Vol. 8, No. 3 (December 2000)
By Gary J. Scott
St. Mary's University
November 20, 1999
ABSTRACT The contributions of a school and pupil to learning are isolated with a unique interpretation of the education production function. Variance in pre-test scores and study time is then discovered to constrain efficiency and equal opportunity within schools. This dispersion creates the potential for Pareto exchange between schools resulting in higher and more equal educational opportunity among pupils across several schools. Finally, a voucher policy empowers persons possessing the necessary circumstantial knowledge for recognizing these Pareto exchanges to execute them.
United States policymakers are asking whether pre-college, school choice might help pupils, especially disadvantaged pupils. A general equilibrium model of schooling is presented that leads to two optimistic conclusions: (i) More school efficiency is complementary with more equal opportunity, increased overall learning, more integration, stable teacher salary schedules, and fewer course preparations per teacher. (ii) A voucher policy is a practical means for securing these ends.
School Versus Pupil Contribution to Learning
Educational opportunity, produced by schools, and pupil effort are distinguished using equations one and two (Carroll, 1963; McKenzie and Staaf, 1974; Scott, 1997). Education at any time is the sum of pre-test achievement and present learning. Present
(1) A = a + K T
(2) q = K T = A - a
A = education or final achievement.
a = pretest achievement.
K = average rate of learning during formal instruction and homework.
T = time-on-task or total time concentrating during classroom instruction and homework.
q = learning from present instruction.
learning is the product of the learning rate and time-on-task. Schools' provision of educational opportunity is the learning rate, K, or learning per time-on-task, q/T.
However, the learning rate remains incomplete for measuring educational opportunity since it omits production cost. To understand this, suppose schools X and Y offer equal learning rates, making a pupil indifferent between the two schools. Further suppose
school X incurs $6,000 of expenditure per-pupil while school Y incurs only $4,000 to produce the same annual learning rate. The pupil/family now prefers school Y because the same learning rate can be obtained while saving $2,000 in taxes or tuition. These savings could be used for future education or spent at the present school to produce an even higher learning rate.
So a more accurate measure of educational opportunity, school efficiency (c), is calculated in equation three. It measures how well the school maximizes K subject to the
(3) c = e / K
e = expenditure per-pupil during the instructional period.
c = cost or price per-unit of learning rate.
budget constraint of $e per-pupil. Therefore, as c decreases, school efficiency increases.
Finally, three determinants of learning are summarized in the fourth equation, which is obtained by substituting q/T for K in equation three and then solving for q. e is expenditure per-pupil or voucher value. Pupil effort is measured by time-on-task, T, and
(4) q = (e T) / c
schools' contribution of efficiency is measured by the cost per-unit of learning rate, c.
Higher and More Equal Learning Opportunity
A voucher policy is expected to increase learning opportunity for the typical pupil by reducing the average c. Learning opportunity is also expected to become more equal
because the specific pupils enjoying this reduction in c are also those starting with the highest c's. Evident from equation three, these lower and more equal C's is equivalent to higher and more equal learning rates, provided expenditure per-pupil is constant. Finally, lower and more equal C's in equation four causes higher and more equal learning, if expenditure per-pupil and time-on-task remain equal.
The Potential for Pareto Exchange
These gains of the pupils benefiting from vouchers will also not impose any cost on remaining pupils. Four simulations of a general equilibrium model demonstrate this probable outcome by focusing on the specific means for improving school efficiency.
1) Unequal Achievement. To illustrate the potential gains arising from pupils having unequal present achievement, three nearby schools, each enrolling five pupils, appear in Graph 1. Each pupil is denoted by a number and occupies a unique position on the scale for present mathematics achievement. Consistent with present government-school policy, pupils are assigned schools by geographic residence. For simplification, assume all pupils enjoy equal expenditure per-pupil and commit identical amounts of time-on
Each school offers a single mathematics course for its five pupils. Also, teachers target the median pupils' achievements when planning the difficulty of lectures, textbook, and homework. This targeting of the median pupil arises from the political equilibrium attained through majority voting in these democratically controlled schools (Enelow & Koehler, 1984; Mueller, 1989). To illustrate these targets, the median pupils are denoted with asterisks above their numbers. So school Z's teacher targets pupil 13's achievement
and thus emphasizes algebra, as opposed to the emphasis on arithmetic in school X and calculus in school Y.
