Instructor’s Guide to PProblem SSSolving
This is a fun course to teach, especially if you are willing to put into practice the techniques you will be espousing. Teaching is, after all, a problem-solving exercise. Of all the classes I have taught (since 1964 when I entered graduate school), this one has taught me the most about teaching mathematics. You may not think of solving puzzles as doing mathematics, but to me, it feels exactly like doing mathematics. Moreover, the techniques that we mathematicians use can be learned and transported – to other classes and to life outside the classroom – even by students who claim to “hate” math.
The suggestions that follow are only suggestions. I would be surprised if anyone who got involved in the course even came close to following the suggested outline. As far as I’m concerned, the enjoyment in this class derives in part from the ability to be flexible, to go in the direction of maximum learning.
Put a “Stretch” on the board at the beginning of each class period (see
Chapter 9). You can use the stretches in order, or choose ones that emphasize
a point you hope to make that day. If nobody in the class can get the stretch,
I leave it unsolved and award 5 extra credit points for anyone who turns
in a solution at the next class period. Start with the first three stretches.
Go to class early so you can chat with students and learn their names.
Course Structure. See the home page. Discuss this carefully. Emphasize the importance of coming to class. You will need to repeat the important aspects of grading, handing in assignments, reading the book, etc. Having a web site with assignments posted goes a long way toward promoting student responsibility. On the first day you can mention some of the material from the first chapter of the book, and ask the students to read that chapter.
Assign an Automathography. Have students write you a letter describing themselves and their math backgrounds. I always ask that they include some personal detail that will help me get to know them as individuals. This is due the second class period. Sometimes I ask them to recall the best teacher they ever had, and list three things that made her or him outstanding. You can also do a worst teacher version.
Play Heap of Beans. I like to get them going on a problem as soon as possible, so I try to start this game on the first day. I recommend buying a bag of split peas (they don’t roll off desks) at the store to use in class. There are a number of games that employ peas or beans. Have the students pair up to play the game. Give beans to each pair and walk around observing or asking questions as they play. You can read through the book’s discussion of the game in Chapter 2, so you can help them begin to think systematically about solving problems. If there is not enough time for most of the pairs to solve the game, tell them to think about it for next time.
Go over PSSSP. This is a good thing to do after they have worked on a problem such as Heap of Beans. Discuss briefly each of the five strategies, and in particular how they apply to Heap of Beans. After I discuss the solution to the game, I ask the students to write a letter to a friend or family member explaining how to win the game. The purpose is to begin to emphasize the connection between clear writing and clear thinking. On this first writing assignment I ask them to rewrite the assignment if I don’t think it gives a clear explanation. No credit is given until a satisfactory version is turned in. Point out the Checklist for Writing Projects in Chapter 10.
Additional assignments. If there is extra time, you can have the class work on the problems Two Bean Heaps or Four Bean Heaps from Chapter 11. Or you can pick other problems that might help introduce the PSSSP techniques.
Work on learning names. You can refer to the automathographies of the students whose names you forget. The details will help you remember them. Always encourage ideas, even incorrect ones. Good problem solvers learn more from mistakes than from successes. Good teachers use incorrect answers to promote learning.
Day 1. Stretch #4 (The sequence lists the initials of one, two, three, etc.) Discuss Proactive (Chapter 3 up to p. 19). Have them work on Changing 50 cents (in class in groups of two)
Day 2. Stretch #5. Discuss the rest of Chapter 3, including the gamesworth. You can give a short assignment for in-class work. Assign a Gamesworth on Seven Elevators (or some other problem), due next time.
Day 3. Stretch #7. Discuss Seven Elevators (or other Gamesworth). Then
begin the chapter on See it (Chapter 4). Green Party (in class, or due
Discuss See it - the usefulness of pictures, tables, charts and models in problem solving. For example, using a model (beans) and charting the solutions to simple versions of the problem helped solve the Heap of Beans game. Similarly, a good picture goes a long way toward solving the Commuter problem. You might also mention that the Wright Brothers beat all their competitors into the air partially because they were so meticulous about keeping tables and charts of all the data they collected.
