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Modern Education Productivity Research: Emerging Implications for the Financing of Education

David H. Monk
Cornell University

Jennifer King Rice
University of Maryland

About the Authors



The advent of high performance standards has renewed efforts to understand and enhance the productivity of educational systems. Analysts are struggling to grasp the resource implications of these standards in the face of inadequate conceptualizations of educational productivity, imperfect data, and inadequately developed statistical tools and research methods. Despite these difficulties, progress is being made, and analysts are beginning to move from relatively simple input-outcome examinations to studies that explicitly tie outcomes to costs. The purpose of this paper is to assess this progress and to make suggestions for next steps.

Our starting point is the premise that the education production function is a real and potentially very useful tool for those concerned with improving the performance of schooling systems. Closely related to the education production function is the education cost function, and some of the most interesting contemporary education productivity research is being conducted from the cost perspective (e.g., Duncombe, Ruggiero, and Yinger 1996 and Imazeki and Reschovsky 1998). Therefore, it is important to understand the dual nature of the relationship between productivity and cost, and this paper begins with a conceptual model of these elements. It is particularly important to distinguish among the various sources of cost in a productivity framework. These distinctions are important from a policy perspective because school finance adjustments tend to evolve in a piecemeal approach in which the goal is to address a particular element of cost (e.g., the marginal cost associated with educating students with special needs, or costs associated with geographical differences in the cost of living). Some recent cost studies are more oriented around the development of comprehensive cost measures that subsume the various components. The interplay between the emerging comprehensive cost indices and the existing panoply of source-specific school finance adjustments needs to be examined, and a primary purpose of this paper is to prompt additional work of this kind.

The conceptual model also serves as a useful organizing device for the subsequent review and critique of cost and productivity studies. We show how the various studies differ with respect to the elements of cost-productivity that are being examined, and we are able to assess the progress that is being made toward developing a set of credible indicators of effectiveness and cost that will be of use to policymakers. We begin with a focus on the various attempts that have been made to estimate costs and then turn to the work that has been done on the productivity of a key educational input—namely, teacher quality. We reason that any satisfactory attempt to grapple with the resource implications of high performance standards will need to deal explicitly with the existing knowledge about the available indicators of teacher quality and learning outcomes for students.

One of the dilemmas facing policymakers is the design of appropriate responses to evidence of inefficiency within the educational system. Suppose, for example, cost-effectiveness analysis progresses to the point at which clear judgments can be made about which district, school, or unit is inefficient with respect to the production of desired learning outcomes. Such a finding on its face provides relatively little insight into the correct policy response. Punitive measures need to be considered carefully, but it makes little sense to reward units for an inefficient operation. We turn to these considerations for policymakers in the final section of the paper. Our goal is to understand how the results of research dealing with education costs and productivity can be translated into improved policies, particularly with respect to raising and distributing revenues for education.

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A Micro-Level Model of Resource Allocation

It is useful to think of resources as instrumentalities that work in tandem with one another to generate desirable results. Resources come in many forms and exist within many contexts, but the trait they have in common is a potential to be configured in ways that give rise to something new, an outcome or result of some kind. Much of the policymaking significance of resources lies in their potential to shape and define desired ends as well as in the hope that steps can be taken to better realize their imbedded potential.

The notion of "potential" is important because it suggests variability in the degree to which outcomes are realized. The variability arises from at least two areas. On the one hand, resources in combination can vary dramatically in their potential ability to generate a given result. There have been many efforts to estimate the magnitude of these maximum or ideal productivity levels, often under the rubric of production function research (for overviews of this research as it has been applied to education, see Hanushek 1979, 1994, and Monk 1992). Some resource combinations simply have higher productivity potentials than do others.

On the other hand, circumstances intervene that can affect the selection of one resource combination versus another and can ultimately affect the ability of an organization to realize the full potential of its resource base. These circumstances take many forms and much contemporary debate involves trying to distinguish between circumstances that are externally imposed as opposed to those that arise out of complicit behavior on the part of actors at the local levels, be they administrators, teachers, other educators, students, parents, or others.

The circumstances giving rise to whatever discrepancies exist between ideal and actual resource allocation practices are of great interest to policymakers. In the following analysis, we explore these ideas by drawing a sharp distinction between ideal and actual practice. We are particularly interested in understanding the reasons for departures from ideal resource allocation practice and thinking through the implications for the design of education funding systems.

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Ideal Resource Allocation Practice

We seek a heuristic that conveys key features of the stockpile of knowledge about the productivity of educational resources. More specifically, we are interested in capturing what is known about what works for students in what ways and under what conditions.

We proceed by approaching the phenomenon from the outcome side at a decentralized, micro level. Let us think in terms of individual students for whom outcome standards have been set such that there is the ith student with the jth outcome standard. Questions quickly arise about what creates these outcome standards. These are important questions, but they are not central to the task at hand. For now, we simply recognize that these standards are generated and articulated by some body that may or may not be external to the educational system (e.g., a state or local board of education, or a legislature). Presumably these standards are set on the basis of beliefs that their realization has salutary social and economic effects over the long run and/or because their realization fosters the fulfillment of whatever social obligation there might be with respect to guaranteeing fundamental human rights. We might also wish to recognize that standards need not be set by a single board or decision-making unit. Indeed, it is possible for central boards to set standards that can be raised by local units responding to higher demands for outcomes that may exist in particular communities. Locally set standards can also be exceeded by decisionmakers (e.g., site-based councils, building administrators, and teachers) located within individual schools and classrooms. There are important implications for how fiscal responsibilities attach to these nested standards, and we return to this point later in the analysis.

Regardless of who is setting the standards, we are conceiving of them at the level of the individual student, and this invites questions about the degree to which the standards vary across students. Again, this is an important matter, but one that need not detain us. Outcomes like a fundamental ability to read and write have a more universal appeal than, for example, more specialized or sophisticated outcomes like the ability to compare and contrast literature from different historical periods or the ability to repair automobiles. Moreover, once we introduce the idea that outcome standards may vary across students, questions arise about the basis and means by which students are best sorted across the standards.

The prevailing debate suggests that the magnitude of these sorting problems can be diminished by raising the base level of the standards so that expectations for pupil performance become more universal, particularly with respect to conventional forms of academic capabilities. Indeed, advocates of "systemic reform" see a universal raising of performance standards as a powerful means of diminishing the adverse effects of having schools sort students into alternative learning tracks. As we shall see, the resource implications of this approach are significant and need to be addressed by those advocating reforms along these lines.

