Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
You have accessResearch article

Why do students quit school? Implications from a dynamical modelling study

Bechir Amdouni

Bechir Amdouni

Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625, USA

Google Scholar

Find this author on PubMed

,
Marlio Paredes

Marlio Paredes

Simon A. Levin Mathematical, Computational & Modeling Sciences Center and School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, USA

[email protected]

Google Scholar

Find this author on PubMed

,
Christopher Kribs

Christopher Kribs

Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA

Google Scholar

Find this author on PubMed

and
Anuj Mubayi

Anuj Mubayi

Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625, USA

Simon A. Levin Mathematical, Computational & Modeling Sciences Center and School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, USA

Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, USA

Google Scholar

Find this author on PubMed

    Abstract

    In 2012, more than three million students dropped out from high school. At this pace, we will have more than 30 million Americans without a high school degree by 2022 and relatively high dropout rates among Hispanic and African American students. We have developed and analysed a data-driven mathematical model that includes multiple interacting mechanisms and estimates of parameters using data from a specifically designed survey applied to a certain group of students of a high school in Chicago to understand the dynamics of dropouts. Our analysis suggests students' academic achievement is directly related to the level of parental involvement more than any other factors in our study. However, if the negative peer influence (leading to lower academic grades) increases beyond a critical value, the effect of parental involvement on the dynamics of dropouts becomes negligible.

    1. Introduction

    A dropout is usually defined as ‘a person who stops going to a school, college, etc., before finishing: a person who drops out of school.’ It is also defined as ‘a person who stops being involved in society because he or she does not believe in its rules, customs, and values’ [1]. The term dropout in the United States is used in three different ways: event dropout, status dropout and cohort dropout [2]. The event dropout rate measures the percentage of students who left school during a specific period, such as a year, without graduating. The cohort rate measures the dropout for a specific age group such as for students in ninth grade. The status dropout is the proportion of the population without diplomas who are between 16 and 24 years old and are not in school. We use the status dropout as a definition for dropouts in this study. However, there may be differences in the use of terminology around the world such as those collected in table 1. In this study, we primarily use terms common in the USA because of our focus and sampled population.

    Table 1.Some terminology used in the USA and other countries.

    United States some other countries
    freshman (∼14–15 years old) grade 9
    sophomore (∼15–16 years old) grade 10
    junior (∼16–17 years old) grade 11
    senior (∼17–18 years old) grade 12
    high school secondary school
    dropout early school leavers
    charter schools autonomous/free schools (receiving reduced state funding)

    According to the National Center for Education and Statistics [3], more than three million students dropped out of high school in 2010 in the USA. This means around 17 000 students drop out daily, considering a school year of 180 days. If this trend continues, there will be approximately 31 million dropouts, roughly 10% of US current population, by 2020.

    Dropouts have negative effects on various aspects of our society. According to the US Census Bureau [4], the median earnings in dollars of a non-graduate from high school was $27 470, compared to $34 197 for graduate high school earner. This difference results in about $350 000 less income over the 35 years of work between a graduate and non-graduate high school student. This will imply less taxes and revenue for the government which results in lower purchasing power and a low level of productivity. In addition, non-high school graduates are less likely to be employed versus the high school graduate [5]. Relative to those who graduate, those who do not graduate will each cost the economy around $250 000 in terms of reliance on Medicare or Medicaid, higher reliance on welfare and lower tax contributions [6].

    Because the high school dropout has an enormous effect on the society as a whole, many studies and reports have analysed various sets of high school data with varied success. Even politicians have spoken about this topic and many plans and strategies have been put in place in order to understand the problem. In 2010, President Obama said: ‘This is a problem we can't afford to accept or ignore. The stakes are too high—for our children, for our economy, for our country. It's time for all of us to come together—parents and students, principals and teachers, business leaders and elected officials—to end America's dropout crisis’. [7]

    Researchers have identified numerous factors for dropping out from high schools. Some of the factors include socio-economic status, academic disengagement, behavioural disengagement, family dynamics, attitudes, values and beliefs about education, school structures, school resources, student body characteristics, school environment, academic policies and practices, supervision and discipline policies and practices, location and type of school, demographic characteristics and community environment [810] as well as challenges associated with learning certain subjects and concepts. Most of these factors are interrelated and it is hard to determine which factor causes the others. Moreover, time and again, high parental involvement and guidance has been shown to have a major impact on a student's life, reducing the likelihood of dropping out. However, parents reported having limited time to share and difficulty of involving children in household activities [11]. Many families include two breadwinners and some are single-parent families with conflicting roles and responsibilities to support their families. Existing policies and programmes for reducing dropout rates are often not designed to meet needs of low-income and working poor families. There are trade-offs between the longer working hours of parents, the high cost associated with the development of children and better environments for children [12]. This research reviews factors driving dropouts and helps to suggest a wide range of components that may be likely to make a difference. The components include evaluating patterns of potential dropout rates, identifying at-risk students and changing the factors that schools may control to reduce dropouts.

