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Classroom Diversification: A Strategic View of Educational ProductivityCorporation for Public School Education K16, Round Rock, Texas
This article advances a theory of educational productivity based on a paradigm of classroom diversification that defines a strategic view of the education production process. The paradigm’s underlying premise is that classroom student performance, and the instructional interactions that produce such outcomes, depend on economies derived from the learning relationships that exist across and among students in a classroom and on the technological fit between students’ learning needs and a teacher’s capacity. In addition to the conceptual classroom diversification framework, measures of classroom student diversity and teacher capacity are presented, followed by a discussion of the implications of the proposed classroom diversification paradigm for educational research, policy, and practice.
Key Words: classroom diversification • student diversity • teacher capacity • technological fit • production function Researchers began over 50 years ago to investigate productivity in schools with the intent to understand the relationship between the supply of school resources and the level of educational outcomes. They studied data collected through large-scale surveys of students, schools, and teachers. The central dependent variables were standardized measures of academic student achievement, and the independent variables were measures of teacher, student, and school characteristics. Using correlation and regression analysis, researchers assumed that well-specified equation models of the schooling process could estimate the effects of school resources, student background, and particular factors, such as teacher training and experience, on student performance. Eric Hanushek’s comprehensive reviews of these studies indicate that educational productivity research has not realized such possibilities. The initial synthesis (Hanushek, 1981) included 29 major studies that covered 130 separate analyses of the relationship between student performance and school and teacher variables. Hanushek analyzed the regression coefficients of the input variables to determine if they were positive or negative and if the effect measured was statistically or not statistically significant. He included a fifth category for coefficients that were not statistically significant and for which the coefficient’s sign could not be determined. Using a simple vote-counting method to compare data, Hanushek found no systematic, positive relationship between schooling inputs and educational outcomes. Hanushek (1986, 1989, 1991) updated the synthesis of the educational productivity research, but more contemporary investigations only further supported his conclusions in the original review. Nonetheless, some researchers challenged Hanushek’s reinforced conclusions. Hedges, Laine, and Greenwald (1994) used a variety of meta-analytic techniques to reanalyze Hanushek’s study sample and found that teacher education, ability, and experience, along with small schools and lower teacher/student ratios, are all positively associated with student achievement. Using effect-size analysis, they also found a substantially positive, significant relationship between student outcomes and per pupil expenditures, but not for the other factors. In his rejoinder to their study, Hanushek (1994) argued that the evidence still strongly indicates that there is no systematic relationship between schooling inputs and educational outcomes. He stated that although the evidence suggests that some districts use their expenditures more effectively than others, this does not ensure that all districts will follow a similar course. Rather, these studies insinuate either that the natural inclinations of school systems do not systematically lead to the effective use of resources or that other components of spending must be more effective in improving educational outcomes. In retrospect, one must view the conclusions derived by Hanushek (1981, 1986, 1989, 1991, 1994) and by Hedges et al. (1994) with caution. A lack of quality data, publication selection bias, and less effective research designs were basic limitations inherent in the studies reanalyzed for these reviews. Since the publication of those studies, researchers have attempted to address a number of technical issues in the hope of improving the utility of the production function concept in studying educational productivity. For example, researchers in one recent study examined the specification of input variables commonly used in educational productivity research, eliminating input variables they identified as incorrectly specified with apparent improvement in the analysis of their data (Dewey, Husted, & Kenny, 2000). Such errors in specifying the input variables create serious statistical problems, limiting the ability of analysts to conclude what school inputs are significant. Meanwhile, other researchers added new input variables to see if they could more accurately model the schooling process (e.g., Lamdin, 1996). Researchers have also applied the economic production function model in randomized field experiment designs (e.g., Mosteller, 1995) rather than in the nonexperimental research design commonly used in most educational productivity studies. Nonexperimental research designs make studies open to errors in specifying or in omitting input variables that may be important determinants of educational outcomes because researchers’ subjective assumptions about the characteristics of the research population form the basis of the design. In comparison, randomized field experiments avoid these flaws, because if researchers randomly select treatment and control groups from large enough populations, they can assume equal distribution of characteristics in the sample population (Ludwig, 2001). Thus, randomized field experiment designs produce results that are more reliable regardless of errors in specifying or in omitting input variables that may be relevant to educational outcomes (Rothstein, 2000). Still other researchers have attempted to improve the reliability of using the economic production function concept by changing its functional form from an additive, homothetic linear model to a nonadditive, nonhomothetic, nonlinear model (e.g., Figlio, 1999). Such a model allows for systematic differences in the way that schooling inputs affect heterogeneous student populations and student performance relative to the scale of production and for the possible existence of nonlinear effects of school inputs on educational outcomes. Researchers have also used hierarchical linear modeling (HLM) techniques that allow more advanced educational productivity modeling than conventional analytic techniques based on ordinary least squares methods (e.g., Huang, 2000). For example, researchers using HLM can examine how environmental variables at different levels of the schooling process—classroom, school, or district—affect individual educational outcomes. They can also investigate cross-level interactions. Hierarchical modeling takes variation to create objects of inference at different levels of the hierarchy. The result is that HLM allows researchers to draw more accurate inferences from the data about the population means at any level: classroom, school, or district (Osborne, 2000). In light of the potential promises to educational productivity research these technical improvements suggest, a more fundamental issue persists. Levin (1980) first proposed that a major problem in the study of educational productivity was the use of the economic production function model to link a list of school resources with a particular educational outcome without the support of any relevant theory. Thus, he argued that educational productivity research had not contributed to an accumulation of knowledge with reasonable consistency and predictive power. Most recently, Cohen, Raudenbush, and Ball (2000) proposed a "theory of instructional resources" that views educational productivity through the instructional interactions of teachers, students, and content, in learning environments. They contended that educational productivity depends almost entirely on how teachers and students use resources in instruction. Essentially, what teachers and students do with resources is what matters most to student learning, not simply if resources are available. This article builds on the recent work of Cohen et al. (2000) and presents a theory of classroom-level educational productivity based on a paradigm of classroom diversification that defines a strategic view of the education production process.1 The paradigm’s premise is that classroom student performance, and the instructional interactions that produce such outcomes, depends on economies derived from the learning relationships that exist across and among students in a classroom and on the congruity between students’ learning needs and a teacher’s capacity. The classroom diversification paradigm defines students’ learning needs as those resources that students must have to develop, to the extent of their innate capabilities and limitations, in the various domains of human growth (e.g., cognitive, social, emotional, physical, aesthetic). Classroom diversification presumes that the more students’ learning needs in a classroom are related to one another by the kinds of technology and resources available for them to use in learning, the more economies can be exploited. This results in higher overall levels of learning productivity across all students. The classroom diversification paradigm defines teachers as the primary agents who regulate these resources through their teaching capacity, by using classroom management skills, teaching strategies, instructional methods, and planning processes: technologies that augment, facilitate, and/or guide students in the various domains of human development. When a teacher’s capacity is relevant to students’ learning needs, classroom diversification theory assumes that the teacher uses technology and allocates available resources to students effectively, thus maximizing student learning. When a teacher’s capacity is not relevant or is insufficient to students’ learning needs, classroom diversification theory assumes that the teacher uses technology and allocates available resources to students ineffectively, thereby reducing student learning. Therefore, instructional interactions among teachers, students, and content knowledge in learning environments are important to student learning. However, when, how well (i.e., effectively and efficiently) and to what extent (i.e., seldom or often) these instructional interactions occur in ways that allow teachers and students to use resources to maximize their learning depends both on economies derived from the learning relationships that exist across and among students in a classroom and on the congruity between students’ learning needs and the teacher’s capacity. Consequently, how principals assign teachers and students to classrooms makes a difference to student learning. I have organized the remaining parts of this article into four sections. The first section develops a conceptual framework for understanding the classroom diversification paradigm of educational productivity. I begin with an overview of limitations and criticisms of the economic concept of the production function, based on the neoclassical economic theory of the firm, the same theory that, intentionally or not, has served as the foundation for the productivity model used in educational productivity research. I then follow this discourse with a presentation of theoretical constructs in forming the classroom diversification paradigm, proposing it as a new foundation for examining the production function in education. The section concludes with a summary of the inherent differences in the primary attributes and assumptions between the neoclassical educational productivity model and the