However, even if high universal performance standards were established, there still remains a point beyond which differentiation can occur. There are, after all, returns to specialization. The educational system, even in the context of a serious and successful pursuit of systemic reform, is not relieved of having to deal with a differentiation of outcome standards. Decisions need to be made about what these differing outcome standards are and how students will be distributed across them.

The result for our purposes is a student-outcome specific matrix in which each student is depicted in terms of the appropriate mix of performance capabilities. We can think in terms of a two-dimensional Matrix A shown in figure 1 in which the elements of this matrix (Oij) indicate whether or not the ith student is expected to reach a performance target in the jth area or domain. Matrix A is a binary matrix in the sense that each element is either 0 or 1. In the case of a universal outcome standard, the jth column will consist exclusively (or almost exclusively) of ones. In the case of a specialized area of capability, zeros will be common and perhaps only a few of the rows (i.e., individual students) will have ones. Matrix A is an outcome-standard matrix. It is the starting point for this analysis of costs and constitutes the anchor for the entire system.

Matrix A corresponds to one aspect of the demand society makes on the educational system to produce results. These demands need not be fixed over time nor exogenously determined, but for our purposes the idea is that they are in place.

Recall that we can differentiate between certain "base" standards that may be set centrally and "add-on" standards that are set locally. Thus, we can distinguish between Matrix A (Central) and Matrix A (Local) and recognize that the only difference will be the number of ones relative to the number of zeros. As we are conceiving it, Matrix A (Local) can have more ones than Matrix A (Central). It follows that there may be several different Matrix As, each corresponding to the standards set at a particular level of the system. Our goal is to link the establishment of a given Matrix A with the associated cost.

Notice that Matrix A does not provide insight into the level of learning expected. Instead, it simply provides an inventory of who is expected to develop capability in a particular area. The "degree of accomplishment" dimension is the second aspect of the demand society makes on the schools to produce results and has a direct bearing on costs. In order to handle this second aspect, we need to broaden the analysis as follows.

For each Oij, we wish to conceive of every known response, treatment, or what we shall call an "educational service" that can be drawn on to facilitate the kind of learning associated with the jth area of learning. These educational services can be conceptualized in terms of discrete configurations of purchased, hired, and donated inputs that are combined with a student's time. We need to differentiate between the quantity of a given service and the quality of the service in question. Differences in quality correspond to differences in the service being provided (i.e., one discrete configuration of resources compared with another), whereas changes in quantity correspond to doing more or less of the same sort of treatment. For example, a school might decide to offer students more classtime during a typical week to help them enhance performance on a new and more demanding learning standard that has been put into place. As long as the treatment falls under the heading "more of the same" resources that were previously being supplied, we are dealing with differences in quantity of a given service. In contrast, if the district made a substantive change in the configuration of resources, perhaps by adding a teaching assistant or reducing class size, then the service in question has changed its character and we are faced with the challenge of figuring out how many units of each of the two conceptually different services must be provided to meet the outcome standard.

For the sake of simplicity, we assume that educational outcomes are produced using fixed proportions of discrete inputs and are subject to constant returns to scale. This means that a doubling of every input (i.e., a doubling of the quantity of the educational service being provided), is associated with a doubling of the learning gain for the student in question. Let the letter S represent each of these possible educational services (i.e., configurations of inputs), and let us define each Sijk so that it is specific to the ith student and the jth learning standard with the letter k serving as an index that orders the various alternative educational services that might be employed. We can think in terms of Sijk in which the various Ss are arrayed along a vertical axis that grows out of the two-dimensional plane on which Matrix A is placed. Thus, for each combination of i and j, there exists a vertical column of Ss shown in figure 2 that represents the alternative ways of meeting the jth learning standard for the ith student.

We introduce the degree of learning dimension into the framework by conceiving of each Sijk as the level of resources required using the kth configuration of inputs for the ith student to reach the stipulated level of learning associated with the jth standard. For now, let us assume that these degree of learning standards are fixed so that either a student is expected to reach the standard (i.e., the corresponding cell entry in Matrix A is a 1) or not (i.e., the corresponding cell entry in Matrix A is a 0).1 Each Sijk can be thought of as the total cost associated with realizing the jth learning standard for the ith student using the kth service or configuration of inputs.

There are two reasons why Sijk can differ from Sij(k+1). First, the intrinsic productivities of the inputs being combined can vary. Some inputs are more productive than others. Second, the unit costs of the various inputs can vary. Some inputs are more readily available than others, and the relative degree of scarcity in the face of the prevailing demand will establish price.

Consider the case of two alternative ways of achieving a given learning outcome, one that involves the time of a well-prepared teacher and one that involves the time of a poorly prepared teacher. Assuming teacher preparation is positively related to both the teacher's productivity with the student in question (we return to this topic in the Teacher Quality Research section) and the cost of the resource (i.e., the teacher's salary), we cannot deduce a priori how the two Sijks will compare with one another. This will depend on the relative strength of the two effects that are pulling in opposite directions.

We use the term "cost" deliberately because we are interested in the resources required to reach the identified outcome using the stipulated configuration. We recognize that some of the configurations will be more attractive (i.e., less costly than others) and that some "resources" are less than optimal (e.g., time from an unmotivated or poorly prepared teacher). Our intent is to have an exhaustive compilation of all the possible ways that the identified learning outcomes can be produced. Suffice it to say that the three-dimensional matrix we have envisioned and labeled Matrix A* in figure 3 will be very large.

There remain potentially important interdependencies across learners that need to be considered. Learning in school settings is not a private affair. The resources required for one student to learn can be affected by the characteristics of fellow learners. However, if we conceive of fellow learners as resources that may or may not be available to the student in question, the various possibilities can be provided for in the vertical columns of Matrix A*.

We have now conceived of the costs associated with achieving each of the identified learning standards for each of the identified students using all imaginable educational treatments or services. In this sense, the formulation is context or circumstance free. For the moment the only constraints we have introduced stem from the set of outcome standards that anchors the system and the characteristics of the learners in question. Out of this universal set of possibilities we wish to identify the ideal resource allocation practices in which "ideal" is conceived as being synonymous with "least cost," to reach the pre-specified outcome standard and in which "cost" is measured in terms of the various Sijks that comprise the various columns of Matrix A*.

We can accomplish this result by traveling up and down each vertical column of Matrix A* searching for the configuration of inputs with the best (i.e., smallest) Sijk. This will be the most desirable or idealized combination of resources for meeting a specific learning standard for a specific student. In other words, this is the least costly option possible given the attributes of the learner, the prevailing state of knowledge about the production of learning, and the absence of geographical as well as organizational context. For each combination of the ith student and the jth learning standard, one element of the vertical column vector will be identified. If these identified "idealized best practice" elements are projected onto a two-dimensional plane, there will emerge a new two-dimensional matrix in which each entry conveys information about the nature of the best practice and the cost associated with realizing the defined outcome standards.

