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Educational policy analysis archives.
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c July 24, 1996
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National education 'goals 2000' : some disastrous unintended consequences / Robert H. Seidman.
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1 of 37 Education Policy Analysis ArchivesVolume 4 Number 11July 24, 1996ISSN 10682341A peerreviewed scholarly electronic journal. Editor: Gene V Glass,Glass@ASU.EDU. College of Educ ation, Arizona State University,Tempe AZ 852872411 Copyright 1996, the EDUCATION POLICY ANALYSIS ARCHIVES.Permission is hereby granted to copy any a rticle provided that EDUCATION POLICY ANALYSIS ARCHIVES is credited and copies are not sold.National Education 'Goals 2000': Some Disastrous Unintended Consequences Robert H. Seidman New Hampshire College (Southern New Hampshire University) ABSTRACT : "Goals 2000: Educate America Act" aims to, among other things, increase the high school graduation rate to at least 90% and eli minate the graduation rate gap between minority and nonminority students. However well in tentioned, this goal is doomed to failure. Powerful systemic forces converge to stabilize the high school graduation rate at about 75% where it has been since 1965 and where no tradition al national policy will be able to advance it very much. Even if education policy could succeed i n increasing the rate to 90% or beyond, undesirable consequences of potentially great magni tude, especially for the targeted minority groups, would result.Goals 2000: Educate America ActSec. 102 National Education Goals.(2) SCHOOL COMPLETION. (A) By the year 2000, the high school graduation rate will increase to at least 90 percent. (B) The objectives for this goal are that(i)the Nation must dramatically reduce its school d ropout rate, and 75 percent of the students who do drop out will successfully complete a high school degree or its equivalent; an d (ii) the gap in high school graduation rates betwee n American students from minority backgrounds and their nonmi nority
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2 of 37 counterparts will be eliminated. (Public Law 103227, 1994)I. Introduction The purpose of the "Goals 2000: Educate America Act is to promote "coherent, nationwide, systemic education reform." (Public Law 103227, 20 USC 5801) However well intentioned such an attempt at reform may be, one a spect is doomed to failure. With respect to School Completion (Goal 2), legislators and educati on policy makers ignore the laws and dynamics of the educational system at their own and our peril. The "system of education" is a vast and complex ent erprise comprising all of the many and different ways society educates it citizens. It is useful to distinguish it from the educational system which possesses a logic and laws of behavior of its own and which can be shown to be highly intractable to attempts to reform it by educ ation policy. This is particularly true with regard to "Goals 2000: Educate America Act." The theory of the logic and behavior of the educati onal system illustrates how powerful systemic forces converge to stabilize the high scho ol attainment rate at about 75% where it has been since 1965 and where no traditional national e ducation policy will be able to advance it very much. Even if education policy could succeed in inc reasing the rate to 90%, or beyond, undesirable consequences of potentially great magni tude, especially for the targeted minority groups, would result. One undesirable consequence is economic disaster fo r those who cannot or choose not to complete high school. They will be shut out of impo rtant noneducational social benefits (e.g., good job opportunities) unless alternative routes a re opened for them. Another consequence is the potential reduction of these very same social b enefits for those who do complete high school. A third consequence manifests itself as an unintend ed, but cruel hoax perpetrated upon the very minorities the Act seeks to help. By virtue of thei r being the last identifiable group to attain the high school diploma in proportion to their numbers in the age cohort, the high school diploma will not have the same power to secure social goods as it did with previous groups. Several policy alternatives are explored: 1) push t he high school attainment ratio to 100% quickly; 2) reduce the high school attainment rate to the 5560% level; 3) abandon the normative principle connecting the educational and socioecono mic systems. Part II presents a brief outline of a comprehensive and general theory of the logic and behavior of national educational systems (Green, 19 97). Certain of its laws and resulting dynamics are exposed. Part III presents a noncausal a priori aggregate model that illustrates certain systemic dynamics. Part IV presents an individual probabilistic utilit y model that extends the aggregate model. Both models illustrate systemic theory with respect to the Congressional Act and serve to locate critical stages in the growth of the educati onal system where education policy is most and least effective. Part V draws conclusions from the analyses of the t wo models and discusses several education and noneducational policy alternatives. Part VI is an analysis of the results of two models from Raymond Boudon which support the results reported here. Appendices A and B contain the mathematics of the I ndividual Utility model. Appendix C contains the mathematics behind the Aggregate model Appendix D contains an
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3 of 37educational attainment table.(Note 1)II. Theory of the Logic and Behavior of the Educati onal System A student who leaves school in the middle of the sc hool year in one part of the country and who enters the same grade in a distant part of the country can generally find nearly identical curricula, procedures and facilities. It appears th at some sort of system exists. Education policy is after all, policy for the educa tional system. But what is the educational system? What are its features? What are the laws of its behavior that set the system in motion? Answers to these questions can help us to assess th e potential impact of the Congressional Act. Primary Features. The primary features of the educational system are threefold: The set of schools and colleges, but not all school s and colleges. 1. These schools and colleges within the system are co nnected by a medium of exchange which includes those certificates, degrees, diploma s, and the like, that allow one to leave the Nth level of the system in one locality and ent er the Nth level in another. They are all instruments by which activities carried out in one place can be recognized and "exchanged" for similar activities of a school or college in so me other place. Certain schools and colleges will fall outside of t he educational system although they will be within the system of education. Certain propriet ary schools may not have their transcripts and diplomas recognized or accepted at other schools that are within the system. 2. The schools and colleges that make up the education al system and that are connected by a medium of exchange are arranged by a principle of s equence: the system of colleges and schools are organized into levels so that if a pers on has attained (i.e., completed) level N, then he or she has attained level N1, but not nece ssarily level N+1. 3. This principle allows us to speak of persons progre ssing through the system and seems to be a necessary property of any educational system d ue in part to differing levels of skill accomplishment, knowledge acquisition and the cogni tive development of individuals. Secondary Features. The system also has certain secondary or derivativ e elements. They are: size, a system of control and a distributive functi on. Distribution. Every society makes some sort of arra ngements for the distribution of its goods (i.e., benefits). The educational system dist ributes educational goods such as knowledge, skills, and certain kinds of taste, amon gst others. In addition to these goods, the system distributes their surrogates, or secondorder educational goods such as grades, diplomas, certificates and the like. 1. The derivative element of "control" is less relevan t for the present analysis than the others. It turns out that size is of central import since e ducation policy that is effective for one stage of systemic growth may be wholly ineffective at another. 2. System Size. The educational system has eight disti nct ways that it can grow (Figure II1). The present analysis focuses upon "growth in attain ment" not only because this is what the Act addresses, but because this mode of growth play s a crucial role in the dynamics of the system which in turn dooms Goal 2 of the Congressio nal Act to certain failure. (Note 2) 3.
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4 of 37 Figure II1. The Modes of Growth The system may expand in response to increases in t he schoolage population either by increasing the numb er of units in the system, or by increasing the number ofstudents in the units of the system, or both. 1. Growth in attainment. The system may expand byincreasing rates of attendance and survival. 2. Vertical Expansion. The system may expand by adding levels either at the top or at the bottom. 3. Horizontal expansion. The system may expand by assuming responsibility for educational and socialfunctions that are either new, that have been ignor ed, or that have been carried out by other institutions. 4. Differentiation. The system may expand either by differentiation of programs or institutions or both 5. Growth in efficiency. The system may expand by intensification, that is, by attempting to do more in the same time of the same in less time. 6. The system may expand by extending the school year or the school day. 7. The system may expand by increasing the number of persons needed to staff it independently of the number of students and number of its units, the magnitiude of the schoolage population, rates of attendance, surviva l. 8. (Green, 1997, p. 10) There are, however, two more pieces to the system t hat need to be developed before we can address the notion of growth and size. One is a normative principle connecting the social system with the educational system and the other is the systemic Law of Zero Correlation that relates the strength of the normative principle to system size. Normative Principle. It is true that some persons, for whatever reason, will come to possess a larger share of educational goods than ot her persons. This may be due to ability (however it is defined within the system), tenacity acuity of choice and any number of other reasons. If noneducational social goods such as income, ear nings opportunities and status are distributed by the socioeconomic system on the basi s of the distribution by the educational system of educational goods (through the instrument ality of secondorder educational goods), then there exists a normative principle that connec ts the educational and socioeconomic systems. This normative principle can be rendered as those h aving a greater share of educational goods merit or deserve a greater share of noneduca tional social goods. See Figure II2. The importance and power of this normative principle is as we shall see, a function of the size of the educational system as measured by the rate of high school attainment. It varies over different stages of systemic growth.
