Posts Tagged ‘Cooperative Learning’

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?


April 19, 2012

My elementary education students are creating vlogs that explain why different algorithms work for different operations. They have been creating roughly one video per week, posting them, and then getting feedback from the course grader. The only graded part of this is at the end of the semester after many drafts.

This week, we did a jigsawing-type activity to improve the videos (like most everything else, this idea was inspired by Andy Rundquist. On Tuesday, I split the students into four groups: one for addition, one for subtraction, one for multiplication, and one for division. The students came to class having watched all of the videos on their particular operation, and the class period was spent deciding what makes for a good explanation for that operation. At the end of the class, we split into new groups where one member of the group had just studied addition, one subtraction, one multiplication, and one division.

Today, we spent the entire class period reviewing videos in these teams. One team member was an “expert” on each operation from Tuesday, and they made suggestions on how to improve the explanations.

I asked everyone if this was useful enough to repeat on our fractions algorithms, and every student said that it was (most were emphatic). This appears to be a success.

My one reservation: although I am not sure, it appears that some students are trying to memorize a good explanation rather than understand. I know that I will be able to tell which students really understand from the oral exams, but I am wondering if it will be clear from the videos. Does anyone have any experience with memorizers?

Semester Summary

December 17, 2010

I outlined a plan at the beginning of the semester here, here, and here. Now, we reflect. (This is really a first stab at what I will present at the JMM).

  1. The Cooperative Learning was a tentative success. Anecdotally, the students this semester probably did slightly better as a whole than my classes have done in previous semesters. Slightly, but not a lot. I think, though, that there is great potential for improvement as I become a better wielder of CL. In many ways, I was a new teacher this semester. I made a lot of mistakes, and figured a lot of things out.

    Importantly, my students seemed to be much happier this semester. Will just a few exceptions, my students really liked working together. However, most of my students would have preferred a little more lecture from me. I speculate that part of this is habit, although I think that I should include slightly more lecture—perhaps 5-10 minutes at the beginning and end of each class.

  2. Students loved the Khan Academy videos.
  3. The new grading system was a smashing success…by which I mean that it was an improvement over the old system. Students enjoyed it, and it really seemed to create a less adversarial relationship with the students. Moreover, there were a handful of students who made remarkable turnarounds. One student went from an very low F to a C, one went from a low D to a C, and one went from a low D to an AB. In my decade or so of teaching, I may have seen people make such a turnaround once or twice; it happened three times this semester. I believe that the grading system helped the students understand what they needed to do AND was forgiving enough that it was worth the time to make up the work.

    That being said, it still has many of the same drawbacks as the traditional way of grading. Students were a little too motivated to “fill in a box” rather than to learn the material. I offered team quizzes during many of the classes: when a team felt like they all learned the material, a randomly-chosen team-member could take a quiz. If he/she got the question correct, the whole team got credit for the topic. It turned out that students would sacrifice learning time to take the quiz. Next time, I will either skip these or make it into a wager—students will get credit if they succeed on the quiz, but will have to eventually make up an extra question if they get one wrong. I am not sure what I will do.

    So I will definitely keep this system, at least until I can figure out how to get closer to eliminating grades.

  4. I ran to work four days each week (with some exceptions) until it got too dark in mid-November. I miss running to work already.
  5. GTD is still working wonders. I am on the hiring committee for my department, and it is helping me keep everything organized.
  6. My Seinfeld-method of keeping track of my research time seemed to work in non-November months. I spent at least an hour on research for 59 of 78 days this semester. Of the 19 that I missed, 15 were in November.
  7. I now wake up at 4:30 most mornings. This was a monumental help in keeping up with my life. I scheduled all make-up quizzes, looked at job applicant files, and did research from 4:30 to 6:45 each morning.

    It was a great semester with great students. Now if only my 59+ hours of research had been productive.

Cooperative Learning Partial Solution

October 1, 2010

I recently wrote about problems that I am having with CL. One such problem was that I was not providing enough closure for the students—I would like to have a large group discussion to go over the solutions.

This is difficult, though, since I am down to only a little more than 30 minutes for the students to work. One solution would be provide fewer, but more meaningful, problems for the students to work on. This is the long-term goal, but I need a short-term hack. I found it yesterday: I am going to start providing solutions at the front of the classroom. Students can then check their answers at their own pace.

This is in addition to my usual interacting with the teams. I will provide two sets of solutions, so there are really three ways teams can check their answers. Since I have 8-9 teams per class, I think this should work.

Cooperative Learning

September 29, 2010

Recall that I am teaching Essential Calculus and Linear Algebra this semester. I have been using Cooperative Learning (CL) this semester with mostly positive, but still mixed, results.

First, my students have been great. They have all worked hard and well with each other.

Next, I think that my Essential Calculus class has been working extremely well with CL. This is a mostly-conceptual class, and this lends itself well. I was thrilled when I had all 27 students—all non-majors—working hard on graphing derivatives; it made me realize how rare it was for me to have literally every student engaged when I used more lecture. Both of my Essential Calculus classes are much more engaged than my non-CL calculus class (for majors and the like) from two years ago. It seems like CL is working wonders here.

My Linear Algebra class is tougher to gauge. The students are all great and work hard. However, it seems like everything about this class is comparable to my non-CL Linear Algebra from two years ago; I was told that the non-CL class was pretty exceptional by my colleagues, so this is definitely not an insult to my current class. Still, I think that I have a lot of room to improve. By asking better questions in class, I could more fully take advantage of what CL has to offer.

Additionally, I have only superficially addressed group skills. This is one place where I could improve both classes, although I somehow do not believe it yet. Not only that, but the students (including the Essential Calculus students) do not seem invested in it, as they have not been doing the self-assessments at the end of class. This is likely largely because I am not terribly invested in it, although I think that there are other factors. For one, students still seem to be anxious to leave class—frankly a little too anxious, since they were initially trying to leave class 2-3 minutes early. To fix this, I changed the classes schedule from:

  1. First 5 minutes: Team warm-ups
  2. Next 15 minutes: Mini-lecture and questions about reading/videos
  3. Next 45 minutes: Team work
  4. Last 5 minutes: Team self-evaluation


  1. First 5 minutes: Team warm-ups
  2. Next 10 minutes: Mini-lecture and questions about reading/videos
  3. Next 45 minutes: Team work
  4. Next 5 minutes: Team self-evaluation
  5. Last 5 minutes: Mini-lecture to preview the next class

Having a preview for the next class is probably pedagogically a good thing to do, anyway, but it also keeps students thinking about the content until the end of the class. Still, this seems to avoid the cause of the problem: I am not invested enough in explicitly team group skills and giving time for group processing than I should be.

One final thing: I have not been doing a good enough job with large-group processing. Ideally, we would discuss the solutions to the problems the teams work on in class, but I have not been doing a good job at making sure this happens. So my 45 minutes of team work will likely turn into 35 minutes of team work plus 10 minutes of large group reflection.

This is still a work in progress, but it has mostly been positive.