I find it helpful to Pause and Reflect with the class at this time to remind them that late homework is not acceptable without advance notice and that they are responsible for everything that goes on in class. Students, especially freshmen, need all the help they can get in being Proactive about their learning.
Day 1: Stretch #10. Emphasize pictures and charts, assign the Green Party (if not yet done) and the Prom Problem (in groups). Another possibility is to pass out a worksheet on Seven Elevators (they can use lots of help on this one). Due in two class periods
Day 2: Stretch #6. Discuss mental images (p. 31) Read the seashore paragraph out loud and see if anyone knows what it’s about. Read it again after saying it’s about kite flying and emphasize again the importance of forming mental images to help understand and solve problems. Then assign “Acrobats” (groups)
Day 3: Stretch #9. Discuss Memorization, p.33. Do the “One is a bun” exercise in class, using a few volunteers who claim to have a poor short-term memory. Then have the students do the exercise with each other in pairs. The problem “Parking Lot” is a good one to assign if you need extra work for the students.
Make sure the class knows when the first exam will be. Do you know everyone’s name? We’ll finish with See it, and start on Simplify it.
Day 1: Stretch #8. Discuss visualization (p. 35). Maybe you have an example from your own experience to relate. If not, read the autobiography of a famous athlete. You’ll almost certainly find an example there. Assign Achievement Scene (p. 36). The Largest Lake problem is a good one (even for a calculus class). Discuss the Clear Writing = Clear Thinking section of the See it chapter as a way to emphasize that answers must be explained clearly and neatly on all assignments and exams. Remind them about the Checklist for Writing Projects in Chapter 10.
Day 2: Stretch #13. Simplify it. Discuss the use of simple cases, as in the Heap of Beans game, for example. Do you know of a famous problem that was solved by starting with a simple case? Changing Places (p.38) is a wonderful example of a game that is very difficult unless you start with a simple version. Have them work on the game in pairs. You can also assign Handshakes (p. 39).
Day 3: Stretch #15. Simplify it. If you haven’t assigned it already, Parking Lot (p. 41) is a good one for them to work on in groups. They need to make assumptions, as in the Commuter Problem, and start with simple cases.
Emphasize again the highlights of simplification. If a problem contains a large number (like 1000 or 40 or 10), try it using a smaller number such as 0,1, or 2. This technique can be used repeatedly throughout the 5th week, including exam day!
Day 1: Stretch # 20. Simplify it. Have them work on the Forty Thieves problem (p. 81) in groups of 3. Starting with just a few thieves can clarify the situation. But they will still need significant help. A pile of coins and a dramatization of the problem with 2,3 and 4 thieves chosen from the class can be amusing and instructive.
Day 2. Stretch #12. Simplify it. Finish the Forty Thieves
problem (if possible) and review for the exam. You can make Forty Thieves
due after the exam.
I ask three kinds of exam questions.
(i) Problems that are like, or almost like, ones we did in class – Heap of Beans, Commuter, Elevators, Green Party, and various stretches.
(ii) Explain the PSSSP techniques and give examples of their use.
(iii) Work on a gamesworth type problem, where you are not expected to get the answer, but merely make a good attack.
Day 3. Exam 1.
Day 1. Stretch. To lead off the discussion of Stir it up, I use some kind of problem with two equations and two unknowns, such as #43 in Chapter 9. You can solve the problem using linear algebra, but the easiest method is just to guess the answer, check your guess and the make a new guess. Even the most recalcitrant of problem solvers should get the answer in 3 or 4 guesses. Next I put the Cryptarithmetic problem on p. 43 on the board and we work it out as a class. Then I assigned Cryptarithmetic 1-5 (p. 78) to be worked on in groups. Note that there is a typo on the 3rd problem. It should be SEED + ICED = SPICE. This assignment is due on Day 3.
Day 2. Stretch #16. Emphasize that the only way to do this is guess and check. It helps to by systematic and record your wrong guesses. Add Ages and Grandfather to the assignment due on Day 3 of this week.