classroom diversification educational productivity model. In the second section, I begin with a discussion of the sources of classroom student diversity relevant to student learning and follow with a presentation of the types of classroom student diversification strategies operational and observable in the classroom. Classroom student diversity refers to the differences among a classroom’s students with regard to their attributes, unique learning needs, and modes of knowledge and skill acquisition. Classroom student diversification is the process that occurs when a principal assigns students to classrooms from these sources of student diversity. I then present measures of classroom student diversification, followed by graphical illustrations, to show their inherent properties relevant to the classroom diversification educational productivity model. In the third section, I discuss teacher capacity relevant to student learning and follow with an overview of the different states of teacher capacity that are inherent in the classroom diversification paradigm. A teacher’s capacity is the skills, abilities, and knowledge derived from his or her training and experiences relevant to students’ learning needs. I then present measures of teacher capacity. Although the algorithms underlying these measures are the same ones used to measure classroom student diversification, their inherent properties have somewhat different interpretations when used to measure teacher capacity. Graphical illustrations follow this presentation to illustrate the inherent properties of the teacher capacity measures relevant to the classroom diversification educational productivity model. In the fourth and final section, I discuss the implications of classroom diversification for educational research, policy, and practice. I preface this discussion with an overview of the possible limitations in the researchers’ use of the classroom diversification framework to study educational productivity. In light of these possible limitations, I then discuss how the classroom diversification framework may prove more feasible to implement than other lines of inquiry recently proposed by scholars of educational productivity. Next, I outline the implications of classroom diversification for educational policy and practice. I conclude the article with thoughts regarding classroom diversification: a strategic view of educational productivity.
This section develops a conceptual framework for understanding the classroom diversification paradigm of educational productivity. I begin with an overview of the economist’s concept of the production function based on the neoclassical economic theory of the firm and then follow this discourse with a presentation of the theoretical constructs of the classroom diversification paradigm as a new foundation for the concept of the production function in educational productivity. I conclude this section with a summary of the inherent differences in the primary attributes and assumptions between the neoclassical educational productivity model and the proposed classroom diversification educational productivity model.
The Neoclassical Educational Productivity Model In the neoclassical economic theory of the firm, an economist explicitly treats a single participant, an entrepreneur, as a rational individual whose singular goal is to maximize profits by maximizing revenue and minimizing costs. The economist enters into the theory other participants, such as employees, customers, and suppliers, only implicitly as "conditions" to which the entrepreneur adjusts in finding an optimal solution for maximizing profit (Hirshleifer, 1980; Sen, 2002). On the basis of these assumptions, economists represented the neoclassical firm as a simple production function that combines inputs in the most efficient manner to create outputs that result in outcomes such as maximized profits. Thus, the term production function refers to the physical relationship between the input of productive resources (e.g., raw materials, labor, capital, land, managerial talent, etc.) and the output of goods or services per unit of time. A production function is simply a summary description that relates some flow of output to the size of the corresponding flow of input required in a production process (Brown & Saks, 1980; Thompson, 1981). Perhaps the most important of the assumptions of the neoclassical economic theory of the firm is that the singular goal of the firm is to maximize profits by maximizing revenue and minimizing costs (Hirshleifer, 1980; Sen, 2002). On the basis of this primary assumption, the neoclassical economic production function model assumes that given a fixed level of available technology, a firm will pass through three stages in expanding its scale of operation. These stages are (a) a short range of increasing returns to scale, (b) a lengthy range of constant returns to scale, and (c) a range of decreasing returns to scale. Increasing returns to scale occur because of increasing specialization in the use of resources and the exploitation of excess technological capacity in the production process. Constant returns to scale occur when the production process has achieved optimal standardization of production. Decreasing returns to scale occur because of limitations to efficient performance in managing the production unit as it increases in size. Firms can maximize profit in any of the three stages but seek to operate in constant returns to scale assuming they also want to operate at maximum efficiency (Miller & Upton, 1986; Wessels, 2000).2 The neoclassical economic production function model also assumes that labor capacity and the use of such capacity are constant—rigid and consistent—given the available technology and organization of production. The result is that there is a uniform and spontaneous transformation of labor into labor services. The extraction of labor derives routinely from the quantity and quality of labor inputs (i.e., human capital; Thompson, 1981). Furthermore, the neoclassical economic production function model assumes that inputs and outputs are homogeneous in that each unit of input and output is like every other, such that they are perfect substitutes for one other. Thus, inputs and outputs are standardized. Likewise, the model assumes standardized processes among two or more production systems if they use the same inputs to produce the same outputs. The implication is that a production enterprise with single or multiple production systems evenly distributes resources and applies standardized technology in producing each unit of output (Thompson, 1981). Despite its simplicity and elegance, economists have long criticized the neoclassical economic theory of the firm for its failure to account for realities in the firm (Cyert & Hedrick, 1972). Neoclassical economists purposefully did not theorize about what goes on inside a firm. Organizationally, neoclassical economists defined the firm’s production function as an undifferentiated "black box" that mysteriously transformed factors of production into products and services for sale on the market. Thus, the neoclassical economic theory of the firm, although useful for understanding markets and resource allocation efficiency, largely ignored the human dimension: the differing interests, resources, and actions of entrepreneurial owners, boards of directors, shareholders, managers, employees, communities, governments, and numerous other firm stakeholders. The result is that the neoclassical economic theory of the firm has done little to explain how production is organized within the firm or how one firm can differ from another (Nelson, 1991). When educational researchers began to study educational productivity in the 1950s, they borrowed the neoclassical economic concepts and represented the school as a simple production function that combines inputs such as school resources in the most efficient manner to create outputs that result in maximized educational outcomes such as student achievement, as measured by standardized tests. Thus, the neoclassical educational productivity model was born. In so doing, educational researchers also inherited, or perhaps adopted, intentionally or not, the same underlying limitations that economists eventually acknowledged. Yet, educational researchers persist today in using these concepts to study educational productivity. Some scholars of educational productivity over the years have alerted the research community to the fact that these inherent limitations have undermined the potential contributions of educational productivity research. For example, scholars such as Windham and Chapman (1990) have argued that the neoclassical economic production function assumptions are unrealistic when applied to classroom learning because the education process deals more closely with human factors. They contend that although economic theory stipulates productivity relationships individually under conditions in which all other things are held constant (ceteris paribus), an educational decision maker must often determine a mix of input variables simultaneously. Teacher quality and quantity, the availability and use of materials, equipment and facilities, and means of motivating student, parent, and community efforts are some of the major input categories that must be considered jointly in most educational decisions. Windham and Chapman (1990) cited the variability in the nature of schooling inputs as another limitation in using the neoclassical economic production function concept in educational productivity research. Although variety exists among the inputs of teachers, students, schools, and materials, and in the way that they combine, individual human inputs can also vary over time. For example, teacher motivation and effort can fluctuate from day to day, or even within a given day. Likewise, the attentiveness and effort of students also can vary, even during the course of a classroom period. Windham and Chapman (1990) point out that some aspects of the schooling process related to the allocation of teacher resources may also be difficult to incorporate into the conceptual form of the neoclassical economic production function. For example, a teacher in a classroom may spend extra time with slower learners while permitting more capable students to work on their own with textbooks or other materials. The neoclassical economic theory of production rationalizes the teacher’s decision. The teacher is operating on the belief that the marginal product of a unit of his or her time is more valuable to the slower learner than to the more advanced student. Therefore, the greatest relative productivity advantage lies in combining the teacher’s input with the inputs of the most disadvantaged students. In determining the marginal benefits of such a decision, a teacher would need to identify and estimate the effect of his or her individual inputs on multiple students. A neoclassical economic production function model can deal with such multi-student factors, but they add complexity to the analysis and heighten the implicit subjectivity of the valuation process. A teacher would also need to know both the costs of inputs and the relationship (independently, jointly produced, and mutually exclusive) among the inputs and the mix of outputs and outcomes, as well as the value assigned to alternative output and outcome mixes. In reality, teachers make their teaching decisions without the complex information described. Operationally, the allocations and combinations of teacher-student inputs are simply too difficult for educational researchers to measure. These input variables include relationships that are often a product of tradition, law, regulation, or contractual agreement. The teacher’s dominance and role in the classroom are examples of tradition institutionalized by law, regulation, and contract, factors that may not be quantifiable for a neoclassical economic production function model but that are relevant to productivity in education (Windham & Chapman, 1990).
