x. School Discipline

Every day children of all ages, from preschool to high school, get disciplined at school. The form of discipline varies, from being sent to the principal’s office all the way to being arrested or expelled. Who gets disciplined at school, and for which offenses and behaviors? Are offenses and punishments getting better or worse? And what is the phenomenon that people refer to as the “school-to-prison pipeline?” In this module, you will use ratios, percents, data analysis, and probability to try to answer these questions.

x.1 Objectives

By the end of this module, students will be able to:

● Construct ratios and percentages from raw data in multiple forms, including fractions and decimals, that will be used to interpret information related to school discipline.

● Judge the validity of claims related to school discipline and using ratios and percentages.

● Explain the meaning of a probability statement and its connections to school discipline.

● Articulate if two related events are mutually exclusive and apply this skill to make judgments about the school-to-prison pipeline.

● Calculate the probability of an event not happening and apply this skill to make judgments about the school-to-prison pipeline.

● Calculate the sum of probabilities for multiple outcomes and apply this skill to make judgments about the school-to-prison pipeline.

● Calculate the probability of two or more independent events occurring using multiple methods, including a formula and a simulation, and use this skill to make judgments about the school-to-prison pipeline.

x.2 Understanding the Issue

In the wake of racial justice protests in 2020, the public’s attention was also turned to the presence of police in schools, and the disproportionate impact that this presence has on students of color, primarily Black and Indigenous students. According to the National Center for Education Statistics [8], 61.4% of the public schools in 2017-2018 had one or more security staff. The number of police officers in schools especially exploded after the mass shooting in Columbine High School in 1999, as a consequence of which the government began to offer federal grants to schools for hiring police officers in an effort to increase school security.

Twenty years later, over 200,000 students are being referred to law enforcement every year. Some of these referrals are arrests with or without charges filed, while others are citations. Students are referred to law enforcement for offenses as minor as talking back to a teacher or having a fight with a classmate. Because students who have contact with law enforcement are more likely to drop out of school and/or end up incarcerated, this increased criminalization of youth behavior has become known as the school-to-prison pipeline.

In this module, we will barely scratch the surface of this huge problem. After the completion of this module, you are encouraged to do more research on the topic, especially if you have not been personally affected by the overpolicing and overdisciplining in schools. Most of the data used in the module comes from a report by the American Civil Liberties Union, “Cops and no Counselors: How the Lack of School Mental Health Staff is Harming Students” [2]. It is important to acknowledge that neither race nor gender are discrete categories in the real world. However, all the data sources used in this module assume two genders and separate race categories. We will have to work with these limitations, but it is worth asking of all data we encounter: how were these data collected and what assumptions were made in collecting them?

In order to engage with this module, you will need to know some terminology:

School resource officer: School resource officers (SROs) are sworn law enforcement officers who are at a school part-time or full-time to maintain school safety and prevent crime. These officers may educate and counsel students, but otherwise have similar responsibilities as police officers. In particular, they can make arrests on school property.

School arrest: This is an arrest that happens on the school grounds, often, but not always, by the school’s SRO.

In-school suspension: When a student commits an offense that the school considers worthy of suspension, the suspension may be completed at the school. Suspended students do not attend regular classes or interact with their peers but are in a separate room where they are required to complete their assignments.

Out-of-school suspension: This is a more traditional form of suspension where students are physically barred from school grounds for one or more days.

Zero tolerance policies: School discipline policies that require predetermined serious consequences for particular offenses. Originally, these policies specifically addressed possession of weapons and drugs but were also expanded to others, such as defiant behavior or not following the school dress code.

Pushout: This term is often used instead of “dropout,” since it can be argued that students do not drop out of school unprompted, but as a consequence of policies and actions that make it difficult or impossible for them to remain in school.

Students with disabilities: For the purpose of looking at school discipline, students with disabilities are those who require special education services due to a disability. If a student has a disability but does not need accommodations for it, that student is not considered to fall under this category.

Truant: A student with a large number of absences, usually set by an official standard like a school board or regulatory agency.

The descriptions of the following four terms were adapted from [2].

School counselor: These are professional staff members who counsel students and parents, consult with other staff, evaluate students, and implement guidance programs. They are among the first staff members to work with students who are struggling.

Social worker: Social workers assist students and families with issues such as poverty, homelessness, lack of access to healthcare, domestic violence, and other issues that might affect student performance at school.

Psychologist: School psychologists help diagnose problems of a personal and educational nature in students, and evaluate student social, emotional, and intellectual development.

Nurse: School nurses are qualified nurses who assist with the health needs of students at schools.

x.3 Cui Bono – Who Benefits?

Strict school discipline is supposed to make parents feel safer sending their children to school, and it is also supposed to make the classroom environment more conducive to learning, with fewer distractions. Families of students who are not being disciplined might welcome, for example, presence of police in schools or zero tolerance policies. It could also be argued that teachers and administrators can benefit from not having to spend time and energy on students they view as disruptive.

x.4 The Big Problem: School-to-prison pipeline

Why do schools discipline so many students? Why do students get arrested at school? Who is most affected by school discipline policies?

