TITLE: Making Assumptions AUTHOR: Eugene Wallingford DATE: September 01, 2012 10:18 AM DESC: ----- BODY:
a dartboard
Patrick Honner has been writing a series of blog posts reviewing problems from the June 2012 New York State Math Regents exams. A recent entry considered a problem in which students were asked to compute the probability that a dart hits the bull's eye on a dartboard. This question requires the student to make a specific assumption: "that every point on the target is equally likely to be hit". Honner writes:
... It's not necessarily bad that we make such assumptions: refining and simplifying problems so they can be more easily analyzed is a crucial part of mathematical modeling and problem solving. What's unfortunate is that, in practice, students are kept outside this decision-making process: how and why we make such assumptions isn't emphasized, which is a shame, because exploring such assumptions is a fundamental mathematical process.
The same kinds of assumptions are built into even the most realistic problems that we set before our students. But discussing assumptions is an essential part of doing math. Which assumptions are reasonable? Which are necessary? What is the effect of a particular assumption on the meaning of the problem, on the value of the answer we will obtain? This kind of reasoning is, in many ways, the real math in a problem. Once we have a formula or two, we are down to crunching numbers. That's arithmetic. Computer science teachers face the risks when we pose problems to our students, including programming problems. Discovering the boundaries of a problem and dealing with the messy details that live on the fringe are an essential part of making software. When we create assignments that can be neatly solved in a week or two, we hide "a fundamental computing process" from our students. We also rob them of a lot of fun. As Honner says, though, making assumptions is not necessarily bad. In the context of teaching a course, they are necessary. Sometimes, we need to focus our students' attention on a specific new skill to be learned or honed. Tidying up the boundaries of a problem bring that skill into greater relief and eliminate what are at the moment unnecessary distractions. It is important, though, for a computing curriculum to offer students increasing opportunities to confront the assumptions we make and begin to make assumptions for themselves. That level of modeling is also a specific skill to be learned and honed. It also can make class more fun for the professor, if a lot messier when it comes time to evaluating student work and assigning grades. Even when we have to make assumptions prior to assigning a problem, discussing them explicitly with students can open their eyes to the rest of the complexity in making software. Besides, some students already sense or know that we are hiding details from them, and having the discussion is a way to honor their knowledge -- and earn their respect. So, the next time you assign a problem, ask yourself: What assumptions have I made in simplifying this problem? Are they necessary? If not, can I loosen them? If yes, can my students benefit from discussing them? And be prepared... If you leave a few messy assumptions lying around a problem for your students to confront and make on their own, some students will be unhappy with you. As Honner says, we teachers spend a lot of time training students to make implicit assumptions unthinkingly. In some ways, we are too successful for our own good. -----