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# Redrawing attendance boundaries to promote racial and ethnic diversity in
elementary schools

Nabeel Gillani  1∗ Doug Beeferman  2 Christine Vega-Pourheydarian  2 Cassandra
Overney  2 Pascal Van Hentenryck  3 Deb Roy2 1Northeastern University
2Massachusetts Institute of Technology 3Georgia Institute of Technology

###### Abstract

Most US school districts draw ”attendance boundaries” to define catchment
areas that assign students to schools near their homes, often recapitulating
neighborhood demographic segregation in schools. Focusing on elementary
schools, we ask: how much might we reduce school segregation by redrawing
attendance boundaries? Combining parent preference data with methods from
combinatorial optimization, we simulate alternative boundaries for 98 US
school districts serving over 3 million elementary-aged students, minimizing
White/non-White segregation while mitigating changes to travel times and
school sizes. Across districts, we observe a median 14% relative decrease in
segregation, which we estimate would require approximately 20% of students to
switch schools and, surprisingly, a slight reduction in travel times. We
release a public dashboard depicting these alternative boundaries
(https://www.schooldiversity.org/) and invite both school boards and their
constituents to evaluate their viability. Our results show the possibility of
greater integration without significant disruptions for families.

##  1 Introduction

It has been over 65 years since the US Supreme Court ordered the racial
desegregation of schools [10]. Yet segregation by race and income in K12
schools continues to hamper access to quality education for millions of
children across the US [64], despite strong evidence that integration reduces
achievement gaps between lower income students of color and their more
affluent, majority race counterparts [5, 41, 77]. Of course, increasing
diversity by fostering more demographic integration is not a foolproof method
for reducing achievement gaps. Too often, even after addressing segregation at
the school level, segregation persists at the classroom or friendship level
[62, 15, 73, 54], or low income students of color feel unsupported in more
integrated environments [21]. Diversity done wrong can cause more harm than
good. And yet, more diverse schools can serve as a necessary first step toward
providing children from different racial and socioeconomic backgrounds the
chance to mix and learn from one another. This learning and mixing is
important beyond its potential role in reducing achievement gaps: it can also
help increase empathy, compassion, reflective thought [78], and encourage more
welcoming attitudes towards diversity later on in life [77, 23]. There is
evidence to suggest that all students can benefit from racially and
socioeconomically diverse classrooms.

Yet across the US, the vast majority of students attend the schools closest to
their homes by virtue of how “school attendance boundaries”—or catchment
areas—are drawn [50, 67, 68], leading schools to recapitulate neighborhood-
level segregation by race and income. Despite the impact attendance boundaries
can have on racial and ethnic diversity in schools, most school segregation
results from how the lines between districts are drawn (e.g. separating cities
from suburbs), instead of school-specific boundaries within districts [26,
50]. Redrawing district boundaries is arguably a more difficult problem,
however, because it falls under the purview of state legislatures—making it
subject to the whims, frictions, and bureaucratic inefficiencies of similarly-
contentious political issues manifesting at state and federal levels. On the
other hand, changing attendance boundaries within districts generally falls
under the purview of those districts. Indeed, a landmark 2007 Supreme Court
case outlawed the use of individual students’ racial backgrounds as an input
into school desegregation efforts and effectively encouraged districts to
explore the redrawing of school attendance boundaries as a desegregation
policy [75].

The purpose of this paper is to explore to what extent it might be possible to
do this—i.e., redraw attendance boundaries within districts in order to
achieve more diverse schools—without imposing large travel burdens,
overcrowding schools, or fragmenting existing geographic “cohesion” (i.e.,
contiguity). We frame our inquiry as an constrained optimization problem and
ask two overarching questions: 1) how can we re-assign geographies to schools
in order to minimize racial segregation while respecting parents’ travel time
and class size preferences? And 2) how fairly are these reductions in
segregation, and associated costs—namely, changes in travel times and school
switching requirements—distributed across Asian, Black, Hispanic/Latinx,
Native American, and White students? To explore these questions, we focus on
elementary schools for similar reasons as [50]: because their boundaries often
approximately combine to form the boundaries of the middle and high schools
they “feed” to, and hence, are foundational in shaping diverse exposures at an
early age. We use parent input and computational tools to simulate changes
across 98 large school districts across the US with district elementary
schools that are classified as non open-enrollment: that is, attendance at
these schools are entirely a function of which neighborhoods are zoned to
attend them. These schools collectively serve over 3 million students.

Our findings show that alternative attendance boundaries could produce a
relative decrease of 12% in White/non-White segregation across districts.
These boundaries would require nearly 20% of students to switch schools, and
interestingly, a slight decrease of just under one minute in these students’
time spent traveling to school. On average, these “costs” of added diversity
appear to be fairly distributed across different student groups, though
through two case studies, we see that this can vary by district and rezoning.
We release our code, several datasets, and a public dashboard
(https://www.schooldiversity.org/) inviting interested researchers and school
districts across the US to further explore the opportunities and potential
trade-offs involved in changing attendance boundaries to advance integration
objectives. In the subsection below, we offer additional background on the
topics of boundary-based school assignment and attendance boundary changes.

###  1.1 Background on attendance boundary-based school assignment

The expansion of school choice programs has sought to challenge the geographic
determinism of boundary-driven school assignment and thereby also mitigate
school segregation [42]. However, choice too, has been shown in several
instances to perpetuate segregation due to self-selection of certain families
into certain schools [52, 79, 12]. Boundaries continue to play a prominent
role in student assignment: as of 2016, approximately 20% of students in
grades 1-12 participated in some type of public school choice (including 8%
opting for charter schools); 9% attended private schools; and the remaining
71% attended an assigned school, likely determined by geography [56]. Choice
programs have continued to gain popularity in recent years, particularly as
some subsets of families have sought new avenues for mitigating the pandemic’s
effects on their children’s learning [37], yet place-based school assignment
continues to be the norm. Even in choice settings, where students live might
influence the priority they are assigned to attend a certain schools [51], or
even which schools are part of the choice set [11]. This makes attendance
boundaries, and more generally, place of residence a perennially important
factor in school attendance policies. The implications of these boundaries and
resultant segregation can run deep: for example, they have been shown to
demarcate stark gradients in access to gifted and talented programs, quality
teachers, school counselors, and a number of other educational resources [51].

