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Doubling Up: Intensive Math Instruction and Educational Attainment

Kalena E. Cortes*

The Bush School of Government and Public Service Texas A&M University

kcortes@bushschool.tamu.edu

Joshua Goodman John F. Kennedy School of Government

Harvard University joshua_goodman@hks.harvard.edu

Takako Nomi

Consortium on Chicago School Research University of Chicago

nomit@ccsr.uchicago.edu

This version: September 14, 2012

Abstract Success or failure in freshman math has long been thought to have a strong impact on subsequent high school outcomes. We study an intensive math instruction policy in which students scoring below average on an 8th grade exam were assigned in 9th grade to an algebra course that doubled instructional time and emphasized problem solving skills. Using a regression discontinuity design, we confirm prior work showing relatively little short-run impact on algebra passing rates and math scores. We show, however, positive and substantial long-run impacts of double-dose algebra on college entrance exam scores, high school graduation rates and college enrollment rates. The majority of this impact comes from students with average math skills but below average reading skills, perhaps because the intervention focused on verbal exposition of mathematical concepts. These facts point to the importance both of evaluating interventions beyond the short-run and of targeting interventions toward appropriately skilled students. This is the first evidence we know of to demonstrate long-run impacts of such intensive math instruction. JEL Classifications: I20, I21, I24, J10, J13, J15, J48 Keywords: Double-dose intervention, Intensive Math Instruction, Student Outcomes, Postsecondary Enrollment

* We are indebted to Chicago Public Schools for sharing their data with us and to Sue Sporte, Director of Research Operations, Consortium on Chicago School Research (CCSR), University of Chicago, for facilitating this sharing. Special thanks for helpful comments from Richard Murnane, Bridget Terry Long, Jeffrey D. Kubik, Lori Taylor, Jacob Vigdor, Caroline Hoxby, Martin West, Kevin Stange, and Nora Gordon, as well as seminar and conference participants at Harvard University’s Program on Education Policy and Governance, State of Texas Education Research Center at Texas A&M University, the Association for Education Finance and Policy, and the National Bureau of Economic Research Economics of Education Program. Colin Sullivan, Heather Sarsons and Shelby Lin provided outstanding research assistance. The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant #R305A120466 to Harvard University. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. Institutional support from Texas A&M University and Harvard University are also gratefully acknowledged. Research results, conclusions, and all errors are our own.

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1. Introduction

The high school graduation rate for American students has declined since the 1970s to about

75 percent, with black and Hispanic graduation rates hovering around 65 percent (Heckman and

LaFontaine 2010). Poor academic preparation of students entering high school is often cited as a

major source of such high dropout rates. Results from the 2011 National Assessment of Educational

Progress suggest that only 35 percent of students enter high school with math skills considered

proficient. Black and Hispanic students’ proficiency rates are an even lower 13 and 20 percent,

respectively.1 These low academic skills may explain observed high failure rates in 9th grade

coursework, particularly in algebra (Herlihy 2007, Horwitz and Snipes 2008).

Such high failure rates are particularly worrying because of their close association with

dropout rates in later grades. Early course failures prevent students from progressing to more

advanced coursework and from earning the credits needed to graduate (Allensworth and Easton

2007). In the Chicago Public Schools (CPS), the focus of this study, roughly half of high school

freshmen fail at least one course, with the highest failure rates in math courses (Allensworth and

Easton 2005). Concern about this fact and the apparent failure of remedial math courses to correct it

led CPS to implement a double-dose algebra policy starting with students entering high school in the

fall of 2003. Under this policy, students scoring below the national median on an 8th grade math test

were subsequently assigned to two periods of freshman algebra rather than the usual one period.

CPS hoped that this doubling of instructional time, along with an increased emphasis on problem

solving skills, would improve algebra passing rates in the short-run and high school graduation rates

in the longer-run.

1 Similarly large skill gaps by income are also apparent. Students poor enough to qualify for free lunch under the National School Lunch Program have a proficiency rate of 17 percent, compared to a 47 percent proficiency rate among students who do not qualify for such subsidies. See “The Nations Report Card: Mathematics 2011” published by the National Center for Education Statistics.

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To analyze the effect of the double-dose policy, we employ a regression discontinuity design

comparing students just above and below the threshold for assignment to additional instructional

time. Using longitudinal data that tracks students from 8th grade through college, we confirm prior

work showing little short-run impact on algebra passing rates and math scores. We show, however,

positive and substantial longer-run impacts of double-dose algebra on college entrance exam scores,

high school graduation rates and college enrollment rates. Interestingly, the majority of this policy

impact comes from students with average math skills but below average reading skills, perhaps

because the intervention focused on verbal exposition of mathematical concepts. These facts point

to the importance both of evaluating interventions beyond the short-run and of targeting

interventions toward appropriately skilled students. Our study is the first evidence we know of to

demonstrate long-run impacts of such intensive math instruction.

Our work contributes to three strands of the research literature. First, given that the

intervention studied here doubled the amount of time students were exposed to freshman algebra,

our study adds to the literature on the importance of instructional time to student achievement.

Some education reformers have pushed U.S. schools to lengthen school days and years, noting that

students in many academically successful nations, particularly in Asia, spend substantially more time

in school than do American students. Proponents of this view point to evidence on summer learning

loss (Cooper et al. 1996), the impact of snow days (Marcotte and Hemelt 2008), the association

between charter school effectiveness and instructional time (Dobbie et al. 2011, Hoxby and Murarka

2009), and other such patterns linking student achievement to hours spent learning (Lavy 2010,

Fitzpatrick et al. 2011). Another set of studies suggests this evidence is weaker than it first appears,

with Fryer Jr. and Levitt (2004) observing little differential summer learning loss, Goodman (2012a)

showing little impact of snow days on achievement, Angrist et al. (2011) showing little relation

between instructional time and charter school effectiveness, and Associates (2011) showing little

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effect of an intervention that substantially increased schools’ instructional times. The emerging

consensus from this literature is that increasing instructional time is no guarantee of better student

outcomes if such time is not well spent. Our results are consistent with heterogeneous impacts of

increased instructional time by math and reading skills.

Second, our work adds to the literature concerning the short-run impact of curricular

interventions, particularly for students struggling in mathematics. Recent years have seen three main

curriculum approaches tried by American schools. Remediation, which diverts students into basic

courses prior to taking regular courses, has generally had little discernible impact on student

achievement, particularly at the college level where it has most often been studied (Jacob and

Lefgren 2004, Lavy and Schlosser 2005, Calcagno and Long 2008, Bettinger and Long 2009,

Martorell and McFarlin Jr 2011, Boatman and Long 2010, Scott-Clayton and Rodriguez 2012).

Algebra “for all”, which pushes students to take algebra courses in earlier grades than they otherwise

would, actually harms student achievement by forcing students into subjects for which they are not

sufficiently prepared (Clotfelter et al. 2012, Nomi 2012). Double-dosing, which places students in

regular courses but supplements those courses with additional instructional time, has generated

modest short-run gains in some settings and no gains in others (Nomi and Allensworth 2009, Nomi

and Allensworth 2010, Roland G. Fryer 2011, Taylor 2012, Dougherty 2012). Perhaps because of

perceived effectiveness at raising short-run achievement levels, the double-dose strategy has become

increasingly common, with half of large urban districts reporting it as their most common form of

support for struggling students.2

Third, and perhaps most important, we contribute to the literature on the long-run impacts

of curriculum on student outcomes. Nearly all such research point to a close association between

coursework completed in high school and later outcomes such as college enrollment and labor

2 See “Urban Indicator: High School Reform Survey, School Year 2006-2007”, by the Council of Great City Schools, 2009.

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market earnings (Altonji 1995, Levine and Zimmerman 1995, Rose and Betts 2004, Attewell and

Domina 2008, Long et al. 2009, Long et al. 2012). Most such papers attempt to deal with the bias

generated by selection into coursework by controlling for a rich set of covariates, either through

OLS or propensity score matching. However, such methods leave open the possibility that the

remaining unobservables are still important factors. The few papers that use quasi-experimental

methods to convincingly eliminate such selection bias also, however, find strong associations

between completed coursework and long-run outcomes, suggesting that such selection bias is not

generating the central findings (Joensen and Nielsen 2009, Goodman 2012b). This paper is one of

the better identified links between high school coursework and educational attainment.

The structure of our paper is as follows. In section 2, we describe in detail the double-dose

algebra policy. In section 3, we describe the data and offer descriptive statistics about students in our

sample. In section 4, we explain the regression discontinuity underlying our identification strategy. In

sections 5 and 6, we describe the impact of double-dosing on students’ educational experiences,

grades, test scores and educational attainment. In section 7, we discuss robustness, heterogeneity and

spillovers onto other subjects. In section 8, we conclude. We now turn to a description of the

double-dose algebra policy itself.

