PurposesEffective teaching; Student collaboration; Reasoning; Representation

DescriptionMaher and Martino (1996) noted that students, when presented with a problem task in a small-group setting, begin by constructing personal representations of the problem or, in attempting to do so, discover that they cannot. After students have attempted to build personal representations, they frequently initiate conversation with others and compare their ideas.

This analytic focuses on the representations and reasoning used by the sixth-grade students (Stephanie, Ankur, Brian, Milin, Jeff, Michelle, Romina, and Angela) who were challenged to examine their understanding of the fairness of a game. As the students played a dice game, they were asked to record the outcomes and use the outcome data to test their original conjecture about the fairness of the game.

The events of this analytic are selected from the Kenilworth NJ longitudinal study data (Maher, 2010). This was the second session exploring fairness of the game, facilitated by Researchers Carolyn Maher and Alice Alston on March 21, 1994. During the previous day, the students were introduced to the Game for Two Players:

Game for Two Players: Roll two dice. If the sum of the two is 2, 3, 4, 10, 11, or 12, Player A gets 1 point (and Player B gets 0). If the sum is 5, 6, 7, 8 or 9, Player B gets 1 point (and Player A gets 0). Continue rolling the dice. The first player to get 10 points is the winner.

Before playing the game, students were asked to predict whether the game was fair and to explain why or why not. The students conjectured that Player A would win, based on the observation that Player A had more number outcomes (six) than Player B (five). They played the game in class a few times the day prior and some students continued to play the game at home with a family member, attending to the combination of numbers that produced the sum. The next day, the students were asked if the outcomes of the game supported their early predictions. Then, they were asked if the game were unfair, how could it be changed to make it fair.

Brian and Angela set up a way to organize and record the dice outcomes. They decided to draw a table to record their results for each trial. They confirmed that they understood the rules of the game and then played the game several times. Notice that the students began by defining a sample space to explore empirical probability. As they continued with the dice game, Player B was winning in ten trials. These data indicated to them that Player B, rather than Player A, had an advantage of winning the game. As the students were engaged in playing the game; they could visually represent the results and collaborate with each other. They are observed asking and answering each other’s questions and supporting their conclusions with visual representations. In playing the game, the students rolled the dice and observed the outcomes after each rolling. They decided to record these results in a table.

It is interesting that two of the students, Stephanie and Milin who played the game at home, each chose to be player B, rather than A. Stephanie shared a chart showing all possible outcomes of the numbers on the dice. Angela and Brian, worked together. Jeff’s group drew tables to record their results. The students seemed curious and puzzled about the findings reported and wondered about their initial conjecture that Player A had the advantage.

Stephanie shared her chart illustrating the possible outcomes for each sum. She used these results later to draw a histogram, indicating a theoretical distribution to explain and convince her peers that the sum of 7 could result most frequently: (1,6), (6,1), (2,5), (5,2), (4,3), (3,4). She argued that the original game was unfair because originally, both players did not have equal outcomes. Stephanie indicated that if an outcome had high probability of occurring, it did not guarantee that it would occur. She said, “And, seven has the highest probability. That doesn’t mean seven’s gonna come up a lot because I know I rolled, I kept getting eights, and Brian kept getting four, and Amy kept rolling what she said.” The discussion introduced the distinction between theoretical and empirical probability, indicating that results from an experiment may differ from what was theoretically conjectured.

After Stephanie presented her visual representations in terms of a histogram. Milin continued rolling the dice and Ankur observed. This time they did not record the results. The group seemed to accept Stephanie’s representation. Consequently, Ankur and Angela modified their representation. Brian and Angela came up with a way of modifying the game and making it fair by using Stephanie’s chart. Angela reported to Researcher Alston, “I figured this [modification to make the game fair] from Stephanie’s chart.” They made a table indicating that player A should be given a sum of (2, 3, 4, 5, 6) and player B, (8, 9 10, 11, 12) and that in rolling a 7, you lose your turn of rolling. They concluded that this made the game fair because both players had the same number of outcomes, and hence equal chances. The numbers (2, 3, 4, 5, 6) have the same number of chances of occurring as the numbers (8, 9, 10,11,12). The group used Stephanie’s histogram to determine the number of outcomes each player had in this case and they found out that they all had equal chances of winning, making the game fair.

Then, the class discussed their results. They could not agree when it came to the total number of outcomes because some students were arguing that to get a sum of 3, there is only one way (2,1); yet others argued that for the two dice, there were two outcomes, (2,1) and (1,2). Some students did not notice that these two outcomes produced different combinations because they did not attend to the die on which the number was appearing.

Problems with probability theory can be better understood if students are actively engaged in collecting data, collaborating, and recording visual representations, such as Stephanie’s histogram to make conclusions about the problem. Notice that there is little intervention of the researchers as the students engaged in playing the game .By gathering data to test their conjecture about the likeliness of certain events, students were informally introduced to the ideas of theoretical and empirical probability.

References

Maher, C. A. (2010). The longitudinal study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 3-8). Springer: New York, NY.

Maher, C.A. & Martino, A.M. (1996). The Development of the Idea of Mathematical Proof: A 5-Year Study. Journal for Research in Mathematics Education, 27(2), 194-214.

Maher, C. A. & Alston, A. S. & (2003). Modeling outcomes from probability tasks: Sixth graders reasoning together. In N. A. Pateman, B. J. Dougherty & 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.

