Famous Math Problems in the Internet Age.
I am starting a new class at my high school next year called, “Advanced Mathematical Reasoning and Problem Solving.” I have gotten through the curriculum commission and also so far have 18 kids signed up for it. (Hoping for a couple more.) I still have to make sure that we will have enough room in the schedule to make it happen. I have a good idea of what I want to teach in this class, but I have one major concern at this point.
Problem Solving in the Internet Age.
I want students to take a set of 8-10 problems home on Mondays to work on alone for a week (or maybe ask each other a bit in person or by calling or text.) and then have students discuss their solutions to these problems on Monday. My problem is that I think that they can search online for the solution to the problems. When I was in school and given a problem my resources were me, myself and I. Well… I could ask a friend or go in and see the teacher, or maybe go to the math lab as well. None of these options would just have the answer already laid out for me, no questions asked, all in the comfort of my own home. How can I get them to let the problem percolate, or to work on it for awhile on their own? I know this helped me develop my mathematical ability a ton. I don’t want them to be deprived of waking up to an answer of a problem they’ve been thinking about.
I am thinking about using problems from “A Moscow Math Circle,” by Sergey Dorichenko. The solutions to problem sets 0 and 9 are online at the MAA. I’ve also typed in some of the problems into google and got solutions posted on various websites. I would also like to work on the Konningsberg bridge problem, the towers of Hannoi, map coloring, Euler’s formula, the locker problem, and many other famous problems.
How can I have students truly work on these problems for more than one class period without looking up the solutions online? I plan on talking about integrity, but when students get stuck on a problem and really want to know the answer, I think that even the most trustworthy student can catch themselves looking for the solution online.
It reminds me of all the kids that can solve a rubik’s cube. There is not one of them that actually got a rubik’s cube and solved it on their own. They all figured it out online, or from a friend who did the same.
With a rubik’s cube, it’s not likely that they would ever solve it by themselves without a tremendous amount of patience and perseverance. They walk around with a false sense of accomplishment, and everyone praises them for it. With the problems I plan to pose, they will also need some patience. How will I prevent them from taking the short path of looking online for the solutions? One of the goals for this class is for students to develop perseverance in solving problems… How will I make this happen?
To Accumulate a Rate --- Integrate! 
Having taught many courses like this, I have a few suggestions.
First, work to establish a classroom culture that values solutions, not answers. Make student presentation of solutions a regular part of class, and build a routine of interactive dialogue around the work. Anyone can look up the answer to a hard problem, but it takes a lot more to fully consume a solution and present it to others.
Second, celebrate creativity in problem solving. Encourage students to find multiple solutions to problems, and to think of things in creatively different ways. This de-emphasizes the value of the answer while promoting important problem solving skills.
Lastly, make problem creation a significant part of the class. Have students create, write, share, and refine their own questions. Not only is this an extremely valuable mathematical activity in and of itself, but it creates interesting problems whose solutions can’t be looked up. Not yet, anyway.
Great ideas! Thanks for sharing. I will use many or your suggestions in class. I knew that I wanted to make the problem solving process the most important and the answer less so, but I was unsure how I could manage that. We do, after all, get so excited about answers. I love when a student comes up with a unique solution and I’m looking forward to that aspect of this class.
Again thanks for the suggestions.
We don’t post answers to our problems, although they may be too simple for high school.
Where does the pressure to look problems up come from? Is it because they really, really, REALLY want to know the answer, or is it because their grade in the course depends on their figuring out the problems? If it’s the former, then I think regular “soapbox speeches” about not just integrity, but about how much more they will learn, and how much richer the experience will be for them, will have a good effect. If it’s the latter, I think that’s part of the problem!
I totally agree with that the grading of answers would be a motivation to look them up. I also think that another reason is that some student want to look or feel smart or have trouble moving on without getting an answer. How do you give students a grade for a class like this? Participation, discussion, presentations… other ideas?