This matrix, called Matrix IBP (for Idealized Best Practice) with each cell entry labeled IBPij, reflects the available knowledge about what works best to reach the learning standards for each of the identified students. The smaller the cell entry values of Matrix IBP, the better is the knowledge base, the more favorable are the prevailing terms of trade for the resources in question, or both. Over time, we might expect the magnitudes of the IBPijs to diminish (as more is learned about how learning takes place) for a given set of learning standards for a given set of learners. However, this is not necessarily true, because the unit prices for the inputs built into the services represented by the elements of Matrix IBP could rise in real terms. Efforts to reduce the magnitude of the IBPijs can come from the results of research designed to improve the effectiveness of inputs; they may also arise from more grassroots types of gains in which teachers, in effect, discover the nature of the production functions they face and find ways to pool their knowledge so that students can benefit from the results.

We can move from Matrix IBP to a calculation of total cost for reaching the targets for the identified students by summing all of the entries found in Matrix IBP. Recall that the individual cells of Matrix IBP provide the minimum cost figure for each student with respect to the type and degree of learning expected in each identified area. Some of the cell entries in Matrix IBP will be 0, and these correspond to instances in which the student in question is not expected to achieve a learning outcome in a particular domain. Thus, there is a single figure that represents the total cost of realizing the stipulated performance standards in which the intrinsic productivity of inputs is fully realized: TC(IBP).

Let us revisit the question of how and by whom the outcome targets were set. We did not deal with this above other than to note that there could be a role for relatively centralized bodies like state boards of education and there could be a role for local bodies like school district boards of education or perhaps school site councils. The key point is the TC(IBP) is very sensitive to the mix and level of the outcome standards. If we think of the state board as setting minimum standards on which a local board can build, it follows that TC(IBP) for the state board will be less than or equal to TC(IBP) for the local board. It would also seem to follow that the state will have more of an interest in covering the cost of reaching the standards being set by the state board, and this suggests a division of fiscal responsibility that is reminiscent of conventional foundation types of school finance formula.

This formulation provides a useful starting point in the effort to conceptualize and calculate the costs of reaching a finite set of learning standards for a given student population. However, Matrix IBP and the price tag, TC(IBP), is not directly observable given the fact that it is completely divorced from actual local circumstance. The next step in the analysis is to begin introducing elements of local circumstance into the formulation, and, as we shall see, there can be dramatic implications for resources. As we introduce local circumstance to the formulation, we begin to enter the real world of schooling practice in which circumstances can force departures from "ideal" practices. Our attention turns next to what we will call "actual" resource allocation practice.

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Actual Resource Allocation Practice

In contrast to the "ideal" distribution and utilization of resources that is described by Matrix IBP, there is existing practice with respect to the distribution and use of resources across students and outcomes. Existing practice can be represented by returning to the three-dimensional Matrix A*. However, instead of traveling up and down the vertical columns searching for best practice, this time the search is for the nature of the actual educational service that is being provided to each student on an outcome-specific basis. Recall that each cell entry in Matrix A*, Sijk (for every k not equal to 0) represents a measure of the total cost associated with each possible service that would be incurred if the service in question were used to achieve the ith student's jth outcome. Previously, we searched for the best (i.e., lowest) value of Sijk; here we are searching for the kth service that most closely corresponds to the service actually being delivered to the student in question. We have already constructed a two-dimensional Matrix IBP in which the elements (IBPij) corresponded to the total cost associated with meeting the ith student's jth need under the best of conditions. Here we can construct a parallel two-dimensional matrix, call it AP (for Actual Practice), in which each element (APij) represents the total cost of meeting the identified needs using the educational services that are currently in use. We can sum all of the elements of the AP Matrix and thereby obtain the total cost of reaching the stipulated learning outcomes for the identified group of students using prevailing practice: TC(AP). TC(AP) captures what it would cost to reach the standards with no changes being made in how we operate schooling systems. It embodies a "more of the same" approach to reform.

Notice that TC(AP) will reflect all existing circumstances that bear on both the productivity and unit costs of resources. The prevailing use of organizational structures will be reflected (i.e., the existing numbers, size, and composition of districts, schools, classes, and groups). Whatever degree of disaffection, lack of motivation, or outright hostility that is present will also be reflected in TC(AP). The idea is to ask how much of the service in use will be required to overcome whatever lack of motivation there might be on the part of a student, a teacher, or both. Similarly, the place-sensitive nature input prices will be reflected. In other words, input prices may be higher in some regions than in others (see, for example, Chambers and Fowler 1995), which bears on the choice of the relevant Sijk in each of the vertical columns of Matrix A*.

By definition, each IBPij will be less than or equal to each APij and TC(IBP) will be less than or equal to TC(AP). Indeed, some of the APijs may be very large. If a service is not well suited for meeting a particular need, the cost of realizing the outcome target using the ill-suited service could become quite large. The discrepancies between the APijs and the IBPijs are important for policymakers. Specifically, these discrepancies measure the degree to which the system is misaligned in the sense that less than ideal uses are being made of resources in relation to the outcome standards that have been set. The larger the discrepancies, the larger is the misalignment within the system.

Such "misalignments" can occur for good and not so good reasons, and this realization introduces an important distinction into the analysis of productivity and cost. For example, a discrepancy can occur between an APij and an IBPij because of structural realities that limit the ability of administrators and teachers to realize ideal practice through no fault of their own. For example, an administrator might be operating within a school that is either too large or too small to operate efficiently. The administrator may be choosing the best Sijk available given the constraint of existing school size, but this best Sijk could be considerably larger than the idealized Sijk in Matrix IBP. Although school size is a decision variable, it would seem inappropriate to hold a building level administrator accountable for a suboptimal school size. We offer this as an example of what we will call realistic (as opposed to idealized) best practice in which the idea is to introduce a level of tolerance for a certain set of suboptimal resource allocation practices.

Although it seems reasonable to introduce this tolerance, placing bounds on what constitutes acceptable and unacceptable departures from idealized best practice is very problematic. Consider the case of an unmotivated student. If we treat the time of such a student as a given, we will find ourselves choosing an Sijk with a relatively high cost because higher levels of the service in question will be needed to offset the low productivity of the student time input. In contrast, if the student could be motivated, the configuration of inputs would change for the better and we can reasonably presume that fewer outside resources will be needed for the student to reach the standard. Shall we hold the teacher responsible for the student's lack of interest? Is the teacher complicit in the use of a suboptimal resource allocation practice? Who should bear the cost of financing these suboptimal practices? A final example concerns the setting of unit prices for key inputs. Although we recognize that input prices may vary geographically, it is possible that actors within the system contribute to the differentials that are observed. For example, some districts may bargain more effectively with their employees than others, and some of the resulting price differentials may reflect what amounts to complicit behavior on the part of certain officials.