Still, changing attendance boundaries within districts continues to be a
highly contentious topic, especially when issues of diversity are also at
stake [48]. Parents may fear that rezoning students will increase travel times
through longer “busing” [27], reduce quality of education [81]—which they
often define vis-á-vis test scores [1] and class sizes [30], produce unsafe
school environments [72], drop property values [43, 9, 8], fragment
communities [9, 72], and require a number of other sacrifices.

These concerns, while sometimes reasonable, often impede practical paths
towards achieving more diverse and integrated schools—e.g., by sparking “white
flight” in response to unfavorable school assignment policies [65] and souring
public opinion towards desegregation efforts as a result of concerns about
long-distance busing and other inconveniences [25]. Furthermore, despite
parents increasingly expressing support for school integration through polls
and surveys [27, 74], they continue to “vote with their feet”, deciding where
to live and send their children to school in ways that reflect racialized
preferences [4, 33, 34, 18, 38]. Such preferences, especially when aggregated
and compounded across families, can yield extreme levels of segregation across
neighborhoods, cities, and schools [14, 70]. Shifting these underlying
preferences is one of the greatest challenges of our time, and is critical for
the implementation of sustainable school desegregation efforts that persist in
the face of changing legal mandates [5]. Alongside this deeper work, however,
it is also critical to identify if there are pathways to achieving more
diverse and integrated schools today—in the case of our focus, through
alternative attendance boundaries—that families may earnestly consider and not
immediately dismiss because they significantly disrupt and decrease day-to-day
quality of life.

This is empirical question is what motivates our current study, which is a
simulation-based exploration of how much alternative attendance boundaries
within districts might reduce racial and ethnic segregation, subject to
various constraints. While several studies have explored relationships between
attendance boundaries and school segregation [50, 68, 67], we have found few
that have explored actually changing school boundaries—with the exception of
[55, 16, 45, 20]—yet these have not focused on achieving greater racial and
ethnic diversity across schools as the main objective of their approach.
Larger districts may hire external vendors to explore alternative boundary
scenarios; however, their exact tools and methods are often opaque, and
diversity is rarely, if ever, a primary objective—though it is sometimes
included as a constraint or post hoc measure [53]. To our knowledge, our work
is the first to simulate alternative attendance boundaries optimized to
achieve racial and ethnic desegregation across a large number of US school
districts. Simulations alone are not sufficient to drive policy change,
especially in the face of parents and others who might oppose such change, but
may help illuminate possible paths to integration “within reach” that both
districts and families may not have previously explored.

We focus on White/non-White segregation as our primary quantity of interest
given its historical significance within the US and abroad; its association
with other family-level factors that have been shown to correlate with
educational outcomes, like socioeconomic status [64]; and the precision and
reliability with which racial/ethnic data is available at the granularity of
schools and small geographic units like Census blocks (as opposed to measures
of socioeconomic status among parents, which are also critical in the
discussion about school segregation, but less reliably and precisely defined
and available [36]). White/non-White segregation does not perfectly capture
patterns of segregation across all school districts: for example, in some
district settings, White and Asian students may be more likely to attend
schools together, segregated away from their Black and Hispanic/Latinx
counterparts [17]. Nevertheless, across most districts, including those in our
sample, White, Black, and Hispanic/Latinx students constitute the vast
majority of the population, rendering White/non-White segregation an important
dimension of analysis.

##  2 Data and Methods

###  2.1 Optimization model

We explore the extent to which we might reduce segregation across three
different metrics: the widely-used Dissimilarity index ($D$); the related Gini
index ($G$); and the Variance Ratio index ($V$). All three metrics are
presented and discussed in [47], with formal definitions below:

$D=\frac{1}{2}\leavevmode\nobreak\ \Sigma_{s\in S}\leavevmode\nobreak\
|\frac{w_{s}}{w_{T}}-\frac{t_{s}-w_{s}}{T-w_{T}}|$ (1)

$G=\frac{\Sigma_{s1\in S}\leavevmode\nobreak\ \Sigma_{s2\in
S}\leavevmode\nobreak\ ({t_{s1}t_{s2}})\leavevmode\nobreak\
|\frac{w_{s1}}{t_{s1}}-\frac{w_{s2}}{t_{s2}}|}{2T^{2}\left[\frac{w_{T}}{T}\right]\left[1-(\frac{w_{T}}{T})\right]}$
(2)

$V=\frac{\left[\leavevmode\nobreak\ \Sigma_{s\in S}\leavevmode\nobreak\
(\frac{w_{s}}{w_{T}})(\frac{w_{s}}{t_{s}})\right]-\left[\frac{w_{T}}{T}\right]}{1-(\frac{w_{T}}{T})}$
(3)

Here, $s$ is an elementary school across all district elementary schools $S$;
$t_{s}$ and $w_{s}$ indicate the total and total White students at $s$,
respectively; and $T$ and $w_{T}$ indicate the total and total White students
across the district, respectively. Perfectly integrated districts—where the
proportion of White/non-White students in each school reflects district-wide
proportions—would receive a score of 0 under these measures, while perfectly
segregated districts would receive a score of 1.

The purpose of exploring several different measures of segregation is that
each describes something slightly different about how students from different
racial and ethnic backgrounds are distributed across schools. Furthermore,
each has its own merits and pitfalls. Dissimilarity has historically been the
most widely used measure of segregation and represents the proportion of White
students in the district who would need to switch schools in order to achieve
perfect integration [39]. Yet it also suffers from a number of shortcomings,
namely, its 1) failure to fully respect the “transfers/exchanges” principle,
whereby movement of students from schools with a higher proportion of other
same-race students to a school with a lower proportion may not decrease
dissimilarity unless one school is over-represented, and the other under-
represented, with respect to the group’s district-wide prevalence [40]; and 2)
potential equal treatment of changes that lower the index, even if some may
have more normative value than others (like reducing a school’s demographic
population of 100% to 90% belonging to a certain group, vs. 60% to 50%) [80].
The Gini index is closely related to the dissimilarity measure but respects
the transfers/exchanges principle. The variance ratio index (also known as the
normalized exposure index in multi-group settings [58]) is essentially an
isolation index that accounts for the underlying demographic distribution of
the given district. In our context, it indicates how much higher the fraction
of White students is in the average White student’s school compared to the
average non-White student’s school. The variance ratio index also respects the
transfers/exchanges principle, but does not respect the principle of
“composition invariance”: e.g., doubling the number of non-White students in
each school would decrease the variance ratio index (because it would increase
White/non-White exposures), even though it may not necessarily change the
extent to which school-level proportions differ from district-wide proportions
(which $D$ and $G$ more closely measure). Nevertheless, it is a popular
measure of segregation in part because it offers insight into what the
average-case encounters between students from different backgrounds might be.

