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As is typical in mathematics, we define certain terms in order to establish a common vocabulary. Any discussion of social justice first defines the problems and norms that exist in a given field. In "(Re)Defining Equity: The Importance of a Critical Perspective", associate professor of mathematics education Rochelle Gutierrez gives a widely quoted definition of dominant math as "mathematics that reflects the status quo in society, that gets valued in high-stakes testing and credentialing, that privileges a static formalism in mathematics, and that is involved in making sense of a world that favors the views and perspectives of a relatively elite group", and critical math as math that "squarely acknowledges the positioning of students as members of a society rife with issues of power and domination". The rest of this guide is based on the acceptance of these assumptions about dominant and critical math ad addresses strategies to move mathematics education from one to the other.

Privilege and Oppression While most people have some understanding of discrimination, there is often little said about the flip side of discrimination, which is privilege. What is privilege? It is the unearned benefits afforded the dominant culture. These benefits come in many forms. Examples of Privilege:

• Automatic respect
• Automatic benefit of the doubt
• Job opportunities
• Mentoring
• Identification with people who are powerful and/or intellegent and/or strong
• Identification with leaders

What is often cited as the greatest privilege afforded the dominant culture (white, male) is the benefit of not having to realize one's own privilege. Because the dominant culture benefits from the system of privilege, they are not forced by circumstance to study it, and they can choose to believe that all their successes are purely a result of their own merit. They have the privilege of believing that their achievements are based on their own merit. In contrast, those on the other side of privilege often need to invest much of their time and energy understanding oppression. Examples of Oppression:

• Perception of incompetence at every level (as students, faculty, staff)
• Perception that they are successful because of Affirmative Action
• Harassment by peers, teachers, supervisors, co-workers.
• Less mentoring, less access to education.
• More responsibilities at home (child care, elder care)

Research shows the surprising result that both the privileged and oppressed are incentivized to ignore, or maintain the system of privilege, but for different reasons. The privileged want to believe that their benefits are well-deserved and the oppressed want to believe that they can overcome their diminished status. This "Just World" view is part of a general preference for people who want to believe that the world is a good, safe, just place instead of an unjust place that needs to be changed.

Implicit bias refers to the attitudes and beliefs that unconsciously affect our thoughts and actions. These biases occur without awareness or control. Both direct (such as life experience) and indirect (such as TV) influences form certain associations to people based on gender, race, age, and other characteristics.

Scientists studying unconscious cognition have shown that people do not always have conscious control over perception, impression formation, and judgment. The traditional paradigm about discriminatory behavior is that people discriminate due to conscious positive and negative thoughts about certain groups of people. According to this belief, discriminatory behavior is the problem of only a few racists, sexists, homophobes, etc. Although conscious discrimination exists, cognition scientists at Harvard, Yale, the University of Virginia, Tufts, the University of Washington, MIT, and others, have shown that implicit bias can be a much bigger factor in our thoughts and actions than conscious beliefs. Implicit biases can produce behavior that completely diverges from a person's avowed or endorsed beliefs so that even those who are actively fighting discrimination are prone to bias. Moreover, even those who suffer from discrimination can be biased against their own oppressed group.

The for the study of race and ethnicity, at Ohio State University lists some characteristics of Implict Bias:

1. Implicit biases are pervasive. Everyone possesses them, even people with avowed commitments to impartiality such as judges.
2. Implicit and explicit biases are related but distinct mental constructs. They are not mutually exclusive and may even reinforce each other.
3. The implicit associations we hold do not necessarily align with our declared beliefs or even reflect stances we would explicitly endorse.
4. We generally tend to hold implicit biases that favor our own ingroup, though research has shown that we can still hold implicit biases against our ingroup.
5. Implicit biases are malleable. Our brains are incredibly complex, and the implicit associations that we have formed can be gradually unlearned through a variety of debiasing techniques.

The list of American universities actively studying and addressing Implicit Bias is growing and includes Harvard University, the University of Virginia, the University of Washington,University of Illinois, Stanford Universit, UCLA, University of Florida, and Ohio State University. The list of international universities actively studying implicit bias is too numerous to mention. A growing both of implicit bias research indicates that implicit bias may be a major factor in preventing women and girls from pursuing or continuing careers in math and science, play a role in evaluation of work done by women in STEM, and influence overall treatment of women and girls' in STEM.

Science, and in particular, mathematics, is considered highly objective. A common thought in and out of academia holds that if molecules and equations aren't prejudiced, nor are scientists. Research sheds some light on this myth:

• White Americans, on average, show strong implicit preference for their own race and bias against women, the elderly, African Americans, Asians, Latinos, and other ethnic minority groups (see [7], [2],[22],[34],[6],[9]).
• Women, minorities, and other oppressed groups are as likely to have biases against their own, or other, oppressed groups. (see [7],[15] ).
• Gender and racial bias are rife in academia, even among those who believe themselves to be progressive and actively involved in nondiscrimination efforts (see [17],[11], [9],[12]).
• Scientists are among the most prone to bias (see [36], [17] ,[13], [28], [24],[4]).
• Gender bias negatively affects female students and female faculty, especially those in leadership positions (see [12], [17], [18],[14]).
• Diversity training has been found to be either ineffective or even worsen gender bias (see [21]).