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5 of 37 Figure II2. Normative Principle Components Educationally Relevant Attributes Such as ... ability, tenacity, acuity of choice, and the like ...are linked to... Educational Benefits Such as ... knowledge, skills, taste, manners, standards of civility and the like ...are linked to... Surrogate Benefits Such as ... certificates, diplomas, transcripts, licenses, letters of recommendation, prestige andthe like ...are linked to... Noneducational Socioeconomic Benefits Such as ... income, employment, opportunities, status and the like Law of Zero Correlation. To understand this law, let us posit a uniform gro wth curve. Suppose that the educational system grows at a uniform rate over a one hundred year period. That is, there is a uniform increase (10% each decade) in the prop ortion of each successive agecohort attaining the 12th level of the system. (The actual growth da ta is shown in Appendix D.) When the high school attainment rate is low (e.g., 10% ) the socioeconomic meaning of high school attainment is likely quite negligible. Employers, all things being equal, would have little reason to choose a high school graduate over a nongraduate especially when there are so many of the latter. In the aggregate, high school a ttainers do not monopolize economic opportunities simply because of attainment. Thus th e strength of the normative principle is low. To be a high school drop out when most of your agecohort drops out presents no serious personal or social problem. See Part A of Figure II 3.
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6 of 37 Figure II.3. Uniform Growth Curve and Social Benefi ts of Attainment As the size of the educational system increases, th e power of the normative principle also increases. Employers now utilize high school attain ment as a selection criterion and social goods, such as status and jobs, begin to be preferentially distributed to high school graduates. See Part B of Figure II3. However, when the attainment rate reaches 100%, the mere possession of the high school diploma can have no socioeconomic meaning whatsoeve r. That is, no social goods can be distributed on the basis of high school attainment because everyone has the diploma. It is at this point (and at 0%), that the power of the normative principle is completely destroyed although its power may be weakened well before this point is rea ched. See Part C of Figure II3. The Law of Zero Correlation is a logical tautology. See Figure II4. It is a priori true. For instance, a society could not distribute any of its goods based upon eye color if everyone had the same color eyes. The actual shape of this curve and its inflection points is an empirical matter. However, the models presented here give us some gui dance in locating the theoretical inflection points. Figure II4. The Law of Zero Correlation There is a point of growth of the system at which t here is no longer any correlation between educational attainme nt and either the distribution of educationally relevant attribut es in the population or the distribution of noneducational s ocial goods associated with educational attainment. (Green, 1997, p.91) Law of Shifting Benefits and Liabilities. This is one of the many corollaries of the Law of Zero Correlation. This corollary assures that high schoo l attainment will have a declining social value
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7 of 37and that concomitantly, failure to attain the high school diploma will have an increasing social liability, as the attainment rate moves toward the 100% zero correlation point. Thus, as zero correlation is approached, the aggregate social ben efits of the attainment group and the aggregate liabilities of nonattainment both increase (Figure s II3 and II5) On the liability side, where school leaving was onc e a possible and viable alternative, it now becomes an evil to be avoided at all costs. The se shifting benefits and liabilities make high school attendance and attainment "compulsory" in wa ys that were surely never meant to be. The personal and social consequences of dropping out of high school can be devastating. The Law of Shifting Benefits and Liabilities does n ot specify the points in systemic growth (Sections A, B and C in Figures II3 and II5) wher e the benefits and liabilities of high school attainment shift. However, the two models presented in Parts III and IV do show that when 55% of the 17 yearold agecohort attains the high scho ol diploma, that group will receive the greater share of social benefits due to the moderate power of the normative principle. Figure II5. Shifting Liabilities of NonAttainment At this point in the growth of the educational syst em, high school attainment is efficacious in obtaining a disproportionate share of social goo ds. Thus, a high school diploma becomes a highly sought after good. This corresponds, in the actual growth of the system, to the year 1948. (See Appendix D) In addition, the models show that when the system b ecomes fairly large (i.e., 76% high school attainment in 1965), the power of this norma tive principle begins to decrease even though, historically, the personal and social belief in it remains high. This is prior to zero correlation setting in and may explain why the system has stabi lized at around 75% attainment and why it has been so resistant to attemps at education refor m. This is also the point at which the liabilities of nonattainment appear to increase dramatically and where the "drop out problem" becam e, politically, a problem to be dealt with. Figure II6 shows the combined effects of the Law o f Shifting Benefits and Liabilities and exposes a peculiar paradox: as zero correlation is approached, the aggregate social benefits once associated with high school attainment decline and the associated social liabilities of
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8 of 37 nonattainment increase. Figure II6. Shifting Benefits and Liabilities of A ttainment If one posits that Section C of Figure II6 represe nts the part of the growth of the system where the effects of these laws are maximally felt, then what would befall the minorities that the Congressional Act seeks to help? To address this qu estion, consider two more systemic principles: the Law of Last Entry and the Principle of the Moving Target. These two principles speak to the "Goals 2000" goal of closing the attai nment gap (and presumably, the social benefits gap) between minorities and nonminority students. The Law of Last Entry states that "as we approach t he point of universal attainment at any level of the system, the last group to enter and co mplete that level will be drawn from lower socioeconomic groups." See Figure II7. However, un like the Law of Zero Correlation, this law is neither tautological nor a priori but can be considered to be an empirical generali zation. The basis for this claim is given in much more detail e lsewhere (Green, 1997). Figure II7. The Law of Last Entry It appears to be true that no society has been able to expand its total educational enterprise to include the lower s tatus groups in proportion to their numbers in the population until the system is "saturated" by the upper and middle status groups. (Green, 1997, p.108) A corollary of the Law of Last Entry is the Princip le of the Moving Target, which states that as the group of last entry reaches its target of proportional 12th grade attainment rate, the target will shift. Note, that if the group of last entry pushes the attainment rate to 100%, then the high school diploma cannot, in and of itself, be us ed to distribute social benefits to anyone, much
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9 of 37less to this last group. Zero correlation will have set in and the target will have shifted to attaining a higher level of the educational system: postsecondary. However, even if the attainment rate does not reach 100% with the group of last entry (in this case, minority groups), this group will still not reap the same benefits of the high school diploma that previous groups reaped due to the Law of Shifting Benefits and Liabilities. The point in the attainment growth where this occurs is an empirical point. However, the models presented in this paper give us some theoretical gu idance. "Goals 2000" seeks to set and carry out a national policy to increase the high school attainment rate from its present level to at least 90%. If the rate stays below 100%, zero correlation would be avoided. I contend, however, t hat the effects of merely approaching zero correlation will be felt well before the 90% attain ment level is reached (if it ever could be reached!). As the theoretical models which follow s how, the felt effect could be one reason why the attainment rate has stabilized for so long at a bout 75%. Empirical confirmation can be found in (Green, 1997).III. THE AGGREGATE MODEL AND APPLICATIONSA. The Model The following Aggregate Model rests upon three ide alized assumptions: Noneducational social benefits are always normally distributed in the population under consideration and remain so over time a change in the high school attainment ratio does not affect the overall normal shape of this distrib ution; 1. This distribution encompasses those who have attain ed the high school diploma, but who have not gone on in formal schooling (attainers), a nd those who have not attained the high school diploma (nonattainers); 2. Society allocates its social benefits in such a way that the attainers monopolize the upper end of the normal distribution. 3. The first assumption fixes the overall shape of the distribution and offers a particular view of distributed justice. This distribution can be th ought to reflect some overall normally distributed attribute or attributes in the total po pulation under consideration. The second and third assumptions tell us that the high school atta iners can be found, as a group, lumped at the upper end of the distribution. The third assumption which admittedly represents an overly rigid meritocratic society, will be altered in the model presented in Part IV, These three assumptions are realized in Figure III2, which is a normal distribution in standardized normal form having a grand median () of zero and a standard deviation () of one. Each asymptote is truncated, for computational purp oses, at 3.9 standard deviations from the mean. The high school attainment ratio is represe nted by the shaded area under the curve. This is the proportion of the total population under con sideration that has attained the high school diploma. The median value of the social benefits of this group is (). The unshaded portion under the curve is the proport ion of the total population that has not attained the high school degree (~) and is equal to (1). The median value of the social benefit for this group is (~).