2. Implementing Double-Dose Algebra

Since the late 1990s, Chicago Public Schools (CPS) has been at the forefront of curriculum

reform designed to increase the rigor of student coursework and prepare students for college

entrance. Starting with students entering high school in the fall of 1997, CPS raised its graduation

requirements to align with the New Basics Curriculum.3 CPS eliminated lower-level and remedial

3 The new basics curriculum was a minimum curriculum recommended by the National Commission of Excellence in Education in 1983, which consists of four years of English, three years of each mathematics, science, and social studies, and one-half year of computer science. The CPS requirements are actually slightly higher than the New Basics

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courses so that all first-time freshmen would enroll in algebra in 9th grade, geometry in 10th grade

and algebra II or trigonometry in 11th grade. Soon after these reforms, CPS officials realized that

students were unable to master the new college-prep curriculum. Passing rates in 9th grade algebra

were quite low, largely because students entered high school with such poor math skills (Roderick

and Camburn 1999).

In response to these low passing rates in 9th grade algebra, CPS launched the double-dose

algebra policy for students entering high school in the fall of 2003. Instead of reinstating the

traditional remedial courses from previous years, CPS required enrollment in two periods of algebra

coursework for all first-time 9th graders testing below the national median on the math portion of

the 8th grade Iowa Tests of Basic Skills (ITBS).4 Such students enrolled for two math credits, a full-

year regular algebra class plus a full-year algebra support class.5 Three student cohorts, those

entering high school in the fall of 2003, 2004 and 2005, were subject to the double-dose policy. Our

analysis focuses on the first two cohorts because the test score-based assignment rule was not

followed closely for the final cohort. We will refer to these as the 2003 and 2004 cohorts.

Prior to the double-dose policy, algebra curricula had varied considerably across CPS high

schools, due to the fairly decentralized nature of the district. Conversely, CPS offered teachers of

double-dose algebra two specific curricula called Agile Mind and Cognitive Tutor, stand-alone lesson

plans they could use, and thrice annual professional development workshops where teachers were

Curriculum, which includes two years of a foreign language and specific courses in mathematics (i.e., algebra, geometry, advanced algebra, and trigonometry). 4 All CPS high schools were subject to the double-dose algebra policy, including 60 neighborhood schools, 11 magnet schools, and 6 vocational schools (Nomi and Allensworth 2009). 5 Double-dose algebra students received 90 minutes of math class time every day for a full academic year. The first math course, regular algebra, consisted mostly of class lectures. The second math course, algebra with support, focused on building math skills that students lacked and covered materials in a different than the textbook. Double-dose teachers used various instructional activities, such as working in small groups, asking probing and open-ended questions, and using board work (Wenzel et al. 2005, Starkel and Price 2006).

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given suggestions about how to use the extra instructional time.6 Though it is difficult to know

precisely what occurred in these extra classes, Nomi and Allensworth (2010) surveyed students to

learn more about the classroom learning environment. They found that students assigned to double-

dose algebra reported much more frequently: writing sentences to explain how they solved a math

problem; explaining how they solved a problem to the class; writing math problems for other

students to solve; discussing possible solutions with other students; and applying math to situations

in life outside of school. The additional time thus focused on building verbal and analytical skills

may have conferred benefits in subjects other than math.

CPS also strongly advised schools to schedule their algebra support courses in three specific

ways. First, double-dose algebra students should have the same teacher for their two periods of

algebra. Second, the two algebra periods should be offered consecutively. Third, double-dose

students should take their algebra support class with the same students who are in their regular

algebra class. Most CPS schools followed these recommendations in the initial year (Nomi and

Allensworth 2009). For the 2003 cohort, 80 percent of double-dose students had the same teacher

for both courses, 72 percent took the two courses consecutively, and rates of overlap between the

two classes’ rosters exceeded 90 percent. By 2004, schools began to object to the scheduling

difficulties of assigning the same teacher to both periods so CPS removed that recommendation.

For the 2004 cohort, only 54 percent of double-dose students had the same teacher for both courses

and only 48 percent took the two courses consecutively. Overlap between the rosters remained,

6 The district made the new double-dose curricula and professional development available only to teachers teaching double-dose algebra courses, but there was a possibility of spillover effects for teachers in regular algebra. However, the professional development was geared towards helping teachers structure two periods of algebra instruction. Moreover, based on CPS officials and staff members’ observations of double-dose classrooms, they found that even teachers who taught both single-period and double-dose algebra tended to differentiate their instruction between the two types of classes. Specifically, teachers tended to use new practices with the double-period class, but continued to use traditional methods with the single-period class. Teachers told them that they did not feel they needed to change methods with the advanced students (i.e., non double-dose students), and that they were hesitant to try new practices that may be more time-consuming with just a single period. The double period of algebra allowed these teachers to feel like they had the time to try new practices (e.g., cooperative groups).

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however, close to 90%. In our analysis, we also explore whether the program’s impacts vary by

cohort in part because of this variation in implementation.

The treatment under consideration here thus had many components. Assignment to double-

dose algebra doubled the amount of instructional time and exposed students to the curricula and

activities discussed above. As we will show, the recommendation that students take the two classes

with the same set of peers caused tracking by skill to increase, thus reducing classroom

heterogeneity. All of these factors were likely to, if anything, improve student outcomes (Duflo et al.

2011). We will also show, however, that the increase tracking by skill placed double-dosed students

among substantially lower skilled peers than non-double dosed students. This factor is likely to, if

anything, hurt student outcomes. Our estimates will, therefore, capture the net impact of all of these

components.

3. Data and Descriptive Statistics

We use longitudinal data from CPS that tracks students from 8th grade through college

enrollment. These data include demographic information, detailed high school transcripts, numerous

standardized test scores, and graduation and college enrollment information. Our sample consists of

all students entering 9th grade for the first time in the fall of 2003 and 2004. We include only

students who have valid 8th grade math scores and who enroll in freshman algebra. We include only

high schools in which at least one classroom of students was assigned to double-dose algebra. For

binary outcomes, students who leave the CPS school system for any reason are coded as zeroes. CPS

attempts to track students’ reasons for leaving. In our sample, students who leave CPS are about

evenly divided between those who are known dropouts, those who leave for other schools (private

schools or public schools outside of Chicago), and those whose reasons for leaving are unknown.

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The summary statistics of the analytic sample are shown in Table 1. Column (1) includes the

entire sample and column (2) includes only students within 10 percentiles of the double-dose

threshold, our main analytic sample. Columns (3) and (4) separate that sample by cohort. As seen in

panel (A), about 90 percent of CPS students are black or Hispanic, with 20 percent in special

education. Though not shown here, over 90 percent of CPS students are low income as indicated by

participation in the federal subsidized lunch program. We will therefore use as controls more

informative socioeconomic and poverty measures constructed for each student’s residential block

group from the 2000 Census and standardized within the full sample.7 We also include as a control

each student’s 8th grade reading percentile.

The first row of panel (B) shows our instrument, each student’s 8th grade score on the math

portion of the Iowa Test of Basic Skills (ITBS), which all CPS 8th graders are required to take. The

mean CPS 8th grade student scores are between the 45th and 46th percentiles on this nationally

normed exam. As shown in column (1), 55 percent of CPS students score below the 50th percentile

and thus should be assigned to double-dose algebra, though the transcript data reveal that only 42%

actually are assigned to this class, suggesting imperfect compliance with the rule. As a result, the

average CPS freshman in our sample takes 1.4 math courses freshman year.

The transcript data also allow for detailed exploration of the treatment itself. We construct

variables, shown in panel (B), showing the extent to which schools were complying with CPS’

guidelines for implementing double-dose algebra. The average student attended a school in which 63

percent of double-dosed students had their two algebra courses during consecutive periods, in which

71 percent of double-dosed students had the same teacher for both courses, and in which 90 percent 7 Our data are linked to the 2000 U.S. Census at the block level corresponding with each student’s home address. Indicators of students’ socioeconomic status and concentration of poverty were derived from the census data about the economic conditions in students’ residential block groups. More specifically, socioeconomic status variable is based on percentage of employed persons 16 years (or older) who are managers and executives and the mean level of education among people over 18. The concentration of poverty variable is based on the percent of males over 18 who are employed one or more weeks during the year and the percent of families above the poverty line.  

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of double-dosed students’ regular algebra classmates were themselves double-dosed. Consistent with

schools’ complaints that scheduling double-dose algebra according to these guidelines was quite

challenging, compliance was substantially lower in 2004 than in 2003, as can be seen in columns (3)

and (4).

We focus on two primary sets of outcomes. First, in panel (C), we explore whether double-

dosing helps student’s academic achievement by constructing a variety of variables measuring

grades, coursework and standardized test scores. The grades and coursework variables reveal that

only 62 percent of the full sample pass algebra (i.e., receiving a D or higher), while even fewer pass

higher level courses such as geometry and trigonometry. Given that grades are subjective measures,

we construct a variety of test scores standardized by cohort, including the PLAN exam, which all

CPS students take in September of both their second and third years in high school, and the ACT

exam, which all CPS students take in April of their third year and is commonly used in the Midwest

for college applications.