To understand what the school-to-prison pipeline is, we need to learn some history of police presence in schools and of strict school discipline policies. We will use ratios and percentages to understand how school discipline has changed over time. We will use these same mathematical topics, along with probability, to see how school discipline does not affect all students the same, and in particular that there are certain groups of students that are disproportionately affected by these policies.

x.5 Mathematics Topics

x.5.1 Ratio

A ratio is a mathematical tool for comparison. Ratios can be used either visually or numerically to compare quantitative data. With a ratio, you can either compare two parts of a group to each other, or you can compare a part of a group to the whole group. For this section, the focus is on the comparison of parts of the group to each other, specifically using the context of school support personnel.

First, there is a need to explore the different forms that a ratio can take. There are three main ways a ratio can be expressed:

1. A verbal form (the ratio of part to part)
2. A colon form (part: part)
3. A fraction form ()

Each of these is read the same way as the verbal form, the ratio of part to part. The main significance is recognizing the notation when you come across it in readings. To avoid confusion, most writers will present ratios using a colon form (part: part) and reserve fraction form for comparisons against the whole. The same convention will be followed in this module and other portions of this text (?).

x.5.2 Percents

Percents will be used heavily in this module. There are only two formulas you need to know:

x.5.2.1 Percent formula

part/whole = percent/100

You can use this formula when you are missing the part, the whole, or the percent.

x.5.2.1.1 Example: In 2013-14, there were approximately 49 million students enrolled in public schools. Out of those students, 5.3% were suspended or expelled. How many students were suspended or expelled?

In this case, we know the whole (49 million) and the percent (5.3), but we do not know the part. Therefore,

part/49 = 5.3/100 = 0.053, so

Part = 0.053*49=2.597, so about 2.6 million students were suspended or expelled in 2013-14.

x.5.2.1.2 Example: In 2015-16, 43.42% of the students, or 21,700,551 students, were in schools with police officers. How many total students were enrolled in schools in 2015-16?

Now we know the part (21,700,551) and the percent (43.42), but not the whole, so in this case,

21,700,551/whole = 4342/100=0.4342, and

Whole = 21,700,551/0.4342 = 49,978,238, so there were 49,978,238 students enrolled in public schools in 2015-16.

x.5.2.1.3 Example: In 2015-16, 61,812 students were arrested out of 49,977,268 who were enrolled nationally that year. What percentage of the student population was arrested that year?

We know the part (61,812) and the whole (49,977,268), but not the percent, so

61,812/49,977,268 = percent/100, or

0.00124 = percent/100, which means that

Percent = 0.00124*100 = 0.124%, so 0.124% of all students were arrested.

Note that the part does not have to be smaller than the whole. For example, 130 is 130% of 100. While in real life we cannot give more than 100% effort, there are situations in which percentages are over 100, for example if comparing two numbers (your pay may be 115% of your friend’s pay) or calculating change (for example, the price of coffee may have gone up 200%).

x.5.2.2 Percent change

Percents are often used to represent change. For example, you may read that the number of suspended students in a school increased by 13% in a year. How is this change calculated?

x.5.2.2.1 Example:There were 60,170 school arrests in 2013-14 school year, and 61,812 arrests in 2015-16.

The total change in the number of school arrests between these two years is 61,812-60,170=1,642. To know what percent increase this is, we need to compare to the original number, in this case 60,170:

1,642/60,170=0.027= 2.72%, so the number of arrests went up by 2.72%.

More generally, the formula for percent change from old value to new value is

where end value - start value is total change, and dividing by start value gives percent change. Note that end value does not have to be bigger than start value, in which case we have a decrease and not an increase. For example, if the number of arrests went down from 61,812 to 60,170, then the total change would be -1,642, which is a 1,642/61,812=2.66% decrease. Note that in this case we divided by 61,812 because that was the starting value, and the percent decrease is smaller than the percent increase, because the change of is the same in both cases, while the starting value is different.

x.5.3 Probability

Probability is a specific type of ratio that allows the comparison of specific outcomes with the entire group of outcomes. We see probability expressed in three ways: as a fraction, as a decimal, and as a percentage. All three of these expressions allow us to see a part and a whole, even if in the case of decimals and percentages where the whole is an adjusted quantity. We have two main ways that we think about probability. One is what is called a theoretical model. In the theoretical model, probabilities are driven by expectations of what we know. One classical example of a theoretical model is a coin flip where we know that there are two specific outcomes. The second model is called an empirical model, and it is based on directly counting the occurrences of outcomes. The process of selecting objects to model a process is called a probability experiment. A probability experiment offers a way to explore different outcomes, which we may also call events. We can manually calculate the outcomes of a probability experiment, or use a tool like a simulator to help us understand a situation that is represented in a statement. Our choice to do this depends on the outcome or event that we might be looking at.