The Classroom Diversification Educational Productivity Model Classroom diversification presumes that the more students’ learning needs in a classroom are related to one another by the kinds of technology and resources available for them to use in learning, the more economies can be exploited. This results in higher overall levels of learning productivity across all students. Therefore, classroom diversification presumes that students’ learning processes are not standardized. Thus, schooling inputs and educational outcomes are heterogeneous under the perspective of classroom diversification; each unit of schooling input and educational outcome is different, and consequently they are not perfect substitutes for one other. The implication is that the classroom teacher distributes schooling resources in an uneven manner but one that is most relevant to students’ learning needs in the classroom. Classroom diversification defines how efficient and effective the teacher distributes these resources to students in terms of the technological fit, defined as follows, between students’ learning needs and the teacher’s capacity. From the perspective of classroom diversification, the congruity between students’ learning needs and the actual technology used to transform schooling inputs into educational outcomes defines a given level of technological fit. The classroom diversification’s notion of technological fit embodies Windham and Chapman’s (1990) concept of technological efficiency: a state of production in which a desired level of output maximizes a given level of inputs. Under such conditions of learning productivity, classroom diversification assumes that the technology used in the student learning process is appropriate and at an optimal level for achieving a desired or expected level of student learning. When technological inefficiencies occur, classroom diversification assumes that the technology being used to transform a given level of inputs into student learning may not be appropriate or at an optimal level for achieving a desired or expected level of student learning for a given group of students. Under classroom diversification, a moderating variable in the success of using technology and allocating resources to students is the classroom teacher’s capacity as defined by the teacher’s knowledge, skills, and experience relevant to student learning. Therefore, technological fit depends largely on the teacher’s capacity available and relevant to students’ learning needs in the classroom. Several configurations of technological fit can exist between the teacher’s capacity and students’ learning needs. To facilitate the discussion of these configurations, let TC(A) represent the teacher’s capacity available to students in the classroom and SL(N) represent students’ learning needs. When a teacher’s capacity is relevant to students’ learning needs, classroom diversification assumes that the teacher more effectively uses technology and efficiently allocates resources to students, thus maximizing student learning. Under such conditions, technological fit is optimal if the classroom generates constant returns to scale, such that available teacher capacity is fully used by students in the classroom, TC(A) = SL(N).
A teacher could also have an oversupply of capacity relevant to the students’ learning needs in the classroom, TC(A) > SL(N). The classroom in this situation would generate, in the short range, increasing returns to scale; excess teacher capacity could be exploited given the entry into the classroom of more students with learning needs matching that teacher’s surplus capacity. Similarly, a teacher could have some portion of available capacity, tC(a1), fully used by students in the classroom, and another portion, tC(a2), not relevant to any student learning needs in the classroom, tC(a1) = SL(N), tC(a2)
Conversely, a teacher could have a shortage of capacity relevant to a particular student learning need in the classroom, TC(A) < SL(N). The classroom in this situation would be generating decreasing returns to scale in that the teacher’s limited capacity would most likely result in poor or inappropriate use of technology and inefficient allocation of resources to students. In such a case, student learning would not be maximized. By contrast, when teacher capacity is available but not relevant to any of the learning needs of the students, TC(A)
From the perspective of classroom diversification, technological fit is essentially the critical factor, to the extent that a teacher effectively uses appropriate technology and efficiently distributes available resources most relevant to the learning needs of each individual student in the classroom. Therefore, classroom diversification assumes that a classroom can operate simultaneously in more than one scale of operation—increasing, constant, and decreasing—providing that the classroom contains several groups of students with different learning needs. The implication is that classroom diversification also assumes the existence of labor in the form of teacher capacity and the use of such capacity to be dynamic—flexible and adaptive—given the students’ learning needs. If the latter conditions exist, the result is a diverse and purposeful transformation of the teacher’s capacity into teaching services. Such extraction of teacher capacity derives routinely from the quantity and quality of the teacher’s human capital. Table 2 summarizes the inherent differences in the primary attributes and assumptions between the neoclassical educational productivity model and the classroom diversification educational productivity model.
Figure 1 visually depicts the production function on the basis of the neoclassical educational productivity model. The neoclassical educational productivity model assumes that the level of output or learning productivity is defined as a function of the quantity of schooling inputs available and the extent to which a given constant level of technology can transform these inputs into educational outcomes. The teacher is one example of a schooling resource that provides inputs into the students’ learning process. The neoclassical educational productivity model assumes that increasing inputs such as teacher resources (e.g., experience, knowledge, ability, skills) available in the classroom should result in higher levels of outputs such as student learning.
In comparison, the classroom diversification educational productivity model also assumes that student achievement is affected by schooling inputs. However, the classroom diversification paradigm does not define the teacher as a school resource that inputs into the student learning process, as shown in Figure 1. Rather, the teacher is defined as a fundamental component of the student learning process itself, as shown graphically in Figure 2.
