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A new twist in the educational tracking debate
Economics of Education Review 22 (2003) 307–315 www.elsevier.com/locate/econedurev
A new twist in the educational tracking debate Ron Zimmer ∗ RAND Corporation, 1700 Main Street, Santa Monica, CA 90407-2138, USA Received 5 December 2000; accepted 18 June 2002
Abstract Recently, the practice of tracking has been receiving more attention by both educators and researchers and some have questioned the policy merit. One of the strongest arguments against tracking is that it creates homogenous classes according to ability and, therefore, reduces the positive spillover effect referred to as a peer effect. While peer effects have been found to be an important input into the production of education no study has specifically examined whether these effects are more or less prevalent in classes where tracking occurs. Utilizing individual student level data, this current research examines whether the peer effect occurs in cases in which tracking is present. The results suggest that the use of tracking diminishes the impact peers have on student achievement for low- and average-ability students while the peer effect is unaffected by tracking for high-ability students. 2003 Elsevier Science Ltd. All rights reserved. JEL classification: I21 Keywords: Educational economics
Recently, the practice of tracking has been receiving more attention by both educators and researchers, and some have questioned the policy’s merit. In the education arena, those who advocate tracking argue that all students, regardless of ability, would learn more in a tracked class relative to a nontracked class (Hallinan, 1994). In tracked classes the teacher can tailor the curriculum to the ability level of the students, thus creating the optimal level of educational gains for all students. However, opponents argue against tracking for three primary reasons. First, tracking leads to a different set of resources being allocated to high-tracked versus low-
tracked classes (Oakes, 1990).1 Second, tracking breeds social inequities as minority and low-income groups are over-represented in low-track and under-represented in high-track classes (Braddock & Dawkins, 1993; Gorman, 1987; Oakes, 1985, 1990). Third, tracking creates homogenous classes according to ability, therefore reducing the positive spillover effect, referred to as a peer effect (Betts & Shkolnik, 2000). While further research is needed to address the first two issues, this current research examines the effect tracking has on the peer effect. That is, does tracking diminish the positive spillover effect from high-ability students to low-ability students? In addition, the effect from tracking on student achievement is examined while holding the peer level constant. The literature increasingly suggests there are differen-
∗ Corresponding author. Tel.: +1 310 393 0411; fax: +1 310 451 7059. E-mail address: [email protected] (R. Zimmer).
1 However, Betts and Shkolnik (2000) found that neither class size nor teacher characteristics vary much whether the student is placed in a tracked or an untracked class.
1. Introduction
0272-7757/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0272-7757(02)00055-9
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tial effects from tracking for students of different abilities (Argys, Rees & Brewer, 1996; Hoffer, 1992; Kerckhoff, 1986)2. However, this literature has failed to determine whether this effect is from curriculum and resource effects or a differential peer effect from the organizational structure of grouping students. In the past, researchers have found strong support for a peer effect in the production of education (Hanushek, Kain, Markman, & Rivkin, 2001; Henderson, Mieskowski, & Savageau, 1978; Summers & Wolfe, 1977; Zimmer & Toma, 2000; Hoxby, 2000). However, no study has examined whether these peer effects are more or less prevalent in classes where tracking occurs and what effect they are having on student achievement in tracked classes3. Zimmer and Toma (2000) interacted the variance of ability in a classroom with the mean ability (peer-effect variable) of students and found that greater variance within a classroom actually reduces the peer effect. Therefore, it could be the case that peer effects are less prevalent in schools that are not tracked as opposed to schools that are. That is, students of different ability levels who are mixed together in non-tracked schools would not be able to interact effectively to create the peer effect. This current research adds to the literature by examining the impact tracking has on the peer effect by including an interaction term. The results suggest that the use of tracking diminishes the impact peers have on student achievement for low and average-ability students while the peer effect is unaffected by tracking for high-ability students. A secondary result suggests that the institutional process of tracking, when controlling for the peer level, has no effect on high-ability students,
2 Kerchoff (1986) examined grouped versus non-grouped students in British schools and found that high-ability students learnt more than non-grouped students while low-grouped students learnt less. Hoffer (1992) examined grouped versus nongrouped students using the Longitudinal Study of America (LSAY) and found that tracking had no effect for averageability students, weak positive effects for high-ability students, and strong negative effects for low-ability students. Argys et al. (1996), using the National Longitudinal Study of 1988 (NELS), suggest that tracking creates educational gains for high-ability students, whereas low-ability students experience educational losses. In contrast, Betts and Shkolnik (2000) suggest only weak differential effects between ability groups, with tracking having no effect on low-ability students but a small positive effect for high-ability students and a small negative effect on averageability students. In a more recent study, Figlio and Page (2002), using the NELS data, support the finding from Betts and Shkolnik as they control for the endogeneity of track placement and finds no evidence that low-ability students are hurt by tracking. 3 White and Kane (1995) suggest that one of the main shortcomings of the existing literature is that it does “not separate the effects of course content and instruction from the effects of homogenous grouping” (p. viii).
whereas, surprisingly, the institutional practice of tracking, when controlling for the peer level, has a positive effect on low-and average-ability students. Only if the reduction in the peer effect is combined with the effect from tracking does the positive effect from low-ability students wash out. In other words, the loss of exposure to more able students offsets the potential of tracking to improve educational performance for the low-ability students.
2. Data To examine the impact tracking has on the peer effect, and ultimately, the impact peers and tracking have on student achievement, a data set with characteristics of the individual students is necessary, including whether or not the student was in a tracked classroom. One such data set is from the second study (SIMS) from International Association for the Evaluation of Educational Achievement (IEA)4. The IEA, through carefully constructed surveys, collects family, school, classroom, and peer characteristics of individual students for the purpose of cross-country comparison, including the US5. This data set includes characteristics of the students’ families and teachers, the organizational structure of the schools, including whether or not the classes are tracked, and preand post-year mathematics test scores for individual students, which allow for a value-added estimate6. In the data, tracking is designated as a dichotomous variable (1 if there is tracking in the classroom and 0 if there is not) and is interacted with a variable representing the ability of peers (the mean test score of classmates). If the resulting coefficient is negative for the interaction term, then it can be concluded that tracking diminishes the peer 4 The most recent IEA data set was not used because it did not have the pre-school-year test score to do a value-added model. 5 In 1981, the IEA carefully selected a sample of schools and administered a broad array of questionnaires answered by students, teachers and administrators (Robitalle & Garden, 1989). The schools selected for the survey were designed to reflect the socioeconomic characteristics of each country. The students in the analysis were in the eighth grade.. The survey asked questions in regard to the student’s family, schools, teachers, and peers within the classrooms (Robitalle & Garden, 1989) The IEA went to great lengths to ensure the quality of the data through rigorous quality control steps including creating a manual for inputting the data and auditing them once they were collected. 6 The mathematics tests, known as SIMS (Second International Mathematics Study), were developed through the input of mathematics experts and measurement specialists and the tests were to reflect the curriculum of each country. The tests went through a review process that included several pilot tests before being implemented.
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effect, whereas a positive coefficient suggests that the peer effect is enhanced. Using the US public school portion of the data, this study examines the impact of tracking on the peer effect and student achievement for high-, average-, and low-ability students using a standard education production model. Together, the information provided by the surveys and the test scores renders a robustness to the results.