    As for the proposed solutions, The National Dropout Center/Network identified fifteen different strategies to mitigate the dropout rate [13]. These strategies consist of an ongoing evaluation system of the goals and policies; community participation and support; safety for students; parental engagement; focus on early childhood education, especially in reading and writing; continuous support through tutoring and mentoring; engaging students within the community; providing after-school and summer programmes especially for at-risk students as well as an alternative school for dropout students; maintaining an ongoing professional development for teachers who are engaged with high-risk students; student engagement through technology and individualizing students' instruction.

    As mentioned above, the suggested factors for high school dropouts and the remedies are numerous and interrelated. We focus on understanding the impact of some significant primary factors such as parental involvement, demographic factors and peer influences between students, some identified from our survey, on the dynamics of high school dropout prevention. This does not undermine the importance of the rest of the factors. The reasons for limiting the model to these main factors are the feasibility of analysing the mathematical model on the one hand, and the lack of studies done on the dynamics of students on and off campus on the other hand.

    Despite the complexity in dropout dynamics [14,15], Mathis et al. [16] argued that the United States government has been implementing policies that are not research based, hence, the urgent need for the present work. There are multiple purposes for this study including testing of a mathematical model based on parental involvement and contact between students in order to understand the dropout dynamics. Dropping out of high school has unfavourable consequences on the individual as well as the society as a whole [17]. It has been linked to numerous factors that can be divided into internal (within the school) and external(outside the school). Despite the large number of suggested solutions, the number of dropouts seems to be on the rise especially within large cities. We study the impact of parental involvement and social contacts between students inside and outside of school on the dynamics of dropout. Here, we focus on school students and their activities, collectively referred to here as school environment. It represents school time during which social relationships between current students develop within a campus, and potential off campus relationships between students and non-students or their parents. Our goal is to focus on factors of the school environment and see how the number of failing students can be decreased and, in consequence, reduce the dropout rate. Our approach consists of the development and analysis of a dynamic mathematical model using an analogy with an SIR-type infectious disease model via a set of differential equations, its analysis and simulations.

    In particular, using our carefully designed study, we focus on (i) identification of factors that may be critical to students' academic achievement and quantification of the impact of parental involvement on it via a survey, (ii) development of a dynamic model using factors identified through the survey specifically designed for the model, and (iii) analysis of the model to understand the role of parental involvement and peer interactions on the dynamics of dropouts.

    2. Methods

    (a) Data sources

    We designed and carried out a survey in a group of students of one sampled school in the Chicago Public School (CPS) system. The group contained students from 9 to 12 grade levels. An approval for conducting the survey was obtained from the institutional review board at Northeastern Illinois University in 2013. The survey was distributed to 145 participants among students of the sampled school. However, five students did not respond to the survey; therefore, 140 surveys were eventually collected. Fifteen additional surveys were considered unusable because they were incomplete. With 125 usable survey responses out of 145, the response rate was 86.2%. The survey contained 16 questions and was taken online by students. Questions in the survey resulted in quantifying metrics such as the number of failing students in different core subjects (Mathematics, Science, English and Social Sciences), degree of parental involvement in their life and number of failing and dropout friends of each student during last year. Sampled students were also asked questions related to last year's performance levels and interests of their friends in education. Additional data, which cannot be obtained from our survey, was collected from the National Center of Education Statistics [2] and CPS online databases. These last two datasets were used to estimate dropout rates and the total enrolment numbers from 2005 to 2013 for the city of Chicago and for the USA [18]. We compared demographic characteristics of national/Chicago city school students with corresponding statistics of students from our sampled school (figure 1). We also collected and relied on the information from previous studies in the literature that surveyed dropout youths in the USA [14,19,20].

    Figure 1.

    Figure 1. Demographic comparison of average statistics from Chicago public schools (CPS) and the sampled school (for this study) in Chicago. Note that this school has around 90% minorities (includes primarily African Americans and Hispanics) similar to the average minorities population in a CPS school.

    CPS district is the third largest school district in the USA [18]. It encompasses a total of 681 schools, out of which 472 are elementary and 106 are high schools. Moreover, it has 96 Charter schools and seven Contract schools. CPS hosts a total of 400 545 students from which 87% come from low-income families. The CPS student population is diverse, but Hispanics and African-Americans make up around 85% of the total (figure 1).

    We carried out a survey at the sampled high school in Chicago which was chosen because it is one of the most representative (falling in the lowest 5% of all schools) schools in south Chicago based on socio-economic factors related to students and the neighbourhood where they live. The total student enrolment in the sampled school in 2013–2014 was 1866 [18], where about 94% of students come from low-income families and 73.3% were Hispanic (figure 1).