There are many other valid measures of segregation, including multi-group
measures like Theil’s Entropy Index [63], and we invite interested readers to
build upon our code (which we release with this paper) to explore these and
other measures further. Critically, we note that all of these measures of
segregation are naive in that they do not account for within-school
segregation and sorting [54, 73]—including levels of “friending bias” [19]
that may manifest within schools and subsequently affect who connects with
whom, how social capital is shared, and ultimately, the extent to which more
diverse schools translate into more engagement across lines of difference.

With these considerations in hand, we design a rezoning algorithm that seeks
to re-assign Census blocks to elementary schools within each district in order
to minimize each of the above measures. Rezoning problems are generally
computationally challenging because of the many geographic units they operate
over, and the sometimes large number of constraints (e.g., in the case of
contiguity constraints) they impose. Much redistricting work to date has
focused on congressional redistricting, and many approaches to this have used
mixed-integer programming (MIP) as a core building block [3], often augmented
with problem-specific search strategies [32]. To compute these combinatorial
optimization problems—which are “NP-hard” and lack efficient, polynomial-time
solutions—we use constraint programming [76] via the CP-SAT model in Google’s
Operations Research (OR) Tools library [57], which has been shown to perform
extremely well on a number of different types of combinatorial optimization
problems [61]. Constraint programming enables us to more flexibly express
constraints and nonlinear objective functions that may otherwise be difficult
to encode. While CP-SAT is able to find high-quality solutions to these
notoriously difficult geographic rezoning problems, given the size of most
districts, it is generally unable to prove that the discovered solutions are
optimal. This means that it may be possible to improve upon the reductions in
segregation we report, perhaps through additional computational resources
and/or alternative model and solver specifications.

The algorithm factors in the following constraints, given they represent
topics that are often top of mind for parents and district officials when
exploring boundary changes [48, 53]:

  1. 1.

Maximum travel time increases. We use the OpenRouteService API [31] to
estimate driving times from Census blocks to schools (see more below), and
require that re-assignments of blocks to new schools do not increase estimated
travel times by more than X% for any given family.

  2. 2.

Maximum school size increases. We use the total population at a given school
as a proxy for a quantity parents often care about in their children’s
schools—class sizes [30]—and require that this total does not exceed Y% of its
current population.

  3. 3.

Contiguity. Unlike most US states’ requirements for Congressional districts
[22], states do not legally mandate school attendance boundaries to be
comprised of contiguous geographic units. Still, while they exist in many
districts, non-contiguous boundaries are often difficult to justify to
families [53]. We define block $b$ to be contiguous with respect to its
assigned school $s$ if a line can be drawn on a map from $b$ to the block
containing school $s$ without crossing through blocks zoned for any other
schools. We enforce contiguity similar to [49], with further details available
in S1 of the online Supplementary Materials. The contiguity constraint
requires blocks that are contiguous with respect to their currently-zoned
school must remain contiguous with respect to their zoned school under any
hypothetical rezoning. Contiguity, of course, is only a proxy for “community
cohesion”, or a desire for parents to preserve existing geographic and social
networks when faced with intra-district boundary changes [9].

To identify plausible values for X% and Y% above—i.e., the travel time and
school size constraints—we use the survey platform Prolific
Prolific111https://prolific.co/. to conduct a survey of 250 US-based public
school parents. We design the survey to better-understand parents’ attitudes
towards school diversity and the trade-offs they are willing to make to
achieve more diverse schools, if any. We gather baseline information about the
parents’ attitudes towards diversity, as well as information about the child’s
current school—including current travel times to school and average class
sizes. We then ask parents questions like the following: “Let’s say that by
changing the school zones in your district, an additional [PERCENT] of your
child’s classmates would come from different [CATEGORY] backgrounds. Imagine
this requires traveling further to school. How many more minutes would you be
ok with your child traveling to school in order for them to experience this
increase in diversity?”. We randomly select values for [PERCENT] and
[CATEGORY] to account for different diversity scenarios (see the online
Supplementary Materials for additional details). Importantly, we acknowledge
the possibility of social desirability bias in parents’ responses [59] as an
important limitation of our survey, and one that may mask several of the
underlying racialized preferences for schooling described earlier.

Acknowledging these limitations, we find that the median increase in travel
times that parents would be willing to accommodate is approximately 60% (or
approximately 6 minutes, given the reported median travel time to school 10
minutes), and the median increase in class size is 15% (or approximately 3
students, up from a reported median class size of 22). Based on these values,
we set the max travel time increase threshold to be 50% (a conservative lower
bound) and the max school size increase to be 15%. We do not accommodate other
modes of transport, e.g. requiring students who currently walk to school to be
able to continue doing so. This may still occur under our current
configurations: e.g., a student’s 10 minute walk may translate into a 2 minute
drive, which could increase to a max of 3 minutes under our 50% threshold. In
the event there is such an alternative nearby option available, and the
algorithm reassigns the student to it, it may still be walkable—though not
guaranteed to be. Therefore, modeling alternative commute options is an
important direction for future work, especially in collaboration with school
districts, who may have different transportation options and profiles.

Finally, in general, survey respondents skew more White, affluent, and
suburban than national averages, with details on how respondents compare to
national averages for US public schools available in S3 of the online
Supplementary Materials. These representational disparities limit the validity
of the survey as a robust indicator of the preferences of families across
public education systems in the US. At best, the survey offers us a starting
point for grounding our models, but one that must be refined through more
participatory, community-centric efforts (a topic we return to in the
discussion).

Figure 1: Input data, objective function, constraints, and outcome measures
from our optimization model.

Figure 1 provides an overview of the problem setup, including the data and
parameter inputs into our optimization model (with the input datasets
described in more detail below), and our main outcome measures of interests:
expected changes in 1) levels of segregation, 2) travel times, and 3) school
switching. Section S2 in the online Supplementary Materials contains a more
detailed description of the optimization model and constraints, including our
implementation of the contiguity constraint. Given the computational
intensiveness of each rezoning task, we use only one CPU core per rezoning
simulation while setting a solver cutoff time of five hours and thirty
minutes. Instances are run on a parallel computing research cluster.

