The Locker Problem: Collaborative Problem Solving
Imagine a school with thirty students, each with one locker. The lockers are arranged neatly in a row down one hallway. One day, the students are bored. The first student walks down the hall and opens every locker. The second student walks down after her and open ever other locker. The third student walks by and stops at every third locker. If the locker is open, he closes it; if it is closed he opens it. The fourth student does the same with every fourth locker, and so on and so on, until every student has passed. When all 30 students are done, which lockers will be open? What if their were 100 students and 100 lockers? 1000?
Students were tasked to solve this problem and chart their solutions, as well as any patterns they noticed. This problem (one of my favorites) asks kids to come up with an organized way of solving a somewhat tedious problem. By pushing them past the original 30 lockers, though, students quickly realize that they cannot rely simply on a good system; they have to notice patterns, figure out the properties of different numbers, and use all their little wonderings and thinking along the way to extend the solution. In the process of doing this, students collaboratively uncover truths about factors, multiples, prime numbers, square numbers, odds, evens, and much more. It lays the groundwork for our unit that begins to introduce students to number theory and the concept of proof in math.
The students were so immersed in the work, that they spent two full days solving it and teasing out every possible pattern. One student liked the experience so much, he came over on day three and asked, "Can we do another problem where you have to figure out the answer, but really there's no answer, there's just a lot of cool things?"