    In order to quantify levels of parental involvement in the school, we asked students questions related to involvement of parents in their daily activities during the previous semester. The parental involvement was defined and measured using inputs of students on three primary activities of their parents: attending parent teacher conferences, attending after-school activities, and checking grades at least once a month. We defined high parental involvement as a student's response mentioning parents being present for all the above three activities. A moderate parental involvement means parents fulfilled two out of the three activities. A low parental involvement means parents fulfilled one of the three activities. Finally, no parental involvement means parents failed to participate in any of the three activities of the student (figure 2a).

    Figure 2.

    Figure 2. Parental involvement, academic achievement and an aspect of social influence. The definitions of these terms and process of data collection are explained in Data sources section. The quantification of parental involvement metric was based on answers to a question on parents' participation in a student's activities, academic achievement was based on students' performance in core courses and the social influences metric was computed based on students' opinion of their friends (about their friends' education and performance levels in the school). (a) Degree of parental involvement, (b) academic achievement of a student and (c) social influence on a student as his/her opinion on his/her friends.

    Academic achievement of a student in the school was used to categorize them into passing, vulnerable and failing categories. A student is grouped in a passing category if he or she passed all the core subjects; in a vulnerable category, if s/he failed only one core subject and in a failing category if s/he failed two or more core courses (figure 2b).

    Peer influences that eventually lead to a student dropping out from the school were also defined and measured. These influences were termed as negative social influences and were manifested through multiple facets such as but not limited to the number of failing friends, the number of dropout friends, and illegal and criminal activities in the region. One way to capture the negative social influence is to quantify the attitude of friends of an individual towards school (figure 2c). Hence, students in the survey were asked about their friends' performance levels in the education (‘among your close friends in the school, how many are failing two or more classes?’) as well as their friends' opinion about school, for which question the answer choices were: if school is a waste of time for your friend, if sometimes it is a waste of time and if not a waste of time? (figure 2c).

    The data were collected primarily to estimate parameters of the dynamic model discussed in §2c. We also investigated the correlation between students' achievement and negative social influence and explored the impact of parental involvement on students' achievement in the school. A typical dropout individual leaves the modelled population at the age 24 (i.e. youths leave the community/neighbourhood of the school at the rate 1/(2418)=16 per year). However, the average number of years a typical student spent in the school is assumed to be around 4 years, though this number could be high for a school in a poor neighbourhood with a vulnerable (to dropout) population of students.

    (b) Model description

    We developed a model based on socio-demographic characteristics of our sampled school. The total population (T) in the model is divided into two subpopulations: normal (N) and dropout (D). The normal population is stratified into passing (P), vulnerable (V) and failing (F) groups of students (table 2). Our model structure follows an SIR-type epidemic compartmental model framework [2125]. Similar types of models have been derived for other social problems such as for fanatic behaviours [26] and alcohol drinking [27]. The model considers multiple mechanisms including rates of incoming students (μN+μ1D), outgoing students (μ1D), parental involvement (σV), the school effectiveness (αP), social influence inside the school between individuals in V class and in F class (β1V (kF/T)), social influence outside the school between failing and dropout individuals (β2(D/T)F), and self-dropout owing to non-school factors (γF), which is ignored in this research because of lack of data for it. From here onwards, we refer to the passing and vulnerable populations as the non-risky population and call the failing and dropouts the risky population (table 3). Table 4 summarizes the definition and units of parameters and table 2 contains the definition of variables. The interpretation of the model parameters is clarified in §3, where parameter estimation is carried out using data from our survey.

    Table 2.Definition of variables in the model.

    variables definition
    T total population of the community
    N population of students in the sampled high school
    P number of passing students in all core classes (mathematics, sciences, English and social sciences)
    V number of vulnerable students, who are failing one core class
    F number of failing students, who are failing two or more core classes
    D number of dropout students

    Table 3.Potential equilibrium states of the model.

    equilibrium definition (v) (f) (d)
    E0 completely passing state (only passing students' type exist) v0=αα+σμ 0 0
    E1 absence of dropouts v1=μβ1k f1=αα+μ(R011R01) 0
    E1 dropout endemic equilibrium v1 f1 d1
    E2 dropout endemic equilibrium v2 f2 d2

    Table 4.Description of the parameters used in the model.

    parameter description value reference
    α per capita transfer rate from passing to vulnerable owing to the school ineffectiveness 0.59 per year estm.
    γ per capita dropout rate because of personal reasons n.a.
    p probability of successful passing of negative influence given a contact (dimensionless) 0.2 estm.
    β1=c1p transmission efficiency (i.e. average effective negative influences) of dropouts (D) on vulnerable (V) students outside the campus 0.51 per year estm.
    β2=c2p transmission efficiency (i.e. average effective negative influences) of dropouts (D) on failing (F) students outside the campus (part of δ(D) function in the model) groups 0.42 per year estm.
    k decrease in intensity of transmission efficiency (negative social influences) of F (as compared to D) on V (dimensionless quantity). Note, β1k=c1p is the transmission efficiency between F and V 0.6 estm.
    σ per capita students' average improvement rate because of enhancement of parental involvement 0.41 per year estm.
    1/μ average number of years a typical high school student stay in the school 4 years estm.
    μ1 per capita local community exit rate of dropouts 0.17 per year assumed