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10 of 37 Figure III1. Standardized Normal Curve for the Dis tribution of Social Benefits ( = high school attai nment ratio; ~ = nonattainment ratio; = grand median = 0; ()= median social benefit for attainer group ; (~) = median social benefit for nonattainer grou p; standard deviation = 1)Table III1 Median Social Benefits, Their Differences, and Thei r Rates of Change For Attainer and Nonattainer Groups by High School Att ainment Ratio(1) Size of Attainment Group: (2) Attainer Median:() (3) NonAttainer Group Median:(~) (4) () (~) (5) Rate of Change of () (6) Rate of Change of (~) 0.012.575 0.0122.587 ..0.05 1.960 0.0632.0230.23884.25000.10 1.645 0.1261.7710.1607 1.00000.15 1.440 0.1891.629 0.1246 0.50000.20 1.283 0.2531.536 0.1090 0.33860.25 1.150 0.3191.469 0.10370.26090.30 1.037 0.3851.422 0.0983 0.20690.35 0.935 0.4541.389 0.0984 0.17920.40 0.842 0.5241.366 0.09950.15420.45 0.755 0.5981.353 0.10330.14120.50 0.675 0.6751.350 0.10600.12880.55 0.598 0.7551.353 0.11410.11850.60 0.524 0.8421.366 0.12370.11520.65 0.454 0.9351.389 0.13360.11050.70 0.3851.037 1.422 0.15200.10910.75 0.3191.150 1.469 0.1714 0.10900.80 0.2531.283 1.536 0.20690.11570.85 0.1891.440 1.629 0.2530 0.12240.90 0.1261.645 1.771 0.3333 0.14240.95 0.0631.960 2.023 0.5000 0.1915
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11 of 37 0.99 0.0122.575 2.587 0.8095 0.3138 Note that the attainer and nonattainer medians cha nge as a function of the attainment ratio. When the ratio () is zero, the nonattainer median is equal to the grand median. When the ratio approaches its limit of one, the attainer med ian approaches the grand median and the nonattainer median approaches 3.9 standard deviat ions from the grand median. We can easily calculate the values of the attainer and nonattain er medians for different values of the attainment ratio.(Note 3) Table III1 shows their values, thei r differences and their rates of change for attainment ratios ranging from 0.01 to 0.99. Figure III2 is a plot of the attainer and nonattainer medians by the attainment ratio. Figure III2. Median Social Benefit of Attainer Gro up (()) and NonAttainer Group ((~)) by High School Attainment Ratio (%) () (fr om Table III1, Columns 2 and 3) B. An Income Disparity Analysis A conventional analysis of high school attainer and nonattainer income disparities considers whatever is gained by the attainers to be the magnitude of the liability experienced by the nonattainers. If, for example, the median inco me of the attainer group is 150% of the nonattainer median income (at a particular attainm ent ratio), then the benefit to the former group is 50% while the liability to the latter group (in foregone income and earnings opportunities, etc.) is 50%. This approach tends to conceal the full imp act of the shifting benefits and liabilities of educational attainment. Table III1 and Figure III1 display another approa ch to this situation. Here we find the difference between the median benefit of the attain er group and the median benefit of the entire population under consideration (Table III1, column 2). We do the same for the nonattainer group (Table III1, column 3). The difference betwe en these two grandmediandispersions is a measure of the relative position of one group with respect to the other (Table III1, column 4). If we think of such social benefits as income, sala ry and wages, then a conventional supply and demand analysis suggests that as the supply of high school graduates increases, the relative
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12 of 37social benefits realized by these graduates, with r espect to those with no high school degree, will decline (given a constant market demand for attaine rs). This is just what happens in the Aggregate Model as the attainment ratio grows from 0.01 to 0.50. However, as the attainment ratio exceeds 50%, the relative advantage of the at tainers over the nonattainers increases.(Note 4) See Figure III2. These latter results of the model are consistent wi th certain empirical findings. Timeseries U.S. Census data for 18yearold to 24yearold mal es from 1939 (when the national high school attainment ratio was 50%) to 1990 display this phen omenon. (Note 5) A U.S. Senate report which examined the incomes of 24to 34yearold ma les expressed surprise at the "paradox" of increasing relative income for high school attainer s over nonattainers. (Note 6) The interaction between the Law of Zero Correlation and the Law of Shifting Benefits and Liabilities has certain explanatory power when the data are examined as illustrated in the Aggregate Model. The "paradox," cited above, evapor ates in light of these systemic dynamics which show the declining benefits associated with a ttainment and the increasing liabilities associated with nonattainment as the zero correlat ion point is approached. (Note 7) C. Stabilization of the High School Attainment Rati o What is the meaning of the "intersection" of the be nefit and liability curves in Figure II6? Although the two curves do not actually intersect ( they have different vertical axes), the "intersection" shown in Figure II6 does illustrate certain interactive systemic effects. This "intersection" can be viewed as an equilibrium poin t in the growth of the system beyond which it no longer pays (in aggregate social benefit terms) to finish high school but is quite a serious social disaster not to do so. In a way, it is an ag gregate recognition of the Law of Zero Correlation and the Law of Shifting Benefits and Li abilities. This phenomenon is illustrated by the Aggregate Model. Figure III3 is a plot of the rate of decline of th e social benefits of attainment generated by the model. Note that after an attainment ratio of 0 .20 the median value declines at a fairly constant rate until the high school attainment rati o reaches 50%. At this point in the growth of the educational system, the rate of decline increases a nd increases sharply at 75% attainment. High School Attainment Ratio
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13 of 37Figure III3. Rate of Change of Attainer Group Medi an (Ordinate) by High School Attainment Ratio (%) (from Table III1, Column 5)Figure III4 is a plot of the rate of decline of th e nonattainer median. Here the median declines at a decreasing rate until 75% attainment at which poi nt the rate begins to increase and then increases sharply at 80% attainment. Figure III4. Rate of Change of NonAttainer Group Median by High School Attainment Ratio (from Table III1, Column 6) Thus, the two curves shown in Figure III2 can be s aid to contain inflection points which occur in the growth of the system where the high sc hool attainment ratio is about 75%. The stabilization of the national attainment ratio at a round 75% may be the social recognition of the phenomenon described by the model.(Note 8) Is it purely coincidental that the inflection point s in the model occur where the national high school attainment ratio has stabilized: at abo ut 75%? Nevertheless, the model does serve to illustrate the phenomenon of systemic "equilibrium" reflecting the interactive dynamics between certain systemic laws. The interaction between thes e laws offers an account of certain systemic phenomena. The behavior of the educational system described ab ove is based upon these systemic features: the Principle of Sequence, the distributi on of secondorder educational goods and the size of the system as measured by the attainment ra tio at the twelfth level. Systemic behavior was driven by the power of a logical tautology, its cor ollary and a normative principle linking the educational and social systems. It is ironic that t he "successful" growth of the system, as measured by an increasing high school attainment ra tio, appears to sow the seeds of a particularly harsh and peculiar brand of failure. (Note 9)IV. THE PROBABILISTIC UTILITY MODEL
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14 of 37 The idealized society reflected in the three assump tions underlying the Aggregate Model is a rigidly meritocratic one. By altering the first a nd third assumptions, (see Section IIIA), we can build a model that reflects a society that distribu tes its noneducational social goods in a somewhat more flexible manner. Like the Aggregate Model, let us assume that the po pulation under consideration is dichotomized into those who have attained the high school diploma (and nothing beyond it) and those who have not attained the degree. Furthermore let us assume two independent normal distributions of social goods, one for the attainme nt group and the other for the nonattainment group. This state of affairs is illustrated in Figu re IV1. Now let us assume that both of these normal distrib utions have identical standard deviations. Thus, we can normalize each of the dist ributions and leave them superimposed, one upon the other, on the social benefits axis. Note t hat the relative position of the two normal curve means remains unaffected by the standardization (i. e., the standardized and unstandardized means remain stationary). These standardized distri butions are shown in Figure IV1. A. The Standardized Normal DistributionsConsider the two standardized normal distributions shown in Figure IV1. Let curves X() and X(~) represent the distributions of earnings opport unities of high school attainers and nonattainers, respectively. Both curves have their asymptotes truncated, to facilitate the computations to follow, at 3.0 standard deviations above and below their respective means of zero and are superimposed upon a common axis, X, sh owing an apparent overlap area, E: that area under both curves which has a common Xaxis ra nge. Figure IV1. Two Overlapping Standardized Normal Cu rves We let stand for the ratio of high school attaine rs to the total population under consideration and let stand for the meritocratic parameter. This parameter represents those in the total population, and in particular that propor tion of distribution X() which monopolizes the highest values of X. It is clear from Figure IV 1 that this parameter imposes an upperbound on the range of distribution X(~) (i.e., I(A)) and concomitantly places a lowerbound on the range of X() (i.e., I(D)). Except where = 0, the ranges of X() and X(~) differ. Let us assume that despite changes in the size of the original nonstandardized normal distributions retain their normal shapes and contin ue to have identical standard deviations and unchanged means. The X() mean remains forever fixe d and thus for any given only a change in can shift the X(~) curve. A mean/medium analysis of these curves is presented in Appendix B. Unlike the Aggregate Model, individuals in X() (i. e., high school attainers) are no longer guaranteed an advantage over persons in X(~) (i.e., nonattainers), with respect to some value of X (level of social benefit). The question now sh ifts from one of absolute advantage (as in the Aggregate Model) to one of relative advantage. We n ow ask, what is the probability that an
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15 of 37individual will be advantaged with respect to X, ov er changes in and in ? The symbols in Figure IV1 refer to proportions and are explained in Table IV1, below.Table IV1 PROPORTIONAL VALUES OF SECTIONS IN FIGURE IV1Section Symbol Meaning A The proportion of the population which is in X() and whichmonopolizes the highest X values.This is the value of the meritocraticparameter. B1 The proportion of the population which is in X() and which does notmonopolize the highest X values. C1 The proportion of the population which is in X(~) and which is notrelegated to the lowest X values. D The proportion of the population which is in X(~) and is relegated tothe lowest X values. E The area of "intersection" of Section B of X() and Section C of X(~). The above conceptualization allows us to calculate the probabilities of persons falling in any of the five sections of Figure IV1 as a functi on of and . These probabilities are conditional probabilities of independent events. Ta ble IV2 gives the formulae for these calculations.Table IV2 PROBABILITIESSection ProbabilityMeaning APr(AX())= The probability of residing inSection A is the conditionalprobability of residing in A giventhat one already resides in X(). BPr(BX())=(1) The probability of residing inSection B is the conditionalprobability of not residing in Agiven that one resides in X(). CPr(CX(~))=(1)(1) The probability of residing in Section C is the conditionalprobability of not residing in Dgiven that one resides in X(~).