Second, in panel (D), we explore whether double-dosing improves educational attainment by

constructing measures of high school graduation and college enrollment. Students are coded as high

school graduates if they received a regular CPS diploma within four or five years of starting high

school. About 50 percent of CPS students in our sample graduate high school within four years,

with another 5 percent graduating in their fifth year. CPS has matched its data on high school

graduates with the National Student Clearinghouse (NSC) data on college enrollment, allowing us to

observe initial college enrollment for any CPS student with a high school diploma.8 We construct

indicators for enrollment in college by October 1 of the fifth year after starting high school. Only 29

percent of the full sample both graduate from a CPS high school and enroll in college within this

8 The NCS collects information on students’ postsecondary education, which includes semester-by-semester college enrollment and attainment. NCS covers 91 percent of colleges (more than 2,800 postsecondary institutions) in the United States. CPS graduates mostly enroll in local colleges that participate in NCS. We therefore have excellent college enrollment records for about 95 percent of CPS graduates.

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time frame, more than half of whom enroll in two-year colleges. We also construct the number of

semesters of college a student has enrolled within eight years of starting high school.9 The average

student in this sample has enrolled for 1.8 semesters of college, nearly half of which occurs at two-

year colleges. We cannot explore college completion rates because students in our sample have not

yet had sufficient time to graduate from four-year colleges and because NSC has little graduation

data from two-year colleges in this sample.

4. Empirical Strategy

Comparison of the outcomes of students who are and are not assigned to double-dosed

algebra would likely yield biased estimates of the policy’s impacts given potentially large differences

in unobserved characteristics between the two groups of students. To eliminate this potential bias,

we exploit the fact that students scoring below the 50th percentile on the 8th grade ITBS math test

were supposed to enroll in double-dose algebra. This rule allows us to identify the impact of double-

dose algebra using a regression discontinuity design applied to the two treated cohorts. We use the

assignment rule as an exogenous source of variation in the probability that a given student will be

double-dosed.

We implement the regression discontinuity approach using the regressions below:

itititititit mathlowscoremathlowscoreY 88 3210 (1)

itititititit mathlowscoremathlowscoreDoubleDose 88 3210 (2)

itititititit mathlowscoremathDoubleDoseY 88 3210 (3)

where for student i in cohort t, lowscore indicates an 8th grade math score below the 50th percentile,

math8 is each student’s 8th grade math score re-centered around the 50th percentile cutoff,

DoubleDose is an indicator for assignment to the extra algebra period, and Y represents an outcome

9 This outcome is based on weighting half-time enrollment as 0.5 semesters and less than half-time enrollment as 0.25 semesters. Note that NSC data contains no information on credits completed, just the number of semesters enrolled.

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of interest. The lowscore coefficient ( 1 ) from equation (1) estimates the discontinuity of interest by

comparing the outcomes of students just below and just above the double-dose threshold. This

reduced form equation produces an intent-to-treat estimate because of imperfect compliance with

the assignment rule. The lowscore coefficient ( 1 ) from the first stage equation (2) measures the

difference in double-dose rates between students just below and just above the threshold.

Our ultimate estimate of interest is thus the DoubleDose coefficient ( 1 ) from equation (3), in

which DoubleDose has been instrumented with lowscore using that first-stage. This approach estimates

a local average treatment effect (LATE) of double-dose algebra for students near the 50th percentile

of 8th grade math skill. Here, students just above the threshold serve as a control group for students

just below the threshold, so that these estimates will be unbiased under the assumption that no other

factors change discontinuously around the threshold itself. The density of 8th grade math scores is

quite smooth around the cutoff, suggesting little scope for manipulation of such scores by students

or teachers and little impact of the threshold on the probability of appearing in our sample.10

Our preferred specification will fit straight lines on either side of the threshold using a

bandwidth of 10 percentiles, and will also control for gender, race, special education status,

socioeconomic and poverty measures, and 8th grade reading scores. We will show that our central

results are robust to alternative choices of controls and bandwidths. In all specifications,

heteroskedasticity robust standard errors will be clustered by each student’s initial high school to

account for within high school correlations in the error terms.

5. The Treatment

We first explore the treatment itself to learn more about how the double-dose algebra policy

was changing students’ freshman year experiences. Before turning to regression results, we look at

10 See Figure A.1.

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visual evidence. Figure 1 plots the proportion of students double-dosed for each 8th grade math

percentile. If compliance with the double-dose rule were perfect, we would expect to see 100%

remediation rates to the left of the 50th percentile threshold. Instead, panel (A) shows imperfect

compliance, with assignment rates reaching a maximum of about 80 percent for students in the 20-

40th percentiles. Students in the lowest percentiles have lower double-dose rates because they are

more likely to be supported through special education programs (who may be taking low-level

courses that do not technically qualify as double-dose algebra). Also, some students above the

threshold are double-dosed, perhaps because schools cannot perfectly divide students into

appropriately sized classes by the assignment rule. Panel (B) reveals that compliance for students just

below the threshold is substantially lower in the 2004 cohort than the 2003 cohort, providing further

motivation to analyze program impacts separately by cohort.

Table 2 shows the first-stage results using a low 8th grade math score indicator as an

instrument for assignment to double-dose algebra. Panel (A) pools the cohorts while panel (B)

allows the estimate to vary by cohort. Column (1) implements equation (2), including only cohort

indicators as controls. Column (2) adds demographic and test score controls as described in the

table. Column (3) includes those controls and high school fixed effects. Column (4) expands the

bandwidth from 10 to 20 percentiles and fits quadratic functions on either side of the threshold

instead of linear functions.

The first-stage discontinuity estimates suggest that students just below the threshold were 40

percentage points more likely to be double-dosed than students just above the threshold. Consistent

with Figure 1, this discontinuity was much larger in 2003 (48 percentage points) than in 2004 (32

percentage points), further motivating our decision throughout the paper to explore differential

impacts by cohort. These estimates are highly robust to inclusion of controls, inclusion of high

school fixed effects, and to alternative bandwidth and functional form assumptions. Table 3

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explores heterogeneity by reading skill, by gender and by race/ethnicity. Each column replicates the

second column from Table 2 with the sample limited to the identified sub-group. There is little

indication of differential compliance by reading skill, gender or race/ethnicity, with the exception of

lower compliance rates among the small number of white students in our sample. These patterns

hold true for both the 2003 and 2004 cohorts.

The double-dose treatment involved multiple changes to the freshman algebra experience.

Figures 2, 3 and 4 show visual evidence of the channels through which double-dose may have

affected student outcomes. Figure 2 shows the most obvious impact of double-dose algebra, namely

the substantial increase in the number of class periods devoted to algebra. Doubling of instructional

time was not, however, the only impact of the intervention. Because double-dose students were also

supposed to be placed into classes with similarly double-dosed students, tracking in freshman math

courses increased. The next two figures show that the CPS guideline that double-dose students

should be in regular algebra classes with their double-dose classmates increased tracking by academic

skill. Figure 3 shows that double-dosed students’ regular algebra classes were more homogeneous, as

measured by the standard deviation of 8th grade math scores in those classes. Lastly, figure 4 shows

that double-dosed students regular algebra peers had substantially lower mean math skills than did

students not assigned to double-dose.

Table 4 explores the impact of assignment to double-dose algebra on the freshman academic

experience. All of the coefficients come from regressions in which assignment to double-dose

algebra has been instrumented by eligibility, as in Table 2. As such, these are treatment-on-the-

treated (TOT) estimates of the impact of double-dosing on those actually double-dosed. In panel

(A), column (1) shows that being double-dosed increased the number of freshman math courses

taken by one, as would be expected from the double-dose strategy. The policy thus doubled

instructional time. Columns (2)-(4) show that students fit this additional math course into their

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schedule not by replacing core academic courses in English, social studies and science, but by

replacing other courses such as fine arts and foreign languages. The result was a positive but small

change in the total number of courses taken freshman year.

Columns (5)-(8) highlight channels other than instructional time and measure various

characteristics of students’ regular (i.e., not double-dose) algebra courses. The increased skill tracking

implied by CPS’ guidelines meant that double-dosed students took algebra classes with peers whose

8th grade math scores were substantially lower than the peers of non-double-dosed students. The

estimates in column (5) imply that double-dosing lowered the mean peer skill of double-dosed

students near the threshold by over 19 percentiles. Column (6) suggests that double-dosed students

near the threshold were in more homogeneous classrooms than their non-double-dosed peers, with

the standard deviation of math skill roughly 3 percentiles lower. Column (7) implies that double-

dosing increased the distance of students to their peers’ mean skill by about 3 percentiles, which

could have negative repercussions if teachers focus their energies on the mean student. Column (8)

suggests that, near the threshold, double-dosed students were in regular algebra classes 2.4 students

larger than non-double-dosed students. Thus, the double-dose policy thus doubled instructional

time in math by replacing other coursework and increased homogeneity of algebra classrooms, but

lowered peer skill, increased distance from the class mean and increased class size. Panel (B) suggests

that none of these effects varied substantially by cohort. We now turn to analysis of the overall

impact of these various changes on coursework, test scores and educational attainment.