Both theoretical probability and empirical probabilities have similar building blocks. The sample space is the collection of all possible outcomes. The probability is a ratio of the specific outcome of interest over the total number of outcomes. In this module, while we recognize the importance of understanding the underlying formulas for probability, we rely on simulations to develop skills that link initial statements of probability and the outcomes of experiments.

x.5.3.1 Exploration 1

Consider the statement: “All students feel safe to be themselves at school.”

● What outcomes are possible?

● Are there reasons that might cause students to feel less safe about being themselves?

● Do you think a student being themselves could make another student uncomfortable? Why or why not?

The possible outcomes are {students feel comfortable, students do not feel comfortable}, and that makes sense. The probability of a student feeling safe to be themselves should result in 1 out of 2 outcomes. Ideally, students feeling safe to be themselves at school should be the only outcome we expect to see. In reality, we find that students do not feel safe to be themselves at school for any number of reasons. Consider the following quote:

“The teachers … they thought we were selling weed in school, they thought that me and her were both selling weed ‘cause like, the way we were dressing, ‘cause we were the only girls at that middle school that dressed like boys. So it was like “now we’re bad.” (non-gender conforming youth from Arizona) [9]

In this quote, the student was exposed to a negative experience, accusations of illegal activity, because of choices associated with how she chose to express who she believed herself to be.

● Were there incidents like this in your school?

Making Sense: Think back to your experiences in school. Identify if there were similar incidents to the incident described above in your community. Do you think that how people were seen by others influenced their behavior, meaning if a label was applied to a person did they adapt to the label? Or do you think that the person stayed the same?

The second type of probability that we use is the empirical model. This comes from raw data. We tend to see this in the form of tables with numeric values, instead of the summary statistics. The basic use of the empirical model is very similar to the ideas from theoretical probability, and when we think about how to make sense of data from a table of data we think about what the theoretical model might have to look like.

x.5.3.2 Exploration 2

The data in the table below is taken from a report by the American Civil Rights Union (ACLU) [2]. The report shows that students in these three states have the same general probability of being arrested per 10,000 students. For many states being arrested as a form of school discipline aligns with zero tolerance offences that qualify for expulsion. For this reason, we may make a reasonable assumption that students may not be arrested multiple times as a form of school discipline.

State Total Students Total Arrests

Missouri 915,033 1,487

Arkansas 480,300 751

Rhode Island 141,210 231

Our first step to exploring the claim in the document is to think about the probability that we want to build. If you are a student going to school, when it comes to being arrested, you either get arrested at school or you don’t get arrested at school. So we have two groups of students, those that get arrested and those that don’t get arrested. We also happen to know about the number of students that get arrested, as well as how many students there are. This means that we have everything we need.

To find the probability of being arrested, we find the ratio of arrests to students for each state. For the Exploration, we will model with Missouri and you can test your ideas on Arkansas and Rhode Island.

arrests/(total students)=1487/915033

We can reduce this to a decimal figure, as finding common factors or a reduced form fraction is highly unlikely. This yields approximately 0.001625.

Now we want to know how this fits with thinking about the probability of being arrested per 10,000 students. In other words, if the probability of getting arrested is 0.001625, this is about 0.1625%; how many arrests would that be among 10,000 students?We use this previously calculated value and solve an equation, to find a value out of 10000:

0.001625=p/10000

Solving this equation gives a value of 16 whole students in Missouri per 10000 based on the overall probability. The report claims that you have the same probability per 10000 in all three states. Can you verify the claim?

x.5.4 More Complex Probability Pieces

Those are the basics of probability. We can use these basic ideas of probability to help us build more complex models that can help us build more complex structures.

x.5.4.1 The complement

The first of these complex structures is the complement. We generally define this mathematically as the probability of an event of interest not happening. If we revisit the exploration above in which we determined the probability of getting arrested. We identified that there were two specific groups, students that got arrested and students that did not get arrested. The probability of not getting arrested can be found in two ways: either by calculating it directly or by subtracting the opposite event from one.

x.5.4.1.1 Demonstration 1: To find the probability of not getting arrested in school in Missouri:

We know that there are 915,033 total students, and we also know that only 1,487 were arrested. So we know that the rest of the students were not arrested. So, we can subtract the arrested students from the total students to find the not arrested students, which gives us 913,546 students that were not arrested. We then find the ratio of not arrested students to total students: 913,546/915,033=0.998375, or around 99.3875%.

We can also find this by using the fact that we know the probability of getting arrested. We start with the group of all students (1), and then we subtract out the group that represents getting arrested: 1 - 0.001625=0.998375, or 99.8375%.

We get the same mathematical answers from calculating the complement using either method. It is important to understand both methods because we have to think about the different types of information that we might have available.

