Tracing Justina’s Argument of Fairness: Investigating a Journey of Probabilistic Reasoning
Creators: name: Faulborn, Nicholas A.
Publisher: name: No Publisher Supplied
Year: 2024

description: The purpose of this analytic is to examine the development of probabilistic reasoning and representations shown by Justina, a sixth-grade student, when working with Adanna, a sixth-grade student, on both dice games (see problem statements below). The video narrative is a compilation of events that documents Justina’s reasoning on what makes the dice games fair as well as the representations that she used to support her argument.The first event showcases the structure and format of dice game #1 that both Justina and Adanna are using to analyze the game. Specifically, they both have pieces of paper and pens to document their results and a die to roll on the desk. Thereafter, after establishing that the dice game #1 is “unfair,” the next 2 events document how Justina and Adanna created “fair” versions of dice game #1. In particular, they create two distinctive versions where they both have an “even” set of different numbers from 1 to 6 (i.e., one player had 1, 3, 5 and the other had 2, 4, 6). During Events 4 and 5, the idea of “fairness” is questioned by Researcher Asante when asking Justina and Adanna about their changes to dice game #1. In other words, Justina and Adanna thoroughly discuss all of the versions of dice game #1 that they worked on when prompted by Researcher Asante. Finally, the last 2 events highlight dice game #2 where Justina and Adanna play a slightly different variation of dice game #1. Specifically, they are using a pair of dice and rolling for the sums of the dice to determine which player will score a point. Altogether, the purpose of these events in relation to the overall goal is that it focuses on Justina’s journey of probabilistic reasoning and construction of appropriate representations to support her reasoning. In particular, the analytic shows how Justina’s journey of reasoning develops over time when playing both dice game #1 and #2. Explicitly, Justina reasoned how the dice games were unfair and created ways to make dice game #1 fair along with her partner, Adanna (Events 2 and 3). In each dice game, Justina used experimental data (within her representations) and occasionally theoretical data to support her claims, but for dice game #1, these conjectures were challenged by Researcher Asante, which sparked more mathematical discussion and made Justina elaborate on her reasoning (Events 4 and 5). Moreover, for each dice game, Justina used representations that were composed of drawing two columns with her and Adanna’s name, documenting the number that was rolled on the die, and indicating if the game they were playing was fair or unfair (Events 3 and 7).When looking at the dice games, a goal for students can be to develop ideas of probability prior to high school and make connections between theoretical and experimental probability. According to Kathleen Shay, “probability requires a way of thinking that does not consist of procedures to be followed to reach a predetermined solution” (2008, p. 5). With this in mind, Justina’s growth of probabilistic reasoning can be aligned with this goal as she makes explicit connections with her representations and knowledge of probability (Event 5).Overall, there are some factors that may have impacted Justina’s journey and claims that she made in relation to the dice games. First, the task design of the dice games gave Justina an opportunity to reason what it means to be a “fair” game and make the games fair while communicating with her partner Adanna (Events 2 and 3). Likewise, the task design allowed Justina to reflect on her mathematical and personal beliefs of probability (Events 4 and 5). We see that “through sharing, questioning, and debate, children put forth hypotheses, and explore their reasonableness” (Alston &amp; Maher, 2003, p. 32). Furthermore, the supply of materials, such as dice, pens, and paper permitted Justina to document her results and create representations, which resulted in determining that the dice games were unfair (Events 2 and 7). Nonetheless, the task design demonstrates how “engaging with complex tasks offers opportunity to work through the confusion and make strategic choice” (Alston &amp; Maher, 2003, p. 25).In addition to the task design, the involvement of researchers, including Researcher Asante and Alston, challenged Justina to explain her argument on what makes the dice games fair. In particular, Researcher Asante asked deep questions about the idea of fairness using the experimental data that Justina and Adanna both had, which significantly impacted Justina’s reasoning as she had to explain in her own words how the dice games were fair with appropriate reasoning to support (Events 4 and 5). Alston and Maher (2003) explained that the IML study’s environment/design of “student-to-student and student-to-researcher dialogue reveal[ed] an emphasis on sense making and collaboration in the quest for meaning. This result[s] in richer and deeper understanding of the conditions of the problem and the varying interpretations developed by the students” (p. 25).The analytic will show how task design coupled with minimal researcher involvement may have prompted students to make representations about probability (Events 3 and 7) and construct arguments on what it means to have play a fair game (Events 4 and 5). This is because the students are encouraged to record their results and make conjectures about what it means to have a fair game between two players (Events 1 and 6). Also, they are given an opportunity to explore and communicate their ideas with another classmate (Events 2 and 3) as well as with researchers (Events 4 and 5).Problem Tasks:Dice Game #1: A Game with One Die: Roll one die. If the die lands on a 1, 2, 3 or 4, Player A gets one point (and Player B gets 0). If the die lands on 5 or 6, Player B gets one point (and Player A gets 0). Continue rolling the die. The first player 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?Dice Game #2: 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?Video References:B84, 42b, Probability problems: Dice games for two players part 1 of 2 (Student view), Grade 6, April 29, 2004, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/t3-2wn3-1z65B85, 43b, Probability problems: Dice games for two players part 2 of 2 (Student view), Grade 6, April 29, 2004, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/t3-feqd-sm07 References:Alston, A., &amp; Maher, C. (2003). Modeling Outcomes from Probability Tasks: Sixth Graders Reasoning Together. In International Group for the Psychology of Mathematics Education (Vol. 2, pp. 25-32). International Group for the Psychology of Mathematics Education.Shay, K. B. (2008). Tracing middle school students’ understanding of probability: A longitudinal study. Retrieved from https://doi.org/doi:10.7282/T3BR8SGM