3. Model To examine the effect from tracking, the peer effect, and the interaction of tracking with the peer effect, I followed a standard education production function model employed by Betts and Shkolnik (2000). The model examines the impact of various inputs, including family and school resources, the student’s peers, and student’s individual characteristics, including a proxy for ability7. The value-added model is displayed as follows8: At,j ⫽ f(Ft,j,St,j,Pt,j,At-1).
(1)
By using the model represented above, the valueadded impacts of various inputs are measured as student achievement for the period t–1 to t9. In this formulation, the effects of all prior inputs are captured in achievement from the previous period, or At-110. Formally, output is measured by individual student j’s educational achievement (At) in the current period as a function of inputs that include student j’s family characteristics (Ft) over period At-1 to At, school inputs (St) for student j over
7 Conceptually, the ideal model (as displayed below) includes family inputs (F), school inputs (S), environmental inputs of peers (P), and the innate ability of the student (I) in period t:At,j = f(Ft,j,St,j,Pt,j,It,j).However, it is difficult to gain an accurate proxy for the innate ability of the students (Hanushek, 1979) and the lack of an ability variable is generally assumed to bias upward the effects of family background on achievement. Therefore, researchers often use a value-added model to mitigate the negative effect of not including such a variable. By estimating the value-added model, the biases are minimized because only the growth effect of innate ability is omitted. 8 Argys et al. suggests that “if there are unobservable student or school characteristics that affect both achievement and track placement, than any association between achievement and tracking may simply be due to these characteristics” (1996, p.624). To control for the selection, Argys et al. model the process through which students are assigned to a particular track and then include these selectivity corrections in the main achievement model. Unfortunately, I was unable to find a suitable instrument in the data set for this present analysis. 9 Betts and Shkolnik (2000) argue that it is critical to control for the student’s initial achievement if accurate estimates are to be obtained. 10 In this case, achievement in all other periods, such as t–1, is a function of the cumulative inputs in t–1.
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period At-1 to At, peer influence (Pt) of student j over period At-1 to At, and student j’s achievement in the previous period (At-1)11. To apply this model I measure achievement, both in the current and previous period, as the number of questions answered correctly by student j on a mathematics test with a possible range of 0 to 40. The achievement level (At) is the students’ end-of-the-year test score and is labeled POSTTEST, while the beginning-of-the-year test (At-1) is labeled PRETEST12. Other inputs include a vector of family inputs (F) and a vector of school inputs (S), including the characteristics of the student’s teacher and school13. (A complete list of variables with their summary statistics is included in Table A1 in the appendix.) Two variables of primary importance in this current research are the tracking and peer variables14. The tracking variable is simply a dummy variable represented by a 1 when tracking is used in the classroom and a 0 when it is not. The peer variable is defined as the mean test score of students in a classroom and is represented by the mean of the scores at the beginning of the school year of all students in the observed student’s classroom15. As consistent with the literature, I also square the mean score to capture nonlinear as well as linear effects on fellow students16. This peer variable is also interacted with the tracking variable. If tracking is utilized within the classroom, then tracking is equal to 1 and the coefficient of the interaction term represents the impact tracking has on the peer effect. Because the peer level is also included, the impact of tracking on the peer effect is measured while holding the peer level constant. A negative and significant coefficient suggests that tracking diminishes the peer effect, while a positive and significant coefficient suggests that tracking enhances the peer
11 For further information on the benefits of using the valueadded model, see Boardman and Murnane (1979). 12 It should be noted that students that did not appear in both the pre- and post-test scores were deleted from our sample. 13 Argys et al. (1996) suggest that it is critical to control teacher and school characteristics if low-ability tracks and highability tracks are systematically assigned different resources. 14 It should be noted that the model suffers from endogenity problems. Part of the purpose of tracking is to put students in more homogenous classes. Therefore, the peers of a student are partially a function of whether a student is in a tracked class. No suitable instrument could be found to perform an IV model to correct for this problem. 15 This is one of the definitions used by Zimmer and Toma (2000) for peers. 16 Henderson et. al. (1978) along with Zimmer and Toma (2000) squared the mean effect to capture a peer effect that is increasing at a decreasing rate (i.e., the squared term is expected to be negative).
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effect. Finally, an insignificant coefficient suggests that tracking has no effect on the peer effect. To examine the effect from tracking, the peer variable, and the interaction of tracking and the peer variable, four samples of data from US public schools were utilized. The first sample uses the full data set of students as the observations. Therefore, the first model examines the impact on achievement from tracking, peers, and the interaction of tracking with the peer effect, along with the student’s other school, family, and the past inputs. In addition to using all students in the sample, three additional samples are used for analysis: one that is restricted to high-ability students, one that is restricted to low-ability students, and one that is restricted to averageability students17. Therefore, the second estimated model uses all the same inputs, but restricts the data to include only high-ability students. A student is classified as highability if he or she is placed in the upper 20th percentile of all students across the nation on the math test at the beginning of the year. Again, the peer-effects literature assumes that a student placed in a class with a higher mean peer group will experience a positive achievement effect. A third model uses the same inputs but restricts the data to include only low-ability students. A student is classified as low-ability if he or she placed in the lower 20th percentile of all students across the nation on the same mathematics test. A fourth model uses the same inputs but restricts the sample to average-ability students. A student is classified as average-ability if he or she scored between the upper 20th percentile and the lower 20th percentile on the same mathematics test. As in Betts and Shkolink (2000) and Zimmer and Toma (2000), each model includes variables that represent family and school inputs. The variables that represent the family socioeconomic characteristics (F) of a student is the occupation level of the father and the mother and the education level of the father and the mother18. The variables that represent school inputs (S) include the size of the classroom, the general experience of the teacher, the math experience of the teacher, the teacher’s pedagogy training, the number of classes of mathematics training for the teacher, the teacher’s age, and the teacher’s gender. Finally, the student’s inputs include the student prior achievement and student’s gender. The following section will present the results of the
model focusing primarily on the interaction term, the peer effect, and tracking.
17 The restricting of data in this way is consistent with the work of Kerckhoff (1986); Hoffer (1992), and Argys et al. (1996). 18 The variables for the education level of the father and mother are limited education measures defined as a binary variable that is equal to one when the father’s or mother’s education is at least secondary, respectfully. Because of the limitations of these variables, readers should interpret the results with caution.