What is something that we notice in this scenario? Can a person be both arrested and not arrested at the same time? That leads us to our second special scenario.

x.5.4.2 Mutually exclusive events

One consideration in probability is the consideration of if events can occur at the same time. One example of events that cannot occur at the same time are an event and its complement, so thinking about our prior example a student cannot be both arrested and not arrested at the same time. This is an example of mutually exclusive events. But when we start to think about more complicated events, like can you be a specific race and be suspended from school, the events become more complicated. Understanding if events are mutually exclusive or not, allows for better and more accurate methods of counting events. Having more accurate probability calculations can help us make more informed judgments about issues.

x.5.4.2.1 Demonstration 1: Determine if you think the following event pairs are mutually exclusive or not. Explain to yourself why you think they are or are not mutually exclusive. What information did you use to make your decisions?

Event Pair A: Being Black and Getting suspended from school

Event Pair B: Being truant and Having regular school attendance

Event Pair C: Making good grades and Being disciplined in school

Following up: Based on models we have looked at, like the simulation at the beginning of the module, we realize that being Black and getting suspended can happen together, so those events are not mutually exclusive. When I know that being truant means not attending school regularly, I realize that event pair B are complements and have to be mutually exclusive. Event pair C is the most difficult to really take apart. Common sense wise and experience wise it seems like maybe the people who got good grades didn’t get in real trouble at school, but then again there is no reason why they couldn’t get into trouble. This means that they have to be not mutually exclusive events.

x.5.4.3 The Addition Rule

As mentioned above, sometimes there is a need to calculate the probability of a combined event. In this case, the event that we are looking at is the case where multiple outcomes satisfy our conditions. Before we start considering answers we have to think about if we have mutually exclusive events. Let’s explore two demonstrations based on the same empirical data adjusted to percentages to help us understand how to use the addition rule with mutually exclusive events and non-mutually exclusive events.

The addition rule allows us to combine multiple events or outcomes to find a total probability. There are two ways to think about this. One perspective is to think about how one event influences a group. We are thinking about school discipline, so we might think about how suspension impacts different racial groups. We would see student race as an outcome and suspension as a separate outcome. The other view of the addition rule is to think about the total probability that a Latin@ student was disciplined at school for a single offence. This would involve finding the probability that a student could get detention, finding the probability that the student could get in-school suspension, finding the probability that the student could get suspended, finding the ways that the student could get arrested, and finding the ways that the student could get expelled. There are several different probabilities that we have to manage. We also have to realize that there is overlap between getting arrested and getting expelled. Once we know the probabilities and the overlaps, we can add the probabilities of the different events together to find the total probability for the individual. Thes

x.5.4.3.1 Demonstration 1: Use the data from the following table to help think about how to answer questions. Then solve the probability problem to the best of your ability.

One issue that has been explored is the relationship between suspension and high school completion. A national survey of 20,774 students in middle school was conducted. From this survey data, 1673 students were identified as having their first suspension within one year of the survey. 684 of these students were Black. In the study, 526 black students who experienced their first suspension within one year of the survey successfully completed high school. What is the probability that a randomly selected student in the survey experienced a first suspension within one year of the original survey and graduated high school if a student is Black? [adapted from 10]

Solving the problem: One of the first steps we have to follow is to think about the events and data that we have. We have the total number of students surveyed. We can think about this as an addition rule problem, because we are taking groups and removing pieces from our groups. We know the total number of students that experienced a first suspension within the first year after the survey, and we also know the number of those students that are Black. Further, we know how many of these students that were Black and suspended but also graduated high school. We can use this to help us build a ratio.

= = 0.0253

Making Sense: Because we reference the original sample from the survey, the whole for the ratio is the whole for the survey, the 20,774 students. By having very specific criteria, we created a very small group of the original sample of students. This makes the probability very low. Do you think the probability is an accurate depiction of what actually happens in the school systems?

Now let’s examine a more complicated simulator related to school discipline.

x.5.4.3.2 Demonstration 2: Let’s consider how to complete a calculation by hand using data from a graph. [Taken from 11]

One theory that has been posed is that students that use a heritage language in their home,meaning a non-English language associated with their heritage, may be considered to have behavior problems because they lack a common understanding of the English language. What is the probability that a student is mechanically restrained and belongs to a racial group associated with a heritage language?

Solving the problem: We have to decide which column is relevant, students served by IDEA or students that have been mechanically restrained. Since the question specifies that we are interested in restrained students we are looking at the second column in the graph. Next we want to think about groups that would be more likely to be associated with a heritage language. Groups that identify as “Natives” tend to have a heritage language, which means that the Native Hawaiian/Other Pacific Islander and American Indian/Alaska Native groups both belong in this category. Many Hispanic families are bilingual, although it is not always the case. For this problem, we will assume that this will allow us to include the group. Our question now becomes should we include the Asian students using the same reasoning? We would need to include both groups if we are using the same underlying reasoning.

Now, we can start to calculate the probability. The presented data is national data, and the groups we have are distinct groups, meaning that if a student were to belong to two demographic groups they would actually fall into the Two or More Races category.