In this proposed model configuration, classroom diversification assumes that the teacher interacts with the students in the learning process by regulating the entry of school resources into the student learning process, efficiently distributing the resources to students on the basis of their learning needs, and effectively using the resources to produce some level of student achievement output. The classroom diversification paradigm defines students’ learning needs as those resources that students must have to develop, to the extent of their innate capabilities and limitations, in the various domains of human growth (e.g., cognitive, social, emotional, physical, aesthetic). These resources manifest themselves in forms that teachers can distribute to students. Such resources include, but are not limited to, school curricula (e.g., gifted vs. nongifted), materials (e.g., library books), tools (e.g., Internet access), structures (e.g., school-business partnerships), systems (e.g., daily attendance), and services (e.g., after-school tutoring). Teachers also regulate resources such as the students’ time (e.g., additional opportunities to learn), space (e.g., reading centers), places or situations (e.g., field trips), and settings (e.g., primary versus dual language bilingual classroom). The classroom diversification paradigm defines teachers as the primary agents who use their teaching capacity to regulate these resources through classroom management, teaching strategies, instructional methods, and planning processes, technologies that augment, facilitate, and/or guide students in the various domains of human development.3 If instructional interactions of teachers, students, and content knowledge in environments are important to student learning, as Cohen et al. (2000) proposed, the technological fit between the teacher’s capacity available, TC(A), and the students’ learning needs, SL(N) forms the basis of this interaction between teacher and student. Essentially, the technological fit between students’ learning needs and teacher’s capacity determines when, how well (i.e., effectively and efficiently), and to what extent (i.e., seldom or often) these instructional interactions occur when teachers and students do use resources to produce student learning. Operationally, when a teacher’s capacity is relevant to the students’ learning needs, as shown in Figure 2, classroom diversification theory assumes that the teacher effectively uses technology and allocates available resources efficiently to students, thus maximizing student learning. When teacher capacity is not relevant or is insufficient to the students’ learning needs, classroom diversification theory assumes the teacher ineffectively uses technology and allocates available resources inefficiently to students, thereby reducing student learning. Thus, the relevance of the teacher’s capacity to the students’ learning needs essentially is the critical factor with regard to technological fit, a concept that is fundamental to how well teachers use technology and allocate available resources to students. Consequently, how principals assign teachers and students to classrooms makes a difference to student learning.
In this section, I begin with a discussion of the sources of classroom student diversity relevant to student learning and follow with a presentation of the types of classroom student diversification strategies operational and observable in the classroom. I then present measures of classroom student diversification followed by graphical illustrations to show their inherent properties relevant to the classroom diversification educational productivity model.4
Sources of Classroom Student Diversity
Student Race and Ethnicity
Student Gender
Student Ability
Student Metacomprehension
Student Mobility
Student Social Economic Status
Student Language Proficiency
Special Education Inclusion
Types of Classroom Student Diversification In a related diversified classroom, students can be grouped together on the basis of student attributes, unique learning needs, and their modes of knowledge and skill acquisition (i.e., the sources of classroom student diversity defined earlier). Operationally, students are related when they rely on common learning technologies, teacher capabilities, and knowledge-based resources. For example, the type of resource that students grouped together by similar ability might share is a common curriculum.6 Once student differences are identified, teachers can attempt to realize the benefits of relatedness by integrating or grouping new students with others who have similar characteristics, by forging new student relationships, and by creating mechanisms to ensure cooperation across groups of students in the classroom. An important issue in related classroom diversification is identifying any real and meaningful areas of commonality that will affect students’ learning in the classroom. If such a meaningful commonality is lacking, assigning students of diverse backgrounds to the same classroom student groups may still be justifiable, but the rationale will need to be different. Hence, the concept of related classroom diversification is more than a definitional issue. In contrast, students in an unrelated diversified classroom can be unrelated on the basis of their attributes, unique learning needs, and modes of knowledge and skill acquisition (i.e., the same sources of classroom student diversity that define related diversified classrooms). Operationally, students are unrelated because they rely on different learning technologies, teacher capabilities, and knowledge-based resources. For example, a teacher could separate students whose unrelatedness is defined by their differences in English proficiency (English speakers vs. non-English speakers). Here, the teacher’s capacity is his or her ability to communicate and teach in the language of the non-English-speaking students. Unrelated diversified classrooms can provide students with a number of learning benefits often not recognized in related diversified classrooms. For example, public law has fueled a trend toward teaching exceptional students in the regular classroom. Students with disabilities can often benefit from well-planned and organized integration experiences. Additionally, non–special education students can be provided unique opportunities to learn firsthand about human differences and similarities, experiences that can influence their understanding of disabilities and improve their sensitivity to peers (Diamond, Hestenes, & O’Connor, 1994). Unrelated diversified classrooms can also provide teachers with learning production efficiencies not available in related diversified classrooms. A classroom strategy commonly used by teachers under conditions of diverse student ability is cooperative learning, a teaching strategy that involves children’s participation in group-learning activities that can promote positive interaction and learning between students of different ability levels (Johnson, Johnson, Johnson, & Roy, 1984). The result is that students exchange their own knowledge and share among one another the skills and abilities each possesses for learning. From the perspective of classroom diversification, this allows a more efficient allocation of limited teacher capacity and finite classroom resources across classroom students. However, Slavin (1987) maintained that peer interaction in and of itself does not enhance learning. Rather, how the teacher guides those student interactions is what determines students’ learning. Classroom diversification assumes that the extent to which a teacher is able to guide those interactions among students of diverse abilities and to facilitate such exchanges of knowledge and sharing of skills and abilities is dependent on the technological fit between students’ learning needs and the teacher’s capacity.7
The Measurement of Classroom Student Diversification The classroom’s total student diversity (DTc) is the summation of a weighted average of the student diversification from each student diversity source, across all sources of student diversity, N, in the classroom. Thus, the DTc measure incorporates two dimensions of total student diversification: (a) the scope or number of sources of student diversity within the classroom and (b) the relative importance of each of these student diversity sources to a classroom’s total student diversity.
Related Student Diversification Because the classroom can include several sources of student diversity, N, total related student diversification for classroom c, DRc, is a function of GDRj, where j = 1 to N, as shown in Equation 3 in Appendix A1. The total related student diversification measure (DRc) is the summation of the share of the group-related student diversification (GDRj) from each student diversity source j across all sources of student diversity, N, in the classroom. Thus, the DRc measure incorporates two dimensions of related classroom student diversification: (a) the scope or number of sources of student diversity within the classroom and (b) the relative importance of each of these student diversity sources to a classroom’s total related student diversification.
Unrelated Student Diversification However, when a classroom has two or more sources of student diversity, Equation 4 in Appendix A1 defines a measure for the total unrelated student diversity derived from the student membership data for classroom c. The total unrelated student diversification expression (DUc) is the summation of a weighted average of the unrelated student diversification from each student diversity source j across all sources of student diversity, N, in the classroom. Thus, the DUc measure incorporates two dimensions of total unrelated student diversification: (a) the scope or number of sources of student diversity within the classroom and (b) the relative importance of each of these student diversity sources to a classroom’s total unrelated student diversification.
A Graphical Illustration Consider a classroom in which the principal assigns 10 regular students to an elementary fourth grade teacher. For the purpose of this illustration, a regular student is one who has no special learning needs required to do fourth grade schoolwork. Figure 3 shows the classroom student diversification that arises from the principal’s assignment of each student to fourth grade Classroom A. The y axis represents the level of classroom student diversification relative to the number of regular students in the classroom, as shown along the x axis.