19 The sensitivity of these results is tested by alternatively defining high- average- and low-ability students. In the first alternative, high-ability is defined as students placed in the top 25th percentile, average-ability is defined as students placed in the middle 50th percentile, and low-ability is defined as students placed in the lower 25th percentile. In the second alternative, high-ability is defined as students placed in the top 15th percentile, average-ability is defined as students placed in the middle
4. Results Using a model specification similar to that of Betts and Shkolnik (2000), I estimate the effect from tracking, peers, and the interaction of tracking and peers from four segmented data sets. Table 1 displays the results. Column one of the table lists the variables, while columns two and three display the estimated coefficient and tscores of each of the variables from the full set of data. Columns four and five display the estimated coefficient and t-scores of the model that restricts the data to only high-ability students. Columns six and seven display the estimated coefficient and t-scores of the model that restricts the data to only low-ability students. Finally, columns eight and nine display the estimated coefficient and t-scores of the model that restricts the data to only average-ability students. In all the models, the tracking variable is a dichotomous dummy variable, 1 if tracking is present in the classroom, 0 if it is not. The peer variable is the average of beginning-of-year test scores of all students in individual j’s classroom. The interaction term is the interaction of the tracking variable and the peer variable. Included in each of the models, in addition to the peer variable, tracking variable, and interaction term, are family and school inputs, along with the student’s gender and prior achievement (as proxy for past inputs). The dependent variable is the achievement score at the end of the school year. For this study, the primary focus is on the coefficient for the interaction term. Also of great interest are the coefficients of the tracking and peer variable and the total effect from tracking, including the mechanism of tracking, the interaction, and peer effects. First, let us focus on the interaction term. For the full sample, the averageability students, and the low-ability students, the coefficients are negative and significant. However, when the data are restricted to only high-ability students, the coefficient is insignificant, suggesting that tracking neither enhances nor diminishes the peer effect for high-ability students. Therefore, the results suggest that tracking diminishes the positive peer effect for low- and averageability students while neither enhancing nor diminishing it for high-ability students19.
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Focusing on the secondary points, let us examine the results of the peer variable and tracking. To interpret the magnitude (and significance) of the coefficient for the peer effect and tracking requires taking into account the coefficient and joint significance of all variables that include multiplicatives of the tracking and peer variables. More formally, our model is At,j ⫽ B0 ⫹ B1(Pt,j) ⫹ B2(P2t,j) ⫹ B3(P∗t,jTt,j)
(2)
⫹ B4(Tt,j) ⫹ .... where At,j is the post-year level of student achievement for student j in period t and is a function of the student j’s peers (Pt,j or the mean pre-year test score of classroom for student j), student j’s peer squared (P2t,j or the square of the mean pre-year test score of classroom for student j), the interaction term (the mean pre-year test score of the classroom, Pt,j, for student j times the dichotomous tracking variable, Tt,j, for student j), the tracking variable (Tt,j or the dichotomous 1/0 dummy variable for tracking within the classroom for student j), and all other explanatory variables. The estimated peer effect is the partial derivative of At,j with respect to the peer variable (Pt,j) and is ∂A / ∂P = B1 + 2B2(Pt,j) + B3(Tt,j), which is the estimated effect from student j’s peers. The estimated tracking effect is the partial derivative of At,j with respect to tracking (Tt,j) and is ∂A / ∂T = B3(Pt,j) + B4, which is the estimated effect from tracking. To test the significance of the peer effect and tracking, the resulting beta coefficients from the partial derivatives must be jointly tested. Therefore, an F-test was conducted on the joint significance of the peer variable and tracking. Focusing first on the peer effect, the F-test indicates significance at the 0.001 level for each of the models, except the model that restricts the data to high-ability students. In other words, the higher the mean test scores of a student’s classmates, the better the average- and low-ability students will perform20. It is surprising that this effect does not show up for the high-ability students as well. However, Argys et al. (1996) suggest that in many cases, placement in high-tracks is not strictly based upon ability, but on other factors including parental influence on the decision process. In other words, highability students may not always be placed exclusively with high-ability students and therefore, have less of a chance of gaining a positive peer effect.
70th percentile, and low-ability is defined as students placed in the lower 15th percentile. In both cases, the interaction term remained insignificant for high-ability students and negative and significant for average- and low-ability students. This suggests this robustness of the results. These results are available upon request from the author. 20 These results are also true for the alternative definitions of high-, average-, and low-ability as specified in footnote 21. Again, results are available upon request from the author.
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As noted above, to interpret the results of the peer effect, the coefficient of all variables that include the peer variable must be considered, including the coefficient of the interaction term when tracking is present (the interaction term does not have to be considered when tracking is not present because the variable takes on a value of zero). For instance, in the full sample, the peer effect yields an estimated coefficient of 0.22 when tracking is present and 0.38 when it is not21. The coefficient implies that, on average, increasing the class mean on the beginning-of-year scores by one point increases a representative student’s end-of-year math test score by approximately 0.22 points when tracking is present and 0.38 when it is not. As with the peer coefficient, the F-test is employed to test the significance of the tracking coefficient. In these models, tracking is significant for the full sample and the average- and low-ability students, and insignificant for the models in which the data set was restricted to highability students22. The results from the full sample of data and the data restricted to average-ability students suggest a negative effect from tracking, net of the peer effect. However, unlike previous research (which measured the total effect from tracking), the tracking variable, on the margin, has a positive coefficient for low- and average-ability students when controlling for the peer level. The total tracking effects, including the peer effect and the effect from the mechanism of tracking (e.g., curriculum, curriculum pace, etc.) are illustrated in Figs. 1–3. Each figure compares student test scores in a tracked versus a non-tracked class at different peer levels. Fig. 1 suggests that high-ability students are better off in a tracked class at higher peer levels and better in a nontracked class at lower levels. This fact is due to the offsetting effects from tracking and peer variables. In assessing the overall tracking effect, the estimates suggest that the negative effect from the mechanism of tracking (–4.05) is countered by the positive effect of peers (0.38 and 0.2). The figure indicates that at the approximate mean peer level of 18, as measured by the pre-test score, the positive peer effect dominates the
21 For the full sample, the total coefficient of the peer effect is derived by adding the coefficient of the peer variable (0.69) plus two times the coefficient of the squared peer variable (0.01) times the mean value of the peer variable (15.81) plus the coefficient of the interaction term (-0.15). The coefficient of the interaction term is added only when tracking is present. 22 The alternative cases have similar results except for highability students (defined as the top 15th percentile) and averageability students (defined as the middle 50th percentile). In the high-ability case, tracking has a negative and significant effect. In the average-ability case, tracking becomes insignificant when defined more narrowly. Once again, results are available upon request from the author.
b
a
0.28 0.05 –0.004 0.03 –0.05 0.44 7.78
2.72 1.02 –0.82 0.56 –2.33a 2.83a –2.67a 27.96a 6.96a 3706 0.6536
0.50 0.01 –0.002 0.01
–0.03 0.51 –2.80
–4.05 0.38 –0.01 0.20 0.26 –0.28 –0.65 1.86 0.67 –0.04 –0.03
2.61a 6.21a –3.14a –2.96a 1.51 0.74 1.10 1.22 59.66a 1.80b –0.56
–2.06a 1.35 2.02a 0.79 1.67 763 0.2966
0.79 1.90a –0.95 0.95
–1.30 0.97 –1.01 1.15 0.75 –0.79 –0.82 1.92b 14.66a –1.00 –0.95
High-ability students only Est. coefficient t-scores
1.98 0.69 –0.01 –0.15 0.28 0.14 0.37 0.46 0.85 0.04 –0.01
Full sample Est. coefficient t-scores
Indicates significance at the 0.05 level. Indicates significance at the 0.10 level.