We are not able to resolve these questions in this analysis, but it is important to introduce the idea of acceptable departures from idealized best practices into our cost formulation. We shall treat it primarily as a placeholder at this point, but it is a very important placeholder and one whose reality has not been factored sufficiently into debates over how to finance education.

Thus, we can define a new matrix, called RBP (Realistic Best Practice), whose elements correspond to the various best possible Sijks taking account of externally imposed local circumstance over which officials have no direct influence. There is room for considerable disagreement about what counts as an externally imposed local circumstance, and decisions about whether to treat a local circumstance as externally imposed or not has a direct impact on the degree to which Matrix RBP will be different from Matrix IBP. The magnitudes of the various elements of Matrix RBP will lie between the magnitudes of the BP and AP matrices. Similarly, if we sum the elements of Matrix RBP and define that amount as TC(RBP), we will find that TC(IBP) is less than or equal to TC(RBP), which will be less than or equal to TC(AP).

We now introduce a final element of real world circumstance—namely, the adequacy of the resource base that is provided to operate the system. The question becomes one of comparing the magnitude of the resources being allocated into the system (Total Actual Funding—TAF) with the various cost figures that we have conceptualized. Keep in mind that merely spending resources implies relatively little about the level and distribution of learning outcomes being realized. Thus, TAF may be larger or smaller than TC(IBP) but is presumably less than TC(AP).

Figure 4 suggests that movement from actual to best practice involves a significant improvement in the utilization of resources. Figure 4 also suggests that resources currently allocated into the system are not adequate for realizing the performance standards for the identified students, even if idealized best practices were in use. The educational system depicted by figure 4 is clearly underfunded, but it is equally clear that a careless pursuit of the outcome targets in the absence of parallel efforts to promote improvement in practice could lead to a serious erosion of the system's efficiency and a waste of resources, a rather ironic result given the goals of the reform. Finally, figure 4 illustrates the important point that policymakers who seek to achieve a stipulated mix and level of student outcomes need to concern themselves with the alignment of the system as well as the adequacy of the resource base.

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The Costs of Adequacy

If we presume that the setting of performance standards implies a judgment about what constitutes an adequate program, the model provides insight into what needs to be clarified before costs can be attached to adequacy. In particular, the model shows that agreement needs to be reached about:

  • 1) The number and nature of the columns in Matrix A;
  • 2) The overall incidence of "universal" standards (i.e., the incidence of columns of ones with no zeros) in Matrix A;
  • 3) The tolerance for the presence of some zeros in the "universal" standards columns in Matrix A and clarity about what is an acceptable level of "some" (e.g., 1 percent, 5 percent, 10 percent, or other);
  • 4) The relative incidence of ones versus zeros in Matrix A (i.e., interest-willingness in going beyond the setting of "universal" standards); and
  • 5) The willingness to accept differential levels of accomplishment based on student attributes (i.e., the degree to which "accomplishment" in a given area is differentiated among students who are expected to perform in that area).

We turn next to the progress that has been made in estimating the costs of achieving high performance standards. We begin by examining explicit attempts to generate cost estimates and then turn to what has been learned about the productivity and cost of a key educational input: the quality of teachers and teaching.

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Existing Attempts to Estimate the Cost of Educational Outcomes

Researchers are dealing more explicitly with the links between costs and outcomes in a school finance context. Several approaches have emerged, and in this analysis we review each in turn. The approaches vary in terms of their degree of emphasis on economics, and we have ordered the discussion such that we move from approaches with the least to approaches with the most economic content. In particular, we examine the educator judgment model, the unit cost of inputs model, the cost of prevailing best practices model, and the cost function model, with and without adjustments for efficiency.

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Educator Judgments

The goal of this approach is to assess the cost of providing an "adequate" education for students based on a consensus among educators over adequacy's relevant components and realistic best practices. These agreed-upon components and practices are then assessed in terms of their cost and totaled into an estimate of the full cost. The approach takes into account the inefficiencies associated with funding the expansion of actual practice to meet outcome standards (recall how large TC(AP) was presumed to be relative to TC(RBP)), but the search for the relevant benchmark tied to realistic best practices is based on judgments from panels of disinterested educators about what is appropriate under a given set of circumstances. One could argue that this is the default approach states have relied on for years as they have designed school finance formula, but in recent years there have been more explicit attempts to look at these judgments from a cost-resource perspective. The resource cost model that was developed by Jay Chambers and Thomas Parrish for Illinois and Alaska (Chambers and Parrish 1994) is a sophisticated and quite ambitious version of this approach. More recently, Guthrie et al. (1997) developed a version of this approach for Wyoming.

The approach relies heavily on the judgments of educators to ascertain the components of realistic best practices based on years of professional experience in different settings. There is no formal link with outcomes other than the available wisdom based on practice from those participating in the process. Although efforts are made to make the panels "disinterested," questions remain about the accuracy of such judgments as well as about potential conflicts of interest because there is likely to be an understandable underlying agenda to justify additional resources which support unmet educational needs of students. We also note that these efforts have not been informed by clear statements from the states or other standard setting bodies about the features of Matrices A and A*. None of the key elements of what constitutes adequate outcomes (see the list of items 1-5, previously) are specified. In this light, the necessary implicit judgments, estimates, and guesses of the various Sijks are all the more difficult to deduce.

There have also been some recent attempts to "cost-out" innovative programs that purport to reflect realistic best practice that are sensitive to both the underlying circumstances educators face as well as the kinds of outcome standards that are being established. These models include "Success for All," accelerated schools, and the School Development Program, among others. Cost assessments of these models to date include King (1994) and Barnett (1996). Chaikind and his colleagues (Chaikind, Danielson, and Brauen 1993) reported the use of similar approaches to estimate the cost of special education. We include these efforts under the educator judgment heading because the models are adopted because of professional judgments about their appropriateness in a given setting and because they include judgments about how to best adapt the requirements of each model to local circumstance. The resulting cost estimates therefore reflect an attempt to achieve benchmark efficiency standards that lie between the two extremes that we have identified (TC(IBP) and TC(AP)).