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16 of 37DPr(DX(~))=(1) The probability of residing in Section D is the conditionalprobability of residing in D giventhat one resides in X(~). E1 Pr(ECX(~))=(1)(1) The probability of residing inSection E given that one isalready in X() is theconditional probability of residing in E (i.e., ) given thatone resides in X(~) and residesin Section C. E2Pr(EBX())=(1) The probability of residing inSection E given that one isalready in X() is the conditionalprobability of residing in E giventhat one resides in X() andresides in B B. Interpretation of Area E The move from proportions in Table IV1 to probabil ities in Table IV2 is a crucial one. Recall that each distribution represents one part o f the dichotomized total population under consideration. The overlapping area E, is not a sha red population between the two groups. It simply illustrates the common range of X shared by area B in X() and C in X(~). Each person in the total population under considera tion has a probability of ending up in one of the two distributions. Since is the propor tion of the total population that has attained the twelfth level, any individual has probability of falling under distribution X() (all other things being equal). Similarly, the probability of not att aining at level 12 is equal to (1). Of course, + (1) equals 1.0, which is the total population under consideration. All of this follows from the laws of proportions. Consider Figure IV1. As Section A changes in size, X(~) shifts to the left or to the right (recall that we have assumed that changes in do n ot affect the shape or position of the distributions). The entire area under any one of th e two distributions is equal to 1.0. Thus, if represents the value of the area of Section A, then 1 is the area of Section B. From this we can see that the conditional probability of an individu al being an attainer and being a monopolizer of the higher values of X is . The laws of symmetry make Section D equal to Sectio n A. Thus, the probability of an individual being a nonattainer and being relegated to the lowest values of X is (1). Similar arguments can be made for Sections B and C. The pro babilistic interpretation of Section E is a more complicated matter, however. Although Sections B and C do not actually have an a rea in common, they do share the common Xaxis range, I(D) to I(A). It is useful to think of Section E as if it is the area of overlap between the two distributions. Recall that the prob ability of being in C is simply (1)(1). Now, the probability of being in C and at the same time being within the scope of distribution X() is just the probability of being in C times th e area of Section E. Similarly, the probability of being in B is (1). The probability of being in B and within the scope of distribution X(~) is just the probability of being in B times the area o f Section E. It should now be clear that Pr(ECX(~)) is the pro bability of any individual nonattainer falling in the same range with and being under the same scope as an attainer. Likewise, Pr(EBX()) is the probability of any individual a ttainer falling in the same range with and being under the same scope as a nonattainer. These two p robabilities need not always be equal. In fact,
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17 of 37 they are equal only when = 0.50. What remains is to calculate the area of Section E (i.e.,). This is done in Appendix A. C. Results of the Analysis Tables IV3 and IV4 give the probabilities of fall ing in Section E given attainment and of falling in Section E given nonattainment, respecti vely. These Tables are derived from the probability formulae in Table IV2. To obtain the p robabilistic marginal utilities of attainment, we simply perform a matrix subtraction, Table IV4 minus Table IV3. The results of this subtraction are shown in Table IV5. Note that the marginal utilities decrease for const ant and increasing , and decrease for constant and increasing . Furthermore, each colu mn, reflects about the row where = 0.50 so that each column below this row is the negative con verse of the column above. An inspection of Table IV5 shows that it is not in dividually advantageous to obtain the high school diploma until 55% of the population und er consideration (17year old age cohort) does so. The row where =0.50 can be considered to be the indifference level. However, a mean/median analysis shows that, in the aggregate, it is always advantageous to be an attainer rather than a nonattainer. This is so because for all values of , () is greater than (~) (except when they are equal, when = 0). A complet e mean/median analysis is given in Appendix B. See columns 4 and 6 in Table B1. This analysis of the Probabilistic Utility Model ex poses an interesting paradox: in the aggregate it is more advantageous to be an attainer no matter what and are; individually this is not always the case. Furthermore, Table IV5 ind icates that the marginal disutility of not attaining the high school degree increases as attai nment increases and also increases as the meritocratic parameter decreases! This phenomenon c an be vividly seen in the lower lefthand quadrant of Table IV5. This quadrant corresponds to the decreasing power o f the normative principle as the attainment rate increases toward 100%. As we move f rom the upper righthand to the lower lefthand corner on the quandrant diagonal, disutil ities can be seen to double, triple and even quadruple at various steps. Table IV3 PROBABILITY OF FALLING IN SECTION E GIVEN ATTAINMEN T OF LEVEL 12 Meritocratic Parameter () *0.10 0.200.300.400.500.600.700.800.900.950.010.00350.00220.00150.00100.00070.00040.00020.00010.0 0000.00000.050.01760.01120.00760.00510.00330.00210.00120.00050.0 0020.00010.100.03530.02240.01520.01010.00670.00420.00230.00110.0 0030.00010.150.05290.03360.02280.01520.01000.00620.00350.00160.0 0050.00020.200.07060.04480.03040.02030.01340.00830.00470.00220.0 0060.00020.250.08820.05600.03800.02540.01670.01040.00580.00270.0 0080.00030.300.10580.06720.04560.03040.02000.01250.00700.00330.0 0100.00030.350.12350.07850.05320.03550.02340.01460.00820.00380.0 0110.00040.400.14110.08970.06080.04060.02670.01660.00940.00440.0 0130.00040.450.15880.10090.06840.04560.03010.01870.01050.00490.0 0140.00050.500.17640.11210.07590.05070.03340.02080.01170.00550.0 0160.0005
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18 of 37 0.550.19400.12330.08350.05580.03670.02290.01290.00600.0 0180.00060.600.21170.13450.09110.06080.04010.02500.01400.00660.0 0190.00060.650.22930.14570.09870.06590.04340.02700.01520.00710.0 0210.00070.700.24700.15690.10630.07100.04680.02910.01640.00770.0 0230.00070.750.26460.16810.11390.07600.05010.03120.01760.00820.0 0240.00080.800.28220.17930.12150.08110.05340.03330.01870.00880.0 0260.00080.850.29990.19050.12910.08620.05680.03540.01990.00930.0 0270.00090.900.31750.20170.13670.09130.06010.03740.02110.00990.0 0290.00090.950.33520.21300.14430.09630.06350.03950.02220.01040.0 0310.0010* Proportion of 12th Level Attainers Table IV4 PROBABILITY OF FALLING IN SECTION E GIVEN ATTAINMEN T BELOW LEVEL 12 Meritocratic Parameter () *0.100.200.300.400.500.600.700.800.900.950.010.34930.22190.15040.10040.06610.04120.02320.01090.0 0320.00100.050.33520.21300.14430.09630.06350.03950.02220.01040.0 0310.00100.100.31750.20170.13670.09130.06010.03740.02110.00990.0 0290.00090.150.29990.19050.12910.08620.05680.03540.01990.00930.0 0270.00090.200.28220.17930.12150.08110.05340.03330.01870.00880.0 0260.00080.250.26460.16810.11390.07600.05010.03120.01760.00820.0 0240.00080.300.24700.15690.10630.07100.04680.02910.01640.00770.0 0230.00070.350.22930.14570.09870.06590.04340.02700.01520.00710.0 0210.00070.400.21170.13450.09110.06080.04010.02500.01400.00660.0 0190.00060.450.19400.12330.08350.05580.03670.02290.01290.00600.0 0180.00060.500.17640.11210.07590.05070.03340.02080.01170.00550.0 0160.00050.550.15880.10090.06840.04560.03010.01870.01050.00490.0 0140.00050.600.14110.08970.06080.04060.02670.01660.00940.00440.0 0130.00040.650.12350.07850.05320.03550.02340.01460.00820.00380.0 0110.00040.700.10580.06720.04560.03040.02000.01250.00700.00330.0 0100.00030.750.08820.05600.03800.02540.01670.01040.00580.00270.0 0080.00030.800.07060.04480.03040.02030.01340.00030.00470.00220.0 0060.00020.850.05290.03360.02280.01520.01000.00620.00350.00160.0 0050.00020.900.03530.02240.01520.01010.00670.00420.00230.00110.0 0030.00010.950.01760.01120.00760.00510.00330.00210.00120.00500.0 0020.0001* Proportion of 12th Level Attainers Table IV5 PROBABILISTIC MARGINAL UTILITIES OF ATTAINMENT OF L EVEL 12 Meritocratic Parameter ()