Because Classroom A has students from only one source of student diversity (regular), the classroom’s total student diversification equals its total related student diversification, DTc = DRc. Total unrelated student diversification does not exist (DUc = 0). Thus, Equation 1 in Appendix A1 produces the curvilinear pattern shown in Figure 3. This overall curvilinear pattern is of special importance. Classroom diversification interprets this curvilinear relationship in terms of economies of scope inherent in having students from one student diversity source in the classroom. For example, by having all the students of one student diversity type, or a very homogeneous classroom, a teacher can design and administer one common curriculum to students in the classroom. Thus, the teacher’s joint cost of learning among two or more students is less than the sum of the production cost of learning for each student. Operationally, as the principal assigned one regular student after the other, the amount of demand on the teacher’s capacity in Classroom A did not increase proportionally by each additional student but by decreasing incremental amounts less than one, as shown in Figure 3.8 The student diversification equations also contain other inherent properties. Consider a fourth grade classroom, Classroom B. Suppose the principal assigned five bilingual students to this classroom after assigning five regular students. Figure 4 shows the student diversification generated from these classroom student assignments.
Note that after the principal’s assignment of the first five students, there was only one student diversity source in the classroom: regular students. The result is that Classroom B’s total student diversification equals its total related student diversification, DTc = DRc. Total unrelated student diversification does not exist (DUc = 0). However, with the principal’s sixth student assignment, a bilingual student, the classroom’s total student diversification decomposed into related and unrelated student diversification. Figure 4 shows this change in classroom’s student diversification, whereby the total classroom student count (Sc) equals six, as shown along the x axis. Note that there are two bars. The first bar is the regular student diversification on the basis of the five regular students (R = 5) the principal first assigned to Classroom B. The second bar is the bilingual student diversification on the basis of the principal’s assignment of the first bilingual student (B = 1) to Classroom B. Note that each bar also consists of two bar segments. The top bar segment and bottom bar segment define the unrelated student diversification and the related student diversification, respectively, for each student diversity source, nonbilingual and bilingual, in Classroom B. A closer inspection of these bars in Figure 4 shows another special characteristic inherent in the student diversification equations. When the principal assigned the first bilingual student to Classroom B, the classroom’s bilingual unrelated student diversification was significantly greater than its related student diversification (DUb >> DRb). This condition demonstrates how a teacher’s capacity may be strained in a classroom when the learning needs of a single student from one student diversity source are so different from the learning needs of students from another student diversity source. Meanwhile, as the principal assigns more bilingual students to Classroom B, concurrently, its
These changes in related and unrelated student diversification reflect the relative importance of each source of student diversity to Classroom B’s total student diversification with each student the principal assigns to the classroom. Moreover, the measures of classroom student diversification predict the formation of economies of scope in a classroom. For example, the changes in bilingual related and unrelated student diversification shown in Figure 4 suggest that economies of scope for the group of bilingual students are emerging in Classroom B. The measures also identify when a classroom should realize full economies of scope. Essentially, a classroom realizes full economies of scope when all its sources of student diversity have student groups of the same size. For Classroom B, full economies of scope were realized when its sources of student diversity, regular and bilingual students, were encapsulated in two groups of equal size. In Figure 4, this occurs when Classroom B’s student count (Sc) of 10 consists of 5 regular students and 5 bilingual students. However, how well the equations in Appendix A1 model a classroom’s actual economies of scope depends on its sources of classroom student diversity. In the Classroom B illustration, what if the regular students and/or bilingual students were also migrant students? The equations in Appendix A1 provide for classroom situations in which a student can hold membership in more than one source of student diversity. In such a classroom, the number of total student memberships (Tm) across these student diversity sources would be greater than the total number of actual students (n) in the classroom, Tm >> n. Figure 5 compares the classroom student diversification for two such fourth grade classrooms, Classroom C and Classroom D, with that of the fourth grade classrooms from the prior two illustrations.
In each of the four classrooms, the principal has assigned 10 students; however, these fourth grade classrooms are not equal in their levels of classroom student diversification. Classroom A has only regular students. Classroom B has equal numbers of regular and bilingual students. Classroom C has equal numbers of regular and bilingual students, but 5 students from these two student diversity groups also are migrant students. This defines a third source of student diversity in the classroom. Similarly, Classroom D has equal numbers of regular and bilingual students, but all 10 students in this classroom also are migrant students. Classroom diversification theory maintains that with regard to classroom student diversity, Classroom D is more diverse than Classroom C, which is more diverse than Classroom B. Classroom A is the least diverse classroom of the four classrooms in Figure 5. Thus, the complexity associated with a classroom’s diverseness of students reflects in its total student diversification measure and in the related and unrelated student diversification measures for each student diversity source. Operationally, for the classrooms in Figure 5, their total student diversification consists of related and unrelated student diversification that defines economies of scope for each source of student diversity in the classrooms: regular, bilingual, and migrant. The classroom diversification model maintains that the technological fit between students’ learning needs and the teacher’s capacity determines how well the teacher exploits these economies of scope in maximizing students’ learning in the classroom. The result is a classroom student diversification model that researchers can use to explore empirically for economies of scope using classroom student diversity membership data relative to the teacher’s capacity. Table 1 shows the assumed effects on student learning performance on the basis of the technological fit between students’ learning needs and the teacher’s capacity.9
In this section, I discuss teacher capacity relevant to student learning and follow with an overview of the different states of teacher capacity that are inherent in the classroom diversification paradigm. A teacher’s capacity is the skills, abilities, and knowledge derived from his or her training and experiences relevant to students’ learning needs. I then present measures of teacher capacity. Although the algorithms underlying these measures are the same ones used to measure classroom student diversification, their inherent properties have somewhat different interpretations when used to measure teacher capacity. Graphic illustrations follow this presentation to illustrate the inherent properties of the teacher capacity measures relevant to the classroom diversification educational productivity model.
Examples of Teacher Capacity Essential to Student Learning Researchers have studied the teacher skills, abilities, and knowledge essential to the learning needs of special student populations. Because of the complex problems associated with poverty, family composition, and mobility, for example, a teacher must have the capacity to provide a supportive, positive atmosphere in which students can feel productive and accepted and in which role model identification can be established (Natriello, McDill, & Pallas, 1990; Rasmussen, 1988; Wolverton, 1988). Researchers have noted similar teacher capacity for teaching students with disabilities, particularly in inclusion-based classroom settings (Sirvis, 1988; Ware, 1990). Specific teacher skills for managing the classroom learning enterprise, affected by social and student behavior embodied in student gender, include a number of effective teaching techniques that enhance student learning (Hadderman, 1987). Yet the greatest challenge for teachers attempting to affect students’ learning may come from student diversity associated with students’ cognitive needs. A teacher must have the capacity to recognize and program for students with diverse ability levels, learning styles, modes of expression, and interests (Interstate New Teacher Assessment and Support Consortium, 1992). Moreover, a teacher must have the experience to know when and how to manage specific classroom teaching strategies that facilitate students’ learning, including ability grouping (Slavin, 1986) and cooperative learning (Johnson et al., 1984). The teacher’s capacity to help students acquire metacognition strategies also is essential for those who differ in their metacomprehension of the curriculum. Teachers must have the capacity to help students become aware of their own understanding of the material or curriculum, or lack thereof, for students to anticipate or recover from problems in comprehension (Baker & Brown, 1984). For example, teachers may need to help some students develop reading study skills, such as focusing, recognizing, and retaining main points; rereading important sections; making adjustments in reading rate; and self-testing to monitor the success of learning activities. Students’ awareness of the understanding and use of these skills is necessary to their metacomprehension (Stewart & Tei, 1983). Moreover, the teacher may need to provide students with frameworks that help them link new facts and ideas encountered in the school with their everyday learning experiences for schema construction to occur (Bransford, 1985). Teachers who monitor and apply their knowledge, deliberately modeling metacognitive behavior, help students become aware of their own thinking so they discover which thinking processes improve their learning (Blakey & Spence, 1990). A teacher’s personal background may also affect classroom student learning (Windham & Chapman, 1990). Researchers have noted distinct differences in teaching style and class management between male and female teachers, as well as differences in the way the students respond to such teacher behaviors (see Grossman & Grossman, 1994, for an overview). In classrooms in which students come from different cultures or backgrounds or use different languages or dialects, teachers must have the capacity to recognize students’ developmentally equivalent patterns of behavior (Bowman, 1989); they also must have the skills to bridge between students’ different language contexts (Anderson & Gipe, 1983; Barnitz, 1986; Hudelson, 1987). A teacher with such capacity who shares with students the same cultural, language, ethnic, or racial background may be better in meeting their learning needs because he or she can also provide students with more relevant, positive role models, as well as be able to relate more closely to students to encourage them to perform better (Arends, Clemson, & Henkelman, 1992; Saracho & Spodek, 1995). Such a teacher also may be able to understand and counsel students better. Thus, a teacher’s personal background provides inherent capacities that may facilitate other sources of teacher capacity and modulate student learning in the classroom.