Tracking Peer variable Peer variable squared Interaction Father’s occupation Mother’s occupation Father’s education Mother’s education Pre-Test Teacher’sexperience Teacher’s math experience Teacher’s gender Teacher’s age Teacher’s education Teacher’s math education Class size Student’s gender Intercept F-Value for peer effect F-value for tracking Sample size R-square
Variables
Table 1 Public school results
–0.03 0.07 –2.03
0.85 0.02 –0.01 0.01
3.56 0.83 –0.02 –0.30 0.93 0.16 0.91 –0.08 0.41 –0.01 0.01
–0.03 0.65 –4.94
0.54 ⫺0.002 –0.001 0.002
2.19a 0.62 ⫺0.28 0.03 –0.10 0.18 –2.46b 8.51a 7.80a 703 0.1711
2.36 0.73 ⫺0.01 ⫺0.17 0.12 0.25 0.28 0.42 0.97 0.07 –0.01
–1.60 2.62a –2.99a 11.89a 4.38a 2238 0.3602
2.11a ⫺0.08 –0.38 0.10
2.08a 3.85a –1.91b –2.22a 0.50 0.96 0.58 0.77 27.96a 2.56a –0.49
Average-ability students only Est. coefficient t-scores
2.77a 2.83a –1.47 –2.96a 2.34a 0.42 1.66b –0.14 3.87a –0.06 0.18
Low-ability students only Est. t-scores coefficient
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R. Zimmer / Economics of Education Review 22 (2003) 307–315
Fig. 1.
High-ability sample.
Fig. 2.
Low-ability sample.
Fig. 3.
Average-ablility sample.
negative mechanism of the tracking effect and the overall effect is positive. Fig. 2 suggests that low-ability students are better off in a tracked class at lower peer levels and better off in a non-tracked class at higher peer levels. In assessing the overall tracking effect, the estimates suggest that the positive effect of tracking (3.56) is countered by the negative effect of a reduced peer effect from tracking, as highlighted by the coefficient of the interaction term (–0.3). The figure indicates that at an approximate mean
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peer level of 12, the reduction of the peer effect dominates the positive mechanism of tracking and the overall tracking effect is negative. Finally, Fig. 3 suggests that average-ability students are better off in a tracked class at lower peer levels and better off in a non-tracked class at higher peer levels. In assessing the overall tracking effect, like the low-ability sample, the estimates suggest that the positive effect of tracking (2.36) is countered by the negative effect of a reduced peer effect from tracking, as highlighted by the coefficient of the interaction term (–0.17). The figure indicates that at the approximate mean peer level of 14, the reduction of the peer effect dominates the positive mechanism of tracking and the overall tracking effect is negative. Therefore, in each of these cases, whether a student is better off in a tracked versus a non-tracked class depends upon the level of peers and suggests contradictory choices. For instance, low-ability students perform better in tracking classes only when they have lowability peers, which is more likely when classes are tracked, and perform better in a non-tracked class when they have high-ability peers, which is more likely when classes are not tracked. Other patterns from the analysis should also be noted. Father’s occupation and education is significant (and positive) only for low-ability students, while mother’s education is significant (and positive) only for highability students and mother’s occupation is insignificant in all cases. School inputs of the teacher’s general experience, math experience, gender, age, general education, math education, and also the class size have mixed results. The teacher’s experience is positive and significant for the full sample as well as for averageability students, whereas the teacher’s experience in teaching mathematics, as well as the teacher’s training in mathematics, is insignificant in all cases. The teacher’s gender is positive and significant for low- and averageability students and the teacher’s age is positive and significant for high-ability students only. For the class-size variable, the coefficient is negative and significant in two cases (high-ability students and the full sample). The student’s gender is positive and significant for low-ability students only. Finally, the control variable for previous educational attainment (pre-test) is positive and significant in each of the cases.
5. Conclusions In the introduction to the paper, I noted that there has been a brewing debate over the effects of tracking on the educational achievement of students. Opponents argue against tracking for a number of reasons including a reduced peer effect for low-ability students. In this study, I found that tracking does indeed reduce the posi-
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tive peer effect for low- and average-ability students. However, as a secondary point, I also found that the mechanism of tracking, at the margin, when holding the peer level constant, has a positive effect on these same students. In total, however, tracking, including both the mechanical and peer effect of tracking, can have a positive effect on low- and average-ability students, but only when these students have lower-level peers, which is more likely to occur with tracking. Therefore, it is hard to advocate tracking when the results suggest that it reduces the peer effect for low- and average-ability students and in total has an insignificant effect on highability students. It also has a positive effect on low- and average ability students only when these students are surrounded by lower-level peers, which tracking is more likely to create.
Acknowledgements I thank Dominic Brewer, Richard Buddin, Dan Goldhaber, Eric Eide, Eugenia Toma, and two anonymous reviewers for their helpful comments.
Appendix Table A1 Descriptive statement Variable Peer variable
Description
Mean
Mean beginning-of-the15.81 year test score (out of possible 40) of students in a classroom Tracking Binary variable=1 if the 0.59 student is in a tracked class; 0 if not Father’s Binary variable=1 if 0.39 occupation father’s occupation is professional or skilled; 0 if unskilled Mother’s Binary variable=1 if 0.62 occupation mother’s occupation is professional or skilled; 0 if unskilled Father’s Binary variable=1 if 0.88 education father’s level of education is at least secondary; 0 otherwise Mother’s Binary variable=1 if 0.91 education mother’s level of education is at least secondary; 0 otherwise Student math score at beginning of the year. Pretest 15.81 Student’s Binary variable=1 if gender student’s gender is male; 0 if female Teacher’s Binary variable=1 if 0.49 gender teacher’s gender is male; 0 if female Teacher’s Number of years of 14.30 experience teaching experience. Teacher’s Number of years of 9.78 math experience teaching experience mathematics Teacher’s age Age of teacher 38.94 Teacher’s Teacher’s number of 28.68 education semester units of general pedagogy in postsecondary education Class size Total number of students 26.64 enrolled in the class Teacher’s Teacher’s number of 10.59 math training semester units of math training Post-test Student math score at end 19.79 of year.
SD 4.70
0.49
0.49
0.49
0.32
0.28
7.93
0.49
8.11 8.92
11.08 40.75
6.99 6.49
9.18
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References Argys, L. M., Rees, D. I., & Brewer, D. J. (1996). Detracking America’s schools: Equity at zero cost? Journal of Policy Analysis and Management, 15(4), 623–645. Betts, J. R., & Shkolnik, J. L (2000). Key difficulties in identifying the effects of ability grouping on student achievement. Economics of Education Review, 19(1), 21–26. Boardman, A. E., & Murnane, R. J. (1979). Using panel data to improve estimates of the determinants of educational achievements. Sociology of Education, 52, 13–121. Braddock, J. H., & Dawkins, M. (1993). Ability grouping, aspirations, and achievement: evidence from the National Longitudinal Study of 1988. Journal of Negro Education, 62(3), Figlio, D. N., & Page, M. E. (2002). School choice and the distributional effects of ability tracking: does separation increase inequality? Journal of Urban Economics, 51, 497–514. Gorman, A. (1987). The stratification of high school learning opportunities. The Sociology of Education, 60, 135–155. Hallinan, M. (1994). Tracking: from theory to practice. Sociology of Education, 67(2), 79–84. Hanushek, E.A., Kain, J.F., Markman, J.M., & Rivkin, S.G. (2001). Does peer ability affect student achievement? NBER Working Paper 8502. Henderson, J. V., Miezkowski, P., & Sauvageau, Y. (1978).