6. Grades, Test Scores and Educational Attainment

A visual preview of our results is available in Figures 5, 6 and 7. Using a bandwidth of 10

percentiles on either side of the threshold, each of these figures plots the mean of the given outcome

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by 8th grade math score, as well as fitted straight lines that come from the regressions we will run.11

Figure 5 suggests little clear impact of double-dosing on algebra passing rates but a noticeable

impact on the fraction of students receiving at least a B. Figure 6 suggests little impact of double-

dosing on math scores the fall after freshman year ended, but a clear impact on such scores a full

year after that. Figure 7 suggest impacts on both high school graduation and college enrollment

rates. We now turn to regression analysis to measure more precisely the magnitude and statistical

significance of these discontinuities.

Table 5 explores the impact of the double-dose policy on math coursework and grades.

Double-dosing increased the proportion of students earning at least a B in freshman algebra by 9.4

percentage points, a more than 65 percent increase from a base of 13.8 percentage points. Though

passing rates for freshman algebra increased by 4.7 percentage points, the increase is statistically

insignificant. Double-dosed students were also no more likely to pass geometry. They were,

however, substantially more likely to pass trigonometry, a course typically taken in the third year of

high school. Mean GPA across all math courses taken after freshman year increased by a marginally

significant 0.14 grade points on a 4.0 scale. As a whole, these results imply that the double-dose

policy greatly improved freshman algebra grades for the upper end of the double-dosed distribution,

but had relatively little impact on passing rates. This latter fact is one of the primary reasons that

CPS has since moved away from this strategy. There is, however, some evidence of improved

passing rates and GPA in later math courses, suggesting the possibility of longer-run benefits

beyond freshman year. Though coursework and grades matter for students’ academic trajectories,

the subjective nature of course grading motivates us to turn to standardized achievement measures

as a potentially better measure of the impact of double-dosing on math skill.

11 Our preferred specification will include student-level demographic controls but these lines come from OLS regressions with no controls other than 8th grade math scores.

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Table 6 explores the impact of double-dosing on mathematics test scores as measured by the

PLAN exam taken in September of a student’s second year, the PLAN exam taken in September of

a student’s third year, and the ACT exam taken in April of each student’s third year. The PLAN

exams contain a pre-algebra/algebra section and a geometry section, which we analyze separately

given that double-dose classes focused on algebra skills. Column (1) suggests impacts on the first

PLAN exam of 0.09 standard deviations. Though statistically insignificant, it is worth noting that the

positive coefficient comes almost entirely from an improvement in algebra, while the point estimate

for geometry is almost exactly zero. Double-dosing does, however, improve scores on later exams.

Double-dosing increases algebra scores by a statistically significant 0.15 standard deviations and has

a slightly smaller though statistically insignificant impact on geometry. Double-dosing thus raises

overall math scores in the fall of students’ third years by 0.16 standard deviations. Perhaps more

importantly, a nearly identical effect is seen on the math portion of the ACT, with double-dose

algebra raising such scores by a statistically significant 0.15 standard deviations on an exam used by

many colleges as part of the admissions process. These effects are nearly identical across the two

cohorts. The insignificant coefficient in column (7), which uses as an outcome an indicator for

taking the ACT, shows that these results are not driven by selection into exam-taking. Results for

the PLAN exams are similarly insignificant. Table 6 thus suggests that double-dosed students

experienced medium-run achievement gains that persisted at least two years after the end of double-

dose classes.

Table 7 explores the impact of double-dosing on educational attainment. In panel (A),

columns (1) and (2) show that double-dosing increases 4- and 5-year graduation rates by 8.7 and 7.9

percentage points, respectively. This represents a 17 percent improvement over the 51 percent of

non-double-dosed students at the threshold who graduate within 4 years. Panel (B) shows that these

improvements in high school graduation rates were larger in magnitude for the 2003 cohort, but not

18

in any statistically significant way. Columns (3)-(7) show that double-dosing also dramatically

improve college enrollment outcomes. Double-dosed students are 8.6 percentage points more likely

to ever enroll in college within five years of starting high school, a 30 percent increase over the base

college enrollment rate of 29 percent. As columns (4) and (5) show, nearly all of this college

enrollment increase comes from two-year colleges, with the majority of that coming from part-time

enrollment in such colleges. Given the relatively low academic skills and high poverty rates of CPS

students at the double-dose threshold, it is unsurprising that double-dosing improved college

enrollment rates at relatively inexpensive and non-selective two-year postsecondary institutions.

Columns (6) and (7) show that, within eight years of starting high school and thus within four years

of expected high school graduation, double-dosed students have enrolled in 0.6 additional semesters

of college, 0.4 of which are in two-year colleges.

7. Robustness, Heterogeneity and Spillovers

Our primary results suggest double-dose algebra improved students’ math achievement, high

school graduation rates and college enrollment rates. We turn now to questions of the robustness of

these results, heterogeneity of the policy’s impacts, and spillovers into subjects other than math.

Table 8 shows robustness checks for the central results of the previous tables. Panel (A) fits straight

lines on either side of the threshold, using a bandwidth of 10 percentiles. The top row includes no

controls other than cohort indicators, the second row (our preferred specification) adds

demographic controls as described in previous tables, and the third line adds high school fixed

effects. All three specifications yield very similar results. Panel (B) replicates the second row of panel

(A) using successively smaller bandwidths. The magnitudes of the coefficients remain quite similar

or, if anything, increase as the bandwidth is reduced, though their statistical significance diminishes

in some cases due to the reduced sample size. Panel (C) includes three placebo tests, each of which

19

replicates the reduced form version of the second row of panel (A). The first and second placebo

tests use the 40th and 60th percentiles as the double-dose threshold, while the third uses the

untreated 2001 and 2002 cohorts. Only two of the 24 coefficients are marginally significant, as

would be expected by chance. There is no sign of spurious discontinuities. This table implies that

our central results are robust to alternative specifications and that such discontinuities appear only at

thresholds and for cohorts where we expect them.

Table 9 explores whether the impacts of double-dosing varied by the academic skill of the

double-dosed student. Our primary regression discontinuity results, which we repeat in panel (A),

estimate a LATE for students near the given threshold. In our case, our estimates apply to students

near the 50th percentile of math skill. Such students vary, however, in their reading skills as

measured by their 8th grade ITBS reading scores. We exploit this fact in panel (B), where we divide

students into those who scored above (“good readers”) and below (“poor readers”) the 45th

percentile on the 8th grade ITBS reading test, which represents roughly the median reading skill of

students at the double-dose threshold. We then interact double-dosing (and its instruments) with

indicators for those two categories. The results are striking. For all of the outcomes shown, double-

dosing had larger positive effects on poor readers than on good readers, differences which are often

statistically significant. For example, double-dosing raised below median readers’ ACT scores by 0.22

standard deviations but raised above median readers’ ACT scores by only 0.09 standard deviations.

The impact of double-dosing on high school graduation and college enrollment is driven almost

entirely by students in the lower part of the reading distribution. Similar hetereogeneity analysis by

gender and race yields little evidence of differential impacts along these dimensions.

To explore whether double-dosing’s impact varied by math skill, we implement in panel (C)

a difference-in-difference specification using all students in the untreated 2001 and 2002 cohorts as a

control for all students in the treated 2003 and 2004 cohorts. By controlling for differences between

20

low- and high-scoring students in the pre-treatment cohorts and for overall differences between

cohorts, we can thus estimate how the difference in outcomes between low- and high-scoring

students changed at the time double-dose algebra was introduced. This approach estimates an

average treatment effect (ATE) of double-dose algebra for all students double-dosed because of the

policy. Such students are, on average, lower skilled than students near the threshold itself. These

results suggest that, across all double-dosed students, double-dosing did improve passing rates and

short-run algebra scores, but had no discernible impact on later test scores or high school graduation

rates. Data limitations prevent exploration of college outcomes for the earlier cohorts. Together,

panels (B) and (C) suggest that double-dose algebra had little long-run impact on the average

double-dosed student but had substantial positive impacts on double-dosed students with relatively

high math skills but relatively low reading skills. That the majority of the positive long-run impact of

double-dosing came through its effect on low skilled readers may be due to the intervention’s focus

on reading and writing skills in the context of learning algebra.

Finally, the increased focus on algebra at the cost of other coursework may potentially have

affected achievement in other academic subjects. Table 10 looks at outcomes in other subjects. We

find strong evidence that, rather than harming other achievement, double-dosing had positive

spillovers in reading and science. Double-dosed students scored nearly 0.2 standard deviations

higher on the verbal portion of their ACTs, were substantially more likely to pass chemistry classes

usually taken in 10th or 11th grade, and had marginally high GPAs across all of their non-math

classes in years after 9th grade. If anything, the skills gained in double-dose algebra seem to have

helped, not hindered, students in other subjects and subsequent years.

21

8. Conclusion

Double-dose strategy has become an increasingly popular way to aid students struggling in

mathematics. Today, nearly half of large urban districts report doubled math instruction as the most

common form of support for students with lower skills (Council of Great City Schools 2009). The

central concern of urban school districts is that algebra may be a gateway for later academic success,

so that early high school failure in math may have large effects on subsequent academic achievement

and graduation rates. As the current policy environment calls for “algebra for all” in 9th grade or

earlier grades, providing an effective and proactive intervention is particularly critical for those who

lack foundational mathematical skills. A successful early intervention may have the greatest chance

of having longer-term effects on students’ academic outcomes.