The above research findings explain some of the following statistics:

• The percentage of female full professors is less than 10% in math and less than 13% in computer science (see [1]).
• The percentage of female math faculty at all levels is less than 20%; at elite research universities, this number drops to 10%. (see [1])

Mahzarin Banaji, Professor of Social Ethics at Harvard University, and her colleagues from the University of Virginia and the University of Washington developed an as part of their larger research agenda Project Implicit. .

Since the gender-science bias test was developed in 1998, more than half a million people in the world have taken it, with more than 70% of test-takers more readily associatiting science with male and arts with female. Moreover, women tended to have similar gender biases as men, dismantling the myth that women aren'talso gender biased against their own gender.

Once awareness of implicit bias is established, educators can be more attune to the effect their biases having on their teaching, evaluations, advising of their students and treatment of their peers.

There are many ways that one can bring awareness of social justice issues into the teaching and learning of mathematics.

• Test your own bias by taking an Implicit Association Test.
• Use mathematics to explore problems of social justice. For example, modeling the salary disparities between men and women and the effect of this difference in future wealth. Another example is the analysis of the density of toxic waste facilities, factories, and dumpsters in certain neighborhoods.
• Re-center mathematics education to include the experience of marginalized groups.
• Explore and address power and discriminatory practices both external and internal to the mathematics community. An example of power issues internal to mathematics is observing who has the power to hire, fire, grant tenure, grant awards, schedule courses, etc. Observe who is getting hired, fired, tenure, good schedules, and receiving coveted awards. An example of power issues external to math is observing who is getting social pressure to take care of families (both children and the elderly).
• Research and teach the mathematics of marginalized groups. Investigate why we are studying the mathematics done by a small subset of the world.
• Explore discrimination within the university and classroom. As a teacher, do you treat students differently or have different expectations of students based on categories like gender and race? As a student, do you have different expectations, demands, treatment of your professor based on categories like race and/or gender?

The items above can be divided into two categories - those that involve doing mathematics directly and those that seek to understand the oppression and privilege that have led to the current state of mathematics today. These two broad categories have many intersections, but the distinction is important because of the resistance to teaching something not directly involving the practice of mathematics in a mathematics classroom. You don't have to give up precious classroom time to insert social justice principles into mathematics.

Mathematics is fortunate in its flexibility - it can insert itself into almost any subject, from biology to politics. For example - rates of growth can be taught at many different levels and is extremely important in economics. One can use rates of growth (and therefore exponential and logarithmic functions) to analyze wage disparities between men and women to predict future wealth at any other point of time.

Examples of the social justice topics that can be analyzed using mathematics:

• Prisons, racial profiling, the death penalty
• Poverty, minimum vs. living wage, sweatshops
• Housing, gentrification, homeownership
• War, defense budgets, military recruiting
• Public health, AIDS, asthma, health insurance
• Educational funding and equity, high stakes testing, class size
• Environmental racism, pollution, resource availability
• Credit cards, managing debt, paying for college
• Saving/budgeting money, opening bank accounts
• High-cost loans (rent-to-own stores, check cashers, refund-anticipatory loans, payday, etc.)
• Filing taxes
• Remittance rates

Below are resources for lesson plans:

1. .
2. Who is entering the prison system? How many prisons should be built in the future? Why are prisons overcrowded?
3. Who is benefitting from Lotteries?
4. Not all voting systems are equal (or perhaps even &quot fair &quot) with the .
5. : the Mathematics of Taxes.
7. has a plethora of resources on social justice math.
8. on the mathematics of social justice.

Short Reading List

• Brown-Glaude, W. (2009). Doing Diversity in Higher Education: Faculty Leaders Share Challenges and Strategies. (Rutgers University Press).
• Bystydzienski, J. & Bird, S. (Eds.), (2006). Removing Barriers: Women in Academic Science, Technology, Engineering, and Mathematics. (Bloomington: Indiana University Press).
• Etzkowitz, H., Kemelgor, C., & Uzzi, B. (Eds.), (2000). Athena Unbound: The Advancement of Women in Science and Technology. (Cambridge: Cambridge University Press).
• Gutstein, G. and Peterson, B. (2006) Rethinking Mathematics: Teaching Social Justice by the Numbers. (New York: Routledge).
• Margolis, J. and A. Fisher. (Eds), (2002). Unlocking the Clubhouse: Women in Computing. (Cambridge, MA: Cambridge University Press).

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