PurposesStudent collaboration; Reasoning; Representation

DescriptionThis analytic examines how collaboration between students can lead to deeper understanding and support student reasoning. Reasoning in mathematics can take many forms and has varying definitions in the mathematics education community (Van Ness, 2017). Regardless, major authorities on mathematics education, such as the NCTM, all see reasoning as essential to mathematical learning and development. As students deepen their understanding of mathematical content, their ability to justify their thinking increases (Tarlow, 2010). It can be challenging to get students to the depth of understanding required for this reasoning.

It has been established that student collaboration can lead to deeper understanding (Tarlow, 2010). Collaboration allows students to function in the classroom like authentic mathematicians, justifying and critiquing reasoning. When collaborating, students must be able to clearly communicate their mathematical thinking and ideas (Uptegrove, 2015). Students often attempt to make sense of a problem on their own, establishing individual representations. The need to share their individual representations with others during collaboration leads to students refining their representations so they can be understood and utilized by others (Uptegrove, 2015). As students share their existing ideas and representations with each other, the ideas evolve and become more sophisticated (Uptegrove, 2015). This allows students to further both their own and the group’s understanding of a given problem or task (Tarlow, 2010) through their collaboration. It is important to note that according to Tarlow (2010), students only deepen their understanding by being active participants in the learning environment. Active participants are those creating representations and engaging in conversation with peers which may include listening to the ideas of others to evaluate their merit. Passive participants may listen to the ideas of others, but do not contribute to the task at hand in meaningful ways. Thus, those that do not engage in the thinking and conversations happening, will not reap the benefits of student collaboration.

Students in this analytic were tasked with examining the fairness of a dice game. The rules of the game and accompanying instructions were as follows: “Roll two dice. If their sum is 2, 3, 4, 10, 11, or 12, player A gets one point (and player B gets 0). If their sum is 5, 6, 7, 8, or 9, player B gets one point (and player A gets 0). Continue rolling the dice. The first person to get ten points is the winner. (1) Is this a fair game? Why or why not? (2) Play the game with a partner. Do the results of playing the game support your answer? Explain. (3) If you think the game is unfair, how could you change it so that it could be fair? Does it matter which numbers are assigned to each player?”

Throughout this analytic, you will see mostly the work of a group of 5 students, Stephanie, Ankur, Milin, Angela, and Brian, attempting to reason through the task outlined above. Initially, the students had varying ways of making sense of the problem. This included referencing specific games played and conceptualizing the number of possible outcomes for each player. As the students shared their work with each other, they established common understanding from which they were able to move forward with the task. As the group continued to collaborate and understand the problem more deeply, their chosen representations for the problem were refined. This ultimately led to the group justifying their work to the whole class when a disagreement regarding the number of ways to roll a 3 or a 6 appeared in the whole class conversation. Given the group of 5 had established deep understanding and thorough representations of the task through their collaboration, they were able to clearly communicate their reasoning to the whole class and convince others of their solution.

Analytic events 1 and 2 shows the start of collaboration. In those events, Stephanie shares her individual representations with others leading to a common understanding of multiple group members. Event 3 sees Angela utilizing Stephanie’s representation to justify a claim showing how collaboration can be used to drive student reasoning and understanding. In Event 4, the whole class debates an aspect of the problem task. Ankur synthesizes the work of his group, including the work of Stephanie and Angela, and communicates it to the class. Ankur’s participation in the class discussion shows what shared understanding from collaboration looks like. Finally, event 5 shows the collimation of collaboration leading to the deepening of understanding, starting with the individual and ending with the whole class.

References

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

Tarlow, L. D. (2010). Pizzas, towers, and binomials. In Combinatorics and Reasoning (pp.121-131). Springer Netherlands.

Uptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. The Journal of Mathematical Behavior, 40, 106-130.

Van Ness, C. K. (2017). Creating and using VMCAnalytics for preservice teachers’ studying of argumentation (Doctoral dissertation, Rutgers University-Graduate School-New Brunswick).

It has been established that student collaboration can lead to deeper understanding (Tarlow, 2010). Collaboration allows students to function in the classroom like authentic mathematicians, justifying and critiquing reasoning. When collaborating, students must be able to clearly communicate their mathematical thinking and ideas (Uptegrove, 2015). Students often attempt to make sense of a problem on their own, establishing individual representations. The need to share their individual representations with others during collaboration leads to students refining their representations so they can be understood and utilized by others (Uptegrove, 2015). As students share their existing ideas and representations with each other, the ideas evolve and become more sophisticated (Uptegrove, 2015). This allows students to further both their own and the group’s understanding of a given problem or task (Tarlow, 2010) through their collaboration. It is important to note that according to Tarlow (2010), students only deepen their understanding by being active participants in the learning environment. Active participants are those creating representations and engaging in conversation with peers which may include listening to the ideas of others to evaluate their merit. Passive participants may listen to the ideas of others, but do not contribute to the task at hand in meaningful ways. Thus, those that do not engage in the thinking and conversations happening, will not reap the benefits of student collaboration.

Students in this analytic were tasked with examining the fairness of a dice game. The rules of the game and accompanying instructions were as follows: “Roll two dice. If their sum is 2, 3, 4, 10, 11, or 12, player A gets one point (and player B gets 0). If their sum is 5, 6, 7, 8, or 9, player B gets one point (and player A gets 0). Continue rolling the dice. The first person to get ten points is the winner. (1) Is this a fair game? Why or why not? (2) Play the game with a partner. Do the results of playing the game support your answer? Explain. (3) If you think the game is unfair, how could you change it so that it could be fair? Does it matter which numbers are assigned to each player?”

Throughout this analytic, you will see mostly the work of a group of 5 students, Stephanie, Ankur, Milin, Angela, and Brian, attempting to reason through the task outlined above. Initially, the students had varying ways of making sense of the problem. This included referencing specific games played and conceptualizing the number of possible outcomes for each player. As the students shared their work with each other, they established common understanding from which they were able to move forward with the task. As the group continued to collaborate and understand the problem more deeply, their chosen representations for the problem were refined. This ultimately led to the group justifying their work to the whole class when a disagreement regarding the number of ways to roll a 3 or a 6 appeared in the whole class conversation. Given the group of 5 had established deep understanding and thorough representations of the task through their collaboration, they were able to clearly communicate their reasoning to the whole class and convince others of their solution.

Analytic events 1 and 2 shows the start of collaboration. In those events, Stephanie shares her individual representations with others leading to a common understanding of multiple group members. Event 3 sees Angela utilizing Stephanie’s representation to justify a claim showing how collaboration can be used to drive student reasoning and understanding. In Event 4, the whole class debates an aspect of the problem task. Ankur synthesizes the work of his group, including the work of Stephanie and Angela, and communicates it to the class. Ankur’s participation in the class discussion shows what shared understanding from collaboration looks like. Finally, event 5 shows the collimation of collaboration leading to the deepening of understanding, starting with the individual and ending with the whole class.

References

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

Tarlow, L. D. (2010). Pizzas, towers, and binomials. In Combinatorics and Reasoning (pp.121-131). Springer Netherlands.

Uptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. The Journal of Mathematical Behavior, 40, 106-130.

Van Ness, C. K. (2017). Creating and using VMCAnalytics for preservice teachers’ studying of argumentation (Doctoral dissertation, Rutgers University-Graduate School-New Brunswick).

Created on2023-04-10T15:01:19-0400

Published on2023-05-31T10:06:42-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-6q22-dv18