We provide evidence of positive and substantial long-run impacts of one particular form of

intensive math instruction on college entrance exam scores, high school graduation rates and college

enrollment rates. We show that this intensive math instruction was most successful for students with

average math skills but relatively low reading skills, and not successful for the average treated

student. This highlights the importance of carefully targeting such interventions to students most

likely to benefit from them. Also, like other recent studies, we find that the test score impacts of this

policy dramatically understate its long-run benefits as measured by educational attainment (Deming,

2009; Chetty et al., 2011). In our sample, OLS suggests that a 0.15 standard deviation increase in

ACT math scores is associated with a 2 percentage point increase in college enrollment rates. We

observe college enrollment effects four times that size, highlighting the fact that long-run analyses of

such interventions may yield very different conclusion than short-run analyses.

One final piece of evidence suggests that CPS’s double-dose policy impacts could be

replicated in other urban school districts across the U.S. Recall that the district strongly suggested

that double-dose students should have the same teacher in regular algebra and in their double-dose

22

class, that the two algebra courses should be offered sequentially, and that double-dose students

should take their double-dose class with the same set of students that were in their regular algebra

class. Table 11 explores whether the impact of double-dosing varied by the extent to which each

school complied with CPS’ implementation guidelines. For each school we construct the fraction of

double-dosed students whose schedule adhered to each of these guidelines. We average all three

measures of compliance in panel (A) and break them out separately in panel (B). In neither panel is

there any evidence that compliance with these guidelines is related to the impact of double-dosing.

This is consistent with the fact that the impact of double-dosing seems not to have varied by the

2003 and 2004 cohorts, even though schools were less likely to comply with the guidelines for the

second cohort. Districts looking to adopt the double-dose strategy could likely reap its benefits

without needing to radically restructure their school days.

23

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Figure 1: Double-Dosing Rates

0.2

.4.6

.8F

ract

ion

do

ub

le−

do

sed

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

(A) Cohorts pooled

0.2

.4.6

.8F

ract

ion

do

ub

le−

do

sed

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

2003 cohort

2004 cohort

(B) Cohorts separated

21

Figure 2: Freshman Algebra Periods

11.

21.

41.

61.

8F

resh

man

alg

ebra

per

iod

s

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

(A) Cohorts pooled

11.

21.

41.

61.

8F

resh

man

alg

ebra

per

iod

s

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

2003 cohort

2004 cohort

(B) Cohorts separated

22

Figure 3: Standard Deviation of Peer Math Skill

911

1315

17S

D o

f p

eer

mat

h s

kil

l

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

(A) Cohorts pooled

911

1315

17S

D o

f p

eer

mat

h s

kil

l

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

2003 cohort

2004 cohort

(B) Cohorts separated

23

Figure 4: Mean Peer Math Skill

1020

3040

5060

7080

90M

ean

pee

r m

ath

sk

ill

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

(A) Cohorts pooled

1020

3040

5060

7080

90M

ean

pee

r m

ath

sk

ill

0 10 20 30 40 50 60 70 80 90 1008th grade math percentile

2003 cohort

2004 cohort

(B) Cohorts separated

24

Figure 5: Freshman Algebra Grades

.58

.6.6

2.6

4.6

6.6

8P

asse

d a

lgeb

ra

40 50 608th grade math percentile

(A) Passed algebra

.14

.16

.18

.2.2

2A

or

B i

n a

lgeb

ra

40 50 608th grade math percentile

(B) A or B in algebra

25

Figure 6: Math Achievement

−.2

−.1

0.1

.2F

all

10 m

ath

sco

re

40 50 608th grade math percentile

(A) Fall 10 math score

−.2

−.1

0.1

.2F

all

11 m

ath

sco

re

40 50 608th grade math percentile

(B) Fall 11 math score

26

Figure 7: Educational Attainment

.5.5

5.6

.65

Gra

du

ated

HS

in

5 y

ears

40 50 608th grade math percentile

(A) Graduated high school in 5 years

.2.2

5.3

.35

.4A

ny

co

lleg

e

40 50 608th grade math percentile

(B) Enrolled in college

27

Table 1: Summary Statistics

(1) (2) (3) (4)Both cohorts, Both cohorts 2003 cohort, 2004 cohort,

full sample near threshold near threshold near threshold

(A) Controls

Female 0.50 0.54 0.54 0.54Black 0.57 0.56 0.58 0.55Hispanic 0.33 0.36 0.34 0.37Special education 0.20 0.08 0.08 0.08Census block poverty measure -0.00 -0.02 -0.01 -0.03Census block SES measure -0.00 -0.01 -0.02 -0.018th grade reading percentile 43.33 46.60 46.48 46.71

(B) Double-dose

8th grade math percentile 45.69 49.92 50.01 49.82Eligible for double-dose 0.55 0.48 0.47 0.49Double-dosed 0.42 0.41 0.41 0.40Freshman math courses 1.40 1.39 1.39 1.39Consecutive periods 0.63 0.65 0.77 0.54Same teacher 0.71 0.73 0.85 0.60Extent of tracking 0.90 0.90 0.94 0.86

(C) Achievement

Passed algebra 0.62 0.63 0.63 0.64Passed geometry 0.57 0.59 0.58 0.59Passed trigonometry 0.51 0.52 0.51 0.53Fall 10 math z-score 0.00 -0.03 -0.04 -0.02Fall 11 math z-score 0.00 -0.05 -0.05 -0.04ACT math z-score 0.00 -0.21 -0.21 -0.22

(D) Attainment

Graduated HS in 4 years 0.49 0.51 0.50 0.53Graduated HS in 5 years 0.54 0.56 0.55 0.57Enrolled in any college 0.29 0.31 0.29 0.33Enrolled in 2-year college 0.16 0.17 0.16 0.19Semesters of college 1.86 1.93 1.89 1.96Semesters of 2-year college 0.82 0.93 0.91 0.94

N 41,122 11,507 5,734 5,773

Notes: Mean values of each variable are shown by sample. Column (1) is the full sample of students from the 2003and 2004 cohorts. Column (2) limits the sample to students within 10 percentiles of the double-dose threshold.Columns (3) and (4) separate column (2) by cohort.

28

Table 2: Eligibility as an Instrument for Double-Dose Algebra

(1) (2) (3) (4)Y = double-dosed No Demographic High school Quadratic,

controls controls fixed effects bandwidth=20

(A) Cohorts pooled

Eligible for double-dose 0.400∗∗∗ 0.399∗∗∗ 0.406∗∗∗ 0.387∗∗∗

(0.038) (0.039) (0.037) (0.039)F (Z) 108.4 106.4 120.3 97.6

(B) Cohorts separated

Eligible * 2003 0.478∗∗∗ 0.478∗∗∗ 0.486∗∗∗ 0.442∗∗∗

(0.039) (0.039) (0.038) (0.038)Eligible * 2004 0.321∗∗∗ 0.319∗∗∗ 0.325∗∗∗ 0.331∗∗∗

(0.047) (0.048) (0.046) (0.047)p (β2003

= β2004) 0.000 0.000 0.000 0.002

N 11,507 11,507 11,507 21,539

Notes: Heteroskedasticity robust standard errors clustered by initial high school are in parentheses (* p<.10 ** p<.05*** p<.01). Each column in each panel represents the first-stage regression of semester of double-dose algebra oneligibility as determined by eighth grade math score. Panel (A) pools the two treatment cohorts. Below panel (A) isthe value from an F-test of the instrument. Panel (B) interacts the instrument with cohort indicators. Below panel(B) is the p-value from an F-test of the equality of the two listed coefficients. Column (1) fits straight lines on bothsides of the threshold using a bandwidth of 10 percentiles. Cohort indicators are included but not shown. Column(2) adds to column (1) controls for gender, race, census block poverty and socioeconomic status, and eighth grademath and reading scores. Column (3) adds to column (2) high school fixed effects. Column (4) adds to column (3)quadratic terms on either side of the threshold and expands the bandwidth to 20 percentiles.

29

Tab

le3:

Fir

st-S

tag

eH

eter

og

enei

ty

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Bel

ow

Ab

ov

eY

=d

ou

ble

-do

sed

med

ian

med

ian

Mal

eF

emal

eB

lack

His

pan

icW

hit

ere

ader

read

er

(A)

Co

ho

rts

po

ole

d

Eli

gib

lefo

rd

ou

ble

-do

se0.