This analytic focuses on the representations and reasoning used by the sixth-grade students (Stephanie, Ankur, Brian, Milin, Jeff, Michelle, Romina, and Angela) who were challenged to examine their understanding of the fairness of a game. As the students played a dice game, they were asked to record the outcomes and use the outcome data to test their original conjecture about the fairness of the game.

The events of this analytic are selected from the Kenilworth NJ longitudinal study data (Maher, 2010). This was the second session exploring fairness of the game, facilitated by Researchers Carolyn Maher and Alice Alston on March 21, 1994. During the previous day, the students were introduced to the Game for Two Players:

Game for Two Players: Roll two dice. If the sum of the two is 2, 3, 4, 10, 11, or 12, Player A gets 1 point (and Player B gets 0). If the sum is 5, 6, 7, 8 or 9, Player B gets 1 point (and Player A gets 0). Continue rolling the dice. The first player to get 10 points is the winner.

Before playing the game, students were asked to predict whether the game was fair and to explain why or why not. The students conjectured that Player A would win, based on the observation that Player A had more number outcomes (six) than Player B (five). They played the game in class a few times the day prior and some students continued to play the game at home with a family member, attending to the combination of numbers that produced the sum. The next day, the students were asked if the outcomes of the game supported their early predictions. Then, they were asked if the game were unfair, how could it be changed to make it fair.

Brian and Angela set up a way to organize and record the dice outcomes. They decided to draw a table to record their results for each trial. They confirmed that they understood the rules of the game and then played the game several times. Notice that the students began by defining a sample space to explore empirical probability. As they continued with the dice game, Player B was winning in ten trials. These data indicated to them that Player B, rather than Player A, had an advantage of winning the game. As the students were engaged in playing the game; they could visually represent the results and collaborate with each other. They are observed asking and answering each other’s questions and supporting their conclusions with visual representations. In playing the game, the students rolled the dice and observed the outcomes after each rolling. They decided to record these results in a table.

It is interesting that two of the students, Stephanie and Milin who played the game at home, each chose to be player B, rather than A. Stephanie shared a chart showing all possible outcomes of the numbers on the dice. Angela and Brian, worked together. Jeff’s group drew tables to record their results. The students seemed curious and puzzled about the findings reported and wondered about their initial conjecture that Player A had the advantage.

Stephanie shared her chart illustrating the possible outcomes for each sum. She used these results later to draw a histogram, indicating a theoretical distribution to explain and convince her peers that the sum of 7 could result most frequently: (1,6), (6,1), (2,5), (5,2), (4,3), (3,4). She argued that the original game was unfair because originally, both players did not have equal outcomes. Stephanie indicated that if an outcome had high probability of occurring, it did not guarantee that it would occur. She said, “And, seven has the highest probability. That doesn’t mean seven’s gonna come up a lot because I know I rolled, I kept getting eights, and Brian kept getting four, and Amy kept rolling what she said.” The discussion introduced the distinction between theoretical and empirical probability, indicating that results from an experiment may differ from what was theoretically conjectured.

After Stephanie presented her visual representations in terms of a histogram. Milin continued rolling the dice and Ankur observed. This time they did not record the results. The group seemed to accept Stephanie’s representation. Consequently, Ankur and Angela modified their representation. Brian and Angela came up with a way of modifying the game and making it fair by using Stephanie’s chart. Angela reported to Researcher Alston, “I figured this [modification to make the game fair] from Stephanie’s chart.” They made a table indicating that player A should be given a sum of (2, 3, 4, 5, 6) and player B, (8, 9 10, 11, 12) and that in rolling a 7, you lose your turn of rolling. They concluded that this made the game fair because both players had the same number of outcomes, and hence equal chances. The numbers (2, 3, 4, 5, 6) have the same number of chances of occurring as the numbers (8, 9, 10,11,12). The group used Stephanie’s histogram to determine the number of outcomes each player had in this case and they found out that they all had equal chances of winning, making the game fair.

Then, the class discussed their results. They could not agree when it came to the total number of outcomes because some students were arguing that to get a sum of 3, there is only one way (2,1); yet others argued that for the two dice, there were two outcomes, (2,1) and (1,2). Some students did not notice that these two outcomes produced different combinations because they did not attend to the die on which the number was appearing.

Problems with probability theory can be better understood if students are actively engaged in collecting data, collaborating, and recording visual representations, such as Stephanie’s histogram to make conclusions about the problem. Notice that there is little intervention of the researchers as the students engaged in playing the game .By gathering data to test their conjecture about the likeliness of certain events, students were informally introduced to the ideas of theoretical and empirical probability.

References

Maher, C. A. (2010). The longitudinal study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 3-8). Springer: New York, NY.

Maher, C.A. & Martino, A.M. (1996). The Development of the Idea of Mathematical Proof: A 5-Year Study. Journal for Research in Mathematics Education, 27(2), 194-214.

Maher, C. A. & Alston, A. S. & (2003). Modeling outcomes from probability tasks: Sixth graders reasoning together. In N. A. Pateman, B. J. Dougherty & 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.

Created on2016-12-01T13:40:38-0400

Published on2017-08-07T09:40:52-0400

Persistent URLhttps://doi.org/doi:10.7282/T3FT8PVX