Graph I. Present Achievement in Mathematics
|School X|| *
1 2 3 4
|School Y||6|| *
7 8 9 10
|School Z|| *
11 12 13 14 15
Due to the three median pupils' enjoying optimum curricula, their learning rates are highest. The remaining pupils' learning rates decline in proportion to the difference between their present achievement and each school's median achievement. So a pupil's learning rate is partly explained by instructional error, the extent to which a pupil's current achievement is inappropriately matched to the school's curriculum or instructional target. Insofar as a student's achievement is less than the median, then his learning rate declines due to an overwhelming or unintelligible curriculum. Insofar as his achievement exceeds the median, then the learning rate declines due to a redundant curriculum. (Bloom, 1974; Cronbach and Snow, 1977; Bryk, Lee, and Holland, 1993; U.S. Dept. of Education, 1993; Scott, 1997).
Notice that despite equal expenditure per-pupil and equal time-on-task among pupils, equal opportunity is not satisfied because learning rates or c vary among pupils, both within and between schools. Therefore, equal expenditure per-pupil is not a sufficient condition for equal learning opportunity, which corroborates Coleman, et. al. (1966), Hanushek (1986, 1994), Chubb and Moe (1990), and Burtless's (1996) finding of a zero regression coefficient for learning and expenditure per-pupil. Despite equal expenditure per-pupil, the three pupils enjoying the highest learning rate in the school system are 3, 8, and 13. Those enjoying the second highest learning rate are 2, 4, 7, 9, 12, and 14. The third ranked group consists of 1, 10, 11, and 15. Finally, pupils 5 and 6 suffer the lowest learning rate.
Vouchers permit pupils in pursuit of a higher learning rate to transfer to another school. However, only pupils 5 and 6 discover a more appropriate curriculum in another school. Pupil 5 prefers the curriculum in school Y where the instructional target is identical to her current achievement. A successful transfer from school X to school Y results in her learning rate increasing from the lowest to the highest rank. Pupil 6 wants to transfer from school Y to X for a similar reason.
These transfers result in several benefits. First, the exchange of pupils 5 and 6 between schools X and Y is Pareto efficient, since both pupils and both schools gain, while no pupil or school suffers a loss. Specifically, pupils 5 and 6, initially suffering lowest learning rate, now join three other pupils in enjoying the highest rate, while all remaining learning rates remain constant. Therefore, this exchange results in a higher average and lower range of learning rates for the entire system of three schools. Second, since expenditure per-pupil remained constant and the average learning rate increases in the schools X and Y, average c declines in both schools. Third, teacher unemployment is avoided due to the absence of any net loss of revenue from both schools. Fourth, if pupils 5 and 6 represent the unique race or ethnicity of their residential neighborhoods and corresponding schools, then the transfers result in more integration. Finally, the math teachers in schools X and Y benefit from eliminating their achievement outliers who were nuisances, due to their special, unmet academic needs.
Even though school Z does not enjoy any efficiency gain through transfers, it pursues an alternative method for increasing efficiency, thereby securing its enrollment and revenue. Pupil I I would enjoy the highest learning rate if the curriculum was targeted on his achievement rather than on 13's. The result would be decreasing dispersion of achievement over time in school Z since all pupils' time-on-task is equal and the lower achievers enjoy higher learning rates. Despite the short-run, net loss in efficiency, achievement is equalized allowing for equal and maximum learning rates and efficiency in the long run. In general, vouchers create the incentive for schools to remediate pupils in order to increase long run or dynamic efficiency. This perhaps explains the more equal achievement found in private schools by Alexander and Pallas (1985), Coleman, Hoffer, and Greeley (1985), Willms (1985), Chubb and Moe (1990), Bryk, Lee, and Holland (1993), and Scott (1997).