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Peer group effects and educational production functions. Journal of Public Economics, 10, 97–106. Hoffer, T. B. (1992). Middle school ability grouping and student achievement in science and mathematics. Educational Evaluation and Policy Analysis, 14(3), 205–227. Hoxby, C. (2000). Peer effects in the classroom: learning from gender and race variation. NBER Working Paper 7867. Kerckhoff, A. C. (1986). Effects of ability grouping and student achievement in science and mathematics. American Sociological Review, 51, 842–858. Oakes, J. (1985). Keeping track: How schools structure inequality. New Haven, CT: Yale University Press. Oakes, J. (1990). Multiplying inequalities: the effects of race, social class, and tracking on opportunities to learn mathematics and science. Santa Monica, CA: RAND Corporation. Robitalle, D. F., & Garden, R. A. (1989). The IEA study of mathematics II: contest and outcomes of school mathematics. Oxford: Pergamon Press. Summers, A. A., & Wolfe, B. L. (1977). Do schools make a difference? American Economic Review, 67, 639–652. White, T. G., & Kane, M. B. (1995). Ability grouping, tracking, and student achievement: a critical review of research support for policy. Washington, D.C: U.S. Department of Education, Office of Educational Research and Improvement. Zimmer, R. W., & Toma, E. F. (2000). Peer effects and educational vouchers: evidence across countries. Journal of Policy Analysis and Management, 19(1), 75–79.
A new twist in the educational tracking debate Ron Zimmer ∗ RAND Corporation, 1700 Main Street, Santa Monica, CA 90407-2138, USA Received 5 December 2000; accepted 18 June 2002
Abstract Recently, the practice of tracking has been receiving more attention by both educators and researchers and some have questioned the policy merit. One of the strongest arguments against tracking is that it creates homogenous classes according to ability and, therefore, reduces the positive spillover effect referred to as a peer effect. While peer effects have been found to be an important input into the production of education no study has specifically examined whether these effects are more or less prevalent in classes where tracking occurs. Utilizing individual student level data, this current research examines whether the peer effect occurs in cases in which tracking is present. The results suggest that the use of tracking diminishes the impact peers have on student achievement for low- and average-ability students while the peer effect is unaffected by tracking for high-ability students. 2003 Elsevier Science Ltd. All rights reserved. JEL classification: I21 Keywords: Educational economics
Recently, the practice of tracking has been receiving more attention by both educators and researchers, and some have questioned the policy’s merit. In the education arena, those who advocate tracking argue that all students, regardless of ability, would learn more in a tracked class relative to a nontracked class (Hallinan, 1994). In tracked classes the teacher can tailor the curriculum to the ability level of the students, thus creating the optimal level of educational gains for all students. However, opponents argue against tracking for three primary reasons. First, tracking leads to a different set of resources being allocated to high-tracked versus low-
tracked classes (Oakes, 1990).1 Second, tracking breeds social inequities as minority and low-income groups are over-represented in low-track and under-represented in high-track classes (Braddock & Dawkins, 1993; Gorman, 1987; Oakes, 1985, 1990). Third, tracking creates homogenous classes according to ability, therefore reducing the positive spillover effect, referred to as a peer effect (Betts & Shkolnik, 2000). While further research is needed to address the first two issues, this current research examines the effect tracking has on the peer effect. That is, does tracking diminish the positive spillover effect from high-ability students to low-ability students? In addition, the effect from tracking on student achievement is examined while holding the peer level constant. The literature increasingly suggests there are differen-
∗ Corresponding author. Tel.: +1 310 393 0411; fax: +1 310 451 7059. E-mail address: [email protected] (R. Zimmer).
1 However, Betts and Shkolnik (2000) found that neither class size nor teacher characteristics vary much whether the student is placed in a tracked or an untracked class.
1. Introduction
0272-7757/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0272-7757(02)00055-9
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tial effects from tracking for students of different abilities (Argys, Rees & Brewer, 1996; Hoffer, 1992; Kerckhoff, 1986)2. However, this literature has failed to determine whether this effect is from curriculum and resource effects or a differential peer effect from the organizational structure of grouping students. In the past, researchers have found strong support for a peer effect in the production of education (Hanushek, Kain, Markman, & Rivkin, 2001; Henderson, Mieskowski, & Savageau, 1978; Summers & Wolfe, 1977; Zimmer & Toma, 2000; Hoxby, 2000). However, no study has examined whether these peer effects are more or less prevalent in classes where tracking occurs and what effect they are having on student achievement in tracked classes3. Zimmer and Toma (2000) interacted the variance of ability in a classroom with the mean ability (peer-effect variable) of students and found that greater variance within a classroom actually reduces the peer effect. Therefore, it could be the case that peer effects are less prevalent in schools that are not tracked as opposed to schools that are. That is, students of different ability levels who are mixed together in non-tracked schools would not be able to interact effectively to create the peer effect. This current research adds to the literature by examining the impact tracking has on the peer effect by including an interaction term. The results suggest that the use of tracking diminishes the impact peers have on student achievement for low and average-ability students while the peer effect is unaffected by tracking for high-ability students. A secondary result suggests that the institutional process of tracking, when controlling for the peer level, has no effect on high-ability students,
2 Kerchoff (1986) examined grouped versus non-grouped students in British schools and found that high-ability students learnt more than non-grouped students while low-grouped students learnt less. Hoffer (1992) examined grouped versus nongrouped students using the Longitudinal Study of America (LSAY) and found that tracking had no effect for averageability students, weak positive effects for high-ability students, and strong negative effects for low-ability students. Argys et al. (1996), using the National Longitudinal Study of 1988 (NELS), suggest that tracking creates educational gains for high-ability students, whereas low-ability students experience educational losses. In contrast, Betts and Shkolnik (2000) suggest only weak differential effects between ability groups, with tracking having no effect on low-ability students but a small positive effect for high-ability students and a small negative effect on averageability students. In a more recent study, Figlio and Page (2002), using the NELS data, support the finding from Betts and Shkolnik as they control for the endogeneity of track placement and finds no evidence that low-ability students are hurt by tracking. 3 White and Kane (1995) suggest that one of the main shortcomings of the existing literature is that it does “not separate the effects of course content and instruction from the effects of homogenous grouping” (p. viii).
whereas, surprisingly, the institutional practice of tracking, when controlling for the peer level, has a positive effect on low-and average-ability students. Only if the reduction in the peer effect is combined with the effect from tracking does the positive effect from low-ability students wash out. In other words, the loss of exposure to more able students offsets the potential of tracking to improve educational performance for the low-ability students.