###  2.2 Identifying districts and school attendance boundaries

The most recent school attendance boundary survey conducted by the US
Department of Education was in 2015/2016 [29]. Therefore, for this study, we
purchase 2021/2022 school attendance boundaries from the data provider
ATTOM222https://www.attomdata.com/data/boundaries-data/school-attendance-zone-
boundaries/.. Using 2020 US Census block shape files collected from the US
Census website333https://www.census.gov/geographies/mapping-files/time-
series/geo/tiger-line-file.html., we determine that a block is zoned for a
particular elementary school if the centroid of that block falls within the
multipolygon delineating the school’s attendance zone for 3rd graders. We
exclusively use 3rd grade boundaries as our proxy for elementary schools given
that 3rd grade is typically classified as an elementary grade, as opposed to
e.g. 6th grade, which may be elementary or middle depending on the
district/state. In the event a district has overlapping attendance boundaries
for certain schools, we map the block to the school with the smallest
attendance boundary (in terms of overall area). This occurs for approximately
7% of blocks across the districts in our study.

We identify our sample of 98 school districts by applying the following
criteria. First, we remove districts that only have one elementary school (and
hence, for which the notion of a boundary change is undefined), and those that
we do not have 2019/2020 NCES school population counts for (described in the
next section). Next, for computational purposes, we include only those
districts that have 200 or fewer elementary schools. After applying these
filters, we are left with 4,231 school districts in our data. The vast
majority—approximately 94% (3,970)—have elementary schools whose boundaries
are entirely “closed-enrollment”: only those students zoned for the school can
attend it444These values drop to 88% and 86% when considering middle (7th
grade) and high (10th grade) attendance boundaries, suggesting districts are
less likely to have open boundaries at younger grade levels.. Importantly, we
note that families across even those districts with closed-boundary elementary
schools may opt to attend in-district magnet programs, which do not have
attendance boundaries—or opt out of the district altogether to attend an
alternative (e.g., charter) school. We discuss this possibility and its
potential implications for this study further below.

The 6% of districts excluded from our sample tend to have a slightly higher
White population, slightly higher Hispanic/Latinx population, and slightly
higher White/non-White segregation than the remaining 94%. We select the
largest 100 districts (in terms of enrollment) across the 94% of districts
with closed-enrollment elementary schools. Compared to the other 3,870
districts with no open-enrollment elementary schools, these 100 districts are
(by definition) larger, but generally do not have higher levels of White/non-
White segregation. Compared to the excluded 6%, these 100 districts are also
generally larger, and do have a higher level of White/non-White segregation.
S4 in the online Supplementary Materials offers further details on these
differences. Due to memory limitations in our computing infrastructure, we are
unable to simulate alternative boundaries for two districts. The remaining 98
districts constitute our final sample.

###  2.3 Estimating students per Census block

We use the 2019/2020 National Center for Education Statistics Common Core of
Data555https://nces.ed.gov/ccd/files.asp. to estimate the number of Black,
Hispanic/Latinx, White, Native American, and Asian students at each school. In
parallel, we download 2020 Census block-level population counts for
individuals who are less than 18 years of age and considered to belong to one
of the above demographic groups.

With these datasets in hand, we estimate $N_{gbs}$, i.e. the number of
students from group $g$ in block $b$ that attend school $s$, to be:

$\frac{C_{gb}}{C_{gB_{s}}}\cdot s_{g}$ (4)

Where $C_{gb}$ is the count of individuals belonging to group $g$ and living
in block $b$ as estimated from the Census data; $C_{gB_{s}}$ is the total
number of individuals from the Census data belonging to group $g$ across
blocks that are zoned for school $s$ (i.e., $B_{s}$); and $s_{g}$ is the total
number of students from group $g$ at $s$. However, in cases where $s_{g}$ is
large, we find that scaling by $\frac{C_{gb}}{C_{gB_{s}}}$ sometimes leads to
counts per block that exceed the total number of students under 18 in that
block, as defined by Census data. Therefore, when $\frac{C_{gb}}{C_{gB_{s}}}$
exceeds 50%, we replace it with $\frac{C_{b}}{C_{B_{s}}}$—i.e., we simply
assume that the fraction of students belonging to $g$ that attend $s$ from $b$
is proportional to the fraction of total students living in $b$. Finally, we
take the ceil of values and iteratively estimate counts per block (starting
with the blocks with the highest value of $C_{gb}$) until all students at the
school have been allocated to a home block. This helps ensure integer student
counts, and also, that the total number of students per group across all
blocks is equivalent to the number attending the school per the NCES data.
Section S1 in the online Supplementary Materials contains additional details
on key assumptions underlying our estimation procedure.

All Census data is collected from [46]. Our procedures are limited because of
our inability to estimate the precise number of elementary-aged students in
each block who attend their zoned elementary school, as some may attend
charter, private, or within-district options with open enrollment (like magnet
programs). Even though certain demographic groups disproportionately may
exercise school choice in different settings [69, 6, 66], because our
estimates are based on ground truth school enrollments by demographic group,
this differential uptake of school choice is likely to bias our block-level
estimates only if families who are part of the same demographic group and
assigned to the same school have different rates of school choice uptake that
are correlated with the block in which they live. It is not immediately
obvious why this might happen, but there are certainly possible explanations
(for example, a particular block might house a popular charter or other
alternative school option). This could affect the results of our boundary
redrawing by either over or understating how much alternative boundaries might
impact school diversity. For example, districts with a high fraction of non-
White students that have disproportionate numbers of White families opting out
of zoned schools in certain blocks compared to others may overstate how much
alternative boundaries could increase integration; conversely, higher
fractions of non-White families opting out across these blocks (e.g., due to
charter options with lotteries that reserve seats for different demographic
groups) may understate it.

In practice, districts cannot know exactly which students living within their
boundaries opt for charter schools or other out-of-district options. However,
they can know the locations of students attending within-district schools—and
so, through future collaborations with districts, we can much more accurately
determine student counts per blocks. School choice, however, not only impacts
block-level student counts (model inputs); it also impacts how likely families
are to adhere to new boundaries (model outputs), and hence, the extent to
which diversity and integration objectives are actually achieved post-
rezoning. In an ideal world, we would have access to a clairvoyant capable of
perfectly anticipating which students are likely to exercise school choice in
the face of proposed student assignment policy changes before these policies
are implemented. However, modeling and anticipating family demand for schools
continues to be an open research problem [60] and challenging practical task
for districts. In the results section, we present simple opt-out scenarios
based on charter and magnet choice patterns across districts as a first step
towards illustrating how school choice might impact the boundary-based
integration strategies and outcomes we focus on in this study.

