PurposesEffective teaching; Homework activity; Lesson activity; Student collaboration; Student elaboration; Student engagement; Student model building; Reasoning; Representation

DescriptionStudents can discover the principles of probability through collaboration as argumentation. This analytic focuses on a group of sixth-grade students as they take a simple dice game with inherent probability properties and reason through the process with argumentation.

The game: 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. 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 to 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 events in this analytic begin with the work of one small group, Stephanie, Milin, Ankur, Angela, and Brian. 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.

Some students take the game home with them and try 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 the problem is necessary. These students are engaged in problem-solving by formulating their own ideas, questioning each other, and using argumentation to determine the correct solution.

In the evolution of a convincing argument, the student must be able to explain how they attained their solution and visually represent it. Stephanie uses two charts to explain the likelihood of a specific sum coming up as well as why the game is unfair. The group was able to determine that the game was unfair and that player B had an advantage in 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 events then focus the attention of the lesson on the entire class. There is a division 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 provided the opportunity for the groups to engage in argumentation to determine the single correct solution. 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 agreed.

There are many forms of understanding and expression of the probabilistic model in this group. It is clear that a collaborative effort is useful in getting the whole group to the correct understanding. With limited facilitator involvement, the students successfully navigate through the problem. Facilitator involvement can be utilized in student questioning to help students discover the solution independently. Creating or discerning between convincing arguments requires the understanding necessary to enable individual discovery. The result is a durable foundation with which to make future associations and increase knowledge.

Reference:

B49, Dice games for two players (student view), Grade 6, March 21, 1994, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3NG4NR2

The game: 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. 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 to 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 events in this analytic begin with the work of one small group, Stephanie, Milin, Ankur, Angela, and Brian. 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.

Some students take the game home with them and try 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 the problem is necessary. These students are engaged in problem-solving by formulating their own ideas, questioning each other, and using argumentation to determine the correct solution.

In the evolution of a convincing argument, the student must be able to explain how they attained their solution and visually represent it. Stephanie uses two charts to explain the likelihood of a specific sum coming up as well as why the game is unfair. The group was able to determine that the game was unfair and that player B had an advantage in 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 events then focus the attention of the lesson on the entire class. There is a division 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 provided the opportunity for the groups to engage in argumentation to determine the single correct solution. 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 agreed.

There are many forms of understanding and expression of the probabilistic model in this group. It is clear that a collaborative effort is useful in getting the whole group to the correct understanding. With limited facilitator involvement, the students successfully navigate through the problem. Facilitator involvement can be utilized in student questioning to help students discover the solution independently. Creating or discerning between convincing arguments requires the understanding necessary to enable individual discovery. The result is a durable foundation with which to make future associations and increase knowledge.

Reference:

B49, Dice games for two players (student view), Grade 6, March 21, 1994, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3NG4NR2

Created on2023-04-10T16:04:03-0400

Published on2023-05-18T13:52:59-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-sttc-wf96