Fasten Your Seat Belts

Abstract

A teacher and her mentor invite readers along for the ride as they use a fun logic game to introduce sixth-graders to advanced mathematical thinking. Mr. O'Brien and Ms. Barnett discover that this new route allows even students who previously fared poorly in math to arrive at the destination with their peers. WHEN IT CAME time to put a title on this article, we called it Fasten Your Seat Belts for two reasons. First, in the classroom tryouts of these activities, we had no idea what was going to happen next. We were entering uncharted territory for us, both veteran teachers, and for the children. And second, we hope readers will want to try the activities for themselves rather than merely read about them. And in that case, readers are advised to fasten their seat belts. One of the major characteristics of mathematics is that one can derive new information from old information with logical certainty. Put a penny on one hand, make a fist with both hands, show the fists to a child of 5 or so, and ask the child to find the penny. If the child chooses the fist with the penny, she will see it when the fist is opened. If the child chooses the other hand and is shown an open hand without a penny, she can't see the penny in the unchosen hand with her eyes, but she can see the penny with her mind. The fancy word for deriving information is inference. Try the hide-a- penny task with a 2-year-old. No go. Two-year-olds are pre-inferential. Inference is the heart of the mathematical concept of proof. The elementary and secondary school years are the time for the construction of logical operations and the development of tactics and skills that are critical for mature mathematical thinking. Indeed, such skills are central to thinking in everyday life. The Bunny Game At odd intervals once a week over a period of several months, we conducted inference activities with several classes of sixth-graders in a midwestern American elementary school. Our starting point was an inference game that we called The Bunny Game.1 What follows is a series of episodes from those classrooms, along with our comments on the thinking we observed. We alternated in the role of the teacher. Teacher: Imagine that a Bunny is hiding in one of the boxes of a 4 x 4 grid. Your job is to find it. The rules of our game are simple. You can ask about various boxes. If the box you ask about is touching the Bunny sidewise (that is, on any of its four sides), you'll get the answer Hot. Otherwise, the answer will be Cold. For example, if the Bunny is in the box shown, and you ask about box B-1, the box is Hot because it touches the Bunny's box sidewise. Teacher: Can you show me another box that's Hot? [The children cite A- 2.] And what about C-2? [The children say, Cold.] And D-4? [Again the children say, Cold.] Children: What about the box the Bunny is in? Is it Hot or Cold? Teacher: If you select the Bunny's box, you are a winner, and the game is over. Working in friendship groups, the children played the game with the teacher selecting the Bunny's box at random. The children also played the game with one another while we functioned as roving observers. Quickly the children became very proficient, and we introduced Bet your bicycle as an expression of absolute certainty. For example, in Game 2, most children were willing to say they knew for sure where the Bunny was. Walter: The Bunny is in D-1. Teacher: Are you willing to bet your bicycle? Walter: Yes, and my dog and my little sister. After a short period, we changed the rules slightly. To emphasize the act of inferring, we told the children that the Bunny game would now have two phases: gathering information and making a claim. Children could gather evidence as long as they wanted, but when they had enough information to tell with certainty where the Bunny was, they should say, I am ready to make a claim. …