Wednesday, February 26, 2014

Using "The Letter Game" to Introduce Deductive Systems and Proof in Geometry

A new trimester will begin in just a couple of weeks, and I will have a new bunch of geometry students.  Before long, we'll be digging in to geometric proof, which is a concept/activity/process that we will develop and use throughout most of the course.  It is always a tough concept to develop with students, and I am always wondering how I can make it easier for them to grasp.

One really nice, fairly accessible activity that we have used with students at my school is "The Letter Game", which was published by Don Gernes in his article The Rules of the Game (full citation below).  The game amounts to a very simple deductive system, in which students are asked to "prove" various statements.  It gives students an opportunity to practice working within the bounds a deductive system, and to get a sense for the structure of proofs and the process of proving something.  This comes before they have to confront the added complexity of geometry, which is the big deductive system they ultimately will be learning to work within.

Before getting to the "The Letter Game", Gernes suggests having students think about games that they are already familiar with: monopoly, basektball, soccer, etc.  Each of these games has undefined terms, defined terms, postulates, and maybe even some theorems.  These are the makings of a deductive system.  Gernes presents basketball as an example, and here's what he lists in each of these categories:
Undefined Terms

Defined Terms
Field Goal
Free Throw

If a player is fouled, then the player gets to shoot a free throw.
If a player travels, then the other team gets possession of the ball.
If a player makes a field goal, then the player's team gets two points.

The referee objectively applies the rules of the game to each play.

This obviously doesn't represent a complete list of all of the terms and rules that make up the game of basketball, but it is enough to have a discussion about what the undefined terms, etc. are, and how each of them contributes to the structure of a deductive system.

The Letter Game is where students get a chance to try their hand at some proofs.  Gernes keeps the system simple, so students don't have too much to keep track of as they begin proving "theorems".  Here's the deductive system he establishes:
Undefined Terms
Letter M, I, and U

x means any string of I's and U's.

1. If a string of letters ends in I, you may add a U at the end.
2. If you have Mx, then you may add x to get Mxx
3. If 3 I's occur, that is, III, then you may substitute U in their place.
4. If UU occurs, you drop it.

With these four postulates, students take given strings of letters and then "prove" another string of letters, using one postulate at a time to manipulate the string.  For example - Given: MIII   Prove: M.

Even though many of my students end up groaning about geometric proofs, most of them end up enjoying the letter game.  It is accessible enough that most of them are able to make some headway, and after a quick example or two, most of the class is able to work independently.  I usually don't get any students reverting to "shut down" mode because they can't get it.  Sometimes I even have students finished early who create their own new letter "theorems" and challenging one another to prove them.  While the activity doesn't make them masters of geometric proof, it does reinforce the concept of a deductive system, and what it is to work in and prove new theorems in a deductive system.  Every step that is taken or statement that is made needs to be backed up by a definition, postulate, or theorem.

Opening up the much more broad deductive system of geometry, with all of its definitions, postulates, and theorems, always makes geometric proof more difficult for my students.  The Letter Game hasn't eliminated those difficulties for my geometry students, but it has been a good first experience with proof for them.  Gernes' activities build a nice segue from concepts that students are familiar and comfortable with to the way geometry works as a system.  His article is definitely worth a look:

Gernes, D. (1999).  The rules of the game.  The Mathematics Teacher, 92(5), 424-429.

1 comment:

  1. Do you find students making connections back to this game? Or do you bring it up? Do they see it as a real game?

    I like the idea of started with a limited pool. It seems like the letters are more abstract, but also not tied down with previous learning and expectations.

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