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Unit Cost of Inputs

Efforts have also been made to focus on differences in the unit cost of key inputs into the educational system. A number of different approaches have been employed, some relying on a market basket strategy (e.g., McMahon 1996) in which the focus is on how much a given basket of inputs costs in one place compared with another, and others in which the emphasis is on underlying models of supply and demand with allowances for compensating differentials such as the fact that teacher salaries tend to be lower in places with favorable working conditions, all else being equal. These latter models are called hedonic wage models and have been studied extensively by Jay Chambers (1997, 1998). These hedonic models are particularly interesting for our purposes because they include explicit distinctions between influences on the unit prices of inputs that are within and outside of the control of local school officials. Recall that this is the essence of the distinction we emphasized between the realistic and idealized best practice standards.

It is worth noting that the unit cost approach to date has not made an explicit connection to the outcome standards reflected in Matrices A and A*. The question is more along the lines of asking how much more it costs to hire the same input in one place compared with another. Clearly, this is a relevant question. It bears directly on one of the key sources of cost difference between the various services that might be employed to produce a given gain in a given area of learning (i.e., Sijk versus Sij(k +1)). But, it should be clear how far short this approach falls of specifying all of the possible sources of cost difference and contingency that need to be dealt with in a comprehensive calculation of what it will cost in a particular place to reach a prespecified set of outcomes. As we shall see in the discussion about cost functions, it is possible to build unit cost indices into more comprehensive measures of the costs to produce educational outcomes.

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Costs of Observed Best Practices

Under this rubric can be found explicit attempts to link outcome standards to comprehensive conceptions of cost. In other words, costs are not restricted to differences only in the unit prices of individual types of inputs, and the formulation deals directly with learning outcome phenomena. The outcome standards are specified in terms of performance on examinations, and the question becomes one of identifying places that seem to be producing these outcomes with admirable levels of efficiency.

The strategy is intuitively straightforward. Districts are identified that have reached a pre-specified minimally acceptable level of performance, and efforts are made to control for gross differences in the contextual reality of the identified districts. For example, districts with extraordinarily high or low levels of wealth and expenditure might be excluded on the grounds that they are highly atypical. The next step involves carefully reviewing practices that exist within the identified districts with an emphasis on identifying efficient results. For example, it might be found that some districts in the group are able to reach the outcome standard with a given set of class sizes and characteristics of instructional personnel. The costs of these various approaches can be estimated and interest can be focused on those successful places with the lowest level of cost. These districts and their practices can become benchmark standards. The associated cost estimates can then be used as the basis of a school finance system that is designed to cover the costs of adequate programs in which adequacy is rooted in outcome standards. A number of states have explored one version or another of this approach in recent years including Ohio (Augenblick 1997), Illinois (Governor's Commission, 1996), Mississippi (Augenblick, Myers, and Anderson 1997), and New York (Monk, Nusser, and Roellke 1998).

The implicit reasoning within this approach is that if some places can produce the desired test score results at a given (low) level of cost, it follows that it is possible for others to do so as well and that we can scale up the system by providing only those resources that would be necessary if the observed best practices were employed. In other words, the resources commensurate with TC(AP) should not be provided, and policymakers can rest easy knowing that they are not facilitating the kind of internal inefficiency that is suggested by the high cost shown for TC(AP) in figure 4.

Of course, the key piece in this reasoning is whether the prevailing best practices in settings in which they are observed are realistically available to places in which they are not currently in place. The approach includes efforts to adjust for differences in extenuating local circumstances, but the adjustments to date are not very sophisticated.

Our conclusion is that the cost of the observed best practice approach represents an important step in defining the middle ground efficiency benchmark that needs to be established in any serious attempt to estimate the costs of outcome performance standards, but that it is based on a crude set of adjustments. The approach moves the field in the correct direction, but the distance traveled is modest. Although there is progress to report, it needs to be recognized that this progress can have adverse effects at individual sites. For example, a given site may be seriously disadvantaged in its efforts to secure funding if it is expected to achieve an observed best practice that is not realistic for understandable reasons. Of course, this begs the question of what the "understandable reasons" are, but this is the crux of the problem facing analysts who work in this area.

Other problems lie in the approach's heavy reliance on existing test scores as the basis of the outcome standards. The available test score indicators are far removed from the kinds of outcome standards implicit in Matrices A and A*. The approach enjoys the virtue of an explicit emphasis on outcomes, but is limited by the narrowness and crudeness of the available indicators.

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Cost Functions

In defining cost functions, economists have been the most active with ambitious models that attempt to take into account differences in unit prices of inputs as well as differences in how the inputs are being combined, all with effort to provide sophisticated controls for district structural characteristics in order to avoid the criticism we just made of the observed best practice approach.

The cost function approach makes use of econometric estimating techniques including explicit attempts to model simultaneous and other endogenous effects as well as nonparametric techniques such as data envelopment methods. These models represent some of the most promising work to date in the effort to establish the costs of outcomes and warrant a careful assessment. The models are potentially of great interest to policymakers because they establish conceptual links between outcomes and resources, and also because they have the potential to give concrete dollar estimates of the costs of achieving adequacy.

Let us begin with a general overview of the approach. The idea is to estimate a cost function. A cost function, when properly estimated, reveals the minimal cost necessary for achieving a given result. Presuming we can specify adequacy in terms of Matrices A and A*, and in theory we should be able to ascertain the idealized minimal cost of doing so—namely, TC(IBP), thanks to the construction of a cost function.

Of course, there are many difficulties which fall into different categories: (1) we are not very advanced in specifying the properties of Matrices A and A* (i.e., our outcome-oriented adequacy standards); (2) we suspect that there is a substantial amount of prevailing inefficiency in the field such that a survey of randomly selected sites could be misleading in terms of identifying best practices; (3) there is considerable endogeneity inherent in the system (i.e., features that have bearing on costs are related to other embedded characteristics and it can be difficult to disentangle the several different ways that influences on cost are connected); and (4) judgments need to be made about what is accepted as realistic versus idealized best practice.

Analysts have responded in various ways to these challenges, and we review their work in the order of the level of ambition and technical sophistication involved. An important step in the direction of estimating education cost functions was taken by Imazeki and Reschovsky (1998). They used multivariate methods to estimate a cost of education function for Wisconsin and took account of endogeneity by using instrumental variable estimating techniques. Imazeki and Reschovsky also included a teacher input cost index so that their model dealt with important unit cost differences as well as with costs associated with differences in how inputs can be combined to produce outcomes.

