# Environmental and Occupational Health Sciences Institute

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### Chris and Jerel’s Initial Idea of Fairness in Ordinary Dice Games (Part 2 of 3)

This analytic is the second of three analytics that showcase the formulation of two students’ ideas of mathematical fairness. The analytics follow the argumentation of two students, Chris and Jerel, throughout two after-school sessions and one interview session as they investigate what makes a game involving rolling one or two dice fair. The first after-school session occurred on April 29, 2004. The second after-school session occurred on May 5, 2004, with the interview happening directly after the second session on the same day. The first analytic begins with Chris and Jerel discussing the fairness of a game involving a single die during the first session. By the end of the first session, Chris has designed his own game involving rolling two dice. The second analytic looks at Chris and Jerel’s work during the second after-school session and interview session where the pair explore a task investigating the fairness of a game involving rolling two dice. This game is different from the one Chris creates in the first analytic. The third analytic includes an investigation on the probability of rolling a sum of 7 versus a sum of 6 when rolling two dice that occurs during the interview.
In this, the second of the three analytics, sixth-grade students Chris and Jerel are solving a problem about fairness with a game using two fair dice. Gameplay involves rolling two dice and assigning points to either Player A or Player B based on the sum of the rolled dice. Player A gets one point (and Player B gets 0) for sums of 2, 3, 4, 10, 11 and 12. Player B gets one point (and Player A gets 0) for sums of 5, 6, 7, 8 and 9. The first player to accumulate ten points is the winner. The questions that students are asking about this game are: 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? 3) If you think the game is unfair, how could you change it so that it would be fair?
Chris and Jerel’s initial response to this game is to highlight that Player A has three “small” numbers (2, 3, and 4) and three “large” numbers (10, 11, and 12) while Player B has all “large” numbers (5, 6, 7, 8, and 9). This suggests that they are not relying on the number of outcomes assigned to each player (Player A having 6 assigned outcomes to Player B’s 5 assigned outcomes) as the measure of fairness for the game. They are considering the numbers assigned to each player (Player A’s 2, 3, 4, 10, 11, and 12 compared to Player B’s 5, 6, 7, 8, and 9) as important to the fairness of the game, demonstrating they are no longer exhibiting an equiprobability bias in the outcome of rolling two dice (as seen in Part 1). Instead, the pair begins to see that each outcome does not have the same probability of occurring.
During gameplay, Shay (2008) explains that researchers had encouraged students to record the outcome of each roll of the dice. This naturally led to students beginning to list out the sample space for the game. We see in the second event, Chris and Jerel begin writing a list of sums for each outcome (i.e. 5=4+1, 5=2+3). During the interview, Chris and Jerel state they started writing out their sample space to figure out why Player B was winning. It is important to note that Chris and Jerel have a sample space of 21 outcomes rather than 36. Shay (2008) attributes this to students not considering symmetric pairs as separate events (for example, rolling a 3 and 4 versus rolling a 4 and 3). In the complete sample space of 36 outcomes, Player A has a 12/36 (approximately 0.333) chance of winning a single event, while Player B has a 24/36 (approximately 0.667) chance of winning a single event. With their sample space of 21 outcomes, Chris and Jerel conclude that Player B has 13 possible outcomes while Player A has 8 possible outcomes. This would mean Player B has a 13/21 (approximately 0.619) chance of winning a single event while Player A has an 8/21 (approximately 0.381) chance of winning a single event. This explains why the game is unfair in favor of Player B.
Throughout the discussion of the sample space for the game, Chris and Jerel mention that 7 came up the most when they were playing. Researcher Powell asks the pair why 7 comes up more than 6 when Chris and Jerel’s sample space shows an equal number of possibilities for rolling a sum of 6, 7, or 8. Chris suggests that a sum of 7 is rolled more frequently than 6 because 7 is comprised of sums of what he calls “big numbers.” These “big numbers” refers to rolling a 4, 5, or 6 on an individual die, while “small numbers” refers to rolling a 1, 2, or 3 on an individual die. This big and small number theory is explored in Part 3.
Problem Task:
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?
[Note: The game favors Player B with a ⅔ probability of winning a point and a probability of approximately .935 of winning a game.]
Videos Referenced:
Title: B89, Probability strand: Dice games with two players (Student View), grade 6, May 05, 2004, raw footage.
Raw footage link: https://rucore.libraries.rutgers.edu/rutgers-lib/70974/
Title: B90, 46a, Probability problems: Dice games for two players (Student view), Grade 6, May 5, 2004, raw footage
Raw footage link: https://rucore.libraries.rutgers.edu/rutgers-lib/70978/
Title: B91, 46b, Probability problems: Dice games for two players (Work view), Grade 6, May 5, 2004, raw footage
Raw footage link: https://rucore.libraries.rutgers.edu/rutgers-lib/70979/
References:
Shay, K. (2008). Tracing Middle School Students’ Understanding of Probability: A Longitudinal Study. Rutgers University