States of Teacher Capacity: Latent and Patent In economic terms, a teacher’s skills, abilities, and subject matter mastery are defined as human capital (Ehrenberg & Smith, 1991). The classroom diversification definition of teacher capacity encompasses the economic concept of human capital but makes a distinction between two types of human capital, latent and patent, that determine how relevant the teacher’s capacity is to students’ learning needs in the classroom. Under the classroom diversification paradigm, all human capital exists in the form of knowledge that cognitive psychologists have categorized into either declarative or procedural knowledge (Nickols, 2000). Declarative knowledge is factual information and can be articulated to individuals, typically in the form of tables, charts, diagrams, texts, and lectures. The formula for finding the area of a rectangle, history dates, and the periodic table of the elements are examples of declarative knowledge. The procedural aspects of organizing curriculum, grouping students by ability, and instructing in dual-language classrooms also are examples of declarative knowledge. In comparison, knowledge that expresses itself in the doing of something is procedural knowledge; it cannot be articulated or directly communicated to individuals. A person’s abilities to connect with people at a psychological level (e.g., to empathize) involve procedural knowledge that a person cannot articulate directly to another individual. A person proves that he or she possesses such knowledge in the doing. However, having innate abilities or knowing a particular process or content area is not the same as using those abilities, performing a process, or actively transferring content information to students in the classroom. Therefore, the classroom diversification paradigm defines latent human capital as the teacher’s declarative and procedural knowledge that does not manifest itself in the classroom. In comparison, the classroom diversification paradigm defines patent human capital as the teacher’s declarative and procedural knowledge that actually manifests itself in the classroom in the form of experience. As a concept, experience is the embodiment of skills, abilities, and knowledge that accrue over time from formal and informal learning opportunities to which the teacher is exposed (Windham & Chapman, 1990). Thus, a teacher’s patent human capital is produced during learning opportunities that include but are not limited to classroom experiences with different grade levels, special population groups, and rural and urban settings. Classroom diversification assumes that all teacher capacity begins as latent human capital and is transformed into patent human capital through the teacher’s teaching experiences in the classroom and other settings. For example, consider a teacher who recently received a 3-day in-service workshop on some new reading strategy. Classroom diversification defines the teacher’s newly acquired skills, abilities, and knowledge gained from this workshop as latent human capital. The latent human capital begins to transform into patent human capital when the teacher acquires teaching experience from applying those new skills, abilities, and knowledge in the context of the classroom. Similarly, classroom diversification defines a teacher’s procedural knowledge developed prior to entering the classroom, such as the ability to connect with people at a psychological level, as latent human capital. From the perspective of classroom diversification, such latent human capital begins to transform into patent human capital only when the teacher acquires experience using the procedural knowledge teaching students in the classroom.10 However, not all of a teacher’s capacity may transform into patent human capital. For example, consider a teacher certified for Grades 1 through 6 who has taught 10 years in first grade classrooms. Classroom diversification defines the teacher’s skills, abilities, and knowledge gained from the 10 years of teaching first grade as patent human capital but defines the skills, abilities, and knowledge initially gained from the teacher preparation program for Grades 2 through 6 as latent human capital. Stated simply, the teacher has the initial knowledge acquired from the preservice program to teach second grade through sixth grade, but this capacity remains latent because the teacher has no experience in these grade levels.11 From the perspective of the classroom diversification paradigm, the degree of diversity in such teaching experiences enhances the teacher’s capacity to address students’ learning needs in diverse classrooms. The result is that classroom diversification defines teacher experience as the primary source of teacher capacity.12
Other Aspects of Instructional Quality Relevant to Teacher Capacity Classroom diversification assumes that teacher effort within a classroom depends on the technological fit between students’ learning needs and the teacher’s capacity. A teacher with little or no capacity related to students with specific diversity needs would not be able to exert full effort within a unit of instructional time to maximize student learning. In comparison, a teacher who had capacity relevant to these same students would be able to exert full effort within a unit of instructional time and therefore would maximize student learning.13 The amount of time teachers devote to a particular instructional task defines the time allocated to teach. The school attempts to impose a management structure that controls the allocation of resources for the instructional process by organizing the curriculum into discrete areas and the day into discrete time periods. However, even within this structure, teachers may respond differently to time allocations. Some teachers may spend greater time on subjects or activities that they feel are important for students while minimizing the time allocated to other activities. Thus, teachers may have substantial differences in terms of the true time that they allocate to any particular instructional task with students, even though the official time allocations are largely standardized by school practice. Classroom diversification assumes the amount of time teachers spend with students on subjects will vary on the basis of the technological fit between students’ learning needs and the teacher’s capacity. A teacher with no or little capacity relevant to students’ learning needs would not allocate sufficient amounts of time to instruction in particular areas to maximize student learning. In comparison, a teacher with capacity relevant to the learning needs of these same students would allocate sufficient amounts of time to instruction in particular areas to maximize student learning.14
The Measurement of Teacher Capacity
Related Teacher Capacity Under such conditions, related teacher capacity increases with each year of experience teaching students from a student diversity source. Equation 2 in Appendix B1 defines a measure for the related teacher capacity derived from the teacher’s teaching experience data associated with a student diversity source. This group-related teacher capacity (GCRj) is the summation of a weighted average of the teacher capacity from each year of experience y' in teacher capacity source j teaching students from student diversity source j. Because the teacher’s capacity can extend across several sources of teacher capacity, Q, total related teacher capacity for teacher t, CRt, is a function of GCRj, where j = 1 to Q, as shown in Equation 3 in Appendix B1. The total related teacher capacity measure (CRt) is the summation of the share of the group-related teacher capacity (GCRj) from each teacher capacity source j across all sources of teacher capacity, Q. Thus, the CRt measure incorporates two dimensions of related teacher capacity: (a) the scope or number of sources of teacher capacity for a teacher and (b) the relative importance of each of these teacher capacity sources to a teacher’s total related teacher capacity.