Probability of restraint with a heritage language= (restraint and Native Hawaiian/Other Pacific Islander) + (restraint and American Indian/Alaska Native) + (restraint and Hispanic/Latino of any race) + (restraint and Asian)

Probability of restraint with a heritage language= 0.1% + 2% + 12% + 1% = 15.1%

Making Sense: Based on our calculation, it appears that only 15% of the restrained students were likely to have a heritage language. How do you think this influences the theory that students that have difficulty speaking and understanding English are more likely to be disciplined?

x.5.4.4 Independent Events

There is another way to think about multiple outcomes, and this happens when the events happen independently. In independent events, two separate events happen that have the same underlying probability, or at least we would like to think that they should (refer to x.5.3.1 Exploration 1 for more detail). This is the crux of many arguments about school discipline. For an event to be independent, the underlying probability of the event should be the same across groups.

● From your experience, do you think this is always the case?

● Are there things that you think should be independent but are clearly not independent?

Let’s look at an example about how the math of independence works.

x.5.4.4.1 Demonstration 1: Black students have a 17% chance of being suspended. Consider each time a Black student gets called to the principal’s office, what happens if two separate students get called to the principal's office . Let’s explore the probability of both students receiving the same outcome, meaning either both students get suspended or neither student gets suspended.

What is the probability that both students are suspended?

What is the probability that neither student is suspended?

First, we need to know the probability of not being suspended.

Now, we can use this to calculate the probability of neither being suspended when called to the principal’s office.

When we have independent events that occur in a sequence, we use multiplication to help us determine the probability. These are the essential mathematical pieces that we need to help us evaluate claims about the school to prison pipeline. Let’s dig a little deeper.

x.6 Solving for Change

x.6.1 What is the School-to-Prison Pipeline?

In this introduction to the module, you will begin to think about the school-to-prison pipeline through engaging with a quote and an infographic. Rather than give you data or answers, this section gives you an opportunity to begin asking your own questions.

x.6.1.1 Questions

1. What, if anything, do you know about the school-to-prison pipeline?
2. Read the following excerpt from [6]:

“He was always a good boy. Polite,” his grandmother says. “He was raised to be respectful.” So how did he end up in prison? “It’s like they greased the chute. Back when he was in the 9th grade, Kyron got into a fight. Boys fight. Always have. No guns, no knives, just two boys tussling. Next thing I know he is locked up. That’s just crazy! It’s wrong.”

a. What is your reaction to this quote? Write at least three different things.

b. How might the content of this quote be related to mathematics?

3. Consider the following infographic [1]

a. What do you notice?

b. What do you wonder?

There is no right or wrong way to notice and wonder, so please list as many observations and questions as you can. Are any of your noticings and wonderings related to math?

This topic may be new to you, and it also may be one that personally affects you. If you have personal experiences with school discipline and police in schools, please engage with this module at your own pace and comfort level. If you know little or nothing about these topics, the following sections will provide you with more information about the problem.

x.6.2 History of police in schools

In this part of the module you will learn about the presence of police in schools and about the prevalence of school shootings. Before you read on, answer this questions:

1. What do you know about police officers in schools?
2. Did you go to a school with an SRO?
3. Why would schools have SROs?

x.6.2.1 Exploration: School resource officers in schools

Consider the table below, giving the number of school resource officers, number of public schools, and the number of public schools with school resource officers, by full- and part-time school resource officer status, 2003–04 through 2015–16 [7]:

Characteristic	2003-04	2005-06	2007-08	2009-10	2015-16
Public school resource officers	34,000	36,700	46,100	40,200	52,100
Full-time public school resource officers	16,100	19,400	24,500	21,100	28,600
Part-time public school resource officers	18,000	17,300	21,700	19,100	23,500
Public schools^1	80,500	83,200	83,000	82,800	83,600
Public schools with school resource officers	26,000	26,900	29,400	25,700	35,100
Public schools with full-time school resource officers	13,900	15,700	15,900	14,700	18,300
Public schools with part-time school resource officers	15,700	14,900	17,000	13,000	17,600

¹Detail may not sum to totals because schools that reported both full-time and part-time school resource officers are only counted once in this total.

NOTE: Detail may not sum to totals because of rounding. School resource officers includes all career law enforcement officers with arrest authority, who have specialized training and are assigned to work in collaboration with school organizations. In 2003–04, schools reported on school resource officers present at their school on a regular basis. In 2005–06, 2007–08, 2009–10, and 2015–16, schools reported on school resource officers present at their school at least once a week. The number of school resource officers includes both full- and part-time school resource officers. The number of schools with school resource officers is defined as those schools providing a non-zero response to the count of school resource officers question.

1. Give at least three observations you can make about the data in the table.
2. How much did the number of SROs in schools change between 2003-04 and 2015-2016? Give the total and percent change.
3. What percentage of schools had SROs in 2015-2016? How did that change from 2003-4?
4. Why do you think the number of SROs has gone up so much since 2003?