405∗

∗∗

0.38

4∗∗∗

0.38

0∗∗∗

0.41

4∗∗∗

0.41

8∗∗∗

0.40

2∗∗∗

0.23

7∗∗

(0.0

41)

(0.0

46)

(0.0

41)

(0.0

42)

(0.0

38)

(0.0

65)

(0.1

04)

F(Z

)95

.869

.285

.198

.611

9.8

38.6

5.2

(B)

Co

ho

rts

sep

arat

ed

Eli

gib

le*

2003

0.47

5∗∗∗

0.47

9∗∗∗

0.46

2∗∗∗

0.49

1∗∗∗

0.49

8∗∗∗

0.47

7∗∗∗

0.32

2∗∗∗

(0.0

43)

(0.0

46)

(0.0

40)

(0.0

44)

(0.0

40)

(0.0

70)

(0.0

92)

Eli

gib

le*

2004

0.33

5∗∗∗

0.28

8∗∗∗

0.30

4∗∗∗

0.33

1∗∗∗

0.33

6∗∗∗

0.33

1∗∗∗

0.14

8(0

.050

)(0

.056

)(0

.051

)(0

.051

)(0

.052

)(0

.071

)(0

.121

)p

(β2003=

β2004)

0.00

20.

000

0.00

00.

001

0.00

40.

010

0.02

9

N5,

612

5,89

55,

291

6,21

66,

494

4,11

889

5

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.E

ach

colu

mn

rep

lica

tes

the

seco

nd

colu

mn

fro

mta

ble

2w

ith

the

sam

ple

lim

ited

toth

eid

enti

fied

sub

-gro

up

.

30

Tab

le4:

Do

ub

le-D

ose

Alg

ebra

,Fre

shm

anC

ou

rsew

ork

and

Pee

rs

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Mat

hA

cad

emic

Oth

erT

ota

lM

ean

pee

rS

Do

fD

ista

nce

toC

lass

cou

rses

cou

rses

cou

rses

cou

rses

skil

lsk

ill

mea

np

eer

size

(A)

Co

ho

rts

po

ole

d

Do

ub

le-d

ose

d0.

994∗

∗∗

-0.1

69∗

-0.6

69∗∗∗

0.15

6∗-1

9.40

1∗∗∗

-3.1

01∗∗∗

3.20

2∗∗∗

2.44

3∗∗∗

(0.0

22)

(0.0

96)

(0.0

88)

(0.0

89)

(1.4

40)

(0.7

61)

(1.2

16)

(0.7

76)

(B)

Co

ho

rts

sep

arat

ed

Do

ub

le-d

ose

d*

2003

0.97

9∗∗∗

-0.1

71∗

-0.6

75∗∗∗

0.13

3-1

9.44

3∗∗∗

-3.3

45∗∗∗

3.32

7∗∗∗

2.34

9∗∗∗

(0.0

22)

(0.0

89)

(0.0

87)

(0.0

82)

(1.4

11)

(0.8

28)

(1.0

47)

(0.8

48)

Do

ub

le-d

ose

d*

2004

1.02

0∗∗∗

-0.1

66-0

.657

∗∗∗

0.19

7∗-1

9.32

7∗∗∗

-2.6

71∗∗∗

2.97

9∗2.

609∗

∗∗

(0.0

24)

(0.1

23)

(0.1

03)

(0.1

17)

(2.1

16)

(0.8

64)

(1.7

32)

(0.8

83)

p(β

2003=

β2004)

0.00

70.

948

0.77

80.

388

0.95

10.

354

0.76

40.

737

Yat

thre

sho

ld1.

212

3.51

32.

255

6.98

051

.609

15.7

9310

.705

20.8

13N

11,5

0711

,507

11,5

0711

,507

11,5

0711

,501

11,5

0711

,507

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.E

ach

colu

mn

inea

chp

anel

rep

rese

nts

the

inst

rum

enta

lv

aria

ble

sre

gre

ssio

no

fth

eli

sted

ou

tco

me

on

sem

este

rso

fd

ou

ble

-do

seal

geb

ra,

wh

ere

do

ub

le-d

ose

isin

stru

men

ted

by

elig

ibil

ity

acco

rdin

gto

the

firs

t-st

age

giv

enin

tab

le2.

Pan

el(A

)p

oo

lsth

etw

otr

eatm

ent

coh

ort

s.P

anel

(B)

inte

ract

sth

ein

stru

men

tan

den

do

gen

ou

sre

gre

sso

rw

ith

coh

ort

ind

icat

ors

.B

elo

wp

anel

(B)

isth

ep

-val

ue

fro

man

F-t

est

of

the

equ

alit

yo

fth

etw

oli

sted

coef

fici

ents

.A

lso

list

edis

the

mea

nv

alu

eo

fea

cho

utc

om

eat

the

elig

ibil

ity

thre

sho

ld.

Co

lum

n(2

)in

clu

des

all

cou

rses

insc

ien

ce,

En

gli

shan

dso

cial

stu

die

s.C

olu

mn

(3)

incl

ud

esal

lco

urs

eso

ther

than

mat

h,s

cien

ce,E

ng

lish

and

soci

alst

ud

ies.

31

Table 5: The Impact of Double-Dose Algebra on Math Coursework

(1) (2) (3) (4) (5)A or B in Passed Passed Passed Math GPAalgebra algebra geom. trig. 10-12

(A) Cohorts pooled

Double-dosed 0.094∗∗ 0.047 -0.022 0.084∗∗ 0.143∗

(0.042) (0.046) (0.049) (0.041) (0.082)

(B) Cohorts separated

Double-dosed * 2003 0.095∗∗ 0.035 -0.010 0.075∗ 0.149∗

(0.038) (0.045) (0.043) (0.039) (0.079)Double-dosed * 2004 0.093∗ 0.067 -0.044 0.101∗ 0.132

(0.053) (0.057) (0.067) (0.055) (0.107)p (β2003

= β2004) 0.930 0.417 0.432 0.512 0.832

Y at threshold 0.138 0.624 0.576 0.544 1.398N 11,507 11,507 11,507 11,507 10,439

Notes: Heteroskedasticity robust standard errors clustered by initial high school are in parentheses (* p<.10 **p<.05 *** p<.01). Each column in each panel represents the instrumental variables regression of the listed outcomeon semesters of double-dose algebra, where double-dose is instrumented by eligibility according to the first-stagegiven in table 2. Panel (A) pools the two treatment cohorts. Panel (B) interacts the instrument and endogenousregressor with cohort indicators. Below panel (B) is the p-value from an F-test of the equality of the two listedcoefficients. Also listed is the mean value of each outcome at the eligibility threshold.

32

Tab

le6:

Th

eIm

pac

to

fD

ou

ble

-Do

seA

lgeb

rao

nM

ath

Ach

iev

emen

t

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Fal

l10

Fal

l10

Fal

l10

Fal

l11

Fal

l11

Fal

l11

Sp

rin

g11

To

ok

mat

hal

geb

rag

eom

etry

mat

hal

geb

rag

eom

etry

AC

Tm

ath

AC

T

(A)

Co

ho

rts

po

ole

d

Do

ub

le-d

ose

d0.

086

0.08

50.

008

0.15

9∗∗

0.14

9∗∗

0.10

10.

153∗

∗-0

.017

(0.0

67)

(0.0

70)

(0.0

82)

(0.0

64)

(0.0

59)

(0.0

85)

(0.0

71)

(0.0

39)

(B)

Co

ho

rts

sep

arat

ed

Do

ub

le-d

ose

d*

2003

0.07

70.

079

0.02

30.

145∗

∗0.

140∗

∗0.

053

0.14

0∗∗

0.00

4(0

.059

)(0

.064

)(0

.073

)(0

.059

)(0

.056

)(0

.077

)(0

.064

)(0

.036

)D

ou

ble

-do

sed

*20

040.

100

0.09

7-0

.017

0.18

4∗∗

0.16

5∗∗

0.18

2∗0.

173∗

∗-0

.055

(0.0

90)

(0.0

90)

(0.1

07)

(0.0

80)

(0.0

78)

(0.1

10)

(0.0

88)

(0.0

54)

p(β

2003=

β2004)

0.67

00.

726

0.50

40.

375

0.66

30.

043

0.45

50.

146

Yat

thre

sho

ld-0

.023

-0.1

07-0

.101

-0.0

76-0

.119

-0.1

31-0

.211

0.57

6N

8,21

08,

210

8,21

07,

428

7,42

87,

428

6,70

011

,507

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.E

ach

colu

mn

inea

chp

anel

rep

rese

nts

the

inst

rum

enta

lv

aria

ble

sre

gre

ssio

no

fth

eli

sted

ou

tco

me

on

sem

este

rso

fd

ou

ble

-do

seal

geb

ra,

wh

ere

do

ub

le-d

ose

isin

stru

men

ted

by

elig

ibil

ity

acco

rdin

gto

the

firs

t-st

age

giv

enin

tab

le2.

Pan

el(A

)p

oo

lsth

etw

otr

eatm

ent

coh

ort

s.P

anel

(B)

inte

ract

sth

ein

stru

men

tan

den

do

gen

ou

sre

gre

sso

rw

ith

coh

ort

ind

icat

ors

.B

elo

wp

anel

(B)

isth

ep

-val

ue

fro

man

F-t

est

of

the

equ

alit

yo

fth

etw

oli

sted

coef

fici

ents

.A

lso

list

edis

the

mea

nv

alu

eo

fea

cho

utc

om

eat

the

elig

ibil

ity

thre

sho

ld.