Unrelated Teacher Capacity However, when a teacher has two or more sources of teacher capacity, Equation 4 in Appendix B1 defines a measure for the total unrelated teacher capacity derived from the teaching experience data for teacher t. The total unrelated teacher capacity expression (CUt) is the summation of a weighted average of the share of unrelated teacher capacity from each teacher capacity source j across all sources of teacher capacity Q. Thus, the CUt measure incorporates two dimensions of unrelated teacher capacity: (a) the scope or number of sources of teacher capacity for a teacher and (b) the relative importance of each of these teacher capacity sources to a teacher’s total unrelated teacher capacity.
A Graphical Illustration Consider a teacher with teaching experience exclusively in fourth grade classrooms of regular students. As defined in earlier illustrations, a regular student is one who has no special learning needs required to do fourth grade schoolwork. Figure 6 shows Teacher A’s capacity that arises from each year of experience teaching fourth grade regular students. The y axis represents the level of teacher capacity relative to the number of years of teaching experience calculated at the start of a school year, as shown along the x axis. Thus, the teacher’s 1st year of teaching is Te = 0, the 2nd year of teaching is Te = 1, the 3rd year of teaching is Te = 2, and so forth.
Because Teacher A has taught students from only student diversity group (regular), the teacher’s total teacher capacity and total related teacher capacity are equal, CTt = CRt. Total unrelated teacher capacity does not exist (CUt = 0). Thus, Equation 1 in Appendix B1 produces the curvilinear pattern shown in Figure 6. This overall curvilinear pattern is of special importance. Classroom diversification interprets this curvilinear relationship in terms of decreasing returns on experience to teacher capacity resulting from always teaching students from only one student diversity source. In the early part of the teacher’s experience, most of the teaching experiences encountered in the classroom are relatively new, regardless of the diversity source. However, as the teacher accumulates years of teaching experience with students from the same diversity source, the incidence of new experiences decreases. Teachers begin to see reoccurring patterns of classroom student activities, storing and refining their teaching capacity relevant to the students’ learning needs. Operationally, as Teacher A accumulated years of experience teaching regular students, the amount of teacher capacity did not increase proportionally with each additional year of experience but by decreasing incremental amounts less than one, as shown in Figure 6.15 The teacher capacity equations also contain other inherent properties. Consider a fourth grade teacher, Teacher B, who has taught regular students from year to year but who begins a university-based program in bilingual education at the start of the 4th year of teaching (Te = 3). She completes the program and earns a bilingual certificate by end of the 5th year of teaching (Te = 4). At the start of the 6th year of teaching (Te = 5), the principal reassigns Teacher B from a classroom of regular students to a classroom of bilingual students, and the teacher continues to teach bilingual students into the 11th year of teaching. Figure 7 illustrates Teacher B’s capacity from each year of experience teaching these fourth grade students.
Note that in the first 5 years of teaching experience (Te = 0 to 4), Teacher B taught students from only one student diversity source: regular students. The result is that the teacher’s total teacher capacity and total related teacher capacity are equal, CTt = CRt. Total unrelated teacher capacity does not exist (CUt = 0). During this time, the teacher is acquiring bilingual teacher capacity from the university-based program that then culminates in a bilingual certificate. However, the latter capacity is latent. During her initial year in the bilingual classroom (Te = 5), Teacher B’s bilingual teacher capacity begins to transform from latent to patent capacity on the basis of new experiences teaching bilingual students, now the primary source of student diversity in her classroom. Meanwhile, the teacher’s total teacher capacity also decomposes into related and unrelated teacher capacity, CRb and CUb, respectively. Figure 7 shows the result of this transformation and decomposition of teacher capacity, in which the total years of experience (Te) equals 6, as shown along the x axis. Note that there are two bars. These bars indicate the amount and scope of Teacher B’s capacity that is now available to students at the start of the 7th year of teaching. The first bar is the regular teacher capacity on the basis of the teacher’s first 5 years of experience teaching regular students (R = 5). The second bar is the bilingual teacher capacity on the basis of the teacher’s 1st year teaching bilingual students (B = 1) in the prior year of teaching (Te = 5). Note that each bar now contains two bar segments. The top bar segment and bottom bar segment define the unrelated teacher capacity and the related teacher capacity, respectively, for each teacher capacity source. A closer inspection of these bars at Te = 6 in Figure 7 shows another special characteristic inherent in the teacher capacity equations. Note that Teacher B’s bilingual unrelated teacher capacity was significantly greater than the teacher’s bilingual related teacher capacity (CUb >> CRb). This condition reflects the notion that a teacher’s initial experiences teaching students from a new student diversity source will be mostly dissimilar to prior experiences teaching students from other student diversity sources. Meanwhile, as Teacher B accumulates more bilingual teaching experience in subsequent school years, concurrently, the teacher’s
These changes in related and unrelated teacher capacity reflect the relative importance of each source of teacher capacity added to Teacher B’s total teacher capacity with each completed year of classroom teaching. Moreover, the measures of teacher capacity predict the transformation of latent capacity to patent capacity in the form of decreasing returns on experience to teacher capacity. For example, the changes from year to year in bilingual related and unrelated teacher capacity shown in Figure 7 suggest that decreasing returns on experience to teacher capacity are emerging for Teacher B. Each additional year of bilingual teaching experience provides fewer refinements to the teacher’s capacity than the prior year. The multisegment bars in Figure 7 highlight still another inherent property in the teacher capacity measures: the teacher’s storage of teacher capacity. Once a teacher engages with a particular student diversity source, classroom diversification maintains that the teacher’s capacity from that teaching experience remains at some level of aptness throughout the life of the teacher’s career. Essentially, both the degree to which the teacher continues to teach students from the same student diversity source year to year and the degree to which she teaches students from new sources of student diversity determine how much the teacher loses, maintains, or expands returns on experience to teacher capacity. However, how well the equations in Appendix B1 model a teacher’s actual decreasing returns on experience to teacher capacity depends on the sources of student diversity that define the teacher’s capacity from year to year. The situation might change if Teacher B’s regular and/or bilingual students were also migrant students. The equations in Appendix B1 provide for teaching situations in which a teacher’s capacity comes from experiences teaching students who are members from more than one student diversity source. For such a teacher, the summation of years of experience (Te) across these sources of teacher capacity would be greater than the actual number of years (y) teaching in the classroom, Te >> y. Figure 8 compares the teacher capacity for two such fourth grade teachers, Teacher C and Teacher D, with that of the teachers from the prior two illustrations.