###  2.4 Estimating travel times

We use the OpenRouteService API [31] to estimate travel (driving) times
between block centroids and schools in each district. Given the large number
of travel times to compute (millions in some of the larger districts) and the
publicly-hosted API’s rate limits, we compile and run a local instance of the
API on our own server, which enables us to submit an arbitrary number of
queries. Queries are comprised of latitude/longitude pairs for a starting
location (Census block centroid) and ending location (school location). Travel
times do not account for traffic patterns.

##  3 Results

Figure 2: Estimated impacts of attendance boundary changes on levels of
segregation across districts. Plots a)-c) depict the before and after-rezoning
distributions of segregation scores according to the Dissimilarity, Variance
Ratio, and Gini indices for the 98 districts in our sample. In plots d)-f),
each bubble represents a district, and the plots show the strong Spearman rank
correlations between the relative reductions in segregation across districts
as measured according to our three indices. In general, the relative changes
described by each measure of segregation are quite similar across districts.

We begin by analyzing White/non-White segregation scores for our 98 districts
across the Dissimilarity ($D$), Gini ($G$), and Variance Ratio ($V$) indices
described earlier. Figures 2(a-c) illustrate the distribution of these
segregation scores across the districts in our sample before and after
producing our hypothetical rezonings. The median values of $D$, $G$, and $V$
before rezoning are 0.39, 0.51, and 0.15, respectively. Following rezoning,
the median across these metrics decreases to 0.33, 0.46, and 0.13,
respectively—corresponding to a median 12%, 7%, and 14% relative decrease when
computing pairwise changes per district. The post-rezoning scores for $D$ and
$V$ are based on simulations that seek to directly minimize these values as
the core objective functions. While we also simulated rezonings designed to
directly minimize $G$, the $|S|^{2}$ number of terms in the objective function
(where $S$=number of schools) added significant computational complexity,
limiting both the number of districts we could simulate changes for (only 91
out of 98 districts) and the quality of rezonings produced by these
simulations. Given the similarities between how $D$ and $G$ measure
segregation, the results for $G$ presented here are computed across
simulations optimizing for $D$—which, surprisingly, produce a larger median
relative decrease in $G$ across districts (7%) than those simulations directly
optimizing for $G$ (5%), further underscoring potential performance and
solution quality issues.

Figures 2(d-f) illustrate Spearman rank correlations ($\rho$) between the
relative reductions in segregation across districts according to our three
metrics. The correlation between relative reductions in $D$ and $V$ across
districts is $\rho=0.86$; between $D$ and $G$ is $\rho=0.94$; and between $V$
and $G$ is $\rho=0.88$ ($p<<.001$ in all cases). In general, it appears that
regardless of these different definitions of segregation, our rezoning
algorithms produce similar relative reductions in segregation across the
districts in our sample. Therefore, for simplicity throughout the remainder of
the paper, we select one of these metrics as the basis of further
investigating the results of our models. In particular, we select $V$ because
it both respects the transfers/exchanges principal and also offers insight
into how segregation is actually experienced within schools (on average) by
students from different racial and ethnic backgrounds. We note $V$’s lower
score relative to $D$ and $G$, which we hypothesize may be due to the fact
that $V$ also measures exposure—and so, districts that have a larger fraction
of non-White students may increase the chances that White students are exposed
to them in schools. Indeed, computing a Spearman correlation between $D-V$ per
district (before rezoning) and the percentage of non-white students across
districts yields $\rho=0.74,p<<0.001$ (with similar results for $G-V$). The
larger the fraction of non-White students in a district, the less the Variance
Ratio index’s value aligns with the values of evenness-based measures of
segregation like $D$ and $G$, likely because more non-White students in a
district increases the chances that the average White student encounters them
at school. Even with this trend, though, there is noticeable segregation
across districts according to $V$, and hence, an opportunity to explore how
much boundary changes might reduce it.

Figure 3: Results from our rezoning algorithm. a) illustrates pairwise changes
in $V$ across districts; b) shows that segregation scores for all
racial/ethnic groups decrease, albeit marginally, under the proposed
rezonings. c) illustrates that, on average across districts, school switching
under the depicted rezonings are relatively evenly distributed across racial
and ethnic groups. Finally, d) illustrates that the depicted boundary changes
might actually slightly decrease average travel times across school districts
and demographic groups. Together, these findings show that there are pathways
to more integrated schools across districts that may not require large
sacrifices by families.

Figure 3 offers a deeper look at changes in $V$, school assignments, and
travel times produced by our models across districts (error bars depict 95%
bias-corrected and accelerated confidence intervals, computed using the boot
library in R [13, 24]). From 3(a), we observe the aforementioned 14% median
relative decrease in $V$ indices across districts after our hypothetical
rezoning. Districts in our sample range in their values for $V$: the most
segregated district has a value of $V=0.44$, while the least segregated has a
value of $V=0.01$. Conducting exploratory correlational analyses, we observe
no statistically-significant association between a relative decreases in $V$
and urbanicity ($ANOVA\leavevmode\nobreak\ F=0.66,p=0.58$), and only moderate
relationships between relative decreases in $V$ with enrollment size
($Spearman\leavevmode\nobreak\ \rho=0.35,p<0.001$) and initial levels of $V$
($Spearman\leavevmode\nobreak\ \rho=0.35,p<0.001$). These results suggest that
larger and more segregated districts have more scope for intra-district
attendance boundary changes to increase integration, but given the relatively
moderate associations with these variables, that the nuanced geographic and
demographic contexts of each district is likely to play an important role in
how much such boundary changes can foster more diverse and integrated schools.

Figures 3(b)-(d) illustrate the potential costs of achieving these reductions
in segregation. From (b), we see that reducing White/non-White segregation
would not lead to higher segregation levels for other racial groups (i.e.,
Black/non-Black; Hispanic/Latinx non-Hispanic/Latinx; etc). In fact, the other
racial groups would also experience reductions in segregation under the
depicted rezonings. In (d), we see that, on average, approximately 20% of
students from different groups would be required to switch schools, and that
the burden of school switching could be distributed approximately evenly
across student groups. While 20% represents a relatively large fraction of
students, it is less than the nearly 40% of parents in our survey who
expressed a willingness to switch schools if their district redrew attendance
boundaries. From an implementation perspective, districts may also phase
boundary changes in gradually instead of all at once, reducing the number of
students required to switch schools in any given year. The literature on the
impact of school switching on student academic and subjective well being
outcomes is mixed, with some findings illustrating positive benefits
conditional on switching to attend better schools, and others illustrating
adverse consequences [35, 71]. Weighing the potential disruption costs of
school switching alongside the potential gains of more integrated schools is
important when determining when and how to make boundary changes.