Day 3. Stretch # 18. The Stir it up lesson for the day is to experiment. I always have fun with this class. Bring three pieces of paper with scotch tape on them labeled 1,2 3. Tape them on the board to represent the doors on the game show discussed in Should You Switch on page 45. You can have the class cover their eyes while you put a $ symbol behind one of the doors. Have someone guess which door the $ is behind. Then show them a door that the $ is not behind and offer him the chance to switch. You can play the game a few times and then have the students play it in pairs, as described on page 45. They play ten times not switching and ten times switching. Have them keep track of wins and losses using each method and combine the class results on the board. You should end up with the switchers winning about twice as often as the non-switchers, although there will be plenty of “experimental error.” This may take the entire class, and you can discuss the results next time. Sometimes I then ask them to write a letter to a parent or friend explaining why it is best to switch. Another tack is to give them a problem with 4 doors (or 10) where the host shows two (or 8) empty doors after the initial guess. In the 4-door case, the odds change to 3:1 in favor of the switchers.
Day 1. Stretch #23. Again, this problem can only be done by “educated” guessing. Give a brief discussion of brainstorming, using the ideas on pages 45-46 of the text. The story about the Gillette executives is a good one. Then have them do the Eraser exercise on page 46. You can collect the (different) ideas on the board to illustrate the benefit of multiple minds. Have them form groups to come up with solutions to Light Switches by the end of the period. They probably won’t get the “real answer,” which uses the fact that light bulbs get hot when they are turned on.
Day 2. Stretch #19. Assign the problems Faculty Debts and Magic Cuts to be worked on in groups. Both these problems involve “Seeing” in unusual ways. Faculty Debts requires you to write out all possibilities, not just those where there is a “least” debt. Magic Cuts requires you to hold the diagram upside down so that a 6 becomes a 9. You will need to give judicious hints. Due next week on Day 1.
Day 3. Stretch #40. Add Die Hard III and Checkerboard Chase to the assignment for next week. You can discuss the idea of working backwards first, and emphasize it as you help them in their groups.
Day 1. Stretch #18. This week’s theme is Pause and Reflect. Even though #18 (magic square) is a trial and error type of problem, there are two observations (made by Pausing and Reflecting) that will help shorten the work. One is to determine what the sum in each row of the square will be. The other is to think about what the number in the middle should be. Have the class divide up into groups to come up with a Plan of Attack for Multiple Locks and Dating Service (page 50). The Plan should include understanding the problem, and for Multiple Locks, simplifying, Dating Service, trial and error. The point here is to think a little about the problems before plunging in. Once a group turns in a Plan, they can proceed toward a solution.
Day 2. Stretch # 22. Pausing and reflecting on #22 might lead the instructor, if not the students, to notice that the game of Fifteen is analogous to playing Tic Tac Toe on a magic square constructed in Day 1. Have the students work on Grilled Cheese (page 82). They should develop a Plan of Attack that uses models, simplification and brainstorming. Stacking the sandwiches is the key. Put the times obtained by the groups on the board, then ask them to Pause and Reflect by discussing the problem with friends and sleeping on it. Probably no one will get the lowest time on the first day.
Day 3. Stretch #21. Some reflection might show that #21 is like #12. Have the groups continue working on Grilled Cheese. You can walk around and give hints. I let them assume the sandwiches start on the grill, and do not have to be removed at the end. This saves some “putting on” and “taking off” time. Milk and Coffee (page 84) is another good problem if you need one.
This week we pause and reflect for a moment on how the problem-solving skills we have acquired can be applied to interpersonal problems. This is an excellent opportunity to demonstrate the broad applicability of the PSSSP strategies. In particular, we can address the problem of working in small groups.
Day 1. Stretch #63 (humorously interpersonal). I suggest leading a class discussion on how the students think PSSSP techniques might be applied to interpersonal problems. You might begin with a 2-3 minute brainstorming session after dividing the class into five groups, one for each S and P. Eventually, you should be able to work in the 11 boldface ideas in the text:
P: Take responsibility, It’s only a problem
P: Deal with emotions first, Begin at the end
S: Listen with understanding, Visualize, Write it down
S: Think win-win, Say it right, Affirmation and humor
Assign Mott’s Messes, to be worked on in groups.
