Introduction to the Introductory Problems

The Introductory Problem Set consists of twenty problems designed to show you how to use the problem library so you can put more emphasis on problem solving with your students.

To see the list, go to the Problems area on the left and click on Introductory Problems. As usual, click on a problem's number to see it, as well as the detailed discussion of both it and its solution. These discussions show the ways in which heuristics from our list (see below) can be helpful. Another part of the discussions covers the ways in which I have used these problems with students.

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 the list of heuristics that is included here. Heuristics are simply thinking strategies, suggestions to help you out when you're stuck. Most of them are pretty simple: "guess an answer and see if it's right--if not, use it to help you guess a better one"; or "draw a picture of the information"; or "break the problem down into smaller bits--determine sub-goals". These heuristics can be a big help in moving the solver off dead center and starting the mental wheels turning.

Try a few yourself! What do you notice about how you're thinking and how you're feeling?

Another value of the heuristics is this: a difficult problem, especially in mathematics, can be a frightening thing to face. A typical student reaction is to worry that "there's a formula for doing this but I can't remember it, so I have to give up." The Stellas, and the heuristics backing them up, can help students keep calm and get going even if they can't remember "the formula." They can be a big help in easing feelings of frustration and encouraging a buoyant involvement with the problem.

It should be clear that the basic premise of what my students call "Stellas" is that when the students first face a Stella problem, they don't immediately know what to do. The problems are not an immediate application of what students studied in their textbook today, or even last week. They are problems that might not require a specific mathematical technique at all. The point of a Stella problem is that there is no obvious way to get started in finding a solution. Often, there is a certain outrageous quality in the way a problem taunts and teases them with its difficulty.

Finally for now, all of this discussion is meant to be descriptive, not prescriptive. All of us in the Stella Project hope that as you begin to explore Stella's Stunners, you will experience the mental charge that comes from figuring out an elusive problem, from "thinking outside of the box". And we hope that you will feel free to experiment with ways of exposing your students to the kind of creative thinking that these problems evoke, and which our education system needs to encourage.

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

Stella problems can evoke one's ingenuity and creativity. As you grapple with them yourself, notice how you are really and truly thinking, and (admit it) how good it feels to be doing that. And let us know how it goes.