    In order to analyse the model, we assume that students will graduate within 4 years. Those who graduate will leave the community and will not have contact with the current student population in the school. Furthermore, students entering high school are classified as passing students. In addition to that, according to Allensworth & Easton [28] from the University of Chicago's Consortium on Chicago School Research [28], students who fail more than one core class during one whole year are less likely to graduate on time and are called ‘off-track’. On the other hand, students who are failing, at most one core class in a whole year, are three times more likely to graduate on time and are called ‘on-track’. The core classes are mathematics, science, English and social sciences. Our definition of vulnerable students is based on this study and are students who are failing one core class. Students who are failing two or more classes are classified as ‘failing students’. It is also assumed that a low parental involvement will lead passing students to move from passing to vulnerable class. If parental involvement continues to be low, this will lead to a longer number of unhealthy contacts with failing students which will eventually lead to dropout. Figure 3 describes the dynamic of the model and it can be expressed as a system of coupled differential equations (see electronic supplementary material, equation (4.1)).

    Figure 3.

    Figure 3. The flowchart of the mathematical model.

    The model captures the dynamics associated with dropout rates via a system of differential equations. The passing population builds its membership from the new incoming freshmen. Owing to the ineffectiveness of the school, a proportion of students will leave the passing population and move to the vulnerable stage. Vulnerable students may improve their grades as a result of reinforced parental influence (with a rate σV) or may move to failing class under the influence of failing and dropout peers. The rate of transfer from vulnerable to failing is β1V ((kF+D)/T), where β1 measures the effective negative peer pressure associated with the failing students and k is the decrease in intensity of contact of failing individuals compared with the corresponding influence by dropout peers. Hence, the transfer-to-failure rate is proportional (with rate constant β1) to the number of contacts per year between vulnerable and failing. The dropout is replenished via δ(D)=β2(D/T)F, which is the number of dropouts per year.

    (c) Relevant quantities

    For γ=0, the model has two reproduction numbers (see [29]) relative to two distinct dropout-free states, namely E0=(v0,0,0) and E1=(v1,f1,0) (see calculations in electronic supplementary material). The first basic reproduction number corresponding to E0 is

    R01=β1kv0μ=β1k(1μ)v0.2.1
    The quantity R01 represents the average number of newly generated vulnerable students influenced by a typical failing and dropout peer in a completely non-risky population. It is the product of the average number of contacts of an individual and the probability of his/her successful influence β1, the infectious period of a typical F individual 1/(μ+γ), the decrease in intensity of contact k and the proportion of non-risky students v0.

    The second basic reproduction number, corresponding to E1, is obtained considering D as the only influential compartment:

    R02=β2f1μ=β2(1μ)f1.2.2
    The quantity R02 represents the average number of newly generated failing students in a completely non-risky population influenced by a typical dropout peer. This number is the product of the effective (successful negative influence) average number of contacts of an individual β2, the duration of influence by a typical F individual 1/μ and the proportion of failing students f1.

    Depending on the value of R01 and R02, we can have up to four equilibrium points. In addition, mathematical analysis and numerical simulations suggest the existence of another threshold quantity Re=(R01−1)/(R01+k) (see the electronic supplementary material for details).

    Remark 2.1

    In case of γ=0:

    • (i) the completely passing equilibrium E0 always exists for the system irrespective of the values of R01 and R02. If R01>1 and R02<Re<1, then the failing endemic equilibrium E1 exists. If R01>1 and Re<R02<1, then four equilibria E0,E1,E1,E2 exist. Finally, if R01>1 and R02>1, then E0,E1 and E2 exist (figure 4a).

    • (ii) If R01<1, then E0 is locally asymptotically stable (see electronic supplementary material). If R01>1, then E0 becomes unstable and numerical simulations suggest that stability of non-trivial equilibria (E1, E1 and E2) will depend on R02 value (figure 4b).

    Figure 4.

    Figure 4. Bifurcation diagram with respect to the model's two reproduction numbers R01 and R02 for the case γ=0. Panel (a) shows conditions of existence of equilibria. Panel (b) shows condition of potential stability of equilibria via only one component of the equilibria, f* (the proportion of failing students). Continuous line represents local stability and dotted line represents instability in panel (b). (a) Existence conditions for equilibria (E0, E1, E1 and E2) (b) Local stability conditions for equilibria (E0, E1, E1 and E2) depend on both R01 and R02.

    Remark 2.2

    In case of γ≠0, again the completely passing equilibrium E0 always exists for the system irrespective of the value of R01=(β1kv0)/(μ+γ)+ (β1γv0)/μ(μ+γ)=β1k(1/(μ+γ))v0+β1(γ/(μ+γ))(1/μ)v0. If R01<1, then E0 is locally asymptotically stable (see electronic supplementary material). If R01>1, then one endemic equilibrium exists and numerical simulations suggest that this endemic equilibrium is locally asymptotically stable. Because we did not have data to estimate γ and because mathematical complexity (by taking γ≠0) provided immense challenge to obtaining analytical results, in this study, we ignored the interpretations of results for the case γ>0.

