Want to Play Geometry?

Kaufmann, M. L., Bomer, M. A., Powel., N. N., (2009). Want to play geometry? Mathematics Teacher, 103(3), 190-195.

The main idea of the article is that sometimes alternative, fun, and interactive methods in the classroom can be used to explain mathematical concepts and ideas. The authors demonstrate this by describing an approach they took when teaching kids the structure and meaning of a proper proof. They put the class into groups and gave each group a die, 5 marbles, an egg carton, and 15 chips. They asked each group to use the pieces to develop their own game, for which they would write the rules for. They groups then exchanged games and critiqued the games made by other groups, looking for contradictions, incompleteness, and repetition. After this activity the teachers discussed with the class the similarities between the rules of their games and the rules and axioms involved in writing correct geometrical proofs. They argue that this is a quick, inexpensive, and motivating way to teach students about "analysis, synthesis, and evaluation".

I would definitely use something like this in my classroom. It is easy to see from the article that the students 1) enjoyed the activity 2) developed a conceptual, relational understanding of proofs 3) could relate these principles to the world outside the classroom. One student was even able to relate the principles they learned to the United States legislature, and the rules outlined in the Constitution. For each new big concept I wish to convey to my class, I want to develop an interactive activity to first allow them to explore the concept on their own and obtain a relational understanding.

Patterns Jumping out of a Simple Checker Puzzle

Staples, S. G., (2004). Patters jumping out of a simple checker puzzle. Mathematics Teacher, 98 (4), 224-227.

I believe that in her article, Susan G. Staples was trying to convey that unexpected mathematical patterns and problems can be found in simple games and activities, and that we can use these in our classrooms to convey to students the possibility of real-world applications of math. The article revolves around a simple checker game, where three black chips and three white chips lay on the edge of a seven space board, as shown below.

The object of the game is to get the black chips where the white are, and vise versa. The only possible moves are sliding one right or left, or jumping over another chip to the empty space. The motivation of the game is to accomplish this in the least amount of moves. While studying with her class the different possibilities of moves, they found that many patterns arose that could even be written as equations with variables. At the conclusion of the article she describes how her students found this activity "exhilarating" and were fascinated by the patterns they found.

I fully support the author's main idea, and believe that this article is a great demonstration of a good classroom activity for promoting mathematical applications outside of school. This is evidenced by the clear enjoyment conveyed by the students. Throughout the article it is clear to see their amusement and their internal motivation while working with the teacher on these patterns. It's clear also that they are mostly deriving the patterns and equations on their own without the teacher's help, so it seems they are gaining a relational type understanding. If mathematical patterns can be found in this simple game, I imagine there are many other activities we can use in future classroom to get similar effects.