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19 of 37 *0.100.200.300.400.500.600.700.800.900.950.010.34600.22000.14900.09900.06500.04100.02300.01100.0 0300.00100.050.31800.20200.13700.09100.06000.03700.02100.01000.0 0300.00100.100.28200.17900.12200.08100.05300.03300.01900.00900.0 0300.00100.150.24700.15700.10600.07100.04700.02900.01600.00800.0 0200.00100.200.21200.13400.09100.06100.04000.02500.01400.00700.0 0200.00100.250.17600.11200.07600.05100.03300.02100.01200.00500.0 0200.00100.300.14100.09000.06100.04100.02700.01700.00900.00400.0 0100.00000.350.10600.06700.04600.03000.02000.01200.00700.00300.0 0100.00000.400.07100.04500.03000.02000.01300.00800.00500.00200.0 0100.00000.450.03500.02200.01500.01000.00700.00400.00200.00100.0 0000.00000.500.00000.00000.00000.00000.00000.00000.00000.00000.0 0000.00000.550.03500.02200.01500.01000.00700.00400.00200 .00100.00000.00000.600.07100.04500.03000.02000.01300.00800.00500 .00200.00100.00000.650.10600.06700.04600.03000.02000.01200.00700 .00300.00100.00000.700.14100.09000.06100.04100.02700.01700.00900 .00400.00100.00000.750.17600.11200.07600.05100.03300.02100.01200 .00500.00200.00100.800.21200.13400.09100.06100.04000.02500.01400 .00700.00200.00100.850.24700.15700.10600.07100.04700.02900.01600 .00800.00200.00100.900.28200.17900.12200.08100.05300.03300.01900 .00900.00300.00100.950.31800.20200.13700.09100.06000.03700.02100 .01000.00300.0010* Proportion of 12th Level AttainersV. RESULTS, CONCLUSIONS, CONJECTURES and POLICY ALTERNATIVES These models illustrate the theoretical limitations of education policy designed to increase the high school attainment rate to 90% or above and to help minorities share in the "benefits" of educational attainment. They are formal models and are not grounded in empirical results. Like Raymond Boudon's models (see Part VI), they avoid t he crosssectional and variable confounding of survey data. They illustrate the pow er of a logical tautology in conjunction with a normative principle. However, these idealized model s are not without limitations The Aggregate Model seems, on the face of it, too m eritocratic for our present society. The distribution of social benefits may not in reality, be normal and their means (as shown in the Utility Model) may not remain constant with systemi c growth (which is clearly not the case in the Aggregate Model). Nonetheless, these models can ser ve as "benchmarks" against which to compare other logicomathematical models containing different assumptions, and still others based upon empirically derived data. They also add to our database of models. Policy Alternatives. The results of the models developed in this analys is suggest a number of possible alternative education policy scenarios. Th ree such follow. Push the High School Attainment Rate to 100% quickl y.
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20 of 37Given that attempts to reduce social inequalities b y increasing the national high school attainment ratio will fail, what would be the conse quences of entirely eliminating educational attainment inequality at the high schoo l level? That is, push the high school attainment rate to 100% so that the high school dip loma can no longer be the basis for the distribution of noneducational social goods. This approach has two major pitfalls. First, the sy stem had better reach 100% attainment very quickly so as to minimize the hardships that w ill have to be endured by the ever decreasing percentage of nonattainers. Second, eve n if such a result could be achieved, the original inequality problems would remain unsolved since the problems would merely be shifted to the next higher level of the educational system postsecondary. If the normative principle persists (and there is n o reason to assume that it will not) then the distributional instrument of social goods will shift to the postsecondary level. This level is, for the most part, selective. One does no t only choose to go on, one is chosen. Thus, enormous pressures will come to bear upon thi s level to alter its selectivity feature. One can argue that this pressure is already fairly strong. Reduce the High School Attainment Rate to the 5560 % Level. This level is below the "equilibrium point" of the Aggregate Model and close to the "indifference" level of the utility model. This is the point at which the effects of the decline in the social benefits of attainment and the precip itous rise in the social liabilities of nonattainment are (theoretically) thought to begin Of course, careful consideration needs to be given to the provision of ample opportunities for all to continue their education (i.e., pursue l earning). Such a policy must avoid an inequitable distribution of the nonattainers based on educationally irrelevant attributes such as race, class and ethnic background. Admitted ly, a policy of this sort would not enjoy widespread political support. Abandon the Normative Principle. The two previous a lternatives assumed the continued presence of the normative principle. But what would life be like without it? The abandonment of this principle might be the most eff icacious, but a politically and socially difficult, way to reduce educational and socioecono mic inequality. If educational attainment is no longer used as an i nstrument for the distribution of noneducational social goods, then perhaps educatio n could once again be pursued for the benefits that are intrinsic in the educational good s themselves and not for the socioeconomic advantages that disappear and reappea r with ever increasing rates and different levels of attainment. Such a move might signal the end of the illusion th at the educational system is a solution to practically every social ill. I do not claim to know just what new instruments for the distribution of social benefits would arise, nor ho w one could go about judging their desirability as a replacement for educational attai nment. However, a reconsideration of the socioeconomic normative principle that disproportio nately rewards formal educational attainment might prove to be a beneficial exercise. VI. ANALYTICAL POSTSCRIPT: BOUDON'S MODELS OFINEQUALITY OF EDUCATIONAL and SOCIAL OPPORTUNITY Two models created by the French Sociologist, Raymo nd Boudon (Boudon, 1974) support the results of the two models presented here. Boudo n's models are of inequality of educational opportunity (IEO) and inequality of social opportun ity (ISO). He analyses their relationship to one another and to the educational and social syste ms. Some of Boudon's relevant results and analyses follow.