    3. Results

    (a) Survey interpretation

     Table 5 shows the number of passing, vulnerable, and failing students for different levels of parental involvement, where high parental involvement means parents fulfil the three categories (attending parent teacher conference, attend after-school activities and check grades at least once a month). Moderate parental involvement means completing two out of the three categories. Low parental involvement means parents completing one task out of three. No parental involvement means parents fail to execute any of the three tasks. Using data from table 5, we conducted a Chi-squared test to study the relationship between parental involvement and academic achievement of students. The null hypothesis was academic achievement is independent of parental involvement. The test suggested that we do not have enough evidence to reject the null (α=5%, χ62=13.66, p=0.0336), that is, academic achievement may depend on parental involvement.

    Table 5.Degree of parental involvement versus students' performance status in core courses.

    degree of parental involvement passing vulnerable failing total=ni weight=wi
    no parental involvement 8 4 7 19 0/3
    low parental involvement 50 7 14 71 1/3
    moderate parental involvement 21 2 1 24 2/3
    high parental involvement 7 3 1 11 3/3
    total 86 16 23 N=125

    Based on this survey, 48.8% of students have contact with peers who always think school is a waste of time, 27% of them have peers around them who sometimes think school is a waste of time (figure 5a), and 57% of dropouts do not live with their parents (figure 5b). The data suggest that as the degree of parental involvement increases in the vulnerable (failing) student's life, the number of their failing/dropout friends decreases (figure 6); however, this trend gets changed for failing students with increases in parental participation in their life (their dropout friends first decreases and then increases; figure 6). This may occur if students become more rebellious to a sudden increase in parental involvement beyond a certain level while being discouraged from their academic performance in education.

    Figure 5.

    Figure 5. Attitude and opinion of students towards school environment and living arrangements of their friends who dropped out. These data are gathered via questions to each of the students in the sample on their opinions about themselves and their friends. (a) Attitude of students' environment towards school, (b) dropouts' living arrangement.

    Figure 6.

    Figure 6. Average number of failing and dropout friends per (V or D) student based on degree of their parental involvement. Tables 6 and 7 show how estimates are computed for these metrics. (a) For a vulnerable student, (b) for a failing student.

    In summary, survey results suggest that parental involvement in a student's life is strongly related to the student's academic achievement, a large number of students are in frequent contact with individuals who think that attending school is a waste of time, and dropout rates are higher among students not living with their parents. Moreover, the type of friends of a student is positively affected by parental involvement only if the student is performing better than failing in education.

    (b) Parameter estimates

    The parameter estimates are obtained using data from our survey and literature review. Methods similar to those discussed in Romero et al. [30], Mubayi et al. [31] and Kribs-Zaleta [32] were used to derive estimates. However, there are some limitations in carrying out our estimates. Some of the parameters such as σ, β1, k and β2 are to be estimated using temporal data on contacts. Because we have only cross-sectional data (i.e. data from one time point), we considered an ad hoc simple method to show estimation process and provide a crude estimate for these parameters.

     Estimate of σ: in our survey samples, 17 students were freshmen, 30 were sophomores, 50 were juniors and 28 were seniors. We found that 86 students were passing all core classes, 16 were vulnerable (failing only one core class) and 23 were failing (failing two or more core classes). The parental involvement metric is a complex component and, for simplicity, we restricted the definition of parental involvement in our study using three categories: parent–teacher conference attendance, attendance at after-school activities and checking grades at least once a month. The weighted average value for parental involvement in the cohort was based on the last two columns of data collected in table 5, where weights (wi) were computed using the levels (i.e. number) of activities of involvement. The σ estimates were obtained under the assumption that the rate of transition of students from vulnerable (V) to passing (D) state because of parental involvement is directly proportional to the degree of parental involvement in students' life. Hence, σ was estimated as

    σi=03winiN=0/319+1/371+2/324+3/311125=0.41 per year,
    where N is the sample size.

     Estimate of α: the data collected from our sampled school included questions on number of failing core classes at two time points: at the time of the survey and five weeks (35 days) ago. Seven students reported that they were passing all core courses five weeks ago and now failing at least one course. The parameter α represents the per capita transition rate of students from passing to vulnerable state. We obtained the estimates of α by averaging unit changes that occur from P to V over all students as below:

    α7353651125=0.59 per year,
    where N=125.

     Estimate of μ and μ1: a typical student enters high school at 14 years old and graduates when around 18 years old. Therefore, the average number of years spent in high school is around four, making μ=1/4 (=0.25) per year. For μ1, we assumed that a typical dropout leaves the modelled population at age 24 (i.e. students leave the population at the rate ≈1/(24−18)=0.17 per year). This is because dropout students are not part of the school population of students (i.e. P+V +F) but D are in the same community.