33

Tab

le7:

Th

eIm

pac

to

fD

ou

ble

-Do

seA

lgeb

rao

nE

du

cati

on

alA

ttai

nm

ent

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Gra

du

ated

Gra

du

ated

En

roll

ed,

En

roll

ed,

Par

t-ti

me,

Sem

este

rs,

Sem

este

rs,

hig

hsc

ho

ol

hig

hsc

ho

ol

any

two

-yea

rtw

o-y

ear

any

two

-yea

rin

4y

ears

in5

yea

rsco

lleg

eco

lleg

eco

lleg

eco

lleg

eco

lleg

e

(A)

Co

ho

rts

po

ole

d

Do

ub

le-d

ose

d0.

087∗

0.07

9∗∗

0.08

6∗∗∗

0.07

9∗∗

0.06

6∗∗∗

0.59

9∗∗

0.42

3∗∗

(0.0

47)

(0.0

39)

(0.0

33)

(0.0

36)

(0.0

26)

(0.2

39)

(0.1

67)

(B)

Co

ho

rts

sep

arat

ed

Do

ub

le-d

ose

d*

2003

0.10

5∗∗

0.09

4∗∗∗

0.09

4∗∗∗

0.07

0∗∗

0.05

7∗∗

0.65

6∗∗∗

0.32

4∗∗

(0.0

44)

(0.0

36)

(0.0

31)

(0.0

34)

(0.0

24)

(0.2

36)

(0.1

57)

Do

ub

le-d

ose

d*

2004

0.05

50.

052

0.07

20.

093∗

0.08

3∗∗

0.49

80.

598∗

∗

(0.0

60)

(0.0

52)

(0.0

46)

(0.0

48)

(0.0

37)

(0.3

29)

(0.2

41)

p(β

2003=

β2004)

0.17

30.

238

0.51

20.

514

0.35

80.

559

0.15

7

Yat

thre

sho

ld0.

509

0.56

30.

290

0.15

90.

077

1.79

50.

873

N11

,507

11,5

0711

,507

11,5

0711

,507

11,5

0711

,507

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.E

ach

colu

mn

inea

chp

anel

rep

rese

nts

the

inst

rum

enta

lv

aria

ble

sre

gre

ssio

no

fth

eli

sted

ou

tco

me

on

sem

este

rso

fd

ou

ble

-do

seal

geb

ra,

wh

ere

do

ub

le-d

ose

isin

stru

men

ted

by

elig

ibil

ity

acco

rdin

gto

the

firs

t-st

age

giv

enin

tab

le2.

Pan

el(A

)p

oo

lsth

etw

otr

eatm

ent

coh

ort

s.P

anel

(B)

inte

ract

sth

ein

stru

men

tan

den

do

gen

ou

sre

gre

sso

rw

ith

coh

ort

ind

icat

ors

.B

elo

wp

anel

(B)

isth

ep

-val

ue

fro

man

F-t

est

of

the

equ

alit

yo

fth

etw

oli

sted

coef

fici

ents

.A

lso

list

edis

the

mea

nv

alu

eo

fea

cho

utc

om

eat

the

elig

ibil

ity

thre

sho

ld.

34

Tab

le8:

Ro

bu

stn

ess

Ch

eck

s

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Pas

sed

Fal

l10

Fal

l11

Sp

rin

g11

Gra

du

ated

En

roll

ed,

Sem

este

rs,

Sem

este

rs,

fres

hm

anm

ath

mat

hA

CT

mat

hh

igh

sch

oo

lan

yan

ytw

o-y

ear

alg

ebra

sco

resc

ore

sco

rein

5y

ears

coll

ege

coll

ege

coll

ege

(A)

Ban

dw

idth

=10

No

con

tro

ls0.

032

0.06

90.

137∗

∗0.

141∗

0.06

00.

067∗

0.46

7∗0.

368∗

∗

(0.0

46)

(0.0

71)

(0.0

64)

(0.0

74)

(0.0

42)

(0.0

37)

(0.2

55)

(0.1

74)

+d

emo

gra

ph

icco

ntr

ols

0.04

70.

086

0.15

9∗∗

0.15

3∗∗

0.07

9∗∗

0.08

6∗∗∗

0.59

9∗∗

0.42

3∗∗

(0.0

46)

(0.0

67)

(0.0

64)

(0.0

71)

(0.0

39)

(0.0

33)

(0.2

39)

(0.1

67)

+h

igh

sch

oo

lfi

xed

effe

cts

0.04

50.

044

0.12

9∗∗

0.12

7∗0.

071∗

0.07

3∗∗

0.48

6∗∗

0.36

8∗∗

(0.0

45)

(0.0

63)

(0.0

62)

(0.0

73)

(0.0

39)

(0.0

34)

(0.2

36)

(0.1

62)

(B)

Oth

erb

and

wid

ths

Ban

dw

idth

=8

0.09

9∗0.

135∗

0.21

0∗∗∗

0.17

1∗∗

0.09

9∗∗

0.07

3∗0.

538∗

0.34

6(0

.052

)(0

.073

)(0

.075

)(0

.072

)(0

.048

)(0

.042

)(0

.288

)(0

.221

)B

and

wid

th=

60.

069

0.09

40.

291∗

∗∗

0.16

7∗0.

153∗

∗0.

100∗

∗0.

608∗

0.17

5(0

.071

)(0

.084

)(0

.103

)(0

.095

)(0

.069

)(0

.047

)(0

.334

)(0

.264

)B

and

wid

th=

40.

045

0.06

90.

306∗

∗0.

184

0.14

20.

117∗

0.81

30.

352

(0.0

79)

(0.1

26)

(0.1

37)

(0.1

30)

(0.0

88)

(0.0

66)

(0.5

11)

(0.3

26)

(C)

Pla

ceb

ote

sts

Th

resh

old

at40

thp

erce

nti

le-0

.000

0.03

3-0

.003

0.02

50.

036∗

0.00

5-0

.010

0.04

2(0

.017

)(0

.032

)(0

.030

)(0

.025

)(0

.021

)(0

.016

)(0

.100

)(0

.067

)T

hre

sho

ldat

60th

per

cen

tile

0.03

1∗0.

027

-0.0

11-0

.029

0.02

10.

021

0.14

80.

115

(0.0

18)

(0.0

28)

(0.0

31)

(0.0

30)

(0.0

22)

(0.0

21)

(0.1

59)

(0.0

94)

Ear

lier

coh

ort

s0.

019

0.01

8-0

.016

-0.0

010.

013

-0.0

07-0

.064

0.00

8(0

.014

)(0

.030

)(0

.030

)(0

.029

)(0

.020

)(0

.015

)(0

.116

)(0

.066

)

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.P

anel

(A)

fits

stra

igh

tli

nes

on

eith

ersi

de

of

the

thre

sho

ld,

usi

ng

ab

and

wid

tho

f10

per

cen

tile

s.P

anel

(B)

rep

lica

tes

the

seco

nd

row

of

pan

el(A

)b

ut

usi

ng

oth

erb

and

wid

ths.

Pan

el(C

)in

clu

des

thre

ep

lace

bo

test

s,ea

cho

fw

hic

hre

pli

cate

sth

ere

du

ced

form

ver

sio

no

fth

ese

con

dro

wo

fp

anel

(A).

Th

efi

rst

and

seco

nd

pla

ceb

ote

sts

use

the

40th

and

60th

per

cen

tile

sas

the

do

ub

le-d

ose

thre

sho

ld,w

hil

eth

eth

ird

use

sth

eu

ntr

eate

d20

01an

d20

02co

ho

rts.

35

Tab

le9:

Het

ero

gen

eity

by

Aca

dem

icS

kil

l

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Pas

sed

Fal

l10

Fal

l11

Sp

rin

g11

Gra

du

ated

En

roll

ed,

Sem

este

rs,

Sem

este

rs,

fres

hm

anm

ath

mat

hA

CT

mat

hh

igh

sch

oo

lan

yan

ytw

o-y

ear

alg

ebra

sco

resc

ore

sco

rein

5y

ears

coll

ege

coll

ege

coll

ege

(A)

RD

,ov

eral

l

Do

ub

le-d

ose

d0.

047

0.08

60.

159∗

∗0.

153∗

∗0.

079∗

∗0.

086∗

∗∗

0.59

9∗∗

0.42

3∗∗

(0.0

46)

(0.0

67)

(0.0

64)

(0.0

71)

(0.0

39)

(0.0

33)

(0.2

39)

(0.1

67)

Yat

thre

sho

ld0.

633

-0.0

28-0

.067

-0.1

920.

576

0.29

01.

837

0.89

8N

11,5

078,

210

7,42

86,

700

11,5

0711

,507

11,5

0711

,507

(B)

RD

,by

read

ing

skil

l

Do

ub

le-d

ose

d*

po

or

read

er0.

085

0.08

80.

184∗

∗0.