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21 of 37A. Boudon's IEO and ISO Models and the Theory of Ed ucational Systems Boudon's models and his analyses are highly suggest ive in many ways. In addition to a methodological approach which avoids some of the pi tfalls of factorial analysis (i.e., partial accounting for total variance, crosssectional "ill usions," and lack of quantitatively adequate data), Boudon adds an important dimension to the de scription of the normative behavior of the type of educational system spawned in Western indus trial societies. This dimension, system animation, is of fundamental import in helping to p rovide a clear and precise picture of the dynamics of systemic motion. By observing (and modeling) the overtime cumulativ e effects of the various factors affecting the educational system's growth, Boudon i s able to discern the logical limits and consequences of this growth. The ceilingeffect and the exponential mechanism that combine to drive the IEO model help generate a number of obser vations and paradoxes that bear significantly upon the theory of educational system s as presented here. Some Familiar Paradoxes One of the most striking paradoxes generated by Bou don's models is that "other things being equal" (which is seldom the case), educationa l growth has the effect of increasing social and economic inequality. This happens even when the system becomes more egalitarian with respect to educational opportunity (EO). This paradox rests upon the assumption that income is dependent upon educational attainment level. Over time, educational level and socioeconomic status increase with educational level increasing more rapidly the highe r the socioeconomic level. Since both of these factors are "independently" responsible for income differentials, "economic inequality will increase over time along with social inequality, fo r the latter is correlated with the former." (Boudon, 1974, pg. 188) The paradox is completed when we add another import ant conclusion reached by the application of Boudon's model: change in social str atification is the only factor that can substantially affect the model's exponential mechan ism and hence ISO. This leads Boudon to conclude that educational growth can partially expl ain the "persistence of economic inequality in Western societies." (ibid., 188) It is quite remark able that Boudon's model and the models pesented here reach identical conclusions using suc h different but complementary methods. The SuccessBreedsFutility Paradox Another paradox illustrates just how the apparent s uccess of the educational system leads to futility for some participants and how the syste m fuels the fires of its own expansion. Boudon's models indicate that one of the main endogenous fac tors responsible for the increase in educational demand is the overtime change in the s tatus expectations of individuals with respect to educational level. ...as time goes on, the structure of expectations a ssociated with the two highest levels of education is constant; intermediate level s are affected most adversely; the structure of expectations relating to the lowest le vels of education becomes less favorable, too, but it is less influenced by the ov erall educational increase than are the intermediary levels. (ibid., 149) Thus, as IEO decreases over time and the educationa l system expands at all levels, the social status expectations for persons at intermediate edu cational levels decrease and these persons must raise their levels just to maintain constant s ocial status expectations. This treadmill effect
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22 of 37means that while the relation between educational l evel and social status changes very little over time, the number of years of schooling associated w ith each of the educational levels increases. Thus, while the average level of educational attain ment in the population increases, the educational levels that are associated with particu lar status expectations are "simultaneously moving upward." As individuals demand more and more education over time, the individual return tends to be nil, while the aggregate return on this demand is high. The lower socioeconomic classes are compelled to demand more education (especially if the higher classes to do), for not to do so condemns these lower class es to constantly falling social status expectations. However, more educational demand only retards this diminution in status and does not increase the lower class's chances of achieving increased social status. This is a particularly frustrating paradox, for in a meritocratic society where the normative principle holds, an individual seems to have an adv antage in securing as much education as he or she can. However, when many individuals seek additi onal education, the aggregate effects of this demand decrease the social status expectations asso ciated with most of the educational levels. This causes people to demand even more education in the next time period. This paradox lends support to a number of results d ue to the interactions between various systemic principles such as the Law of Zero Correla tion, the Principle of Shifting Benefits and Liabilities, the Law of Last Entry, and the Princip le of the Moving Target. Boudon shows that when expectations associated with some particular e ducational level become reduced, a decrease in expectations at all levels results. (ibid., Tabl e 8.4, 147) Boudon sees evidence that this point has been reach ed at the secondary level in some industrial societies, but "it seems that not even t he most advanced industrial societies have achieved a proportion of college students so large that a severe decrease in the expectations at this level can be observed." (ibid., 150) One wonde rs whether or not the American educational system has moved to a point beyond Boudon's claim? Because of their logicomathematical nature, the models presented here are generalizable over all systemic levels. Already, over 60% of the high school graduates enter higher education al institutions (National Center for Education Statistics, 1994). It may not be long before the sy stem approachs zero correlation at this level! Perhaps in anticipation of zero correlation at the college level, Thurow has called for a "system of postsecondary education for the noncol lege bound student" (Thurow, 1994). However, I suggest that such a "system" (even if es tablished independently of the educational system) would itself be absorbed into the education al system and therefore be subject to its laws and thus perpetuate the paradoxes discussed here. S uch is the power of the dynamics of the educational system.(Note 10)B. Further Observations on Systemic Growth While the paradoxes generated by Boudon's model are important for establishing the boundaries and limitations of educational systems, there are other observations on growth that warrant exploration. Boudon, in his Appendix to Chapter 9, indicates tha t by manipulating the demand for education (i.e., predicating demand in the educatio nal system upon exogenous rather than endogenous factors), equality of educational opport unity (EEO) can be affected. This is the only alternative, other than changing social stratificat ion, that he offers to remedy IEO and ISO. Now, if the number of positions (student slots) in the educational system at the highest level remains unchanged and if the number of positi ons at the middle level is increased by D during time period t to t+1, and if the number of p ositions at the lowest level is decreased by D during this same time period then, how is the nu mber of persons with lowest social background T(t ) who reach at least the middle educ ational level affected by the value of D? Boudon concludes on the basis of this "modified" mo del that T(t) is an increasing function
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23 of 37of time and an increasing function of D. Furthermor e, T(t) increases at a decreasing rate as a function of processphase. According to Boudon, the duration's of the three phases are a function of D ("an increase in D has the effect of shortenin g the first and second phases..."). Thus nonlinear returns in T(t) are associated with incr ease in the value of D. This thesis is presented in expanded form in (Boudon, 1976). This "modified" model (reflecting an "idealtypical planned educational system") results in a decrease in IEO through the manipulation of deman d, while the IEO parameter, "a", remains constant over time. (This IEO parameter has marked similarities to the meritocratic parameter, , presented in the Aggregate and Individual Utility m odels.) The freemarket endogenous educational system creates what appear to be insurm ountable problems (i.e., the paradoxes). On the other hand, the exogenous educational system permits us in theory at least, to correct some of these undesirable effects. Boudon r ightfully questions the high social costs of this remedy. Nevertheless, this "modified" model ma y provide additional insights into the growth mechanism of the system and may have enormous impli cations for policy and planning especially if the demand for education is to be con trolled. It deserves further study. C. A Logistic Growth Curve In an intriguing footnote (ibid., 201, ff.3), Boudo n suggests that in conjunction with the paradoxes cited above, there is a particular point in the freemarket educational system development where "growth is more rapid at the high er level than at the secondary level and thus a decrease in IEO and ISO is curtained." (ibid., 19 9) This growth, fueled by unrestrained demand for more education, may lead to a state of "latent crisis." This runaway exponential growth trend may be checked by a "braking process" that is propo rtional to the trend, leading to a logistic rather than an exponential growth curve. What are the circumstances that would lead to this braking process and would these circumstances be endogenous or exogenous to the edu cational system? The answers to these questions are fundamental to education policy. Thes e answers appear to be intimately related to many of the systemic principles in the general theo ry of educational systems. Finally, what is to be made of Boudon's enigmatic s tatement that "the concern of all industrial societies with shortterm higher educati on can be better understood in the light of the dialectic between the exponential growth of educati onal demand and the (proportional) braking process...?" Perhaps the theory of the educational system and the models put forward here can shed some light on this question.Notes1. Originally presented at the Annual Meeting of th e American Educational Research Association San Francisco, California April 19, 1995 Session 16 .36 The Political Context of Educational Reform (Division G; SIG/Politics of Education). Som e of the ideas and models presented here have appeared in various forms and stages of comple tion in previous works. In particular: Thomas F. Green(1997) with David P. Ericson and Rob ert H. Seidman; Seidman(1982); Seidman(1981). The analysis of Boudon's work has ne ver been published. 2. This paper uses the high school attainment rate as the measure of systemic "size due to growth in attainment." One reason is that this is what the Congressional Act focuses on. Another, is that the 12th grade is the last level of the educational system that is nonselective. For the most part, one not only chooses to go on to postsecondary edu cation, one is chosen. It is this fact, together with certain systemic laws, that illustrates the in herent futility of certain education policies at particular stages of systemic growth.I use the 17 yearold agecohort to measure the hig h school attainment rate. This is the cohort
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24 of 37used by the National Center for Education Statistic s (1995) to track the high school attainment ratio since 1869. The models presented here are bas ed upon a dichotomized population: those who have not completed high school and those who ha ve but have not gone on to the postsecondary level of the system. However, some researchers use a different agecohor t. For example, the National Education Goals Panel uses the 1920 yearold age cohort (Nat ional Education Goals Panel, September 1994). Other studies report high school completion rates amongst various age cohorts, including 2122 yearolds and even 2930 yearolds (National Center for Education Statistics, 1993).The numeric ratios will differ, of course. A standard m easure of high school "completion and school leaving" has been proposed. The "appropriate unit o f analysis" is the graduating class cohort.(Hartzell, 1992).3. A sample calculation can be found in Appendix C.4. It is probably unreasonable to apply the model a t the lower attainment rates where the power of the normative principle is very low. However, th e model does serve to illustrate the idea that the relative benefit disparity between the two grou ps first decreases and then increases. This phenomenon suggests that a particular educational p olicy appropriate for one stage of systemic growth may not be appropriate for another.5. U.S. Bureau of the Census, Decenial Census Repor ts for 1940, 1950, 1960, 1970; Current Population Reports, P60, nos. 85, 90, 92, 97, 101. U. S. Bureau of the Census, Current Population Reports for 1984, 1987, 1990; P70, nos. 11, 21, 32 ("Educational Background and Economic Status").6. See Levin(1972) for a traditional analysis of th e relevant data. 7. For an extended analysis from another methodolog ical perspective, see Appendix C in (Green, 1997).8. See the Table reproduced in Appendix D (National Center for Education Statistics, 1995). It is interesting to note that the U.S. Government projec tion of the high school attainment ratio to the year 2006 keeps it at about 74% (using the 18 yearold cohort). Why? No reason is given. See Tables 26 and B4 (National Center for Education Sta tistics, 1996). 9. This irony (in the form of paradoxes) is address ed by Boudon (1974) and is analyzed in Part VI above. Boudon's models confirm the results of th e Aggregate and Individual models. 10. For an example of such an absorption scenario, see Seidman's (1982) analysis of the "lifelong learning system." REFERENCES Boudon, Raymond (1974) Education, Opportunity, and Social Responsibility. NY: Wiley. Boudon, Raymond (1976) "Elements pour une theorie f ormelle de la mobilite sociale" Quality and Quantity, 5; pp. 3985.Green, T.F., with Ericson, D.P. and Seidman, R.H., (1997). Predicting the Behavior of the Educational System. NY: Educator's International Pr ess (Classics in Education Series). Hartzell, G., McKay, J. & Frymier, J. (1992), "Calc ulating Dropout Rates Locally and Nationally
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25 of 37 with the Holding Power Index." ERIC ED 343 953.Levin, Henry M. et al. (1972) The Cost to the Natio n of Inadequate Education. A Report Prepared for the Select Committee on Equal Educatio nal Opportunity of the U.S. Senate. Washington, DC: U.S. Government Printing Office.National Center for Education Statistics (1993) Dro pout Rates in the United States: 1992. Washington, DC: U.S. Government Printing Office. (E RIC ED 363 671) National Center for Education Statistics (1996) Pro jections of Education Statistics to 2004. Washington, DC: U.S. Government Printing Office.National Center for Education Statistics (1995) Dig est of Education Statistics. Washington, DC: U.S. Government Printing Office.National Education Goals Panel (September 1994) "Na tional Education Goals Report: Building a Nation of Learners" Washington, DC: U.S. Government Printing Office. Public Law 103227 (1994), "Goals 2000: Educate Ame rica Act". 103d Congress. Seidman, R.H. (1981) "The Explanatory Power of Two Idealized Models of Educational and Social Attainment." American Research Association A nnual Meeting, Los Angeles, CA. (ERIC: ED 206068)Seidman, R.H. (1982). The Logic and Behavioral Prin ciples of Educational Systems: Social Independence or Dependence?, chapter in M.S. Archer (Ed.), The Sociology of Educational Expansion: Takeoff, Growth and Inflation in Educat ional Systems (SAGE Studies in International Sociology 27), CA: Sage.Thurow, Lester (1994) "Education and Falling Wages, New England Journal of Public Policy, 10(1), pp. 535.U.S. Bureau of the Census, Decenial Census Reports (1940, 1950, 1960, 1970) Current Population Reports; P60 nos. 85, 90, 92, 97, 101. W ashington DC: U.S. Government Printing Office.U.S. Bureau of the Census (1940, 1950, 1960, 1970) Current Population Reports: P60 nos. 85, 90, 92, 97, 101. Washington D.C.: U.S. Government P rinting Office. U.S. Bureau of the Census (1984, 1987, 1990) Curren t Population Reports: Educational Background and Economic Status; P70 nos. 11, 21, 32 Washington DC: U.S. Government Printing Office.APPENDIX ACALCULATIONS OF SECTION E AREA To calculate , we begin by truncating the asymptot es of the two standardized normal curves (Figure IV1) at 3.0 standard deviations above and below their respective means. As a result, we lose 0.26% of the population of any one curve.