     Estimate of β's and k: for vulnerable students, based on the degree of parental involvement, we calculated the number of friends who are failing and the number of friends who are dropouts (table 6). Similarly for failing students, we calculated the number of failing and dropout friends for students (table 7). We assumed that each friend can contribute to a contact/influence. We also assume that failing and dropout contact can lead to negative social influence. From tables 6 and 7, we can see that as parental involvement increases, the average contact with failing and dropout friends almost decreases for both vulnerable and failing students.

    Table 6.Average number of contacts between vulnerable students, and failing/dropouts based on parental involvement (PI).

    ‘friends’
    vulnerable students failing
    dropout
    type of PI count of V (CV) count (CFf) average (=CV/CFf) count (CDf) average (=CV/CDf)
    no parental involvement 4 13 3.25 5 1.25
    low parental involvement 7 18 2.57 9 1.29
    moderate parental involvement 2 4 2 0 0
    high parental involvement 3 6 2 2 0.67
    total 16 41 16

    Table 7.Average number of contacts of a failing student with other failing students or dropouts structured based on parental involvement (PI).

    ‘friends’
    failing students failing
    dropout
    type of PI count of F (CF) count (CFf) average (=CF/CFf) count (CDf) average (=CF/CDf)
    no parental involvement 7 18 2.57 9 1.29
    low parental involvement 14 32 2.29 15 1.07
    moderate parental involvement 1 1 1 0 0
    high parental involvement 1 4 4 0 0
    total 23 55 24

    As shown in table 6, the 16 vulnerable students have 41 failing friends and 16 dropout friends. That is, on average, every vulnerable student has 2.56 (≈41/16) failing friends and 1.0 (≈16/16) dropout friend. Table 7 shows that the 23 failing students have 55 failing friends and 24 dropout friends. On average, every failing student has 2.4 (≈55/23) failing friends and 1.05 (≈24/23) dropout friends.

    There are three negative social influence parameters, namely, β1, β2 and β1k, where β1 represents the potential effective influence between a vulnerable and a dropout individual, β2 represents the potential effective influence between a failing and a dropout student and β1k represents the potential effective influence between a vulnerable and a failing student. Contacts are assumed to vary over a year, so that units of these parameters are taken as per year. In order to have precise units, temporal data needs to be collected and used to estimate these parameters. Each of these three parameters is equal to the product of the average number of contacts with failing and dropout students and the probability of a successful negative social influence given a contact (table 4). Let c1 and c1 be the average numbers of negative contacts that a vulnerable student has with a failing and dropout friend, respectively. Because of the difficulty of calculating the number of negative contacts inside and outside the school, we use the average number of failing friends as an average number of negative contacts. We use data in table 6 to estimate these parameters c1=41/16=2.6 and c1=16/16=1.0. However, because of limited data our definition of computing average number of contacts is a simple one and could be made more sophisticated if longitudinal data can be collected or available.

    The student population in the city of Chicago faces multiple challenges such as low income, low parental involvement and high crime rate [33]. Students spend on average 8 h (480 min) per day in school considering walking a few minutes to and from school. To estimate the number of minutes a student spends on average with a failing friend, we add the passing periods (7*4=28 min), a lunch period (50 min) and few minutes before (9 min) and after school (9 min) which add to 96 min equivalent to 20% of the whole day in school. Therefore, we assume the probability of a successful negative influence between vulnerable students and failing to be about 20% or higher. Because parameter β1 is a product of average number of contacts and effectiveness of an influence given a contact, this leads to β1=c1*p=2.56*0.2=0.51 per year and β1k=c1p=10.2=0.2 per year.

    Using the formula β1k=0.2, we can compute k=0.2. However, the parameter k, for our purpose, was estimated by using the following criteria. Students spend on average 8 h out of 13 h a day in a school. It is assumed that on average 13 h a day, a typical student interact with others in the similar age group. Even if students are at home, now-a-days, students are in contact with their friends non-stop through social media. However, we assume negligible social media influence, because we focus on schools that are in low socio-economic regions and students in these regions are assumed to have less social media access. Hence, the proportion of influences of a F (they are in school for 8 h) as compared to that of a D (not in school whole day) student is 0.6(=8/13). Thus, we took k≈0.6.

    β2 represents the potential negative social influences between failing students and dropouts and is equal to the product of the average number of negative contacts with dropout for failing students and the probability of a successful impact. Let c2 be the average of number of contacts that a failing student has with a dropout. We use data in table 7 to estimate it as c2=24/23=1.1. Students spend on average 8 h per day in a school out of a 13 h day. To estimate the number of minutes a failing student spends on average with a dropout friend, we subtract 8 h from 13 h, giving an estimate of 5 h (300 min). This 300 min is equivalent to 38.5% of the whole day. Therefore, we assume the probability of a successful negative influence between failing students and dropouts to be about 38.5% or higher. Similar to β1, we have β2=c2*p=1.1*0.385=0.42 per year.

