When I want to assess my students’ problem solving abilities to begin the semester, I ask one simple question. It’s the title of this post. So simple. It requires nothing more than whole number addition and multiplication and a rudimentary knowledge of the calendar. But watching student after student answer the question incorrectly, I lay the groundwork for them to see where they may be deficient in their skills and would want my instruction.
Before we get too involved in the specific mistakes
that they make, let’s talk about what I consider problem solving to be. George Pólya published an important book in 1945 called How to Solve It. It’s a small book, still in print, and it’s still in print for good reason. It provides the basis for the four steps of problem solving:
- Understand the problem.
- Make a plan.
- Carry out the plan.
- Look back.
These four steps (and subnotes that go with each one) are what I drill into their heads for the entire semester I have them. Seriously, that’s all I really teach them. I fill gaps in their knowledge and do some fun stuff too, but following these four steps is the most important lesson I impart. But I don’t open with it.
When I ask them how many days old they are, almost to the last student, they begin with step 3, which is the part where you do your computation. Almost all of them start by multiplying 365 by how many years old they were on their last birthday. But that’s typically where they finish, too. They don’t take into account any extra days beyond this simple computation because that’s what they’re used to.
After everyone has answered to their own satisfaction, I have them write their answers anonymously on the front board. We take a step back once everyone has had a chance to do so, and we look for patterns. Some will have given the problem more thought, but the obvious mode of this data set is typically 4,380. Most of my kids are 12. When I ask for hands up from the students who wrote 4,380, I quickly ask them for their birthday. They usually have different birthdays, so I ask them how they got the same answer. Then, finally thinking occurs! When they realize that there have been days since their birthdays that did not get counted, they want to revise their answers. I encourage that. Always.
Then I start looking for other pairs of answers that are very close to one another. If I find a pair that are two days apart, for example, I check their birthdays, too. If they were not born two days apart, I ask them to pair up and find out why there is a discrepancy. I don’t focus on who’s wrong and who’s right. I just want them to find out why their answers are different. After we take another stab at the problem, we come back together again and I ask them why their answers were different. Usually, it’s because one of them considered leap years, where an extra day was added to the calendar, and the other one didn’t. This revelation brings the 4,380 crowd nearly to tears. “There’s more math?” And yes, they revise again.
Bear in mind here, that I never tell anyone that they’re wrong. I also never tell anyone that they’re right in this process. It’s too early for that. I simply encourage them to go back and revise their work and then find someone to compare answers with. When we come back together again on this second day, I ask them what made this question harder than it first appeared. The consensus is usually that they didn’t really understand the problem.
Now they are ready to learn the first step of problem solving.