School Z serves an additional function. The potential transfer to school Z provides pupil 4 with a realistic goal for advancement. It also serves as a nearby “safety net” for pupil 7 in the event he falls further behind his peers in school Y. It is assumed, by the way, that school X remains optimal for pupil 4 despite the desire to challenge or prod the pupil with school Z's more advanced curriculum. In other words, school Z would overwhelm, rather than challenge pupil 4 and result in a permanently less intelligible curriculum and lower learning rate. To conclude, this first simulation demonstrates that higher and more equal learning rates, or lower and more equal C's, are realistic outcomes with vouchers.
2) Unequal Achievement with Two Ability Tracks. Some recommend ability tracking within schools to solve the problem of instructional error. However, graph 11 helps demonstrate that it is less efficient than transferring pupils between schools.
Even though the schools and pupils in Graph 11 are identical to the previous simulation, achievement now includes all the subjects of language, writing, science, math, computers, and social studies. Pupils are ranked identically in each subject. Also, each school added one ability track to better accommodate pupils' academic needs. For example, school X continues to target pupil 3 in the remedial track, while pupil 5 enjoys exclusively the optimality of the advanced track. Finally, since achievement now consists of all the academic subjects, each track requires one equivalent, full-time teacher who instructs the track in all subjects throughout the school day.
This tracking policy does increase the average learning rate within each school. Specifically, pupil 6's learning rate increases while all his peers' learning rates remain constant, thereby increasing the average. Also, the range of learning rates declined, increasing equality of opportunity both within and among schools. However, each school incurs the cost of hiring an additional full-time teacher for the new track, and thus expenditures per-pupil increase.
Graph II. Composite Achievement in all Academic Subjects
|School X|| *
1 2 3 4
|School Y|| *
7 8 9 10
|School Z||* *
11 12 13 14 15
A voucher policy is superior because it attains even higher and more equal learning rates, while avoiding the cost of an additional track. Notice that school X desires to enroll pupil 6 and eliminate the advanced track designed for pupil 5, resulting in saving the cost of a full-time teacher. School Y, on the other hand, desires to enroll pupil 5 and eliminate the remedial track designed for pupil 6 for the same cost reduction. Schools X and Y and pupils 5 and 6 negotiate this exchange because it frees up a full-time teacher in both schools, while maintaining pupils' learning rates and both schools' revenue.
The two extra teachers are then used to add ability tracks centered on pupils I and 10, presently suffering the lowest learning rates, in order to discourage them from transferring to schools not illustrated in the graph. Hence, adding vouchers to the initial policy of two ability tracks within each school results in even higher and more equal learning rates, while expenditure per-pupil remain constant.
3) Unequal Achievement with Complete Ability Tracking. Perhaps tracking maximizes and equalizes learning rates, if each school provides a track for every achievement. Graphically, an asterisk would now appear above every pupil in Graph 11. This most aggressive tracking policy increases the average learning rate and eliminates any remaining dispersion in learning rates since all pupils now enjoy an optimum curriculum. But again, these benefits require the hiring of three additional teachers within each school, thereby increasing expenditure per-pupil.
A voucher policy remains superior because it attains an even higher average learning rate while maintaining the zero dispersion of learning rates, without increasing expenditure per-pupil. The transfer of pupils 5 and 6 between schools X and Y still renders a full-time teacher superfluous in both schools. The superfluous teaching position then indirectly increases each schools' average learning rate by either sharing and thus reducing remaining teachers' preparations, granting sabbatical leave, designating the position to a teaching assistant or administrator, or awarding higher salaries to the remaining teachers. The superfluous position might also be eliminated for the sake of tax or tuition relief, the alternative for increasing efficiency by maintaining average K and decreasing e.
