy enough to find online as they are!
Bret

3. suevanhattum Says:

Oops! Sorry. Please delete my comment if it was a useful hint.

• bretbenesh Says:

It isn’t a problem, so there is no need to delete. I was just concerned about _me_ saying something that I shouldn’t.

On Sat, Mar 8, 2014 at 12:31 AM, Solvable by Radicals wrote:

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4. mrdardy Says:

Bret

Fantastic stuff! Do you have a source for these problems? I would love to try to design something along these lines for my school. We are on a trimester schedule here. I am a happy man anytime that Polya gets referenced.