You probably notice from doing this exploration that the number of SROs and the number of schools with SROs has gone up. In the next exploration we will consider one possible reason for this increase.

x.6.2.2 Exploration: School shootings as a reason for police presence

According to [2, p. 8],

“Following the 1999 Columbine High School shooting, President Clinton called for the first round of Community Oriented Policing Services (COPS) grants as a response that would allow for school/police partnerships focused on “school crime, drug use, and discipline problems.” COPS is a unit of the U.S. Department of Justice. After the Sandy Hook tragedy in 2012, President Obama allocated another $45 million into COPS to fund additional school police. Federal grants were supplemented by state grants and local monies to sustain SRO programs.”

In other words, highly publicized school shootings were part of the impetus for increased presence of police in schools. But how common are school shootings? Consider the infographic below [3]:

1. According to a previous table, there were 83,600 public schools in 2015-16 in the U.S. Using the infographic above, what is the smallest and largest percentage possible of schools that had a shooting in that school year?
2. How are these percentages similar to or different from your perceptions about how frequent school shootings are?
3. Based on the percentages you calculated and the infographic, do you think that police presence in schools is needed to reduce the risk of school shootings? Justify your answer.

x.6.2.3 Summary

The presence of police officers in schools has significantly increased since 1999 in response to the Columbine High School and other shootings. The majority of public schools have a part-time or full-time SRO on campus. While school shootings get a lot of publicity, they are very rare. In the next section we will look at other supports that have been considered important for preventing school shootings and for improving well-being of students.

x.6.3 Presence of other student supports in schools

It has been argued that to prevent shootings, schools should invest in supporting students’ mental and physical well-being. However, in most states, there is a shortage of nurses, social workers, psychologists, and counselors in schools. In this section, you will compare numbers of school support staff (including SROs) in different states.

Before you continue, answer the following questions:

1. Was there a nurse, social worker, psychologist, and/or counselor in schools you went to?
2. How common do you think these support staff are in schools?

We will first look at Delaware and Washington, DC as examples, and then you will have a chance to look at your own location if it is different from these two.

x.6.3.1 Exploration: School support staff in Delaware

Use the following applet to help you explore the following guiding questions about school support staff in the state of Delaware [2].

1. Which pieces look to be similar in size? Are there relationships between the piece sizes?
2. Show the names of the sections. Use this to help you refine your discussion above.
3. Now select the option that allows you to see the number that corresponds with each section. Do the numeric values change any of the relationships that you developed? Are the numbers closeWindow() to the same values?

https://www.geogebra.org/classic/n6ekzput

In the previous exploration, certain pieces were generally the same size even though they did not hold the same numeric value. Let’s take a minute to explore that difference between the visual representation and the numeric data.

x.6.3.2 Analysis: School support staff in Delaware

The orange slice (security guards, 47) and the golden slice (social workers, 54) looked to be about the same size, but the numbers were different. How do we make sense of the disagreement?

If I construct a ratio that compares the two parts what will that look like?

But I am comparing things, not the pieces of the pie chart.

.

I am working to make sense of the quantities, so I should substitute the quantities.

I see that these numbers are closeWindow(), but not the same. What happens in the ratio when I round each number to the tens place.

I know that I can reduce fractions, so I must be able to reduce ratios, too.

So even though the numbers are different, I can still think of them as being roughly the same. I would say that the ratio of security guards to social workers is approximately one to one or roughly one to one. It is not exactly one to one because the numbers are not the same, so the choice of wording is very important.

x.6.3.3 Exploration: School support staff in Washington DC

Examine the following table with data for the school support staff in the Washington DC school district [2].

Counselors	Social Workers	Psychologists	Nurses	Law Enforcement	Security Guards
235	223	198	154	154	333

1. Look at the numbers for the groups and make comparisons between the groups.
2. Does rounding to the tens place allow for generating any interesting ratios?
3. Are there any exact ratios?
4. Critical thinking: How does the ratio of (Law Enforcement + Security Guards):(Counselors+Social Workers+Psychologists+Nurses) look? Were you expecting it to be closeWindow()r together or further apart? How does this help you think about the school-to-prison pipeline?

In the next exploration you will have a chance to look at your state or a state that you are interested in.

x.6.3.4 Exploration: Numbers in your state

Read the following excerpt from [2, p.11]

Given the importance of these providers, experts and professional organizations provide recommended student-to-SBHM provider ratios. The American School Counselor Association recommends a ratio of 250 students per counselor. The National Association of School Psychologists (NASP) recommends a ratio of 500-700 students per school psychologist, depending on the comprehensiveness of services being provided. School Social Work Association of America (SSWAA) recommends that social work services should also be provided at a ratio of 250 students to one social worker. Several states, along with the American Nurses Association, recommend a ratio of one school nurse to 750 students in healthy student populations.

1. Refer to table below [2, p.17] of the report to find the number of students in your state. Do not look at other columns. You can choose the state where you currently live, the state in which you were born, the state where you went to school, or even just a state that interests you.

2. Calculate the number of counselors, school psychologists, social workers, and nurses that should be employed in that state’s public schools according to the recommended ratios from the paragraph.
3. Compare your calculated answers with actual numbers of counselors, social workers, psychologists, and nurses in the table for your chosen state. How do the numbers compare?
4. What did you learn from this investigation and what do you still wish to know?