2. Data To examine the impact tracking has on the peer effect, and ultimately, the impact peers and tracking have on student achievement, a data set with characteristics of the individual students is necessary, including whether or not the student was in a tracked classroom. One such data set is from the second study (SIMS) from International Association for the Evaluation of Educational Achievement (IEA)4. The IEA, through carefully constructed surveys, collects family, school, classroom, and peer characteristics of individual students for the purpose of cross-country comparison, including the US5. This data set includes characteristics of the students’ families and teachers, the organizational structure of the schools, including whether or not the classes are tracked, and preand post-year mathematics test scores for individual students, which allow for a value-added estimate6. In the data, tracking is designated as a dichotomous variable (1 if there is tracking in the classroom and 0 if there is not) and is interacted with a variable representing the ability of peers (the mean test score of classmates). If the resulting coefficient is negative for the interaction term, then it can be concluded that tracking diminishes the peer 4 The most recent IEA data set was not used because it did not have the pre-school-year test score to do a value-added model. 5 In 1981, the IEA carefully selected a sample of schools and administered a broad array of questionnaires answered by students, teachers and administrators (Robitalle & Garden, 1989). The schools selected for the survey were designed to reflect the socioeconomic characteristics of each country. The students in the analysis were in the eighth grade.. The survey asked questions in regard to the student’s family, schools, teachers, and peers within the classrooms (Robitalle & Garden, 1989) The IEA went to great lengths to ensure the quality of the data through rigorous quality control steps including creating a manual for inputting the data and auditing them once they were collected. 6 The mathematics tests, known as SIMS (Second International Mathematics Study), were developed through the input of mathematics experts and measurement specialists and the tests were to reflect the curriculum of each country. The tests went through a review process that included several pilot tests before being implemented.
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effect, whereas a positive coefficient suggests that the peer effect is enhanced. Using the US public school portion of the data, this study examines the impact of tracking on the peer effect and student achievement for high-, average-, and low-ability students using a standard education production model. Together, the information provided by the surveys and the test scores renders a robustness to the results.
3. Model To examine the effect from tracking, the peer effect, and the interaction of tracking with the peer effect, I followed a standard education production function model employed by Betts and Shkolnik (2000). The model examines the impact of various inputs, including family and school resources, the student’s peers, and student’s individual characteristics, including a proxy for ability7. The value-added model is displayed as follows8: At,j ⫽ f(Ft,j,St,j,Pt,j,At-1).
(1)
By using the model represented above, the valueadded impacts of various inputs are measured as student achievement for the period t–1 to t9. In this formulation, the effects of all prior inputs are captured in achievement from the previous period, or At-110. Formally, output is measured by individual student j’s educational achievement (At) in the current period as a function of inputs that include student j’s family characteristics (Ft) over period At-1 to At, school inputs (St) for student j over
7 Conceptually, the ideal model (as displayed below) includes family inputs (F), school inputs (S), environmental inputs of peers (P), and the innate ability of the student (I) in period t:At,j = f(Ft,j,St,j,Pt,j,It,j).However, it is difficult to gain an accurate proxy for the innate ability of the students (Hanushek, 1979) and the lack of an ability variable is generally assumed to bias upward the effects of family background on achievement. Therefore, researchers often use a value-added model to mitigate the negative effect of not including such a variable. By estimating the value-added model, the biases are minimized because only the growth effect of innate ability is omitted. 8 Argys et al. suggests that “if there are unobservable student or school characteristics that affect both achievement and track placement, than any association between achievement and tracking may simply be due to these characteristics” (1996, p.624). To control for the selection, Argys et al. model the process through which students are assigned to a particular track and then include these selectivity corrections in the main achievement model. Unfortunately, I was unable to find a suitable instrument in the data set for this present analysis. 9 Betts and Shkolnik (2000) argue that it is critical to control for the student’s initial achievement if accurate estimates are to be obtained. 10 In this case, achievement in all other periods, such as t–1, is a function of the cumulative inputs in t–1.
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period At-1 to At, peer influence (Pt) of student j over period At-1 to At, and student j’s achievement in the previous period (At-1)11. To apply this model I measure achievement, both in the current and previous period, as the number of questions answered correctly by student j on a mathematics test with a possible range of 0 to 40. The achievement level (At) is the students’ end-of-the-year test score and is labeled POSTTEST, while the beginning-of-the-year test (At-1) is labeled PRETEST12. Other inputs include a vector of family inputs (F) and a vector of school inputs (S), including the characteristics of the student’s teacher and school13. (A complete list of variables with their summary statistics is included in Table A1 in the appendix.) Two variables of primary importance in this current research are the tracking and peer variables14. The tracking variable is simply a dummy variable represented by a 1 when tracking is used in the classroom and a 0 when it is not. The peer variable is defined as the mean test score of students in a classroom and is represented by the mean of the scores at the beginning of the school year of all students in the observed student’s classroom15. As consistent with the literature, I also square the mean score to capture nonlinear as well as linear effects on fellow students16. This peer variable is also interacted with the tracking variable. If tracking is utilized within the classroom, then tracking is equal to 1 and the coefficient of the interaction term represents the impact tracking has on the peer effect. Because the peer level is also included, the impact of tracking on the peer effect is measured while holding the peer level constant. A negative and significant coefficient suggests that tracking diminishes the peer effect, while a positive and significant coefficient suggests that tracking enhances the peer
11 For further information on the benefits of using the valueadded model, see Boardman and Murnane (1979). 12 It should be noted that students that did not appear in both the pre- and post-test scores were deleted from our sample. 13 Argys et al. (1996) suggest that it is critical to control teacher and school characteristics if low-ability tracks and highability tracks are systematically assigned different resources. 14 It should be noted that the model suffers from endogenity problems. Part of the purpose of tracking is to put students in more homogenous classes. Therefore, the peers of a student are partially a function of whether a student is in a tracked class. No suitable instrument could be found to perform an IV model to correct for this problem. 15 This is one of the definitions used by Zimmer and Toma (2000) for peers. 16 Henderson et. al. (1978) along with Zimmer and Toma (2000) squared the mean effect to capture a peer effect that is increasing at a decreasing rate (i.e., the squared term is expected to be negative).
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effect. Finally, an insignificant coefficient suggests that tracking has no effect on the peer effect. To examine the effect from tracking, the peer variable, and the interaction of tracking and the peer variable, four samples of data from US public schools were utilized. The first sample uses the full data set of students as the observations. Therefore, the first model examines the impact on achievement from tracking, peers, and the interaction of tracking with the peer effect, along with the student’s other school, family, and the past inputs. In addition to using all students in the sample, three additional samples are used for analysis: one that is restricted to high-ability students, one that is restricted to low-ability students, and one that is restricted to averageability students17. Therefore, the second estimated model uses all the same inputs, but restricts the data to include only high-ability students. A student is classified as highability if he or she is placed in the upper 20th percentile of all students across the nation on the math test at the beginning of the year. Again, the peer-effects literature assumes that a student placed in a class with a higher mean peer group will experience a positive achievement effect. A third model uses the same inputs but restricts the data to include only low-ability students. A student is classified as low-ability if he or she placed in the lower 20th percentile of all students across the nation on the same mathematics test. A fourth model uses the same inputs but restricts the sample to average-ability students. A student is classified as average-ability if he or she scored between the upper 20th percentile and the lower 20th percentile on the same mathematics test. As in Betts and Shkolink (2000) and Zimmer and Toma (2000), each model includes variables that represent family and school inputs. The variables that represent the family socioeconomic characteristics (F) of a student is the occupation level of the father and the mother and the education level of the father and the mother18. The variables that represent school inputs (S) include the size of the classroom, the general experience of the teacher, the math experience of the teacher, the teacher’s pedagogy training, the number of classes of mathematics training for the teacher, the teacher’s age, and the teacher’s gender. Finally, the student’s inputs include the student prior achievement and student’s gender. The following section will present the results of the
model focusing primarily on the interaction term, the peer effect, and tracking.
