Here is a simple game that is wonderful for building perseverance and problem-solving skills. There are many ways to play the game. The simplest, and the one show my students first, requires 7 counters. These can be snap cubes, paper clips, or pennies. Anything small, relatively uniform, and easy to pick up will work.
This game is best played with 2 players. The object of the game is to be the player who removes the last item from the pile. Players alternate turns and can remove one or two items each turn.
There is a winning strategy for the game, which most children will figure out after playing the game a number of times. Once they have figured out how to win the game with these original characteristics, change either the number of items that can be removed at a single time, or the total of number of items in the original pile, and play repeatedly until the student figures out the winning strategy for this configuration.
When a student has figured out a winning strategy, help her to put the strategy in words and write it down. You can present the task by saying, “If you wanted to teach a friend how to win at this game, what would you tell her?” As I’ve mentioned in other blog posts, this sort of writing is extremely complex and demanding, so I would recommend that you as the adult write down what it is the student says. Take the student’s original words and make suggestions about how they could be more precise by pointing out any expressions that are unclear or ambiguous. Once you’ve got a precise statement, copy it onto paper, so you can compare it with the winning strategies of other variations (i.e., when you are starting with a different number of blocks or are allowed to take away a different number of blocks).
Once students have played a number of variations using counters, try the variation here in which players advance on a “ladder,” and the winner is the person who reaches the top rung first. You can find another version of the game, this one using playing cards, in Math for Smarty Pants by Marilyn Burns, p. 64.
If you are working with students in middle school or beyond, challenge them to come up with a winning strategy that would apply to any configuration.
This game, as well as the Factor Game, which I wrote about in my previous blog post, are both very engaging and can help students to think analytically. But even more than that, they both nicely model an excellent structure for presenting almost all new material in math, certainly up to and including beginning algebra. In both games I’ve described, students begin by doing a certain activity several times. After they have had some experience with the activity and have a beginning sense of how the activity works, they are asked to analyze it, that is, to figure out in much more detail, exactly how it works, which means, understanding the fundamental relationships at play. The activity is governed by rules that make it clear whether a suggested solution is adequate or not. The student’s analysis will involve some trial and error and will benefit from some way of recording their observations. The teacher’s role throughout this process is primarily to serve as a coach who can help the student record her work, break it into smaller parts, and help her evaluate the adequacy of her solutions. Although previous knowledge is useful, if not essential, to perform the task, the student’s job isn’t simply to parrot some previously learned information, but to apply it to a particular task.