### Chris and Jerel’s Initial Idea of Fairness in Ordinary Dice Games (Part 3 of 3)

This analytic is the third of three analytics that showcase the formulation of two students’ ideas of mathematical fairness. The analytics follow the argumentation of two students, Chris and Jerel, throughout two after-school sessions and one interview session as they investigate what makes a game involving rolling one or two dice fair. The first after-school session occurred on April 29, 2004. The second after-school session occurred on May 5, 2004, with the interview happening directly after the second session on the same day. The first analytic begins with Chris and Jerel discussing the fairness of a game involving a single die during the first session. By the end of the first session, Chris has designed his own game involving rolling two dice. The second analytic looks at Chris and Jerel’s work during the second after-school session and interview session where the pair explore a task investigating the fairness of a game involving rolling two dice. This game is different from the one Chris creates in the first analytic. The third analytic includes an investigation on the probability of rolling a sum of 7 versus a sum of 6 when rolling two dice that occurs during the interview.
In this analytic (the third of three analytics), Chris and Jerel are sixth-grade students discussing a problem about fairness with a game using two fair dice during the interview with Researcher Powell. Gameplay involves rolling two dice and assigning points to either Player A or Player B based on the sum of the rolled dice. Player A gets one point (and Player B gets 0) for sums of 2, 3, 4, 10, 11 and 12. Player B gets one point (and Player A gets 0) for sums of 5, 6, 7, 8 and 9. The first player to accumulate ten points is the winner. The questions that students are asking about this game are: 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? 3) If you think the game is unfair, how could you change it so that it would be fair?
During gameplay, Shay (2008) explains that researchers had encouraged students to record the outcome of each roll of the dice. This naturally led to students beginning to list out the sample space for the game. During the interview, Chris and Jerel show the sample space they developed. It is important to note that Chris and Jerel have a sample space of 21 outcomes rather than 36. Shay (2008) attributes this to students not considering symmetric pairs as separate events. For example, rolling a 3 and 4 versus rolling a 4 and 3. This is significant because in the sample space Chris and Jerel have created 6, 7, and 8 all have a 3/21 (approximately 0.143) chance of occurring, while in the sample space of 36 outcomes 6 and 8 have a 5/36 (approximately 0.129) chance of occurring and 7 has a 6/36 (approximately 0.167) chance of occurring. Due to the incomplete sample space, Chris and Jerel are investigating why the sum of 7 occurs more frequently, leading to a question in the fairness of rolling a single die.
At the end of the second analytic. Chris and Jerel have expressed the idea that rolling a 7 occurs more frequently than rolling a 6 even though, according to their notes, each number can be the result of exactly three outcomes. The boys claim that this is because the sums of 7 have more “large numbers” referring to rolling a 4, 5, or 6 on one of the two dice. During the third analytic Chris and Jerel carry out two trials to investigate the claim that when rolling a single die the “large numbers” (4, 5, and 6) occur more frequently than the “small numbers” (1, 2, and 3). The first trial results in more small numbers being rolled than large numbers, while the second trial results in more large numbers being rolled than small numbers. Combining the results of both trials reveals that 12 small numbers were rolled while 10 large numbers were rolled. This evidence causes Chris and Jerel to tentatively reject their claim that large numbers are more likely to occur than small numbers.
After viewing the analytic, it is worth reflecting on what might be a follow up task to challenge Chris and Jerel to find ALL possible outcomes.
Problem Task:
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?
[Note: The game favors Player B with a ⅔ probability of winning a point and a probability of approximately .935 of winning a game.]
Videos Referenced:
Title: B90, 46a, Probability problems: Dice games for two players (Student view), Grade 6, May 5, 2004, raw footage https://rucore.libraries.rutgers.edu/rutgers-lib/70978/
Title: B91, 46b, Probability problems: Dice games for two players (Work view), Grade 6, May 5, 2004, raw footage https://rucore.libraries.rutgers.edu/rutgers-lib/70979/
References:
Shay, K. (2008). Tracing Middle School Students’ Understanding of Probability: A Longitudinal Study. Rutgers University