219∗

∗∗

0.12

7∗∗∗

0.13

3∗∗∗

0.82

0∗∗∗

0.54

8∗∗∗

(0.0

52)

(0.0

81)

(0.0

73)

(0.0

75)

(0.0

48)

(0.0

41)

(0.2

82)

(0.1

98)

Do

ub

le-d

ose

d*

go

od

read

er0.

009

0.08

00.

136∗

∗0.

093

0.03

10.

039

0.37

90.

299∗

(0.0

43)

(0.0

63)

(0.0

68)

(0.0

77)

(0.0

40)

(0.0

36)

(0.2

70)

(0.1

73)

p(β

below=

βabove)

0.01

80.

884

0.39

20.

015

0.01

50.

019

0.09

80.

112

N11

,507

8,21

07,

428

6,70

011

,507

11,5

0711

,507

11,5

07

(C)

DD

,ov

eral

l

Do

ub

le-d

ose

d0.

064∗

∗∗

0.07

6∗∗∗

0.02

40.

009

0.01

6(0

.019

)(0

.027

)(0

.029

)(0

.032

)(0

.013

)

Yfo

rb

elo

w*b

efo

re0.

551

-0.5

20-0

.512

-0.5

360.

457

N79

,394

53,3

9749

,653

44,5

8579

,394

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.P

anel

(A)

con

tain

sth

ere

gre

ssio

nd

isco

nti

nu

ity

resu

lts

fro

mp

rio

rta

ble

s,b

elo

ww

hic

his

the

mea

nv

alu

eo

fea

cho

utc

om

eat

the

elig

ibil

ity

thre

sho

ld.

Pan

el(B

)se

par

ates

stu

den

tsb

y8t

hg

rad

ere

adin

gsc

ore

,in

tera

ctin

gb

oth

the

inst

rum

ent

and

end

og

eno

us

reg

ress

or

wit

hre

adin

gsk

ill

gro

up

s.B

elo

wea

chco

effi

cien

tin

pan

el(B

)is

the

p-v

alu

efr

om

anF

-tes

to

fth

eeq

ual

ity

of

the

two

coef

fici

ents

.P

anel

(C)

con

tain

sin

stru

men

tal

var

iab

les

reg

ress

ion

su

sin

gth

ed

iffe

ren

ce-i

n-d

iffe

ren

cesp

ecifi

cati

on

des

crib

edin

the

tex

t.B

elo

wp

anel

(C)

isth

em

ean

val

ue

of

each

ou

tco

me

for

stu

den

tsb

elo

wth

eel

igib

ilit

yth

resh

old

inth

e20

01an

d20

02co

ho

rts.

36

Tab

le10

:S

pil

lov

ers

on

toO

ther

Su

bje

cts

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Fal

l10

AC

TF

all

10A

CT

Pas

sed

No

nm

ath

No

nm

ath

ver

bal

ver

bal

scie

nce

scie

nce

chem

istr

yG

PA

,9G

PA

,10-

12

(A)

Co

ho

rts

po

ole

d

Do

ub

le-d

ose

d0.

106∗

0.18

5∗∗∗

-0.0

010.

039

0.09

9∗∗

-0.0

430.

150∗

(0.0

63)

(0.0

65)

(0.0

83)

(0.0

85)

(0.0

44)

(0.0

80)

(0.0

85)

(B)

Co

ho

rts

sep

arat

ed

Do

ub

le-d

ose

d*

2003

0.10

2∗0.

173∗

∗∗

0.05

60.

074

0.09

2∗∗

0.00

60.

152∗

∗

(0.0

57)

(0.0

59)

(0.0

76)

(0.0

79)

(0.0

42)

(0.0

69)

(0.0

75)

Do

ub

le-d

ose

d*

2004

0.11

10.

205∗

∗-0

.101

-0.0

150.

111∗

-0.1

310.

145

(0.0

82)

(0.0

92)

(0.1

11)

(0.1

07)

(0.0

58)

(0.1

13)

(0.1

19)

p(β

2003=

β2004)

0.86

00.

650

0.02

00.

221

0.66

90.

070

0.92

8

Yat

thre

sho

ld-0

.069

-0.1

23-0

.047

-0.1

680.

434

1.92

81.

786

N8,

228

6,70

08,

184

6,69

911

,507

11,5

0710

,509

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.E

ach

colu

mn

inea

chp

anel

rep

rese

nts

the

inst

rum

enta

lv

aria

ble

sre

gre

ssio

no

fth

eli

sted

ou

tco

me

on

sem

este

rso

fd

ou

ble

-do

seal

geb

ra,

wh

ere

do

ub

le-d

ose

isin

stru

men

ted

by

elig

ibil

ity

acco

rdin

gto

the

firs

t-st

age

giv

enin

tab

le2.

Pan

el(A

)p

oo

lsth

etw

otr

eatm

ent

coh

ort

s.P

anel

(B)

inte

ract

sth

ein

stru

men

tan

den

do

gen

ou

sre

gre

sso

rw

ith

coh

ort

ind

icat

ors

.B

elo

wp

anel

(B)

isth

ep

-val

ue

fro

man

F-t

est

of

the

equ

alit

yo

fth

etw

oli

sted

coef

fici

ents

.A

lso

list

edis

the

mea

nv

alu

eo

fea

cho

utc

om

eat

the

elig

ibil

ity

thre

sho

ld.

37

Tab

le11

:C

han

nel

s

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Pas

sed

Fal

l10

Fal

l11

Sp

rin

g11

Gra

du

ated

En

roll

ed,

Sem

este

rs,

Sem

este

rs,

fres

hm

anm

ath

mat

hA

CT

mat

hh

igh

sch

oo

lan

yan

ytw

o-y

ear

alg

ebra

sco

resc

ore

sco

rein

5y

ears

coll

ege

coll

ege

coll

ege

(A)

By

ov

eral

lco

mp

lian

ce

Do

ub

le-d

ose

d0.

052

0.09

10.

160∗

∗0.

151∗

∗0.

078∗

0.09

0∗∗

0.63

8∗∗

0.44

2∗∗

(0.0

46)

(0.0

69)

(0.0

65)

(0.0

72)

(0.0

40)

(0.0

35)

(0.2

53)

(0.1

77)

Do

ub

le-d

ose

d*

Gu

idel

ine

com

pli

ance

-0.1

04-0

.176

∗-0

.023

0.07

20.

020

-0.0

93-0

.823

∗-0

.395

(0.0

80)

(0.1

06)

(0.0

79)

(0.1

03)

(0.0

78)

(0.0

79)

(0.4

99)

(0.3

95)

(B)

By

each

gu

idel

ine

Do

ub

le-d

ose

d0.

052

0.09

30.

156∗

∗0.

153∗

∗0.

080∗

0.08

9∗∗

0.64

4∗∗

0.42

7∗∗

(0.0

47)

(0.0

69)

(0.0

67)

(0.0

72)

(0.0

41)

(0.0

36)

(0.2

56)

(0.1

79)

Do

ub

le-d

ose

d*

Co

nse

cuti

ve

per

iod

s-0

.142

∗0.

070

0.08

30.

155∗

-0.0

690.

009

-0.4

25-0

.572

∗

(0.0

78)

(0.0

80)

(0.0

73)

(0.0

93)

(0.0

75)

(0.0

70)

(0.4

82)

(0.3

11)

Do

ub

le-d

ose

d*

Sam

ete

ach

er0.

013

-0.1

67∗∗

-0.0

66-0

.091

-0.0

29-0

.072

-0.6

250.

260

(0.1

04)

(0.0

81)

(0.1

22)

(0.1

31)

(0.1

11)

(0.0

95)

(0.6

71)

(0.3

62)

Do

ub

le-d

ose

d*

Ex

ten

to

ftr

ack

ing

0.10

5-0

.141

-0.1

07-0

.030

0.25

8-0

.047

0.74

1-0

.062

(0.1

69)

(0.1

92)

(0.1

97)

(0.2

67)

(0.2

05)

(0.1

95)

(1.2

47)

(0.6

64)

N11

,507

8,21

07,

428

6,70

011

,507

11,5

0711

,507

11,5

07

No

tes:

Het

ero

sked

asti

city

rob

ust

stan

dar

der

rors

clu

ster

edb

yin

itia

lh

igh

sch

oo

lar

ein

par

enth

eses

(*p<

.10

**p<

.05

***

p<

.01)

.B

oth

pan

els

rep

lica

teth

em

ain

spec

ifica

tio

ns

fro

mp

rio

rta

ble

sb

ut

inte

ract

bo

thth

ein

stru

men

tan

den

do

gen

ou

sre

gre

sso

rw

ith

the

giv

enm

easu

res

of

gu

idel

ine

com

pli

ance

.T

ho

sem

easu

res

of

gu

idel

ine

com

pli

ance

are

also

con

tro

lled

for

dir

ectl

yb

ut

no

tsh

ow

n.

38

Figure A.1: Distribution of 8th Grade Math Scores

0.0

2.0

4.0

6F

ract

ion

30 40 50 60 708th grade math percentile

(A) 2003 cohort

0.0

2.0

4.0

6F

ract

ion

30 40 50 60 708th grade math percentile

(B) 2004 cohort

39