Day 2. Stretch #67. At this point I assign each member of the class to a permanent group of 4-5 people (5 if some members don’t attend class very often) that will work together for the rest of the semester. I try to mix weak and strong students evenly through the groups. Frankly, assigning groups at random often seems to work better, but this way enables me to force students to work with new partners and contains some elements of fairness. It definitely works better than letting the students pick their own groups. Assign the new groups to work on Why Groups (page 72) and Grappling with Groups (page 75).
Day 3. Stretch #24. This day you can hand out a major group project, such as Sessa’s Wheat (page 89). At the same time, have the groups formulate a Team Performance Agreement (page 74). They can spend any extra time brainstorming about the major project.
There are plenty of other problems and projects that can be assigned around the theme of interpersonal problems. See, for example, My Stuff and Mission Statement on page 60.
This week’s theme is estimation, which should tie in with the larger
project you have assigned. This year I am having the class estimate the
circumference of the earth, but
there are a number of other suitable estimation projects: Sessa’s Wheat, Lease or Buy, Trouble With Tribbles. Be warned that a number of your students will not know the difference between (linear) feet, square feet and cubic feet. If you don’t believe me, ask them how many cubic inches in a cubic foot. As always, pictures and other ways to See It are a big help.
Day 1. Stretch #25. You can introduce confidence intervals for the estimations in the stretch. The project groups should be completing a Team Performance Agreement if they haven’t already. Then I let them work on their Plan of Attack for the project.
Day 2. Stretch #28. You can lead a discussion of confidence intervals after the students work on the stretch. For example, if they do 90% confidence intervals for their guesses, 90% of the class should get the right answer in their confidence intervals. In general, we tend to be overconfident with our guesses. You many find that fewer than 90% get the right answer. Assign Largest Lake (page 83) to be completed by next class period.
Day 3. New Stretch: Estimate the number of pizzas the class eats in a year. Here again you can have them give a 90% confidence interval. If it’s a nice day, the project Maple Tree (page 83) is a nice exercise. Have them estimate the height of a tree in at least two different ways. Have them provide 90% confidence intervals. The Plans of Attack should be collected.
This week is primarily review and examination. I generally put about five questions on the exam.
1. A rehash of a stretch or two.
2. A problem on applying PSSSP to interpersonal problems.
3. A revisit of one of the projects they have worked on.
4. A gamesworth type problem – attack intelligently without necessarily solving.
5. A question about the larger project they are working on.
Day 1. Stretch # 36. Review for the Exam
Day 2. Exam.
Day 3. Stretch # 37. Discuss the exam and the Plans of Attack. Data collection should be well under way. Chair Challenge is a good team project to assign for class time if they aren’t busy with the project.
From now to the end of the semester, the general goal is to finish up the major project and re-emphasize the PSSSP techniques as they are applied to the remaining assignments. I like to start with a Pause and Reflect discussion on what the goals of the course are.
Day 1. Stretch #66. A good See and Stir project is Polyominoes from Chapter 11.
Day 2. Stretch #42. For a fun sort of Stir project, assign House Hunt from Chapter 11. Both projects for the week due on Day 3.
Day 3. Stretch #44. This day can be for completing the two projects above or working on the major project.
Day 1. Stretch #48. Cycling Heaps (Chapter 11) is a good project to reinforce See, Simplify and Stir.
Day 2. Stretch #52. Weights (Chapter 11) emphasizes See and Stir.
Day 3. Stretch # 53. Two previous projects due. Work on major project.
Day 1. Stretch #54. Dollar Auction (Chapter 11) is a good project
for the last week. You can usually get two students in a bidding war so
that they both end up bidding over $1 for your dollar. Excellent analogy
with cold war arms escalation.
Day 2. Stretch #55. Finish major project. You may want to have groups make short presentations of their work.
Day 3. Stretch #57. Finish major project, review for final. I make the final pretty much like a longer version of the two one-hour exams. I include one question about PSSSP techniques and one gamesworth type of question, along with a question on the major project.