### Chris and Jerel’s Initial Idea of Fairness in Ordinary Dice Games (Part 1 of 3)

This analytic is the first of three analytics that showcase the formulation of two students’ ideas of mathematical fairness. The analytics follow the argumentation of two students, Chris and Jerel, throughout two after-school sessions and one interview session as they investigate what makes a game involving rolling one or two dice fair. The first after-school session occurred on April 29, 2004. The second after-school session occurred on May 5, 2004, with the interview happening directly after the second session on the same day. The first analytic begins with Chris and Jerel discussing the fairness of a game involving a single die during the first session. By the end of the first session, Chris has designed his own game involving rolling two dice. The second analytic looks at Chris and Jerel’s work during the second after-school session and interview session where the pair explore a task investigating the fairness of a game involving rolling two dice. This game is different from the one Chris creates in the first analytic. The third analytic includes an investigation on the probability of rolling a sum of 7 versus a sum of 6 when rolling two dice that occurs during the interview.
In this analytic (the first of three analytics) during the first after-school session Chris and Jerel, both sixth-grade students, are solving a problem about the fairness of a game using a single fair die. Gameplay involves rolling a single fair die: if a 1, 2, 3, or 4 is rolled Player A gets one point (and Player B gets 0) and if a 5 or 6 is rolled Player B gets one point (and Player A gets 0). The first player to accumulate ten points is the winner. The questions that students are asking about this game are: 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? 3) If you think the game is unfair, how could you change it so that it would be fair?
Chris and Jerel begin gameplay before making a prediction. They claim that the given game is unfair using the results of their playing the game as support for their conclusion. In this analytic the students provide two indicators of fairness. The first indicator is the number of possible outcomes for each player (in this case Player A has four outcomes and Player B has two outcomes). The second indicator is the score difference of the game. In their case, Player A won ten points to Player B’s two points. When asked to create their own fair game Chris and Jerel create a game where each player has three outcomes: Player A gets a point for a roll of 1, 2, or 3 and Player B gets a point for a roll of 4, 5, or 6.
After playing their fair game, Chris and Jerel play a new game invented by Chris involving two dice. In this game Player A gets a point for rolling a sum that is even and Player B gets a point for rolling a sum that is odd. They assert that this game is fair by the same two indicators, that there is an equal number of outcomes (Player A: 2, 4, 6, 8, 10 and 12, Player B: 1, 3, 5, 7, 9, 11) and that the score of the game is very close (Player B wins 10 points to Player A’s 9 points). In this game, Chris and Jerel exhibit the equiprobability bias, the belief that all outcomes have the same probability of occurring (Shay, 2008). At the end of the session, a researcher suggests to the students that it is not possible to roll a sum of 1 with two dice. The session concludes before the two students were able to explore this further. Chris and Jerel explore further the sample space of rolling two dice in Part 2 of this series.
Problem Task:
One dice problem:
Roll one die. If the die lands on 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 points (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 would be fair?
Chris’ Game:
Roll two dice. If their sum is 2, 4, 6, 8, 10 or 12, Player A gets one point (and Player B gets 0). If the sum of the dice is 1, 3, 5, 7, 9, or 11, Player B gets on point (and Player A gets 0).
Video References:
B82, 42a, Probability problems: Dice games for two players part 1 of 2 (Student view), Grade 6, April 29, 2004, raw footage
Raw footage link: https://doi.org/doi:10.7282/t3-mtf2-xx81
References:
Shay, K. (2008). Tracing Middle School Students’ Understanding of Probability: A Longitudinal Study. Rutgers University Dissertation