Somewhat surprisingly, plot (e) shows that average school switcher would
actually experience a decrease in their travel times to and from school,
despite the fact that our model permitted up to a 50% increase in travel times
for any given family. This is notable because it suggests that 1) long-range
“busing” [25] is not necessarily required to achieve more diversity in
schools, and 2) some existing attendance boundaries may potentially be drawn
(“gerrymandered”) in ways that assign students to schools further from their
homes, resulting in slightly higher levels of segregation as a result [67]. We
note the speculative nature of this latter point, especially given the
existence of research suggesting that irregulary-shaped boundaries may
actually contribute to greater integration [68]. Indeed, it is possible
families may have moved after the implementation of such boundaries precisely
to avoid more integrated schools, producing a net increase in segregation.
Further research is needed to better understand precisely why it appears that
current boundaries could be redrawn to foster integration while also reducing
travel times. Finally, an important observation from Figures 3(c) and (d) is
that, again on average across districts, the potential costs of desegregation
are fairly distributed across the depicted racial and ethnic groups.

###  3.1 Sensitivity analyses

Figure 4: Sensitivity analyses of main results. Plots a)-c) depict how much
segregation, percentage of students who are rezoned, and travel times for
rezoned students might change as a result of changing travel time and
contiguity constraints (while fixing max school size increases to 15%). Values
are medians across the 98 districts in our sample. In general, dropping
contiguity constraints appears to have the largest potential impact on
reducing segregation, but we estimate would also produce much more school
switching and higher travel times for students. Plots d) and e) illustrate
simple models for how much demographic group-specific patterns of choice might
impact reductions in segregation produced by boundary changes. In both of
these cases, we estimate patterns of choice would lead to a median relative
reduction of approximately 10% across districts in our sample, down from the
14% indicated by our models without modeling choice-based opt-outs.

A median 14% relative decrease in segregation across districts represents a
non-trivial step towards more integrated schools, yet on its own is quite
modest and far from achieving full integration. It highlights how, under our
selected constraint values, there are inherent limits to how “sticky” the
issue of segregation is, and the limited extent to which intra-district
boundary changes might promote more diverse and integrated schools. To explore
how segregation might change under different parameter configurations, Figures
4(a-c) fixes maximum school size increases to 15% and illustrates how changing
travel time and contiguity constraints might impact median levels of
segregation, school switching, and travel times for rezoned students across
our 98 districts. For example, setting the max travel increase threshold to
100% (or in the most extreme case, allowing families to experience a doubling
in travel time to school) could yield a median relative decrease in
segregation of nearly 20%, but would require a median of nearly 30% of
students to switch schools and experience a slight average increase in travel
times (though these still appear to be marginal). Keeping the travel time
increase at a maximum of 50% but dropping the contiguity constraint could
yield a median relative decrease in segregation of 35%, but would require
nearly 45% of students to switch schools, and experience a half-minute average
increase in travel to school. Applying such relaxations together—not requiring
contiguity and allowing even larger increases in travel times of a miximum of
200%—could decrease segregation by over 50%, but would also require a median
of over 70% of students to switch schools and experience a median travel time
increase of nearly 3 minutes. Perhaps most salient in these plots is that
requiring contiguity appears to significantly limit the extent to which intra-
district boundary changes might foster more diverse and integrated schools.

A critical limitation of all of our results so far is that they do not factor
in the likelihood of complex system dynamics that could manifest if districts
actually did adopt the rezonings described here—for example, neighborhood
relocation (e.g. “white flight”) in response to unfavorable rezonings [65, 7],
or the disproportionate use of school choice by families to opt for other
district or charter options that enable them to circumvent the effects of
changing boundaries. All of these could affect the extent to which the methods
proposed thus far can actually help districts achieve greater racial and
ethnic integration in schools. Unfortunately, as discussed earlier,
anticipating rates of family opt-out is a challenging task. We explore a
simple model of family opt-out based on existing patterns of charter and
magnet school choice within the districts in our sample to offer a preliminary
investigation of how much school choice might undermine boundary-based student
assignment policy changes seeking to foster more diverse and integrated
schools. We use the NCES Common Core of Data to identify elementary charter
and in-district magnet schools located within at least one of the elementary
school attendance boundaries across the 98 districts in our sample. Of these
98 districts, 84 contain at least one charter or magnet school, totaling 988
total such schools. We estimate that the median ratio of elementary students
per district attending a charter or magnet school compared to a closed-
enrollment boundary-based school is approximately 12.8%. There is, however,
high variance across districts: the ratio in the district where choice is most
prevalent is 67%; in the least-prevalent district, it is less than 1%. In 26
out of 84 districts, White students are more likely to attend choice programs
than non-White students, suggesting that different public school choice
contexts may foster different choice patterns among families from different
demographic backgrounds [6, 69].

For each district, we compute the ratio of elementary students per
racial/ethnic group (White, Black, Hispanic/Latinx, Asian, Native American)
who attend a charter or magnet school compared to a closed-enrollment boundary
school. These estimates serve as the basis of two “opt-out” scenarios we model
to explore how much the prevalence of choice across districts might undermine
boundary-based policy changes seeking to foster greater integration. In one
scenario—“lower choice”—we assume that students who are rezoned under a
hypothetical boundary change opt-out at a rate of $\frac{1}{2}$ their existing
demographic group-specific choice uptake ratio. To make this concrete: if our
algorithm re-assigns a census block with 20 Black students, and district-wide,
the ratio of Black students attending a charter or magnet school located in
the district compared to a closed-enrollment boundary-assigned school is 10%,
then we reduce the count of Black students who actually transition to the new
school by $\frac{1}{2}\cdot 0.1\cdot 20$, or one student. The second
scenario—“higher choice”—works similarly, but simply applies the district-wide
choice ratio: in the earlier example, $0.1\cdot 20$, or two Black students,
would opt out. The key assumption motivating these two scenarios is that the
rate of existing public school choice in a particular district is likely to
influence the extent to which families have and exercise options beyond their
boundary-assigned school. Of course, actual rates of boundary-assigned school
opt-out will vary not only as a function of existing choice options, but also,
where those schools are located relative to families; the programs they offer;
the specific nature of how a rezoning might alter the features of their
boundary-assigned school; and several other factors. Indeed, exploring more
robust models to anticipate such choice in the face of potential boundary
changes is important avenue for future work.