To conclude, transfers and group learning are more efficient than offering optimum, yet duplicate curricula among schools. Also, class sizes in excess of one is more realistic because the tax or tuition required for enjoying a teacher-pupil ratio of one would exhaust most families' income.
4) Unequal Time-on-Task. Unequal time-on-task, as opposed to unequal achievement in the previous simulations, creates a similar possibility for Pareto improvement. Unlike the previous simulations, pupils' planned homework per-day for all academic subjects are measured horizontally in Graph 111. It is assumed that achievement, learning rate, and expenditure per-pupil is identical for all pupils at the beginning of instruction, allowing each school to begin with a single track.
When planning lessons, teachers make an assumption concerning the amount of homework that pupils will realistically complete. Insofar as the teachers overestimate, lessons progress too rapidly, causing learning rates to decline because the curricula become too advanced. Insofar as homework is underestimated, lessons become redundant, which also causes learning rates to decline. Teachers once again pace their lessons optimally for the median pupils, so asterisks appear above the median pupils.
Graph III. Planned Time-on-task for Daily Homework
|One Hour||Three Hours||Five Hours|
The median pupils consistently enjoy maximum learning rates because the paces of the curricula follow their time-on-task and corresponding daily achievements. However, insofar as the remaining pupils' time-on-task differs from the median, then their achievement eventually varies in the same direction. Since achievement eventually diverges from the median pupil's achievement, learning rates decline for all non-median pupils, due to an increase in instructional error.
This instructional error caused by unequal achievement in turn caused by unequal time-on-task is the familiar problem encountered previously. The non-median pupil/families learn that insofar as their time-on-task departs from the school's median, then learning rates decline proportionally throughout the academic year. They seek to avoid this occurrence by enrolling in a school in which their personal time-on-task is nearest the median pupil's. So pupil I prefers school Y, pupil 5 prefers Z, and pupils 10 and I I prefer X.
These transfers also involve several benefits. The four pupils transferring would enjoy an increase in their learning rates from the lowest to the middle rank. Hence, learning rates become higher and more equal. Also, the benefits arising in previous simulations also apply here: more efficiency, more integration, avoidance of teacher unemployment, elimination of outlying pupils, and the presence of “safety net” and “striving for” schools.
These conclusions can be generalized to schools of any scale of enrollment. Each pupil in the model might represent a bloc of pupils with identical characteristics. Teachers then offer multiple sections of a given course and accompanying instructional target. It remains more efficient for a teacher to offer several sections of an identical course using a single preparation than to teach the same topic to several sections with unique preparations targeted for each section's unique achievement. Also, efficiency does increase with more pupils, but only on the condition that dispersion of achievement and/or time-on-task is equal or less than the original dispersion. Similarly, a school with any enrollment can increase its efficiency by decreasing the dispersion of achievement and/or time-on-task.
To conclude, all four simulations demonstrate two mathematically equivalent conclusions. (i) Vouchers increase the average and equality of learning rates (K from equation three) when pupils are initially unequal in achievement or time-on-task. (Ii) Vouchers decrease the average and dispersion of cost per-unit of learning rate (c from equation four) when pupils are initially unequal in achievement or time-on-task.
Knowledge and Educational Planning
The model also confirms that circumstantial knowledge, such as unequal achievement and time-on-task, is as important as theoretical knowledge for increasing school efficiency (Hayek, 1935, 1945; Sowell, 1980; Stiglitz, 1994; Caldwell, 1997). Educational administrators and teachers use theoretical or scientific knowledge to inform their policymaking and teaching. In addition to mastering their teaching specialization, most are experts in pedagogy, finance, child psychology, etc. Since all pupils and most parents do not possess this theoretical expertise which is obtained from university training, they trust the professional judgement of teachers and administrators (Gambetta, 1988; Fukuyama, 1995).