Summary

The majority of public schools have a part-time or full-time SRO on campus, but not counselors, social workers, psychologists, and nurses. This is true nationwide.

In the next section, we will get to the central problem with school discipline: that it affects different populations very differently. We will first look at arrests and then at suspensions.

x.6.4 Who gets arrested in schools

In this part of the module, you will learn about school arrests. You will look at some national statistics to draw conclusions about racial disparities that exist in school arrests.

Before you continue, answer the following questions:

1. How often do you think students are arrested at school?
2. Why do you think students are arrested at school?

x.6.4.1 Exploration: School arrests

According to [2], there were over 230,000 referrals to law enforcement and 61,000 school arrests in the 2015-2016 school year. The report also states that the actual number is likely much higher due to underreporting.

Consider the following table [2, p. 28] that gives numbers and rates of school arrests by race and disability status by state per 10,000.

1. What do you notice about this table? What do you wonder? Remember that there is no right or wrong way to notice and wonder, but try to include mathematical observations.
2. Notice that the first two number columns give you whole numbers while the others give you rates. Explain what the 12 in the Nation/All students cell in the table means.
3. Using the data in the Nation row, can you calculate how many students with disabilities were arrested in 2015-2016?
4. Are there any particular groups of students who seem to be arrested more than others? What is your evidence for your conclusion?
5. Are there any particular states or regions that seem to arrest students more than others? What is your evidence for this conclusion?

x.6.4.2 Exploration: Reasons for school arrests

In addition to the previous table, consider this one [2, p. 38]. This is a table of prevalence of incidents classified as “serious offenses.”

1. Which of the offenses in the table do you think should be considered serious enough to warrant an arrest? Find how many of these offenses there were in 2015-2016. Compare to the actual number of arrests in 2015-2016 (61,812). How do they compare? What does that tell you?
2. Assuming that arrests in schools were made only for “serious offenses” (which is not necessarily true), calculate what percentage of “serious offenses” resulted in an arrest.
3. Again, assuming that arrests in schools were made for “serious offenses” only, show that at least some students were arrested for physical attack, fight, threat, or robbery without a weapon.

x.6.4.3 Summary

Large numbers of arrests are made in schools every year, most often not for dangerously serious offenses. The arrests are disproportionally among boys, Black students, and students with disabilities.

In the next section we will look at a more common form of school discipline: suspensions.

x.6.5 Who gets suspended from school

In this part of the module, you will learn about school suspensions and begin to make connections between school discipline and incarceration.

Before you continue, answer the following questions:

1. Why do you think students get suspended from school?
2. Are all groups of students equally likely to get suspended or are some groups suspended more than others?

x.6.5.1 Exploration: Prison population simulation

In this exploration, you will examine a data table that shows the overall population for men's prisons by race [4]. Then you will examine three different simulations to see which one you think would be most likely to produce the prison population given. In reading the output from the simulations, there is an overall summary column presented at the left. In the center, there are the summaries from each trial. Then in the far right hand column, the raw data is shown. The simulations model the selection of 50 individual prisoners in 25 repeated samples. Before you start working with the simulations (where you only need to select “start” to collect data), think about strategies that you can use to see if you think a model will match.

White	Latinx	Black	Other
29%	18%	34%	19%

Simulation A represents the general population according to the US Census.

Simulation B represents the percentages of students suspended in schools.

Simulation C represents the percentages with all groups being evenly distributed.

1. The three simulations did not present an exact match to the prison population, but which model do you think was the closeWindow()st to the prison population model?
2. What factors helped you make your decision about fitting the model?
3. What attributes does the model hide by only presenting the proportions that would be useful in examining the fit for the models?
4. The simulations automatically calculated the percentages of each population group. Explore the data that was produced in the simulations. How could you start to think about how the sampler connects with the outcomes?

Notes about probability: In the three simulations, the process was modeled by picking balls from an urn. For the Exploration that you just completed, the outcomes of the probability experiment you conducted were the prisoners that were being simulated. There was also a specific group that was counted to create each proportion. These specific groups were the races of interest being counted. The probability is a ratio of the specific outcome of interest over the total number of outcomes. In the simulations, we looked at different races as the specific outcomes. It may be helpful to revisit the exploration with this definition and see if any of your ideas change.

x.6.5.2 Summary

As is the case with arrests, Black students, students with disabilities, and boys are disproportionately disciplined at school.

In the last part of the module, it may become more clear where the term school-to-prison pipeline comes from. As it turns out, students who are disciplined at school are more likely to leave school and have increased interactions with law enforcement.

x.6.6 Where is the pipeline?

In this part of the module, we return to the concept of the school-to-prison pipeline and look at some connections between suspensions, school pushout, and incarceration.