    These parameter point estimates result in R01=1.04>1, R02=0.05 and Re=0.03. Because R01>1 and R02>Re, we have a dropout endemic equilibrium in the system.

    (c) Simulations

    Numerical simulations were carried out to study the impact of parameters on the threshold quantities and dynamics of the system. Figure 7a shows that as social influences increase, R01 increases linearly though the rate of its increase decreases with increases in σ. That is, in the presence of low parental involvement, high efforts will be needed to reduce negative social influences within school in order to reduce the failing student population significantly. The threshold quantities R02 and Re (persistence conditions for dropouts in the population) decrease exponentially past a critical value with decrease in influences occurring on vulnerable students (captured by β1 in the model; figure 7b). However, R01 (persistence conditions for failing students in the population) decreases only linearly with decreases in β1. Figure 7c confirms the asymptotical behaviours of the system analytically computed in §2c; the long timescale reflects changes in the overall dropout levels in the school system across generations of students.

    Figure 7.

    Figure 7. Numerical simulations show variations in threshold quantities and dynamics of failing students with changes in sensitivity parameters (k, σ, β1 and β2). In panel (a), k was varied from 0 to 1 for three σ values (0.01, 0.41 and 0.81 per year), and R01 was computed. In the panel (b), β1 was varied in (0.5,1.5) per year and R01, R02 and Re were computed. In panel (c), the curve R01=1.04 was computed using table 4 parameter estimates, the curve R01=2.57 was computed using k=0.5, β1=1.51 per year and β2=0.2 per year, whereas the curve R01=7.18 was computed using β1=3.51 per year. In these graphs, the rest of the parameter estimates were kept fixed to the values mentioned in table 4. (a) k versus R01. (b) β1 versus the three threshold quantities (R01,R02,Re). (c) Dynamics of proportion of failing students when R01>1 along with (i) R02<Re<1, (ii) Re<R02<1 and (iii) 1<R02.

     Figure 8 shows the impact of social influences within the school on the proportion of dropouts in the community. We fixed the level of the social influence occurring outside the school, β2, and carried out simulations based on low and high levels of parental guidance for a low/high level of social influence inside the school. First, we set a high level of social influence and high level of parental guidance. As the literature indicates, the number of dropouts rises at first then stabilizes at a low level. Second, for the same high level of social influence inside the school, we set a low level of parental guidance. This led to a high proportion of dropouts. Third, we lowered the level of social influence. Regardless of parental involvement, the number of dropouts was very low. We found similar results for the F compartment. Regardless of whether parental involvement is high or low, as long as the level of social influence is low represented by low number of contacts between vulnerable and failing students, the number of dropouts as well as the number of failing students will be very low. Hence, there is a need to find the critical level of social influence to see its effect on the number of dropouts as well as the number of failing students.

    Figure 8.

    Figure 8. Impact of negative social peer influences on the proportion of dropouts in the presence of varying degree of parental guidance. In these simulations, low and high social influences within the school are captured by k=0.3 and k=0.9, respectively, and low and high parental influences are captured by σ=0.1 per year and σ=0.7 per year, respectively. Initial conditions are 75% passing, 16% vulnerable, 8% failing and 1% dropout students.

    From figure 9, we can see that in the presence of high parental involvement, as long as the level of social influence represented in our model by β1 is below 0.8, the number of failing students will be low. From figure 9, in case of low parental involvement, the critical level to insure low number of failing students is 0.4 for β1. Regardless of the level of parental involvement, if β1 is more than 0.8, the number of failing students will rise (figure 9). We found similar results for the D compartment. That is, regardless of the parental involvement, increasing peer influence, within the school, beyond a critical value results in an increasing number of failing students and dropouts. The critical value of peer influence is a function of parental involvement and is smaller for higher values of parental involvement.

    Figure 9.

    Figure 9. The critical level of social influence for the number of dropouts for γ=0. In these simulations, low and high social influences within the school are captured by k=0.3 and k=0.9, respectively, and low and high parental influences are captured by σ=0.1 per year and σ=0.7, respectively. Initial conditions are 75% passing, 16% vulnerable, 8% failing and 1% dropout students.

    4. Discussion

    Dropping out from high school has been a major problem in the USA, especially in low socio-economic communities. There are many factors that may be reasons for student dropout; however, two factors, parents' guidance and peer influences, can drastically shape one's decisions to continue or discontinue school. In this study, we study dynamics of dropouts as a function of parental and social influences occurring both within and outside of school. We developed a mathematical model that captures patterns of students and dropouts with analogy to the susceptible–exposed–infected–recovered disease model [34,35]. We studied this model analytically and then illustrated our model using estimated parameters from our survey. Studies have shown a strong correlation between dropout and socio-economic status, academic disengagement, family dynamics, school culture and numerous other factors [3638]. Our aim here is to focus on some factors that can be managed within the school. In particular, we asked, how can we lower the number of failing and dropout students using school resources, regardless of the outside factors? Primarily, we look at the role of factors related to social influences.

