The Introductory Set of Stella's Stunners

by Rudd Crawford

Here is a set of 21 problems that show the kinds of problems in the Stella library and how the collection can help you put more emphasis on problem solving with students:

The basic idea behind what my students call "Stellas" is that when they first face one, they don't know what to do. The problems are not an immediate application of what the students studied in their textbook that day, or even last week. Further, there is often a certain outrageous quality in the way a Stella problem taunts and teases with its difficulty.

Thus, the central question about Stellas is, "What do you do when you don't know what to do?" A major answer to that question is provided by a list of heuristics, or thinking strategies, suggestions to help out when you are stuck. Most of these heuristics are pretty simple: Guess an answer and see if it is right — if not, use it to help you guess a better answer; or draw a picture of the information; or break the problem down into smaller bits — determine subgoals. These heuristics can be a big help in moving the solver off dead center and start the mental wheels turning.

There is another aspect to heuristics as tools for thinking. A difficult problem, especially in mathematics, can be an unsettling thing to face. A typical student reaction is to worry, "There's a formula for doing this, but I can't remember it, so I have to give up." Or, "Everybody's going to get this before I do; I must be stupid." Heuristics can help students keep going even if they cannot remember "the formula" and can be a big help in easing feelings of frustration and encouraging a buoyant involvement with the problem.

The problems in this introductory set are discussed in some detail. Solutions are written out at length, showing ways in which heuristics can be helpful. Another part of the discussion describes how I have used the problem with students.

But this discussion is meant to be descriptive, not prescriptive. All of us at ORC hope that, as your students begin to explore Stella's Stunners, they will experience the mental charge that comes from figuring out an elusive problem, from "thinking outside of the box," so to speak. We hope that you, as the teacher, will experiment with ways to expose your students to the kind of creative thinking that these problems evoke, and which our educational system needs to encourage.

So give these introductory problems a whirl yourself, and try some of them out on your students. Then, if you like what you see, you can delve into the full library to find other problems to tackle. For ease in getting started, problems have been suggested for a particular course, but most problems are not tied to a single course.

Once you have taken the plunge, I hope you will notice how differently your students are using their minds from the way they typically do when they are simply "learning the material" of a course. And I hope you will find that these problems bring out the ingenuity and creativity that Americans are famous for. I hope you will notice that your students are really and truly thinking, and that they come to enjoy doing that. Stellas have spiced up my math classes whenever I have used them; I hope they'll do the same for yours.

I was chatting with a former student of mine, Ian Yarber, who is now recreation director for the city of Oberlin and a member of the Oberlin Board of Education. He said that a student came in to see him in his office about a math problem he was stuck on. Ian asked the youngster, "Haven't you been doing Stella's Stunners?" The child said he had not, and Ian said, "Well, if you'd been doing Stellas, you'd have figured out this problem by now."