Imazeki and Reschovsky acknowledge the problem associated with uneven amounts of inefficiency across the observations in their data. This unevenness is problematic because it means that high levels of observed spending due to inefficient operation may be misinterpreted as unavoidable high costs of producing the outcomes in question. Concerns about the uneven levels of efficiency among observed schooling units has prompted some analysts to build "efficiency" adjustments in their cost function approach. William Duncombe and his colleagues have worked on this problem and have made use of data envelopment techniques to construct measures of individual school district's efficiency levels using New York state data (e.g., Duncombe, Ruggiero, and Yinger 1996). The logic is that the addition of a control for differences in efficiency across the sites in the sample establishes the long-sought realistic best practice benchmark that will ultimately permit the state to make aid adjustments that are sensitive only to bonafide differences in costs (higher expenditures due to circumstances over which there is no local discretion).

A central question is whether techniques like data envelopment are adequate to the task of generating a trustworthy efficiency adjustment-control. There are good reasons to exercise caution. First, the technique is similar in principle to what underlies the observed best practice method in which districts achieving similar results are compared and those doing so with the least amount of cost are singled out as examples of what, in theory, is possible for the others to achieve. A problem arises if what is possible for some is realistically not possible for others. The data envelopment method attempts to keep the analysis realistic by employing a linear optimization routine that compares districts facing similar exogenously determined environmental factors (e.g., size, wealth, composition of the student population, etc.). Although these environmental features are relevant and permit more sophisticated controls than those shown earlier for the cost of the observed best practice model, there is no doubt that they fall short of controlling completely for the circumstantial influences on what counts as realistic best practice (TC(RBP)).

As an example of how the statistical controls can fall short of the mark, consider the case of two school districts with very similar characteristics with the exception of their size. Suppose that the smaller of the two districts finds itself spending at higher levels to provide a comparable outcome for its students. Suppose also that the reason for these higher costs is a rancorous history of past attempts to reorganize the district into a larger unit. Finally, suppose that there is ample "blame" for this state of affairs, which is widely shared across and within the affected communities. It is hard to conceive of a statistical indicator which is going to capture the rancorous history that could have bearing on a decision to treat the higher spending in the smaller setting as a legitimate higher cost rather than as an instance of inefficiency that should not be offset by the state. We can reach different conclusions about what counts as "realistic" best practice in the smaller school setting, and an efficiency indicator that is based on the data envelopment methods that are currently available is not likely to resolve this question.

Second, the technique depends heavily on the specification of outcomes. Recall the emphasis we placed on the components of Matrices A and A* as the core of an outcome-oriented standard. The existing data envelopment methods are based on standardized test score outcomes and are insensitive to the kinds of important outcome specification questions that are included in these two matrices. This is important because there is a potential for the envelopment comparisons to be made across districts pursuing very different agendas in terms of outcomes. The higher spending that is observed in one place may reflect an efficient pursuit of higher standards (the results of which are not captured by the existing assessments), but the envelopment method could interpret the higher spending as evidence of a serious inefficiency. A clear specification of outcome targets and consensus about what the state-imposed adequacy standards are going to be is essential for the development of an accurate and dependable efficiency index.

Third, there are conceptually distinct degrees of efficiency, and data envelopment methods actually employ a relatively weak efficiency test. Ruggiero (1996) called attention to the difference between Farrell and Koopmans standards of efficiency. Passing the Farrell standard means that there is no way to reduce inputs equi-proportionally and maintain the same level of outcome. In other words, a school district will be Farrell efficient if it is impossible to reduce all inputs by some common percentage amount and maintain the same level of outcome. In contrast, within an inefficient district in a Farrell sense, it would be possible to reduce all inputs by, for example, 3 percent and have no adverse effect on outcomes. Koopmans efficiency requires that all slack be removed from the system such that it is impossible to reduce any input without adversely affecting the level of outcome. Thus, Koopmans efficiency is a more stringent standard in the sense that a district could achieve Farrell efficiency and still be able to make efficiency improvements by reducing the supply of one input relative to the others while holding the outcome constant.

The distinction between Farrell and Koopmans standards of efficiency is significant because data envelopment techniques make implicit use of the Farrell standard. In other words, while the data envelopment approach provides a control for differences in efficiency across the units in the sample, the efficiency of these units may still vary in the Koopmans sense, and it is possible for this variation to be substantial. The problem is that we are still left with a situation in which high expenditure levels in one place relative to another may be due to differences in Koopmans efficiency or differences in real costs. It is worth noting that this problem remains even if the other problems are resolved. Ruggiero (1996) has addressed this problem using a second stage (parametric) canonical regression technique that builds upon the data envelopment method to come closer to the identification of Koopmans efficient school districts.

These reasons for skepticism create a dilemma for policy analysts as well as for policymakers. We might agree that a cost function complete with a Koopmans efficiency index developed according to Ruggiero's specifications is conceptually preferable to a less sophisticated, observed best practice approach or a cost function with no adjustment for efficiency, but this conceptual progress comes with some potentially significant costs. It is more than a matter of making incremental progress toward a fixed goal, because there is real potential for efficient practices to be misinterpreted as inefficient practices. Districts that are working with realistic best practices could find themselves penalized wrongly because of remaining limitations in the techniques being developed. However, conceptual progress should not be discounted, and it is clear that further efforts need to be made to extend this work.

It is particularly important to make progress in terms of the specification of the outcomes (i.e., clarifying the properties of Matrices A and A*). We also see promise in approaches that blend elements of the educator judgment and the cost function approach. It could be possible, and quite desirable, to rely on sophisticated cost functions to generate first approximations of estimated costs with a given set of circumstances to reach the stipulated outcome standards but to then build in an explicit appeals or "clarification" process which would permit informed judgments about particular local circumstances that may make the first approximation results unattainable. We speculate that an iterative process that draws on professional judgments in the context of cost estimates emerging from sophisticated cost function offers the best hope of making progress toward identifying the true costs of adequacy. For an alternative view that places greater relevance on the professional judgement approach, see Guthrie and Rothstein (1999).

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Teacher Quality Research

Although many types of inputs contribute to the production of desired educational outcomes, in this section we narrow our attention to address what is known about the impact of teacher quality. In other words, we shift our focus to studies that trade comprehensiveness for a more focused examination of a specific category of inputs. We have selected studies estimating relationships between teacher quality and educational outcomes for several reasons. First, teacher resources represent a large proportion of the total resources committed to education, and consequently can have a disproportionate effect on the productivity of the enterprise. Because teachers are a key component of almost all of the Sijks in Matrix A, it makes sense to take stock of what is known about the productivity of this cross-cutting input. Second, several elements of this category of inputs, for instance, preservice teacher preparation programs and certification requirements, are particularly interesting from a policymaking perspective.