Figures 4(d-e) show how much we expect levels of $V$ to change (compared to no
opt-out) across both of these scenarios. In both cases, $V$ decreases by a
median of approximately 10%, down from 14% in the original case where there
are no expected opt-outs. This suggests that, at least according to these
simplistic models, choice patterns might undermine integration outcomes by a
factor of nearly one third. Crucially, these scenarios do not account for
potential private school opt-outs, capacity constraints at choice-based
schools, or other real-world complexities, and thus should be viewed as
simplistic early efforts to illustrate how the dynamics of family preferences
and school selection might impact integration objectives that districts might
pursue through boundary changes.

###  3.2 District case studies

Thus far, we have discussed average expected impacts of boundary changes in
integration and other outcomes. Yet these averages likely mask important
heterogeneities across different types of districts. To explore some of these
heterogeneities, we conduct two case studies. The first involves the most
segregated district in our sample, Atlanta Public Schools, which has a
White/non-White Variance Ratio index of 0.44 and serves nearly 23,000 students
across 44 closed-enrollment attendance boundary elementary schools. The second
involves the district closest to the median level of segregation across
districts in our sample: Garden Grove Unified District, California, which has
a Variance Ratio index of 0.15 and serves over 20,000 students across 47
closed-enrollment elementary schools.

Figure 5: Case study for the most White/non-White segregated district in our
sample, Atlanta Public Schools, which has a segregation score of $V=0.44$. The
shapes in a) through c) represent 2020 US census blocks; the colors in a)
delineate 2021/2022 school attendance boundaries for the depicted district
(”status quo”); b) shows the estimated percent of each block’s population
estimated to be white (darker blue implies a higher percentage), with blocks
removed if they are estimated to have a student population of zero. c) shows
the rezoning produced by our algorithm. d) shows the expected change in the
proportion of students at each elementary school in the district who are
White, before and after rezoning. e) through g) show the anticipated changes
in segregation scores, percentage of students needing to be rezoned, and
change in travel times for each demographic group. The results reveal several
notable findings. First, as expected, the most dramatic boundary changes
appear to occur in school boundaries that fall at the interface of White and
non-White parts of the city. Additionally, as shown in d), changes in school-
level distributions tend towards the district White student percentage, with a
handful of schools experiencing the most dramatic changes. However, the share
of White students is also increased at several schools that already have a
White share higher than the district, illustrating trade-offs district leaders
may be faced with when deciding which schools to target with desegregation
efforts. From e) and g), we see that all student groups would experience
reductions in segregation and travel times, respectively, but f) shows
disparities in which students might be rezoned—with the largest fraction found
among Native American (8 out of 16), Hispanic/Latinx (613 out of 1,756), and
Asian (93 out of 336) students. Figure 6: Case study for a typical (median)
White/non-White segregated district in our sample, Garden Grove Unified, CA,
which has a segregation score of $V=0.15$. Plots a)-g) defined analogously to
Figure 4. In general, we observe similar trends as the Atlanta case study,
with fewer disparities in the percentage of students across demographic groups
who would be rezoned, and a marginal increase in travel times for White
students in the district who would be rezoned under the depicted boundary
change scenario.

Figures 5 and 6 illustrate the outputs of these case studies. Maps (a)-(c) in
each figure illustrate the present-day elementary school attendance
boundaries, the relative prevalence of White students per Census block, and
our hypothetical rezoning. Examining (d) illustrates how the fraction of
students in the depicted group would change at each school after implementing
the alternative zoning, compared to before. As expected, rezonings generally
move school-level demographics to reflect district-level proportions as much
as possible, sometimes shifting the percentage of White students at a given
school by several fold, as seen in Figure 5(d). The most substantial changes
also appear to occur within a small subset of schools, with most schools
across the district experiencing little or no change. However, some schools
become more segregated with respect to the district: that is, a subset of
schools which already have a proportion of White or non-White students
exceeding district-level proportions see an increase in their White or non-
White share, respectively, and therefore diverge further from, instead of
converging to, district-level shares of White/non-White students. These
fluctuations are more extreme in Atlanta compared to Garden Grove Unified, and
to be expected given our Variance Ratio index objective function, which
optimizes an aggregate district-level measure without explicitly requiring
that reductions in segregation be evenly redistributed across individual
schools. In practice, the choice of objective function, overall desegregation
goals (including specific schools of focus), and even notions of fairness are
likely to be context-specific and require input and domain expertise from both
districts leaders and families. While our measures of segregation here are
defined at the district level, we return to a point made in the
introduction—that there will be inherent limits to how much racial balancing
at schools can be achieved through intra-district changes alone. For example,
according to our sample, Atlanta Public Schools has nearly 23,000 students at
boundary-based elementary schools, approximately 4,200 of whom are White. Cobb
County, adjacent to Atlanta, has approximately 48,000 students in our sample,
nearly 18,000 of whom are White. Exploring inter-district changes between
districts with stark racial and ethnic gradients such as these, while more
politically challenging, may offer even more hope for fostering more diverse
and integrated schools.

Plots (e) through (g) in each case study depict the changes in segregation,
school assignments, and travel times expected across demographic groups. In
both cases, we see slight reductions in segregation and travel times across
demographic groups, with the exception of White students in the Garden Grove
Unified case, who would experience a marginal ($<<1$ minute) increase in
travel each way. Furthermore, in Figure 6(f) (Garden Grove Unified), we see a
relatively balanced school switching requirement across demographic
groups666With the exception of Native American Students, who constitute less
than 0.1% of the students across the district elementary schools included in
our sample; however, in the corresponding plot for Atlanta (Figure 5(f)), we
see larger disparities across groups. Most notably, White, Hispanic/Latinx,
and Asian students are 2-3x as likely as Black students to have to switch
schools; Native American students are nearly 5x as likely. Notably, however,
Native American, Asian, and Hispanic/Latinx students constitute a small
fraction of the overall student body in Atlanta’s elementary schools. These
differences show that even though in an aggregate sense across districts,
school switching is fairly distributed across groups, these results will
likely vary by district. This highlights both several of the trade-offs
district officials might need to make in order to achieve more integrated
schools, but also, opportunities for more creative approaches to modeling,
constraining, and ultimately addressing the issue of changing policies in
order to mitigate segregation.