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26 of 37Since the two curves are identical (i.e., both are standardized normal curves), the point on the Xaxis ((I) directly below the point of intersecti on, I) lies midway between the X() and X(~) distribution means, () and (~), respectively. Th is follows from the laws of symmetry, since Section D is always equal to Section A in area. Fig ure A1 emphasizes the area of intersection in Figure IV1. Figure A1. Section E Area Emphasized(E() and E(~) correspond to E1 and E2 respectiv ely, in Table IV2)We know by symmetry, that the area to the right of the vertical line Iu(I) to (I) on curve X(~) (i.e., area E(~) is equal to the area to the left o f line I to (I) on curve X() (i.e., area E() ). Thus, twice E(~) or twice E() gives us , the area of Section E() Now we can proceed to develop a pair of algorithms that enable us to calculate area E(~). The area , equals 1.0 when equals zero. In this situation, X(~) and X() are superimposed one upon the other. Since (~) = () their relat ive difference, , is equal to the absolute value of (~) () which is equal to zero. When =1.0, area equals zero. In this case, X(~) and X() are mutually exclusive and equals 6 .0. Between these two extremes, ranges from zero to 1.0.We first examine the case where ranges from zero to 0.5 and then the case where it ranges from 0.5 to 1.0. (Note that 0.5 is used throughout as an approximation to 0.4987, which is used in the calculations due to truncation.) CASE 1: (0 < = = > 0.5) Consider Figure A2. The relative distance, , betw een the two means, (~) and () is equal to the distance on the Xaxis under area A (i.e., t he area corresponding to the value of ).
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27 of 37 Figure A2. Case 1: Where Ranges from 0 to 0.5Note that when = 0, the two means, (~) and (), coincide simply because the two curves, X(~) and X() are superimposed one upon the other As the value of increases, the X(~) curve is shifted to the left, a distance equal to t he distance on the Xaxis under Section A. Call this distance , which is the value of the X(~) cur ve translation. Since (2) = 3.0, we need only find (1) in order t o find (i.e., = (2) (1) ). Area F is equal to 0.4987 G and (1) is found from a standardized no rmal curve table. Once we have computed , we can locate (I) with respect to (~) See Figur e A3. Figure A3. The Parameters for Finding Note that (I) lies /2 above (~) Area G is foun d from a standardized normal curve table. Area E(~) is equal to 0.4987 G. The area , is si mply twice area E(~) The algorithm for this computation is shown in Algorithm A1. ALGORITHM A1CASE 1: WHERE RANGES FROM 0 TO 0.5 (Refer to Figures A2 and A3) Step F = 0.4987 1.
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28 of 37 (1) from standardized normal curve table 2. = (2) (1) 3. (I) = /2 with respect to (~) 4. G from standardized normal curve table 5. E(~) = 0.4987 G 6. = 2(E(~) ) 7.CASE 2: (0.5 < = = > 1.0)Figure A4 depicts the situation for this case, and the algorithm for the computation of is shown in Algorithm A2. Figure A4. Case 2: Where Ranges from 0.5 to 1.0ALGORITHM A2CASE 2: WHERE RANGES FROM 0.5 TO 1.0 (Refer to Figures A3 & A4) Step F = 0.4987 1. (1) from standardized normal curve table 2. = (2) + (1) 3. (I) = /2 with respect to (~) 4. G from standardized normal curve table 5. E(~) = 0. 4987 G 6. = 2(E(~) ) 7. Table A1, gives the values of for values in st eps of 0.l. Table A2 gives the intermediate values of F, (1) , (I) G, (~) for values in steps of 0.1.Table A1VALUES OF AS A FUNCTION OF
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29 of 37 0 1.00000.100.38720.200.27760.300.21240.400.16660.500.13100.600.10060.700.07500.800.05160.90 0.02940.950.01721.00 0Table A2INTERMEDIATE VALUES FROM ALGORITHMS A1 and A2 F(1)(I)GE(~)0..0...0.10 0.39871.2751.7250.86250.30510.19360.20 0.29870.8352.1651.08250.35990.13880.30 0.19870.5202.4801.24000.39250.10620.40 0.09870.2502.7501.37500.41540.08330.50 003.0001.50000.43320.06550.60 0.10130.2553.2551.62750.44840.05030.70 0.20130.5303.5301.76500.46120.03750.80 0.30130.8503.8501.92500.47290.02580.90 0.40131.2904.2902.14500.48400.01470.95 0.45131.6604.6602.33000.49010.00861..6.000...APPENDIX BMEAN/MEDIAN ANALYSIS OF THE PROBABILISTIC UTILITY M ODEL We can set the Model in motion. See Figure B1. Not e that when = 0, the following equalities hold: (B) = (C) = (I) = (~) = () 1. Absolute value of ((A) (I)) = absolute value of ((D) (I)) 2.