Unlike theoretical knowledge, circumstantial knowledge is concentrated in families, rather than among teachers and administrators. This circumstantial knowledge consists of the mundane details of time and place. For example, a pupil and her parents know the degree to which she completed her assigned history reading on Wednesday evening. Even though this fact is critical for designing the optimal history lesson on Thursday morning or placing her in a school with an optimum curriculum, administrators and teachers do not know for certain the extent to which she and her peers complete homework.
So teachers and administrators proceed with less knowledge of circumstances compared to pupils and parents, because it is not easily obtained, aggregated, and summarized in statistics. Vouchers permit families, who are more expertise in circumstantial knowledge, to participate in the enrollment decision, while not dominating it.
Consequences of Ignoring Circumstantial Knowledge
Several examples demonstrate the inefficiency or failure of the current education system of geographically determined enrollment and centralized planning that necessarily proceeds with less knowledge of circumstances. Using the preceding model of educational production and Pareto exchange, each of the following is argued to be a popular fallacy:
Academic standards should be mandated for the sake of maintaining high and equal educational opportunity. Achievement norms are calculated using the historical average achievement for an age cohort or grade. For the sake of planning, this norm is used to set the instructional target from which future learning departs. For example, a state curriculum expert discovers the average achievement for beginning seventh graders to be pupil 13's achievement in Graph 11. Textbooks are selected and lesson plans designed using pupil 13 as the academic standard or reference point.
The curriculum, however, is optimal only for pupil 13 in school Z. Even worse, the curriculum selected by the state planner using the criterion of optimality would in turn be rejected by two-thirds of the teachers using the same criterion. Appearing to be a fair and efficient central planning decision using the mean; it results in highly unequal and minimum learning rates because the details of achievement dispersion were ignored. So average and equal educational opportunity increase insofar as teachers, who know their unique pupils' achievement imperfectly but better than planners, control their annual as well as daily curricula standards.
School quality is measured by pupils' final achievement. It is now commonplace to rate schools using average student performance on standardized tests. However, this measure is discovered to be faulty using the production model described in equations one through three as well as the concept of circumstantial knowledge.
For instance, the lowest achieving school in a state may in fact be the most efficient school. Suppose pupils in school A score an average of 90%, while school B pupils score 40%. But learning in school B is nevertheless greater if the pupils in B began the year at 20% while pupils in school A began at 80%. School B pupils learned 100% more, since their gain was 20% compared to a gain of 10% for school A.
School B remains superior even if both schools produced equal learning, provided students in school A studied twice as much to attain the same increment of learning. Finally, school B remains superior even if learning rates are equal, provided school B incurs less expenditure per-pupil. In conclusion, despite school A pupils scoring higher on their exit exams, school B remains superior for good reason.
School quality is conceptually measured by c in equation three, which accounts for pretest scores, time-on-task, and expenditure per-pupil. Would school A, in the previous example, maintain its 90% achievement if it had school B pupils' pretest scores, time-on-task, and expenditure per-pupil? This contextual data necessary to empirically answer this question is either too expensive or impossible to collect. For instance, time-on-task is difficult to measure accurately while children are at home. Even more difficult is determining whether low time-on-task is caused by uncooperative pupils or teachers' acquiescing to a low norm. So for the purpose of ranking and regulating schools, final achievement scores are theoretically inappropriate and empirically inaccurate.
The invidious ranking of schools by average achievement encourages school improvement. To understand the fallacy of this causal reasoning, notice that school X in Graph II will not cooperate in the Pareto exchange of pupils 5 and 6 if it is being evaluated with average achievement. Despite the increase of overall learning, equality, and efficiency from the exchange, average achievement would nevertheless decline school X from enrolling pupils with lower pretest scores. So average achievement actually declines in school X as it produces more learning, efficiency, and equality within its own school and cooperates in the similar improvement of other schools.