17 The restricting of data in this way is consistent with the work of Kerckhoff (1986); Hoffer (1992), and Argys et al. (1996). 18 The variables for the education level of the father and mother are limited education measures defined as a binary variable that is equal to one when the father’s or mother’s education is at least secondary, respectfully. Because of the limitations of these variables, readers should interpret the results with caution.
19 The sensitivity of these results is tested by alternatively defining high- average- and low-ability students. In the first alternative, high-ability is defined as students placed in the top 25th percentile, average-ability is defined as students placed in the middle 50th percentile, and low-ability is defined as students placed in the lower 25th percentile. In the second alternative, high-ability is defined as students placed in the top 15th percentile, average-ability is defined as students placed in the middle
4. Results Using a model specification similar to that of Betts and Shkolnik (2000), I estimate the effect from tracking, peers, and the interaction of tracking and peers from four segmented data sets. Table 1 displays the results. Column one of the table lists the variables, while columns two and three display the estimated coefficient and tscores of each of the variables from the full set of data. Columns four and five display the estimated coefficient and t-scores of the model that restricts the data to only high-ability students. Columns six and seven display the estimated coefficient and t-scores of the model that restricts the data to only low-ability students. Finally, columns eight and nine display the estimated coefficient and t-scores of the model that restricts the data to only average-ability students. In all the models, the tracking variable is a dichotomous dummy variable, 1 if tracking is present in the classroom, 0 if it is not. The peer variable is the average of beginning-of-year test scores of all students in individual j’s classroom. The interaction term is the interaction of the tracking variable and the peer variable. Included in each of the models, in addition to the peer variable, tracking variable, and interaction term, are family and school inputs, along with the student’s gender and prior achievement (as proxy for past inputs). The dependent variable is the achievement score at the end of the school year. For this study, the primary focus is on the coefficient for the interaction term. Also of great interest are the coefficients of the tracking and peer variable and the total effect from tracking, including the mechanism of tracking, the interaction, and peer effects. First, let us focus on the interaction term. For the full sample, the averageability students, and the low-ability students, the coefficients are negative and significant. However, when the data are restricted to only high-ability students, the coefficient is insignificant, suggesting that tracking neither enhances nor diminishes the peer effect for high-ability students. Therefore, the results suggest that tracking diminishes the positive peer effect for low- and averageability students while neither enhancing nor diminishing it for high-ability students19.
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Focusing on the secondary points, let us examine the results of the peer variable and tracking. To interpret the magnitude (and significance) of the coefficient for the peer effect and tracking requires taking into account the coefficient and joint significance of all variables that include multiplicatives of the tracking and peer variables. More formally, our model is At,j ⫽ B0 ⫹ B1(Pt,j) ⫹ B2(P2t,j) ⫹ B3(P∗t,jTt,j)
(2)
⫹ B4(Tt,j) ⫹ .... where At,j is the post-year level of student achievement for student j in period t and is a function of the student j’s peers (Pt,j or the mean pre-year test score of classroom for student j), student j’s peer squared (P2t,j or the square of the mean pre-year test score of classroom for student j), the interaction term (the mean pre-year test score of the classroom, Pt,j, for student j times the dichotomous tracking variable, Tt,j, for student j), the tracking variable (Tt,j or the dichotomous 1/0 dummy variable for tracking within the classroom for student j), and all other explanatory variables. The estimated peer effect is the partial derivative of At,j with respect to the peer variable (Pt,j) and is ∂A / ∂P = B1 + 2B2(Pt,j) + B3(Tt,j), which is the estimated effect from student j’s peers. The estimated tracking effect is the partial derivative of At,j with respect to tracking (Tt,j) and is ∂A / ∂T = B3(Pt,j) + B4, which is the estimated effect from tracking. To test the significance of the peer effect and tracking, the resulting beta coefficients from the partial derivatives must be jointly tested. Therefore, an F-test was conducted on the joint significance of the peer variable and tracking. Focusing first on the peer effect, the F-test indicates significance at the 0.001 level for each of the models, except the model that restricts the data to high-ability students. In other words, the higher the mean test scores of a student’s classmates, the better the average- and low-ability students will perform20. It is surprising that this effect does not show up for the high-ability students as well. However, Argys et al. (1996) suggest that in many cases, placement in high-tracks is not strictly based upon ability, but on other factors including parental influence on the decision process. In other words, highability students may not always be placed exclusively with high-ability students and therefore, have less of a chance of gaining a positive peer effect.
70th percentile, and low-ability is defined as students placed in the lower 15th percentile. In both cases, the interaction term remained insignificant for high-ability students and negative and significant for average- and low-ability students. This suggests this robustness of the results. These results are available upon request from the author. 20 These results are also true for the alternative definitions of high-, average-, and low-ability as specified in footnote 21. Again, results are available upon request from the author.
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As noted above, to interpret the results of the peer effect, the coefficient of all variables that include the peer variable must be considered, including the coefficient of the interaction term when tracking is present (the interaction term does not have to be considered when tracking is not present because the variable takes on a value of zero). For instance, in the full sample, the peer effect yields an estimated coefficient of 0.22 when tracking is present and 0.38 when it is not21. The coefficient implies that, on average, increasing the class mean on the beginning-of-year scores by one point increases a representative student’s end-of-year math test score by approximately 0.22 points when tracking is present and 0.38 when it is not. As with the peer coefficient, the F-test is employed to test the significance of the tracking coefficient. In these models, tracking is significant for the full sample and the average- and low-ability students, and insignificant for the models in which the data set was restricted to highability students22. The results from the full sample of data and the data restricted to average-ability students suggest a negative effect from tracking, net of the peer effect. However, unlike previous research (which measured the total effect from tracking), the tracking variable, on the margin, has a positive coefficient for low- and average-ability students when controlling for the peer level. The total tracking effects, including the peer effect and the effect from the mechanism of tracking (e.g., curriculum, curriculum pace, etc.) are illustrated in Figs. 1–3. Each figure compares student test scores in a tracked versus a non-tracked class at different peer levels. Fig. 1 suggests that high-ability students are better off in a tracked class at higher peer levels and better in a nontracked class at lower levels. This fact is due to the offsetting effects from tracking and peer variables. In assessing the overall tracking effect, the estimates suggest that the negative effect from the mechanism of tracking (–4.05) is countered by the positive effect of peers (0.38 and 0.2). The figure indicates that at the approximate mean peer level of 18, as measured by the pre-test score, the positive peer effect dominates the
21 For the full sample, the total coefficient of the peer effect is derived by adding the coefficient of the peer variable (0.69) plus two times the coefficient of the squared peer variable (0.01) times the mean value of the peer variable (15.81) plus the coefficient of the interaction term (-0.15). The coefficient of the interaction term is added only when tracking is present. 22 The alternative cases have similar results except for highability students (defined as the top 15th percentile) and averageability students (defined as the middle 50th percentile). In the high-ability case, tracking has a negative and significant effect. In the average-ability case, tracking becomes insignificant when defined more narrowly. Once again, results are available upon request from the author.
b
a
0.28 0.05 –0.004 0.03 –0.05 0.44 7.78
2.72 1.02 –0.82 0.56 –2.33a 2.83a –2.67a 27.96a 6.96a 3706 0.6536
0.50 0.01 –0.002 0.01
–0.03 0.51 –2.80
–4.05 0.38 –0.01 0.20 0.26 –0.28 –0.65 1.86 0.67 –0.04 –0.03
2.61a 6.21a –3.14a –2.96a 1.51 0.74 1.10 1.22 59.66a 1.80b –0.56
–2.06a 1.35 2.02a 0.79 1.67 763 0.2966
0.79 1.90a –0.95 0.95
–1.30 0.97 –1.01 1.15 0.75 –0.79 –0.82 1.92b 14.66a –1.00 –0.95
High-ability students only Est. coefficient t-scores
1.98 0.69 –0.01 –0.15 0.28 0.14 0.37 0.46 0.85 0.04 –0.01
Full sample Est. coefficient t-scores
Indicates significance at the 0.05 level. Indicates significance at the 0.10 level.