The studies included in our review vary in terms of how heavily they rely on formal economic models of production. Some use sophisticated econometric techniques to test hypothesized models of production. Others consider the costs and effects of various alternatives to teacher preparation. Still others test for relationships using simple bivariate correlational analyses. We include a variety of studies along this "methodological continuum" and also present findings from reviews that others exploring this literature have conducted. We contend that although all of this work does not fit squarely into the categories of production or cost function research, it is nonetheless important to consider given the lessons it provides regarding such a key input to the production process.

The category of inputs associated with teacher quality is broad. For instance, teachers' pay scales are generally based on factors which include years of experience and degree level. In addition, characteristics such as course work taken to prepare for the profession, prestige of the institution at which one's degree was earned, and literacy or knowledge measured through the use of tests have been identified as attributes likely to contribute to successful teaching. In this section, we examine the impact of three specific indicators of teacher quality: (1) a teacher's preparation program, including degree level, links to state certification, and the presence of extended or other alternative teacher education programs; (2) the specific course work taken by teachers in preparation for the profession, with attention given to both the amount (e.g., number of courses, number of credits) and the type (e.g., pedagogical, content-specific) of courses taken; and (3)teachers' test scores indicating some aspect of teacher knowledge, proficiency, and level of literacy. All of the studies reviewed focus on preservice preparation rather than in-service professional development. The impact of these indicators of teacher quality has been measured in terms of a variety of educational outcomes including student achievement (general-composite as well as in specific subjects), principals' evaluations of teachers, teachers' perceptions of themselves and the quality and impact of their preparation, and teacher attitudes. We were able to draw several broad conclusions about the relationship between these teacher quality variables and educational outcomes from the numerous studies and research reviews that we examined.

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Teacher Preparation/Certification Programs

One indicator of the quality of teachers concerns the "package" of their educational preparation, without attention to the individual components of that package (e.g., specific courses) or to the skills and knowledge acquired. This input has been studied in terms of the level of academic degree possessed by the teacher, the number of years of schooling, and whether or not the teacher has earned state certification to teach through traditional versus alternative routes. Much attention in the literature on teacher quality and preparation deals with the question of whether the quality of the candidates who are enrolled in, and graduate from, teacher education programs is lower than that of students in other degree programs, thus limiting the quality of the supply of teachers.

In general, the studies we reviewed pertaining to the impact of teacher education programs on teacher effectiveness offer several insights. First, "traditional" teacher education programs seem to make a difference with regard to a variety of measures of teacher quality and performance. Olsen (1985) found that graduates of education programs tend to be equal to or better than noneducation graduates in terms of their high school rank, math and English placement scores, and cumulative grade point averages in a variety of college subject areas. Hawk, Coble, and Swanson (1985) used a matched comparison design to demonstrate that student math achievement scores are higher for students whose teachers were certified in mathematics. Goldhaber and Brewer (1996) also report positive effects of subject-specific training programs on student math and science achievement. Darling-Hammond's (1990) review of the literature on the relationship between teacher education and teacher effectiveness found that fully prepared and certified teachers are generally more highly rated and more successful with students than teachers without full preparation.

In addition, several studies explore the impact of alternative teacher education programs such as requiring graduate education for teachers. Research shows that the relationship between graduate study and teaching effectiveness is modest (Domas and Tiedeman 1950; Goldhaber and Brewer 1996; Turner et al. 1986). Furthermore, several studies address the productivity of alternative routes to teacher certification through cost analyses concluding that alternatives such as extended year programs (Hawley 1987) and master's degrees (Knapp et al. 1991) may not be cost effective.

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Teacher Course Work

Measures of the level and type of course work taken by teachers represent proxies for what teachers know and can do in the sense that course work indicates the degree of exposure individuals have had to particular areas of study (e.g., subject-specific content versus teaching methods). During the mid-1980s, the debate over the importance of subject matter versus education course work in teacher preparation programs took on new life (Ferguson and Womack 1993). This theme surfaces in a number of the studies we examined which consider course work as the indicator of teacher quality.

The studies we reviewed vary in terms of the measures, data, and methods used. Nonetheless, they are rather consistent in their findings. Most notably, they suggest that teacher course work in both content areas and pedagogy contributes to positive educational outcomes, but the relative impact of their effects varies. Subject matter preparation in the subject area taught is shown to be important in several studies (Perkes 1968; Hawk, Coble, and Swanson 1985), but investments appear to have diminishing returns after a certain point (Darling-Hammond 1990; Monk 1994). In contrast, course work in education methods is shown to have consistent positive effects that often outweigh those of content coursework (Ferguson and Womack 1993; Monk 1994). Further supporting this finding are a number of meta-analyses that emphasize the importance of pedagogical verses content course work in the preparation of teachers (Evertson, Hawley, and Zlotnik 1985; Ashton and Crocker 1987; Darling-Hammond 1990).

Several of the more sophisticated multivariate studies reviewed demonstrate some of the complexities associated with the education production function. More specifically, the production process appears to depend on a number of factors including student characteristics, teacher attributes, and subject area (see, for examples, Druva and Anderson 1983; and Monk 1994). In addition, Monk and King (1994) looked at multiple levels of schooling to conclude that it is the cumulative effect of the set of teachers a student has had over time, rather than the subject matter preparation of the entire faculty, that affects student mathematics and science achievement.

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Teacher Test Scores

Test scores are arguably the best measure of what a teacher knows and can do because they go beyond exposure to programs and specific courses to assess the knowledge and skills that individuals have actually acquired. However, test scores are also arguably the least policy manipulable relative to the other indicators of teacher quality discussed in this paper. Although policymakers can require that certain tests be taken and passed by teacher candidates, it is far more difficult to influence the degree to which individuals excel on these tests, particularly broad proficiency assessments like literacy tests. The debate over the role and relevance of teacher test scores received a great deal of attention in the late 1970s through the 1980s. One explanation for this may be that the legality of using the National Teacher Examination (NTE) for certification purposes was upheld by the United States Supreme Court in N.E.A. versus South Carolina in 1978 (Stedman 1984).

Given the role of the NTE as a potential gatekeeper for teachers, the predictive validity of this instrument has been the object of study. Although Ayers and Qualls (1979) found that NTE scores are significantly related to grade point averages and scores on the ACT, correlations between NTE scores and principal and pupil ratings were found to be quite low. Likewise, Quirk, Witten, and Weinberg (1973) demonstrate that NTE scores are not highly correlated with supervisor ratings during the student-teaching period or during the first year of teaching. Pugach and Raths (1983) make several recommendations about the use of the NTE that argue against using this test as an end-of-program criterion for teacher candidates.