    Data analysis suggests that as the degree of parental involvement increases in the vulnerable student's life, the number of his/her dropout friends decreases; however, for failing students, although increases in parental participation in their life result in an initial decrease in their dropout friends, eventually the number of such friends increases. Moreover, a large number of students are contacting friends who think that attending school is a waste of time. We also found that dropout rates are higher among students not living with their parents and, as expected, academic achievement is strongly dependent on parental involvement.

    We model individuals of a community in an age group similar to high school students. Our model population consists of four different compartments: passing (students passing all core subjects), vulnerable (students failing one core subject), failing (students failing two or more core subjects) and dropout (those who dropped out from school and living in the school community). These individuals can transition sequentially from one compartment to another under different social mechanisms. The long-term system behaviour is governed by three threshold quantities (R01, R02 and Re) defined below.

    The parameter R01 represents the average number of newly generated vulnerable students influenced by a typical failing or dropout peers in a population of primarily passing students. This number is equal to the sum of contributions by individuals in the F and D classes. The quantity R02 represents the average number of newly generated failing students in a vulnerable population influenced by a typical dropout peer. This number is equal to the effective (successful negative influence) average number of social contacts of an individual, the duration of influence by a typical failing student, and the proportion of failing students. Re is a critical value of R02 below which there are no dropouts.

    Our mathematical analysis suggests the existence of four different states of the system, namely: the completely passing state (the state with all students passing all core subjects; this state depends on the proportion of students who move from passing to vulnerable, the degree of parental involvement and the time spent in school before graduation), the state with no dropouts (healthy community with all students attending school; it depends on the efficiency of influences of vulnerable and failing students, the degree of parental involvement and time spent in the community) and two dropout equilibria where dropouts persist in a community irrespective of parental guidance. The completely passing state is asymptotically stable only when the average number of newly generated vulnerable students (R01) is less than one. The state with no dropouts exists and is locally stable when R01>1 and R02<Re<1. The two dropout equilibria exist when R01>1 and Re<R02, but only one seems to be locally stable whenever they exist. We estimated model parameters from data that we collected from one public school in Chicago. Our results suggest that the negative social influences within the school leading to dropout and the average degree of parental guidance in a student's life are two major factors that drive the dynamics of dropout. Furthermore, parental guidance is found to be a significant factor only under a low level of negative social influence within the school. However, if the negative social influence increases beyond a critical value, then the impact of parental guidance becomes negligible.

    There can be multiple factors (such as parental involvement, socio-economic conditions, family status, etc.) for creating high rates of dropouts in a community; however, owing to the magnitude of the problem and limited resources, it is challenging to precisely tease out driving mechanisms for the current state of dropouts. The results here provide preliminary understanding of mechanisms and suggest that we can lower or limit the number of dropouts and failing students by managing negative social influences occurring within a school. If schools with high dropout rates can identify early their vulnerable and failing students and can limit their homogeneous mixing (interacting only with other V/F students rather than with students passing all subjects) while fostering parental involvement in students' activities, then they can achieve sustained reduction in the number of dropouts.

    Despite the interesting results that we have obtained from this data-driven study, we recognize some limitations including use of approximate values for some parameters in the absence of precise data to estimate them. The school in this research was selected because it has high dropout rates as well as being in the region of the city that is underdeveloped and crime-affected. This is a preliminary study and an extension of this work (future research) using a different school with similar student demographics can be used to validate model results. The survey was conducted at this school as authors had such access to it. A more realistic approach will be to conduct surveys in a randomly selected group of public schools in Chicago and to estimate parameters by taking averages over all the schools and students. In this study, we assumed influences (on peers) occurring from the dropout youths are primarily negative. However, with analogy to published models of illicit drug abuse, the dropout population may also deter students from dropping out (as a result of observed consequences in their life from dropping out); hence, their influences can also have a positive impact on one's life. In this study, our main aim was to develop and present a theoretical framework where the problem of high dropout rates and a large number of failing students in certain communities can be addressed. We identified some critical trends and it is our hope that this work will inspire further work in developing a quantitative analysis related to failing and dropout epidemics in certain regions.

    Ethics

    An approval for conducting the survey was obtained from the institutional review board (IRB) at Northeastern Illinois University-Chicago in 2013.

    Data accessibility

    The details of the data are presented within the study. Additional materials are collected in electronic supplementary material.

    Authors' contributions

    A.B. and A.M. designed the survey and collected data. All four authors contributed to thematic development of the study, model development, estimation of parameters, mathematical analysis and writing of the manuscript. All authors read and approved the final manuscript.

    Competing interests

    The authors declare that they have no competing interests.

    Funding

    This project was partially supported by A.M.'s grants from the National Science Foundation (NSF-grant no. DMPS-0838705, and -grant no. ACI 1525012).

    Footnotes

    Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3659468.

    Published by the Royal Society. All rights reserved.

    References