Tracking Peer variable Peer variable squared Interaction Father’s occupation Mother’s occupation Father’s education Mother’s education Pre-Test Teacher’sexperience Teacher’s math experience Teacher’s gender Teacher’s age Teacher’s education Teacher’s math education Class size Student’s gender Intercept F-Value for peer effect F-value for tracking Sample size R-square
Variables
Table 1 Public school results
–0.03 0.07 –2.03
0.85 0.02 –0.01 0.01
3.56 0.83 –0.02 –0.30 0.93 0.16 0.91 –0.08 0.41 –0.01 0.01
–0.03 0.65 –4.94
0.54 ⫺0.002 –0.001 0.002
2.19a 0.62 ⫺0.28 0.03 –0.10 0.18 –2.46b 8.51a 7.80a 703 0.1711
2.36 0.73 ⫺0.01 ⫺0.17 0.12 0.25 0.28 0.42 0.97 0.07 –0.01
–1.60 2.62a –2.99a 11.89a 4.38a 2238 0.3602
2.11a ⫺0.08 –0.38 0.10
2.08a 3.85a –1.91b –2.22a 0.50 0.96 0.58 0.77 27.96a 2.56a –0.49
Average-ability students only Est. coefficient t-scores
2.77a 2.83a –1.47 –2.96a 2.34a 0.42 1.66b –0.14 3.87a –0.06 0.18
Low-ability students only Est. t-scores coefficient
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Fig. 1.
High-ability sample.
Fig. 2.
Low-ability sample.
Fig. 3.
Average-ablility sample.
negative mechanism of the tracking effect and the overall effect is positive. Fig. 2 suggests that low-ability students are better off in a tracked class at lower peer levels and better off in a non-tracked class at higher peer levels. In assessing the overall tracking effect, the estimates suggest that the positive effect of tracking (3.56) is countered by the negative effect of a reduced peer effect from tracking, as highlighted by the coefficient of the interaction term (–0.3). The figure indicates that at an approximate mean
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peer level of 12, the reduction of the peer effect dominates the positive mechanism of tracking and the overall tracking effect is negative. Finally, Fig. 3 suggests that average-ability students are better off in a tracked class at lower peer levels and better off in a non-tracked class at higher peer levels. In assessing the overall tracking effect, like the low-ability sample, the estimates suggest that the positive effect of tracking (2.36) is countered by the negative effect of a reduced peer effect from tracking, as highlighted by the coefficient of the interaction term (–0.17). The figure indicates that at the approximate mean peer level of 14, the reduction of the peer effect dominates the positive mechanism of tracking and the overall tracking effect is negative. Therefore, in each of these cases, whether a student is better off in a tracked versus a non-tracked class depends upon the level of peers and suggests contradictory choices. For instance, low-ability students perform better in tracking classes only when they have lowability peers, which is more likely when classes are tracked, and perform better in a non-tracked class when they have high-ability peers, which is more likely when classes are not tracked. Other patterns from the analysis should also be noted. Father’s occupation and education is significant (and positive) only for low-ability students, while mother’s education is significant (and positive) only for highability students and mother’s occupation is insignificant in all cases. School inputs of the teacher’s general experience, math experience, gender, age, general education, math education, and also the class size have mixed results. The teacher’s experience is positive and significant for the full sample as well as for averageability students, whereas the teacher’s experience in teaching mathematics, as well as the teacher’s training in mathematics, is insignificant in all cases. The teacher’s gender is positive and significant for low- and averageability students and the teacher’s age is positive and significant for high-ability students only. For the class-size variable, the coefficient is negative and significant in two cases (high-ability students and the full sample). The student’s gender is positive and significant for low-ability students only. Finally, the control variable for previous educational attainment (pre-test) is positive and significant in each of the cases.
5. Conclusions In the introduction to the paper, I noted that there has been a brewing debate over the effects of tracking on the educational achievement of students. Opponents argue against tracking for a number of reasons including a reduced peer effect for low-ability students. In this study, I found that tracking does indeed reduce the posi-
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tive peer effect for low- and average-ability students. However, as a secondary point, I also found that the mechanism of tracking, at the margin, when holding the peer level constant, has a positive effect on these same students. In total, however, tracking, including both the mechanical and peer effect of tracking, can have a positive effect on low- and average-ability students, but only when these students have lower-level peers, which is more likely to occur with tracking. Therefore, it is hard to advocate tracking when the results suggest that it reduces the peer effect for low- and average-ability students and in total has an insignificant effect on highability students. It also has a positive effect on low- and average ability students only when these students are surrounded by lower-level peers, which tracking is more likely to create.
Acknowledgements I thank Dominic Brewer, Richard Buddin, Dan Goldhaber, Eric Eide, Eugenia Toma, and two anonymous reviewers for their helpful comments.
Appendix Table A1 Descriptive statement Variable Peer variable
Description
Mean
Mean beginning-of-the15.81 year test score (out of possible 40) of students in a classroom Tracking Binary variable=1 if the 0.59 student is in a tracked class; 0 if not Father’s Binary variable=1 if 0.39 occupation father’s occupation is professional or skilled; 0 if unskilled Mother’s Binary variable=1 if 0.62 occupation mother’s occupation is professional or skilled; 0 if unskilled Father’s Binary variable=1 if 0.88 education father’s level of education is at least secondary; 0 otherwise Mother’s Binary variable=1 if 0.91 education mother’s level of education is at least secondary; 0 otherwise Student math score at beginning of the year. Pretest 15.81 Student’s Binary variable=1 if gender student’s gender is male; 0 if female Teacher’s Binary variable=1 if 0.49 gender teacher’s gender is male; 0 if female Teacher’s Number of years of 14.30 experience teaching experience. Teacher’s Number of years of 9.78 math experience teaching experience mathematics Teacher’s age Age of teacher 38.94 Teacher’s Teacher’s number of 28.68 education semester units of general pedagogy in postsecondary education Class size Total number of students 26.64 enrolled in the class Teacher’s Teacher’s number of 10.59 math training semester units of math training Post-test Student math score at end 19.79 of year.
SD 4.70
0.49
0.49
0.49
0.32
0.28
7.93
0.49
8.11 8.92
11.08 40.75
6.99 6.49
9.18
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