##  4 Discussion

Our results demonstrate that there are practical and fair pathways to changing
attendance boundaries in order to achieve more diverse schools, though the
impact these policies have on individual schools and different student
groups—for example, who would be required to switch schools, and how much
school switching might either further or hinder these students’ academic
progress—can vary across districts. This reality calls for a nuanced and
district-specific approach to modeling, evaluating, and eventually adopting
potential boundary changes. Particularly notable is that there exist
alternative boundary scenarios that might reduce segregation and travel times
across districts, highlighting that these are not always at odds, and
contrasting with the public narrative around long-range busing that emerged
among many majority-race members of the population during desegregation
efforts in the 20th century [25].

While we observe a median relative decrease of 14% in the White/non-White
Variance Ratio index of segregation across districts after our hypothetical
rezonings, these improvements are largely a function of the constraints our
models impose, and still constitute only small steps towards addressing issues
of White/non-White segregation. Nevertheless, as our sensitivity analyses
show, changing constraint values (and particularly, dropping contiguity) can
have a multiplier effect on how much alternative boundaries might reduce
segregation. On the other hand, our preliminary modeling suggests that family
opt-outs from boundary-assigned schools in favor of charter or magnet programs
may slightly undermine integration objectives. To support explorations of
these sensitivities and the impact different policies might have on individual
schools and demographic groups, we release a public
dashboard777https://www.schooldiversity.org/. and its underlying code and data
illustrating different boundary scenarios and outcomes for the districts we
explore in this study. We invite researchers to use these resources to explore
new models that capture more of the nuances and specifics individual districts
often consider when making boundary changes (some of which we discuss below).
We also invite districts and families to explore the outputs in the dashboard
and comment on their viability as starting points for informing realistic
policies for fostering more diverse schools.

Parents’ racialized preferences for where they live and send their children to
school will continue to act as formidable headwinds challenging even the most
thoughtful and well-designed efforts to foster more diverse schools. Our study
does not contribute to answering the normative question of how to change these
preferences, or the political one of whether school districts can garner the
will to implement policies that improve integration. However, it offers an
empirical contribution that we believe may be of interest to both researchers
and school districts: that such improvements appear to be possible across many
districts, and that they can be achieved with practical and fair tradeoffs.
Even then, which tradeoffs count as ”practical” and ”fair” will differ across
communities and individuals, and across racial/ethnic and class lines. This
points to a number of limitations in our current study, which in turn open the
door to new and exciting directions for future work. We classify these
limitations as opportunities across three interconnected categories that we
invite researchers and interested practitioners to explore in greater detail:
data, model, and broader relevance to education policy efforts.

With respect to data, our method relies on estimated counts of students per
group, per Census block—which could be improved by obtaining ground-truth data
through district collaborations. We are also unable to factor in other data,
like transportation costs, that districts might weigh as they decide on
rezoning policies. The limited window that Free/Reduced-priced lunch provides
into students’ socioeconomic status [36], coupled with the limited
availability of family socioeconomic indicators at the Census block level,
prevents us from exploring socioeconomic dimensions of segregation, which
several districts seek to mitigate [62]. Working closely with specific
district partners to obtain and incorporate more detailed, historical, and up-
to-date data may help alleviate many of these issues. Finally, we proxy
“community cohesion” with contiguity. In reality, a family’s community, and
students’ friends, are a function of geography along with many other
(potentially unobservable) factors. Developing more nuanced ways of
determining and factoring in notions of community into rezoning models may
open the door to new boundary configurations that relax the constraining
impact of contiguity constraints while still satisfying family and district-
level preferences.

There are also a number of model improvements that may make our results more
useful in practice. Balancing utilization across schools, limiting the
percentage of students who are rezoned (a la [53]), and even more explicitly
factoring in fairness requirements instead of merely analyzing fairness post
hoc serve as important directions for modeling improvements. To ensure
increases in elementary school diversity also propagate to middle and high
schools, exploring objective functions that factor in feeder patterns and
account for the full K12 lifecycle—instead of only the earlier years—may also
produce more practical and desirable boundary changes. As mentioned earlier,
we may also expand our model to incorporate historical data or domain
expertise to predict how a given rezoning might spark families to leave
neighborhoods and/or disproportionately leverage school choice to access other
district or charter options—and factor these possibilities into the
optimization process. Finally, given that diversity may not today be a core
consideration or impetus for redrawing boundaries in most districts, we might
augment our models to aid district policymakers in simulating new boundaries
when exploring questions more germane to their day-to-day, like determining
locations for new schools, or deciding which schools to shut down (for
example, in response to declining enrollment). With minor extensions, the
models we present here can aid with these decisions while still foregrounding
their potential impacts on diversity, travel, and other outcomes of interest.

Perhaps the biggest open question from our study is: how might families and
district leaders respond to these hypothetical rezonings, and how much could
they actually increase diversity in schools? We believe this is an important
avenue for follow-on research, and a critical part of translating this
research into education policies that help promote school diversity. As
discussed throughout the study, boundary design matters, but so does its
interplay with how families opt for choice-based schools (should they exist in
their districts). Therefore, it is critical to explore how the methods
presented here may more accurately anticipate family opt-out rates in favor of
charter or magnet programs—or even private schools; how they might inform the
design of “zones of choice” that define meta-boundaries for clusters of
schools that parents can then choose amongst [11, 2]; and several other
student assignment and school choice policies, many of which are emergent.
Furthermore, as discussed in the introduction, most segregation in schools can
be attributed to boundary delineations between districts, not simply those
within them. Expanding geographic scope to explore between-district boundary
changes (a more challenging computational problem as well) may help yield
policy simulations that produce more practical and effective pathways to
integrated schools. Finally, capturing and factoring in input from both
families and district leaders, for example, through participatory [44] and
value sensitive [28] design methods may further help inform school
desegregation policies that are realistic and practically-achievable.

Changing school demographics does not guarantee more diverse friendships,
sharing of social capital and resources, greater empathy for different life
experiences, and other potential gains that can ultimately benefit all
students [54, 73, 19]. Yet it is a necessary first step towards achieving many
of these downstream outcomes. We hope this study is a useful building block to
support future work on this critical topic.

##  5 Acknowledgements and funding sources

Funding for this project was provided by the MIT Center for Constructive
Communication. The authors declare no competing interests. The research was
approved by MIT’s Institutional Review Board. We thank Peter Bergman, Jennifer
Candipan, Kumar Chandra, Tyler McDaniel, Tomas Monarrez, Laurent Perron,
Halley Potter, Francine Stephens, Nathanial Thomas, participants at Harvard’s
Urban Data Lab, and this paper’s anonymous reviewers for their helpful
comments and guidance.

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