I have this 64-year-old book on my shelf by G. Polya called How to Solve It. In the preface, Polya states:
“A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest: but if it challenges your curiosity and if you solve it by our own means, you may experience the tension and enjoy the triumph of discovery.
… thus, a teacher of mathematics has a great opportunity. If he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for independent thinking.”
For 18 years I have been trying to find the balance between skills (easy to teach), concepts (usually harder to teach) and problem solving (the hardest to teach) in my math curriculum. I find that the systematic use of Problem of the Week (POW) effectively addresses the development of problem solving skills as well as provide opportunities to review concepts and skills that can atrophy without consistent practice.
I assign one POW as homework every other week to both my 7th and 8th grade classes in addition to about 20 minutes of math homework from our text. To help in time management, I provide a 7-day calendar on the back of the problem with bulleted suggestions on what to do each day. A central idea of POW’s is that complex problems often require time to solve and may be best worked on little by little over the length of the assignment. The calendar provides my students a metaphorical “hook” on which to hang their problem solving process. It also serves as a gentle reminder that the “W” of POW stands for Week, not Sunday evening. I assume about 20 minutes of work on the POW each evening, which brings the grand total for homework assigned to approximately 40 minutes.
Since I generally assign POW’s on a Monday, I ask my students to refrain from working together on the problems until Thursday. They are allowed to ask other adults, including parents and teachers, for guidance at any time. I find that the wait time for peer “help” cut down on blatant copying and refocuses the energy on the independent nature of the task at hand.
When the problem is handed out on Monday, we take some time in class to read independently, underline crucial information and ask any clarifying questions. Depending on the nature of the problem, I may offer a hint for a procedure, such as a saying that a diagram would really help or if the problem is similar to previous ones. Often there are students who offer this same information without my prodding, which is preferable.
For the rest of the week I purposely do not bring up the POW to the class again unless students ask about it. I am trying to create an environment for independent problem solving. Of course, one great problem solving strategy is to search out expert help, so when questions about the problem arise, I always encourage my students to ask them in front of the whole class as a way of opening up the topic without my necessarily leading it. At this point, I offer support; hints and comments about the problem that I hope provide appropriate support without actually answering the problem
I expect each student to hand in a one-page, 4 paragraph essay about the POW one week after assigning it. This essay is very structured and highly scaffolded. I want my students to respond concretely and efficiently to four aspects of the problem solving process I emphasize in my curriculum. The paragraphs are as follow: Introduction, Procedure, Solution and Reflection.
The introduction includes a rewording of the problem to show understanding, a list of crucial information gleaned from the problem to help set the stage for solving it and a description of what type of math is involved in order to nudge them to recall past problems that may help them with the current one.
The procedure paragraph is basically a narrative of what they did and why they chose to do it. Depending on the problem and the student, this could include dead ends, outside help, and, hopefully, break through moments. I find that many students have benefited from writing this paragraph sequentially, almost like a list (first this, then this, then this…. finally this).
The solution paragraph, aside from offering an answer (if one was found) should describe it in a clear and unequivocal fashion. It also needs to address the plausibility of other solutions.
The reflection paragraph has become the most valuable insight I have into my students thinking process. This paragraph needs to include what they found challenging about the problem as well as what type of learning they got out of it. It can also include a short description of their week in general and the challenges they face budgeting time, balancing friends, family and sports and so many other interferences on the process. This paragraph has allowed me glimpses into the challenges an adolescent faces not only with math, but also with life in general.
I grade the essays using a 4-point rubric for each of the four paragraphs. The bottom of the rubric includes two spaces for comments, one for strengths and one for suggestions. The POW essay grades greatly inform the math processes grade I give my students in math each. I have about 60 students in a given year, 32 seventh graders and 32 eighth graders. I spend between one and two hours per week reading POW essays. I have found myself reading them almost backwards, which is to say, I tend to read the reflection first, then breeze through the rest of the essay as quickly as I can. I alternate weeks between the two grades so I am not attempting to read 60 essays on any given week.
I try to be very careful when choosing problems. I avoid selecting problems that are topical to the units I am teaching. I do not want to not create a sense of simply practicing skills taught previously in class. I find myself preferring problems dealing with topics in math not generally covered in many curriculums, such as probability, statistics, certain topics in geometry and measurement and discrete math. I do not like “famous” problems because my students are increasingly becoming adept at finding the solutions on the Internet. Here is a typical problem I have assigned to my 8th graders:
The Musicians' Dilemma
Four musicians are late for a gig. They have seventeen minutes to cross a dark and scary bridge to get to the concert. It takes the band's drummer ten minutes to cross, the guitarist five minutes, the trumpeter two minutes, and the singer one minute. They only have one flashlight, and no more than two people can cross the bridge at a time.
Because the bridge is so dark, they have to use the flashlight for each trip. Keep in mind that throwing the flashlight and stringing the flashlight on a line are not permitted (no gimmicks, please). Only a person can carry the flashlight.
How can this be done? Is there an unique solution? Be sure to explain your answers in detail, and answer both questions.
I have several sources for problems, such as: Mathforum.com and a book called Crossing the River with Dogs: Problem Solving for College Students. I often try to personalize the problems to give a sense of place and time that my students might relate to as well as to keep my students guessing as to whether I create the problems myself. Again, I do this to avoid any implication of simply “finding” the answer from a book or a web site.
The systematic use of Problems of the Week is a crucial pillar in my math curriculum. The provide opportunities for my students to explore their thinking with a diverse set of non-standard math problems and develop their problem solving skills. For many students, they are a source of family discussion and sharing of experience at a time when many adolescents are increasingly keeping their school experiences to themselves. The essays provide a pathway to writing clearly and concretely about their learning processes. Finally, they offer me insights into student thinking. Their biweekly essays offer me windows into their lives that goes beyond mathematics. Their reflection paragraphs have given me a much greater sense of compassion for the trials and tribulations my students go through to be successful in school.
Johnson, K; Herr, T; Kysh, J Crossing the River with Dogs: Problem Solving for College Students Emeryville, CA: Key College Publishing, 2004
Math Forum @ Drexel
G Poyla How To Solve It!
Palo Alto: Stanford Press, 1943