Before you continue, answer the following question:

1. What long-term consequences may there be for students who are suspended from school or arrested at school?

x.6.1 Exploration: Data from Texas

In 2011, researchers from the Council of State Governments Justice Center in partnership with the Public Policy Research Institute at Texas A&M University published a report [5] based on data from all public school students in Texas who were in seventh grade in 2000, 2001, and 2002. The researchers followed these students for at least six years, with access to academic and juvenile records. Among the researchers’ findings was that only 3% of the offenses for which students are disciplined are ones where by law schools are required to suspend or excel. The other 97% are at administrators’ discretion.

The study also found that 59.6% of all the students were disciplined one or more times and 40.4% were not. The table below summarizes some of the outcomes for the different groups of students.

	Students with disciplinary actions (59.6%)	Students without disciplinary actions (40.4%)
Held back a at least once	31%	5%
Dropped out	10%	2%
Juvenile justice contact	23%	2%

1. Using what you now know about ratios, percents, and probability, explain whether the term “school-to-prison pipeline” is justified for the students from this study.

2. What else would you like to know to better answer this question?

x.6.2 Summary

Students who have been disciplined at school in Texas between 2000 and 2002 were much more likely to be held back a grade, drop out, or have contact with juvenile justice. These school discipline actions were 97% of the time at the administrators’ discretion and not mandated by school policies; and they had a lasting impact on a large number of students.

x.7 Reading Questions

1. Was there a School Resource Officer (SRO) in your elementary, middle, or high school(s)? If you are comfortable, please share what your experience was like.
2. How does the national SRO data [7] compare to your own experience? (Note that the NCES data is only for public schools and does not separate by grade level. You can look at the elementary, middle school, and high school data if you are interested.)
3. What have you learned about the presence of police in schools from this module? Based on your investigations in this module, is police presence in schools beneficial, detrimental, neither, or other? Please explain.
4. What have you learned about school arrests from this module? Do you think there are too many, too few, or just the right number of arrests in schools? Justify your answer using mathematics.
5. What have you learned about school suspensions from this module? Do you think there are too many, too few, or just the right number of suspensions in schools? Justify your answer using mathematics.
6. Solutions: What are some things that districts and states can do (or have done) to decrease the number of arrests and suspensions, and to closeWindow() the racial gap in school discipline?

x.8 Exercises

1. Using the mathematics from this module along with additional sources you wish to use, make an infographic about school discipline and school-to-prison pipeline.

x.9 References

1. Advancement Project, The School-to-Prison Pipeline: Infographic (September 3, 2013), https://advancementproject.org/resources/school-prison-pipeline-infographic/.
2. American Civil Rights Union, Cops and No Counselors: How the Lack of School Mental Health Staff is Harming Students (2019), https://www.aclu.org/sites/default/files/field_document/030419-acluschooldisciplinereport.pdf.
3. A. Kamenetz, A. Arnold, E. Cardinali, National Public Radio, The School Shootings that Weren’t (August 27, 2018), https://www.npr.org/sections/ed/2018/08/27/640323347/the-school-shootings-that-werent.
4. Bureau of Justice Statistics, Prisoners in 2018 (2020), https://bjs.ojp.gov/library/publications/prisoners-2018.
5. T. Fabelo, M. D. Thompson, M. Plotkin, D. Carmichael, M. P. Marchbanks III, E. A. Booth, Breaking Schools’ Rules: A Statewide Study of How School Discipline Relates to Students’ Success and Juvenile Justice Involvement (2011), https://ppri.tamu.edu/files/Breaking_Schools_Rules.pdf.
6. C. Landry, Oklahoma Policy Institute, No Exit: The School-to-Prison pipeline (Neglected Oklahoma) (August 27, 2014, updated May 2, 2019), https://okpolicy.org/exit-school-prison-pipeline-guest-post-camille-landry/.
7. National Center for Education Statistics, Number of school resource officers, number of public schools, and the number of public schools with school resource officers, by full- and part-time school resource officer status: 2003-04 through 2015-16 (2016), https://nces.ed.gov/surveys/ssocs/tables/tab_my01_2016_all.asp.
8. National Center for Education Statistics, Percentage of public schools with security staff present at least once a week, and percentage with security staff routinely carrying a firearm, by selected school characteristics: 2005–06 through 2017-18 (2019), https://nces.ed.gov/programs/digest/d19/tables/dt19_233.70.asp
9. Snapp, Shannon D., Jennifer M. Hoenig, Amanda Fields, and Stephen T. Russell. “Messy, Butch, and Queer: LGBTQ Youth and the School-to-Prison Pipeline.” Journal of Adolescent Research 30, no. 1 (January 2015): 57–82. https://doi.org/10.1177/0743558414557625.
10. Mallett, Christopher A. “The School-to-Prison Pipeline: Disproportionate Impact on Vulnerable Children and Adolescents.” Education and Urban Society 49, no. 6 (July 2017): 563–92. https://doi.org/10.1177/0013124516644053.
11. U. S. Department of Education Office for Civil Rights. “Civil Rights Data Collection: Data Snapshot (School Discipline).” Data Snapshot, March 2014. https://www2.ed.gov/about/offices/list/ocr/docs/crdc-discipline-snapshot.pdf.