Discovering Probability with Dice Games and the Evolution of a Convincing Argument

Students can discover through collaboration the principles of theoretical and experimental probability. This analytic focuses on a group of sixth-grade students as they take a simple dice game and open a world of questions that they themselves end up finding the answers to. The game seems simple: Roll two dice. If the sum of the two is 2, 3, 4, 10, 11 or 12, Player A gets 1 point. If the sum is 5, 6, 7, 8 or 9, Player B gets 1 point. Continue rolling the dice. The first player to get 10 points is the winner. Before playing the game, the problem asks if this is a fair game and why or why not. After playing the game, the problem asks if the results of playing the game support your answers from the first question. The third question asks how the game can be changed to make the game fair if you think the game is unfair. The game and questions seem very simple to start. Initially, many of the students thought that Player A would have the advantage because that player has six numbers that would count as a point for them while Player B has only five. As the students start playing the game, they all quickly realize that Player B seems to have the upper hand in the game. Before coming to this realization, the groups had to understand the rules of the game. Some of the students were unsure of the scoring and which player should receive the point. As with any task such as this, it is important to read the directions and understand what the task at hand truly is and how to fulfill the requirements. All the students quickly caught on after some time and went ahead and played multiple games. As the dice game proceeded in class, some students even took the game home with them and tried it out with their family members. The teacher or researcher did not direct the students to continue the game for homework but some of the students were so interested in this task that they decided to keep going with it at home. Full student engagement requires student interest. To become fully invested in a task the desire to see out the problem is a necessity. These students were having fun, questioning each other, answering each others questions and successfully collaborating on the problem. In the evolution of a convincing argument, the student must be able to not only explain how they attained their solution but also be able to visually represent it. Many of the students created their own visual interpretations but Stephanie created some very intricate charts. Stephanie uses two charts to explain to her group-mates the likelihood of a specific sum coming up as well as why the game is unfair. After a certain point, the group was able to determine that the game was unfair and Player B had an advantage to winning the game. The group discussed two ways to make the game fair. The first way was Player A would earn a point if a 2 through 6 was rolled, Player B would earn a point if an 8 through 12 was rolled and a 7 would turn into a re-roll and no one would get the point. The second way was one Player would take all the odd sums and the other player would take all the even sums. The group was able to use Stephanie’s chart to prove that each player would have an 18/36 chance of winning the game, therefore making it fair in both scenarios. The video then focuses the attention of the lesson on the entire class. There seemed to be a divide between the groups in the case of how many total outcomes there actually were. Some of the groups felt that situations like rolling a 1 then a 2 was the same thing as rolling a 2 then a 1. The other groups felt that this should be considered two different possibilities. This divide left room for the groups to explain their side of the argument. Stephanie’s group was able to successfully convince the entire class both verbally and visually. Some of the group members that originally disagreed with Stephanie’s group volunteered to present to the class the correct solution. Stephanie and her group’s conclusions were so convincing to the class that everyone was in agreement. There are many phases to approaching and solving a probabilistic model and it is very apparent that a collaborative effort is necessary. With limited teacher involvement we can see that students are able to successfully navigate their way through a problem such as this. Teacher involvement can be utilized in student questioning to help guide students to discovering the solution on their own. It is not until each student can create a convincing argument that a true solution can be reached. References: Maher, C. A. (1998). Learning to reason probabilistically. In S. Berenson, K. Dawkins, M. Blanton, W. Coulombe, J. Kolb, K. Norwood, and L. Stiff (Eds.), Proceedings of the 20th Conference of the North American Group for the Psychology of Mathematics Education, (1), 82-87. Raleigh, NC: North Carolina State University. Alston, A. S. & Maher, C. A. (2003). Modeling outcomes from probability tasks: Sixth graders reasoning together. In N. A. Pateman, B. J. Dougherty and J. T. Zilliox (Eds.), Proceedings of the 27th Annual Conference of the International Group for the Psychology of Mathematics Education, (2), 25-32. Honolulu, HI: CRDG, College of Education, University of Hawaii