Other studies suggest that some test scores seem to predict high levels of teacher performance and desired educational outcomes. More specifically, tests that assess the impact of literacy levels or verbal abilities of teachers tend to show positive effects (Coleman et al. 1966; Ehrenberg and Brewer 1995; Ferguson 1991). In contrast, studies of the impact of the NTE (as noted above) and other state-mandated tests of basic skills, teaching abilities, or both (Guyton and Farokhi 1987) do not appear to be strong predictors of teacher performance. Finally, these studies also reinforce the complexity of the education production process in that the impact of what teachers know and can do as indicated by test scores depends on factors like student attributes (Ehrenberg and Brewer 1995; Strauss and Sawyer 1986).

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Lessons to Learn About Indicators of Teacher Quality

The central goal of this review was to organize the existing research concerning the productivity of teacher inputs to demonstrate how education productivity research can inform policy decisions. We chose to focus on teacher quality given the large proportion of educational resources attributable to this type of input and the relevance that findings in this area have for policy. Indeed, numerous policymakers have called for various reforms related to the preparation of teachers (Bush 1987). For instance, in its call for improved teacher preparation, the National Commission on Excellence in Education (1983), in their report A Nation at Risk, stated "teacher preparation programs are too heavily weighted with courses in educational methods at the expense of course in subjects to be taught." The Carnegie Foundation for the Advancement of Teaching recommended that teacher education programs require a 3.0 grade point average for admission, and that teachers complete courses in an academic core in four years and then spend a fifth year learning about education (Boyer 1983). Likewise, the Holmes Group (1986) advised that all major universities with substantial enrollments of preservice teachers should adopt the four-year liberal arts baccalaureate as a prerequisite for acceptance into their teacher education programs. Most recently, the National Commission on Teaching and America's Future has focused on accreditation, recommending that these issues be left to professional organizations. Clearly, the studies reviewed in this section have implications for these types of policy decisions, and improving the overall productivity of the educational enterprise.

In general, the wide range of studies reviewed here suggest several broad conclusions regarding teacher quality. First, teacher education programs seem to make a difference, but alternative routes to certification such as extended programs or the requirement of a master's degree for certification can be questioned on cost-effectiveness grounds. Second, although teacher course work in both subject matter and pedagogy have been shown to contribute to positive educational outcomes, investments in the area of subject-matter preparation may have diminishing returns after some point. Third, some teacher test scores, particularly those that measure broad qualities like literacy or verbal ability, appear to be associated with high levels of teacher performance. Finally, and perhaps most important, several of the more methodologically sophisticated studies demonstrate the complex nature of the education production process. Factors associated with students, teachers, and courses have been shown to affect the impact of teacher quality variables on educational outcomes. Also, other issues such as the alignment between teacher preparation and teacher assignment have begun to emerge in the literature as important issues that have an impact on the productivity of teacher resources (Hawk et al. 1985, Monk and Rice 1998).

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Implications for Policymakers and Researchers

Education policymakers face contentious choices in a climate of limited resources. They are responsible for making wise decisions about how to get the most productive use of these resources. The resource allocation model we presented in this paper provides a starting point for framing and even guiding the decisions that must be made. The distinction between actual practice and best practice, particularly realistic best practice, is important to maintain as efforts are made to make efficient progress toward attaining new outcome standards. The greater the discrepancy in the cost associated with realistic best practice and actual practice, the more productive the system can become. These considerations are important for policymakers at many different levels of the decision-making structure.

It is heartening to see the progress that has been made toward mapping the path to greater levels of productivity in the education arena. Efforts to estimate the magnitude and nature of the links between education outcomes with their costs are becoming more sophisticated as well as more informative and useful. Furthermore, knowledge about the productivity of key cross-cutting inputs such as teacher quality is becoming more conclusive, providing insights that can lead policymakers toward improved practice.

What are the next best steps to take in the quest to realize more productive use of resources in schools and school systems across the country? We see three promising steps that need to be taken in the near term.

First, further work needs to be done to establish the conceptual link between outcomes and costs. The matrices we introduced provide useful insights, but we recognize that more needs to be done, particularly with respect to efficiencies that can be realized by providing services to multiple students at once as well as by providing services that meet multiple goals simultaneously. The model is built on the premise that different students benefit more or less from different kinds of services with respect to particular educational outcomes, and this has implications for the cost of the service alternatives. However, it is reasonable to expect that what works best for one student may also work well for others in ways that make it possible to realize additional efficiencies. Similarly, the model specifies services with respect to individual educational goals, but clearly some types of services are intended and can be expected to promote multiple outcomes simultaneously. These issues of aggregation are important to consider as we operationalize this model because schools and school systems are bound to serve groups of students and routinely seek to meet multiple goals simultaneously.

Second, policymakers need to be clearer about the content of their high performance standards, particularly with respect to the number and types of standards and, even more important, with respect to the degree to which those standards are expected to be universal. Departures from universality may involve applying the standard to only a subset of the student population, or allowing the standard to be met at different levels for different students. The specification of outcome standards is a key step in the further development of the resource allocation model and the linking of costs to outcomes. Policymakers need to do more than generate and salute vacuous rhetoric. Hard decisions need to be made about the degree to which we aspire to universal versus differential outcomes across students. The resource allocation model makes it clear that answers to questions about the costs of adequacy presuppose clear pictures of what the outcome standards entail.

Finally, we were impressed with the potential for iterative cost calculation methods to generate the most useful estimates of cost. We think the key is to draw upon educator judgments that are informed by the results of sophisticated cost and production function estimations. We were similarly impressed by the progress being made toward estimating the productivity of teacher resources, although we share in the frustration of many regarding the existing limits on the availability of direct indicators of teacher quality. There are important implications for the collection of the next generation of data for cost-productivity research. It is essential to more directly measure the capabilities of teachers. Crude proxy measures for teacher quality, such as degree level, years of experience, and even numbers of courses taken are inadequate substitutes for direct measures of teacher content knowledge and teaching capabilities. It is also essential for the next generation of data for cost-productivity research to include sophisticated measures of pupil outcome gains. Value-added test score measures may be the best that can be expected in the near term, but we dare to hope that progress can be made toward the developing assessment instruments that are well aligned with the outcome standards embodied in Matrices A and A*.

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1 Alternatively, we could introduce the idea that in addition to the 0 versus 1 question that is dealt with by Matrix A, the expected degree of learning might vary among the students who are expected to learn in a given area. In other words, among the 1s in a given column of Matrix A, there may be variation in the degree to which each of the students is expected to perform.

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