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30 of 37When = 1, another set of equalities hold:(C) = (B) = (I) 3. (A) = (~) 4. (D) = (~) 5. Absolute value of ((A) (I)) = absolute value of ((D) (I)) 6. Between these two extremes, it is possible to calcu late the relative differences between medians (() and (~) are the grand means and grand median s of their respective distributions) of the various sections of the two curves shown in Figure B1. Figure B1. Medians/Means for Sections of CurvesAssume that () remains constant and that both cur ves retain their normal shapes as the size of (and concomitantly, ~) and change. We take () as our point of reference, since it remains constant, and calculate the other medians w ith respect to it. 1. Schema's for Median Calculations for Changing Va lues of We begin, as we did in Appendix A, by truncating th e asymptotes of the two standardized normal curves at 3.0 standard deviations above and below t heir respective means. Medians (A) and (B) have already been calculated in the Aggregate Model and can be found in columns 2 and 3 of Table III1.(~) is the distance on the Xaxis under Section A. This distance is the value computed as an intermediate step by Algorithms 1 and 2. See Table A2. (I) is simply one half (~) and is also computed as an intermediate step by Algorithms 1 an d 2. See Table A2. We now develop schemas that compute the values of (C) and (D) for changing values of . Due to the symmetry of the two curves and the equal ity of Sections A and D, median (C) will always be as much to the right of (~) as (B) is t o the left of () Thus, (7) (C) = (~) (B). (7) In a similar fashion, (D) will always be as much t o the left of (~) as (A) is to the right of () Thus, (8) (D) = (~) (A). (8)
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31 of 37 Table B1 displays the results of these computation s. 2. Changing Means (() and (~) ) With Changing and Constant . We have assumed throughout that the size of has n o effect on the means of the dichotomized populations. Furthermore, for computational purpose s, we have assumed that only (~) was affected by changing and that () remains perman ently anchored. It is not unreasonable to assume that both means ch ange with changing and that both means change with changing . However, both of these case s reduce to the analysis that has already been performed for the probability distributions ge nerated by the formulae in Table IV2 (constant () for changing and changing ).Table B1INTERMEDIATE VALUES FROM ALGORITHMS 1 AND 2 (1) (2)(3)(4)(5)(6)(7)(8)()(A)(B)(I)(C)(~)(D)003.000003.00.10 01.6450.1260.86251.59901.7253.3700.2001.2830.2531.08251.91202.1653.4480.30 01.0370.3851.24002.09502.4803.5170.4000.8420.5241.37502.22602.7503.5920.5000.6750.6751.50002.3250 3.0003.6750.6000.5240.8421.62752.41303.2553.7790.7000.3851.0371.76502.49303.5303.9150.8000.2531.2831.92502.56703.8504.1030.9000.1261.6452.14502.64504.2904.4160.9500.0631.9602.33002.70004.6604.7231.0 003.03.03.06.06.0 To construct the probability tables for changing me ans, we can use the probability distributions generated by the formulae in Table IV2. We need on ly know the sizes of and , and the relative difference between the two dichotomized po pulation means (see Appendix A). This relative difference, absolute value of () (~) is a function only of the size of . Thus, if both means change with changing and with changing , and if we know the relative difference between the means, we can calculate the new . We c an then consult the existing probability tables produced by the formulae in Table IV2.3. Nonnormal Distributions with Equal and Unequal Ranges The same sort of mean/median and probability analys es that have been performed for normal distributions can be performed for nonnormal distr ibutions. One must, however, first derive the formulae for the various curves and utilize the cal culus to obtain the areas in questions and their shifting means and medians. The mathematics involve d in this kind of analysis is more complex.
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32 of 37APPENDIX C A SAMPLE CALCULATION FOR THE AGGREGATE MODEL Here is a sample calculation of the median value of the social benefits for high school attainers and nonattainers.Suppose that the attainment ratio stands at 30 perc ent. See Figure C1. We know that the attainer group monopolizes the social benefits ranging in va lue from 0.52 to 3.9 standard deviations from the grand mean.The median benefit for this group is thus () = 1. 037 standard deviations. This is the point under the portion of the total distribution where half of the high school attainers (i.e., 15 percent) lie to the right and where the other half lie to the left. The median social benefits for the remaining 70 per cent of the total population (i.e., the nonattainer group) is (~) = 0.385 This is the point under the portion of the total distribution where one half of the high school nonattainers (i.e., 35 percent) lie to the right and the other half lie to the left.The median social benefit values are derived from t he standardized normal distribution, which represents a particular normal distribution of soci al benefits. If it turns out that, for this particu lar normal distribution, the median of the total distri bution is $8,000 with a standard deviation of $2,500, we can easily calculate the medians (in dol lars) of the attainer and nonattainer groups. Attainer Group Median: $10,593 = $8,000 + (1.037 x $2,500); nonAttainer Group Median: $7,038 = $8,000 + (0.385 x $2,500). Figure C1. Standardized Normal Curve for the Distr ibution of Social Benefits ( = high school attainment ratio; ~= nonattainmen t ratio; grand median=0; () = median social benefit for attainer group; (~)= median soc ial benefit for nonattainer group; standard deviation = 1)It is probably unreasonable to apply the model at t he lower attainment ratios where the power of the normative principle is very low. However, the m odel does serve to illustrate the idea that the relative benefit disparity between the two groups f irst decreases and then increases. This
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33 of 37 phenomenon suggests that a particular education pol icy appropriate for one stage of systemic growth might not be appropriate for another stage.Appendix D Empirical High School Attainment Data*School Year Graduates as Percent of 17yearold Population 1869702.01879802.51889903.5189900 6.4190910 8.8191920 16.8192930 29.0193940 50.8194748 52.6194950 59.019515257.4195354 59.819555663.1195657 63.1195758 64.8195859 66.2195960 69.5196061 67.9196162 69.3196263 70.9
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34 of 37 196364 76.7196465 72.1196566 76.4196667 76.3196768 76.3196869 77.1196970 76.9197071 75.9197172 75.619727375.0197374 74.419747573.6197576 73.7197677 73.8197778 73.019787971.7197980 71.4198081 71.7198182 72.4198283 72.9198384 73.1198485 72.4198586 72.0198687 71.8198788 72.1198889 71.0
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35 of 37 198990 72.4199091 73.2199192 73.1199293 73.2199394 73.1 National Center for Education Statistics (1995). Table 98 in Digest of Education Statistics. Washington, DC: U.S. Government Printing Office.Addendum January 2, 2001 Appendix D Empirical High School Attainment Data*School Year Graduates as a percentof 17yearold population 19901 73.21991273.219923 72.219934 71.719945 70.719956 69.819967 69.119978 68.919989 70.6* National Center for Education Statistics (1999). Digest of Education Statistics. Washington, DC:U.S. Government Printing Office. Table 104http://nces.ed.gov/pubs2000/digest99/d99t104.htmlAbout the Author Robert H. Seidman New Hampshire College[Updated contact information, January 2003: r.seidman@snhu.edu Southern New Hampshire University.]Robert H. Seidman is a professor at the New Hampshi re College Graduate School and the
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36 of 37 Executive Editor of the Journal of Educational Computing Research (http://www.baywood.com/Journals/PreviewJournals.as p?Id=07356331).Copyright 1996 by the Education Policy Analysis ArchivesEPAA can be accessed either by visiting one of its seve ral archived forms or by subscribing to the LISTSERV known as EPAA at LISTSERV@asu.edu. (To sub scribe, send an email letter to LISTSERV@asu.edu whose sole contents are SUB EPAA y ourname.) As articles are published by the Archives they are sent immediately to the EPAA subscribers and simultaneously archived in three forms. Articles are archived on EPAA as individual files under the name of the author a nd the Volume and article number. For example, the article by Stephen Kemmis in Volume 1, Number 1 of the Archives can be retrieved by sending an email letter to LISTSERV@a su.edu and making the single line in the letter rea d GET KEMMIS V1N1 F=MAIL. For a table of contents of the entire ARCHIVES, send the following email message to LISTSERV@asu.edu: INDEX EPAA F=MAIL, tha t is, send an email letter and make its single line read INDEX EPAA F=MAIL.The World Wide Web address for the Education Policy Analysis Archives is http://seamonkey.ed.asu.edu/epaaEducation Policy Analysis Archives are "gophered" in the directory CampusWide Inform ation at the gopher server INFO.ASU.EDU.To receive a publication guide for submitting artic les, see the EPAA World Wide Web site or send an email letter to LISTSERV@asu.edu and include the single l ine GET EPAA PUBGUIDE F=MAIL. It will be sent to you by return email. General questions about ap propriateness of topics or particular articles may be addressed to the Editor, Gene V Glass, Glass@asu.ed u or reach him at College of Education, Arizona Sta te University, Tempe, AZ 852872411. (6029652692)Editorial Board Greg Camillicamilli@zodiac.rutgers.edu John Covaleskiejcovales@nmu.edu Andrew Coulson andrewco@ix.netcom.com Alan Davis adavis@castle.cudenver.edu Sherman Dorn dornsj@ctrvax.vanderbilt.edu Mark E. Fetlermfetler@ctc.ca.gov Thomas F. Greentfgreen@mailbox.syr.edu Alison I. Griffithagriffith@edu.yorku.ca Arlen Gullickson gullickson@gw.wmich.edu Ernest R. Houseernie.house@colorado.edu Aimee Howleyess016@marshall.wvnet.edu Craig B. Howley u56e3@wvnvm.bitnet William Hunterhunter@acs.ucalgary.ca Richard M. Jaeger rmjaeger@iris.uncg.edu Benjamin Levinlevin@ccu.umanitoba.ca Thomas MauhsPughthomas.mauhspugh@dartmouth.edu
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37 of 37Dewayne Matthewsdm@wiche.edu Mary P. McKeowniadmpm@asuvm.inre.asu.edu Les McLeanlmclean@oise.on.ca Susan Bobbitt Nolensunolen@u.washington.edu Anne L. Pembertonapembert@pen.k12.va.us Hugh G. Petrieprohugh@ubvms.cc.buffalo.edu Richard C. Richardsonrichard.richardson@asu.edu Anthony G. Rud Jr.rud@sage.cc.purdue.edu Dennis Sayersdmsayers@ucdavis.edu Jay Scribnerjayscrib@tenet.edu Robert Stonehillrstonehi@inet.ed.gov Robert T. Stoutstout@asu.edu
