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Community Based Mathematics Project of Philadelphia

Stop and Frisk
Submitted By: CBMP

    Display Image
    Silhouette graphic of a police officer stop and frisk
    Grade
    Grade
    7th Grade
    Math Focus
    Math Focus
    Probability Simulations
    Context
    Context
    Social Justice
    Common Core State Standards
    Statistics & Probability
    Use random sampling to draw inferences about a population: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. (Math.7.SP.A.1)
    Investigate chance processes and develop, use, and evaluate probability models. : Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. (Math.7.SP.C.7)Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? (Math.7.SP.C.8c)
    Overview / Lesson Summary

    In this lesson, students will investigate the Stop and Frisk policy which was implemented in Philadelphia in 2008. Students will use a simulation to model the rates at which pedestrians of varied races would be stopped if the stops were made at random, and then compare this to the actual data to explore the consequences of the policy in terms of social justice. Mathematical concepts include theoretical and experimental probability, sampling, the law of large numbers, and simulations. Students also learn that by using mathematics to analyze publicly available data they can demonstrate racial bias.

    Preparation and Materials
    • Math counters, counting cubes, construction paper squares, or similar items in 5 colors. Each group of students will need up to 50 of each color. 
    • Containers or paper bags to hold the counters
    Introducing the Context

    This lesson involves content that may be personally relevant and/or possibly uncomfortable for some students. It is important to allow students to share their personal experiences with racial profiling and how it has affected them, their family and friends.

    Introduce the idea of stop and frisk by building on students' knowledge. 

    • What does "stop and frisk" mean? Establish a definition (a police officer stopping a civilian to pat down their clothes to search for weapons or drugs). 
    • Has anyone ever been stopped and frisked by the police? Do you know anyone who has? (You may wish to have students write about their experiences in advance.) 

    Have students read the facts on the Student Handout  which describes Mayor Nutter's stop and frisk policy. Ask:

    • What do you think about this policy?
    • What questions does it raise for you?
    • How could we use data to understand the impact of this policy better? Compare personal experiences and large data sets for helping to determine if a policy is fair. 

    Have students explore the Data on Stop and Frisk on the second page of the handout and then collect their observatinos and questions. Some important things to notice include:

    • The number of stops increased (more than doubled) in all racial categories from 2005 to 2009.
    • African Americans are stopped much more often than any other race
    • African Americans make up less than ½ of the population of Philadelphia

    If it did not come up in the initial discussion, talk about why it might be important to figure out the percent of the total number of stops that were made in each racial demographic category. Together, use a calculator to determine the percents and discuss the results:

    Race 2005 2009
    Black 68.3% 72.3%
    White 22.0% 18.4%
    Hispanic 8.6% 8.2%
    Asian 1.1% 1.0%
    Other 0.1% 0.1%

    By looking at these percentages we can see that in both years over ⅔ of the stops were made on African Americans, yet they make up only 43% of the population of Philadelphia.

    • Introduce the term racial profiling, defined as the discriminatory practice by law enforcement officials of targeting individuals for suspicion of crime based on the individual's race, ethnicity, religion or national origin.
    Introducing the Mathematical Ideas

    Explain that you are now going to learn how to use a simulation to model what it would look like if the stops were made at random, without regard to a person’s race or ethnicity. In statistics, a simulation is a way to model random events so that the outcomes closely match real-world outcomes. 

    Ask students what it means for a situation to be random. Talk about some examples of situations that are random and some that are not. For example, guessing each answer on a true/false quiz is random. If you guessed the answers on a 100-question true/false test you might expect to get about 50 of them correct. Rolling a die is random. Since there are 6 results that are equally likely, you would expect to roll a 2 about 1/6 of the time.                                      

    To simulate a random stop and frisk situation, explain that you'll use colored tokens to represent people who live in Philadelphia. It would not make sense to create one token for each of the 1.5 million people in Philadelphia, so instead you'll use a sample that is a reasonable size. However, the composition of the sample must match the composition of the population.  So if 43% of the population is African American, you want approximately 43% of the tokens to be of a certain color.

    Decide as a class how many tokens of each color should go in the bag to represent the population, which will allow for more practice with equivalent ratios. Have students record the numbers on the handout.

    • Decide on the total number of tokens and have students figure out the number of each color they should use to approximate the demographics. For 100 tokens you can use the actual percents. For 50 tokens, divide them in half: e.g., use 22 tokens of one color to represent African Americans and 21 tokens as another to represent Whites. Record the number of tokens on page 3 of the student handout. 
    • Discuss how you have just created a sample and ask how this is similar and different from the actual population. Lead them to the idea that the collection of tokens are now a representative sample that reflects the racial breakdown of the population of Philadelphia.

    Have students make predictions before they begin: 

    • If we pick a token at random to represent a single incident of a police officer making a random stop, are we more or less likely to pick a certain color? Why?
    • If we continue to randomly pick tokens one a time, which colors do you expect to come up more often? less often?

    Students may not fully understand the idea that the results should match up with the demographics of the city at this point, but you can return to that after they have experience collecting the data.

    Exploration

    Review the Simulation Procedure described handout:  

    1. Put the correct number of tokens into the bag to create a representative sample.
    2. Randomly pick a token without looking in the bag. This represents one stop and frisk incident. Make a mark on the tally chart for the race that the token’s color represents.
    3. At the end of 10 picks, transfer the results to the Cumulative Table on the next page. The cumulative table is for the "running total" of all the picks you've made so far, up to the final 100. For example, after 30 picks, you should record the results for picks 1-30.
    4. After you have recorded 100 picks, compute the fraction and percentage for each race after every 10 picks.

    Demonstrate how students will pick randomly from the bag and record each pick.

    Have each group of students take or create enough tokens to represent each demographic, and then perform the simulation and record their results. When they finish, they can discuss the follow‐up questions.

     

    Discussion

    Share and discuss the results of each group's simulation. You could also combine the results to create a class set of results.

    • Have students compare their percentages for the first 10 picks with the first 50 picks and the first 100 picks. If you create a class set, compare these to all of the picks across the whole class. Have students describe the trend they see. 
    • The results should illustrate the law of large numbers: the more picks that are done, the closer the experimental probability, or simulation results, will be to the theoretical probability, or the actual demographics of the city.

    Have students discuss their reactions to the simulation: 

    • Did the percent of your random "stop and frisk" picks match what you expected? Did they match the data we looked at in the beginning of class about how often each group was actually stopped? Why not? What does this mean?
    • Review the term racial profiling and its significance in this situation. Do you think the data shows evidence of racial profiling?

    Talk about how the ACLU filed a lawsuit about this situation and used this kind of mathematics to make the argument that more people of color were being stopped by this policy than we would expect given the population demographics in Philadelphia. See the links below for more on the lawsuit, the importance of pubicly available data, and what happenned in other cities.

    Discuss how looking at data can help us to use evidence rather than only personal experiences to understand issues of social justice like this.

     

    Extensions

    Research measures the city has taken to address this situation since the lawsuit. Do you think anything has changed about the way stops are made by the police in the city? What information would we need to explore this further?

    See links below for more on the lawsuit, the importance of publicly available data, and what has happened in other cities. 

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