Publicizing final achievement might encourage more student effort and thus increase final achievement. However, it discourages the Pareto exchanges that result in higher learning rates, the schools' contribution to higher final achievement.
Retaining a pupil for failing a grade is in his best academic interest. Suppose that in the absence of tracking, pupil 6 in Graph 11 is considered insufficiently advanced to proceed with his peers in school Y to the eighth grade. If the upcoming seventh graders about to join pupil 6 are identical to school Z pupils, then retention is academically superior to promotion, even though he will continue to struggle with the lowest learning rate.
Now suppose the pupils in school X in graph II comprise the upcoming eighth graders in school X. The transfer of pupil 6 to join these fellow eighth graders in school X is superior to his being retained in school Y's seventh grade. He would enjoy a higher learning rate and thus learn more, remain with his age cohort, graduate on schedule, and avoid the stigma of trailing his friends in school Y. More generally, the third option of transferring might be superior to retention, which in turn is superior to social promotion.
Special programs are necessary for at-risk and gifted children. The problem of instructional error has recently surfaced as more urgent. Parents complain of curricula being too difficult or insufficiently challenging. Teachers concur that they cannot accommodate all their pupils' academic needs because they are burdened with so many course preparations. This problem has resulted in increased government funding for remedial and gifted programs.
The theory justifying these new programs is misleading insofar as it misdiagnoses pupils. For instance, pupil 6 in Graph I might be diagnosed as a slow learner, needing remedial attention. Admittedly, his learning rate is the lowest in the school. But the diagnosis of “slow learner” is false insofar as it implies a physiological or psychological defect. The problem is not the pupil's physical or emotional constitution, but rather an inappropriate curriculum. Alternatively, pupil 5 in graph one is perhaps diagnosed as a gifted or extremely fast learner. But the diagnosis “fast learner” is misleading because she currently suffers the lowest learning rate in the school. While not denying physiological influences, many of their problems stem from sub-optimum curricula (Collier, 1994; Scott, 1997).
Nevertheless, these pupils benefit from additional government funds earmarked for remedial and gifted programs. However, this spending is wasteful insofar as it is used to solve the problem depicted in Graph II. Pupils 5 and 6, the special need pupils, could have their learning rates maximized with relatively cost-free transfers.
To summarize, how does an urban, disadvantaged youth use her voucher to improve her chances of being accepted into college? Due to early academic effort, she excelled her peers in achievement. A portion of the money that might have been used to implement an advanced track for her in her present school is placed into savings to defray college tuition. Instead of remaining unchallenged in her present school, she uses her voucher to transfer to an alternative school where she joins peers of similar achievement and thus enjoys more learning per hour.
But due to her above-average commitment in time-on-task even in her new school, she finds herself in a new predicament. Rather than acquiesce to the lower homework norms of her new school, she transfers to a third alternative where she joins peers of similar achievement, yet higher perseverance in time-on-task. The first transfer increased her learning rate, while the second transfer allowed her to increase her time-on-task without compromising this higher learning rate. Since both transfers accelerated her accumulation of education and circumvented the need to finance duplicate, advanced tracks, she is academically and financially prepared for college and thus earns acceptance. Hence, with a voucher, the desire to learn more was greeted with the opportunity to learn more in three instances.
None of her advancement was gained at others' expense. All other pupils' learning rates remained maximum and all parents' taxes remained minimum. She also encouraged and cooperated with peers of similar achievement and similar time-on-task in order to enjoy the efficiency of group learning.
On the supply side, vouchers make each school a more independent unit responsible for the average learning rate it offers its unique pupils (Kirzner, 1997). The resulting education system would be characterized as chaotic and unintelligible due to the absence of duplicate tracks among schools. But the new set of schools, offering families more diversity in instructional targets and norms for time-on-task, is a fairer and more efficient response to the inevitable variety of pupil circumstances.
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