### Frequency-domain fluorescence lifetime imaging microscopy for metabolic imaging of autofluorescent coenzymes NADH and FAD in biofilms

Fluorescence lifetime imaging microscopy (FLIM) is an imaging technique that canmeasure fluorophore lifetime in biological cells and tissues. FLIM can provide additional information beyond conventional fluorescence intensity or spectral imaging through knowledge of the fluorescence lifetime of a sample, which is an intrinsic trait of a fluorophore that can be extremely sensitive to the surrounding environment of the fluorophore, including factors such as pH, temperature, oxygenation, viscosity, and binding state.
In this thesis, a prototype frequency-domain FLIM system was evaluated for its ability to measure fluorescence lifetimes of the autofluorescent coenzymes NADH and FAD in biological samples. To produce the most accurate, reliable, and stable fluorescence lifetime measurements, calibration methods using fluorophores with known lifetimes were established, as well as experiments that determined the best combination of parameters (image size (number of pixels), scan size (field-of-view), integration time, and PMT gain) to acquire weak signals on the new FLIM system. To assess whether this FLIM system could image the lifetimes of NADH and FAD within biofilms, FLIM was performed on oral biofilms prepared from human saliva on hydroxyapatite discs. Fluorescence lifetimes in untreated control groups were compared to treatment groups that received anti-bacterial mouthwash.
After quantitative analysis was performed on images of control and treatment biofilms, the mouthwash treatment was found to have an effect on the fluorescence lifetimes, as the mean fluorescence lifetime of the control biofilms differed from the treatment biofilms. The change in fluorescence lifetime measured in NADH was more significant than it was for FAD. Some of the results from the quantitative analysis of the experiments ran were that the NADH lifetime for the control biofilms was calculated to be 1.41± 0.04 ns and the mean FAD lifetime was 1.39± 0.03 ns over three samples imaged at one session.
This thesis describes instrument calibration, image acquisition, and image analysis methods for measuring NADH and FAD autofluorescence in biofilms. These studies provide support for further development of FLIM as a tool that can be used in multiple health-related applications for non-invasive imaging and detection of alterations in metabolism.M.S.Includes bibliographical reference

### Undergraduate women of color in computer science: how social and academic experiences shape sense of belonging

While significant advancements in Science, Technology, Engineering, and Mathematics (STEM) are at the forefront of society, there continues to be a considerable gender and racial gap in the advancement of women of color in these fields (Espinosa, 2011; Ong et al., 2011; Shein, 2018). Low completion rates are even greater when looking at Black and Latina women in computer science (CS). This phenomenological study examined how the academic and social experiences of nine Black and Latina women computer science students shaped their journeys and sense of belonging as developing computer scientists. This study drew upon two interviews and journal entries to unpack how this group of women of color is experiencing their journeys through CS at Rutgers. I used Critical Race Theory (CRT) and Strayhorn’s (2012, 2018) model for sense of belonging for STEM students as frameworks for interpreting the experiences of women of color and what may lead to attrition or unwelcome environments. The findings of this study depict an isolating experience for Black and Latina students in CS. These students faced racist and sexist comments from their peers, which deeply impacted their ability to feel connected. They were conscious of their status as the “only one,” which hindered their ability to develop a sense of belonging. To counteract these feelings, many of the participants found solace in women-centric programming, including a Computer Science Living-Learning Community. Some critical recommendations include funding women-centric intervention programs, implementing inclusive hiring practices to hire more women in CS leadership positions, and building a working group of faculty and staff to examine the issues in the local context. The paper begins with the statement of the problem and social context. Next, I review relevant literature that examines the lack of women of color in STEM and CS. Following the literature review, I describe the methods and findings. Finally, I conclude with implications and limitations.Ed.D.Includes bibliographical reference

### Palliative states: migrant advocacy and necrocapitalist care in Canadian mental health services

The following exploratory project problematizes mental health support as a site of equitable social change through mapping the organizational relations of migrant mental health interventions in Canadian immigration law and social services. Increasing numbers of advocates and activists enter the mental health professions to support precarious migrants in “frontline work.” Frontline service provision is romanticized by the political left without questioning for whom and from what agenda mental health policies, programs, and services operate. In Palliative States, I answer the following questions: What are the frameworks, tools, and outcomes of progressive psychiatry, psychology, and social work? How are professional advocates’ social justice and anti-oppression politics institutionalized and redirected by Canadian state-funded mental health care? Drawing on institutional ethnographic interviews with psychiatrists, legal professionals, and community mental health workers, this dissertation claims mental health interventions are palliative, providing necrocapitalist care that ultimately consigns migrants to exploitation, disablement, incarceration, and premature death. “Palliation” describes a form of reasoning, mode of intervention, and experience of subjectivation, which together form the ideological state apparatus I call the palliative state. Palliation is a capitalist ideology towards mental health care that delivers austere social support masquerading as welfare. Palliative techniques include mitigation, comfort, and distraction, which temporarily suspend suffering and death. Finally, palliation is a type of consciousness. Foremost of these is “salvage mentality,” or the traumatic response to resuscitate life through system navigation and emotional adaptation, optimizing capitalist systems. Chapter One theorizes salvage mentality as leftists’ workaholism, defeatism, and deferral of social change. It also historically situates salvage mentality within Black, Indigenous, and racialized professionals’ calls for reparations from the settler capitalist state. Contemporary state-funded Indigenous and multicultural mental health initiatives promote individual capitalist survival under the guise of social security and well-being. Through a historical materialist analysis of qualitative interviews, organizational literature, legislation, and case law, the remaining chapters investigate the outcomes of salvage work. In attempts to mitigate inaccessible immigration procedures, psychiatrists supply mental health evidence to help secure status, a strategy stymied by the government’s ongoing interdiction of specific refugees. Chapter Two examines psychiatrists’ self-perceptions of their usefulness to argue that the power of mental health report writing has been overstated. Psychiatrists’ sensitive assessments and hyperproductive report writing mediate migrants’ encounters with the state, restoring faith in Canada’s immigration system. Chapter Three demonstrates how mental health reports are palliative distractions undercut by race-specific policies. These include the Canada–U.S. Safe Third Country Agreement (since 2004), Designated Country of Origin system (2012–2019), and Nigeria Jurisprudential Guide (2018–2020). Finally, in efforts to ease the anxiety and depression of newcomers, social workers comfort and shield landed immigrants from distress, helping them accept or overlook employment discrimination, unfair working conditions, and racism. Chapter Four argues that social workers adopt a mental health recovery model towards oppression that builds migrants’ resilience to psychologically overcome oppression, thereby acculturating migrants to poverty. As forms of palliative maternalism, validation, hope, and ignorance work as psychological breakwaters that can inadvertently decapacitate and debilitate migrants.Ph.D.Includes bibliographical reference

### The future of auditing: harnessing blockchain and emerging technology, and understanding the impact of exogenous shocks

This dissertation addresses the persistent concerns of accounting fraud and auditing failures, which can undermine the reliability and transparency of financial reporting and negatively affect the functioning of capital markets. Based on the contention that continuous auditing and monitoring offer potential solutions, this dissertation proposes a novel framework for continuous auditing by integrating blockchain and emerging technologies. The first essay presents the Blockchain for Continuous Auditing (BC4CA) model, which combines Blockchain Technology (BCT), Business Process Management (BPM), and Process Mining (PM) to achieve Continuous Accounting (CA) and improve the reliability and accountability of audit evidence. The second essay proposes a model for re-engineering the property tax process by integrating blockchain, smart contracts, and eXplainable AI (XAI) to increase transparency, accuracy, and efficiency in the property valuation process. The third essay examines the effect of the exogenous shocks from COVID-19 and K-SOX on audit efforts in Korea, focusing on the moderating effect of the Big 4 audit firms. Through its analysis, the essay provides valuable insights into the influence of exogenous shocks on the auditing industry.Ph.D.Includes bibliographical reference