Each of these four teachers has completed 10 years of teaching in fourth grade classrooms; however, they are not equal in their teacher capacity. Teacher A has taught only regular students. Teacher B taught regular students for 5 years and then taught bilingual students for the next 5 years. Teacher C taught regular students for 5 years and then taught bilingual students for the next 5 years, but in 5 of these years, some of these students were also migrant students and thus defined a third source of teacher capacity. Similarly, Teacher D taught regular students for 5 years and then taught bilingual students for the next 5 years, but in all 10 of these years, the students were also migrant students. Classroom diversification maintains that relative to the learning needs in a classroom composed of regular, bilingual, and migrant students, Teacher D’s capacity is greater than Teacher C’s capacity, which is greater than Teacher B’s capacity. Teacher A has the least capacity of the four teachers in Figure 8. Thus, the complexity associated with a teacher’s teaching capacity reflects in both the teacher’s total teacher capacity measure and in his or her related and unrelated teacher capacity measures for each teacher capacity source. However, these teacher capacity measures become meaningful only within the context of the students’ learning needs, as defined by the student diversity sources that exist in the teacher’s classroom. For example, Teacher A would have the most capacity of the four teachers if after 10 years of experience, all four teachers were assigned only regular students in their classrooms. Operationally, for the teachers in Figure 8, their total teacher capacity consists of related and unrelated teacher capacity that defines decreasing returns on experience to teacher capacity teaching year to year students from each source of student diversity: regular, bilingual, and migrant. Classroom diversification maintains that the technological fit between students’ learning needs and the teacher’s capacity determines how well the teacher uses that capacity to exploit economies of scope in maximizing students’ learning in the classroom. The result is a teacher capacity model that researchers can use to explore empirically decreasing returns on experience to teacher capacity. The variables of concern include the teacher’s teaching experience data relative to student diversification in the classroom. Table 1 shows the assumed effects on student learning performance on the basis of the technological fit between students’ learning needs and the teacher’s capacity.
In this final section, I discuss the implications of classroom diversification for educational research, policy, and practice. I preface this discussion with an overview of the possible limitations in researchers’ use of the classroom diversification framework to study educational productivity. In light of these possible limitations, I then discuss how the classroom diversification framework may prove more feasible to implement than other lines of inquiry recently proposed by scholars of educational productivity. Next, I outline the implications of classroom diversification for educational policy and practice. I conclude the article with closing thoughts regarding classroom diversification: a strategic view of educational productivity.
Limitations and Assumptions How researchers define the unit of analysis to measure teacher capacity also may affect the research results of a classroom diversification educational productivity model. The measures are based on teaching experience to estimate the amount and the diverseness of teacher capacity available to students in a classroom. Educational productivity studies that have examined the relationship between teacher experience and educational outcomes have typically measured experience in years of classroom teaching. However, a teacher’s experience with students from a particular student diversity source may consist of daily episodes that take place throughout the day, as in self-contained elementary classrooms. In comparison, a teacher’s experience with students in middle school and high school, in which teachers teach subject areas across multiple classes, may consist of daily episodes lasting up to an hour or weekly intermittent episodes of longer duration, depending on the school’s schedule program (e.g., block vs. traditional). How researchers measure teacher experience—in hours, days, weeks, months, or years—to define teacher capacity in a classroom diversification educational productivity model could affect research results. The advantage to measuring teacher experience in years is that the school system predefines this unit of time, typically September through August, which makes the interpretation of teaching experience data relatively easy and straightforward. Researchers’ measure of teacher experience by trimesters or quarters also would be relatively easy and straightforward. To measure a teacher’s experiences in days, weeks, or months, however, researchers would need to define the start and end of a teacher’s experience. How researchers establish the start and end of a teacher’s experience to define teacher capacity in a classroom diversification educational productivity model could affect research results. The access researchers have to relevant data is yet another possible limitation in the use of the classroom diversification educational productivity model. Some states do collect detail information about the schooling process at the campus level for reporting and accountability purposes to state or federal funded programs (e.g., No Child Left Behind [NCLB] Title Programs). For example, Texas, Arkansas, and Florida are states with detailed student longitudinal databases (Snow-Renner & Torrence, 2002). States or districts that use the Tennessee Value-Added Assessment System are another possible source of relevant data for the study of classroom diversification. Researchers who use this teacher evaluation model require the collection of each student’s test data, accumulated over time and linked to that student’s teacher, school, and school system (Sanders & Horn, 1998). Other data sources may emerge as states continue to implement NCLB (No Child Left Behind Act, 2002). NCLB is the federal government’s largest investment in elementary and secondary education and provides resources to help ensure that disadvantaged students have access to quality public education. To accomplish this goal, NCLB requires that states collect and analyze data about schools and system performance using student assessment and other student-level information (Snow-Renner & Torrence, 2002). Researchers should also consider how well the classroom diversification framework can be generalized to all grade levels. Some caution is needed in extrapolating to other grade levels any significant findings, or lack thereof, from investigating classroom diversification at another grade level. The classroom diversification educational productivity model might be more or less appropriate within educational settings that are specialized (e.g., high school math classes) or less specialized (e.g., self-contained third grade elementary classrooms). In addition, school-level variables such as the size of the school’s support staff, location (rural or urban), parental support, and leadership of the school, may be important intervening variables in the relationship between classroom student diversity and teacher capacity to student performance. Researchers could examine the interaction of classroom student diversity and teacher capacity with other aspects of the learning environment but within a hierarchical diversification model that expands the classroom diversification framework from the classroom to the school, community, and district.16 The availability to researchers of appropriate analytical methods is perhaps the greatest possible limitation to using the classroom diversification educational productivity model. The education production function model is based on regression procedures in which a single output variable is predicted by one or more input variables (e.g., teacher experience) and by intervening variables (e.g., socioeconomic status). Fortune (1993) contended that the correlation methods that form the foundation of the education production function model are inappropriate for the study of educational productivity and endorsed the use of t-test research designs. Classroom diversification does not suggest that researchers abandon the basic concept of the production function, which is to relate inputs such as school resources to a given level of educational outcomes. Neither does classroom diversification suggest that researchers should avoid other statistical methods, such as the t test, for the study of educational productivity. Researchers need to use statistical methods appropriate to their research designs in the study of classroom productivity (see Ludwig, 2001). Rather, classroom diversification provides researchers with an alternative view of educational productivity, one that recognizes the fundamental nature of the classroom learning environment as one based on human relationships. These relationships follow principles of human interaction in the classroom that provide researchers with a theoretical framework that explains why teachers and students are most likely to use resources to maximize educational outcomes. Thus, classroom diversification provides researchers with an opportunity to design production functions—or t-test experiments for that matter—that result in more meaningful findings. Thus, how well researchers use the classroom diversification framework in their studies of educational productivity will affect the research results.
Implications for Educational Research However, an underlying ethical dilemma often precludes the use of learning experiments with teachers and students by modifying their access to resources. If the resources to teachers and students in a learning experiment are increased, the public may criticize a school system on the grounds that the available resources researchers initially withheld from teachers and students were wasted public taxes. Decrease the resources to teachers and students and the public may criticize the school system for placing the researchers’ interests above the welfare of the students’ learning and teachers’ teaching. Furthermore, withholding mandated instructional experiences from children for the sake of establishing a control group for research may in itself be not only unethical but also illegal. Operationally, students’ parents are not likely to support their children’s participation in an experiment that may put the students’ learning at risk simply to accommodate a researcher’s line of inquiry into how resources affect teachers’ teaching. Likewise, teachers are not likely to participate in an experiment in which their teaching may negatively affect their students’ learning because of resource constraints that someone else has intentionally created or predesigned. Researchers could circumvent these problems using the classroom diversification framework and the notion of technological fit. When a teacher’s capacity is relevant to students’ learning needs, classroom diversification assumes that the teacher effectively uses technology and allocates available resources to students thus maximizing student learning. When teacher capacity is not relevant or is insufficient to students’ learning needs, classroom diversification assumes that the teacher ineffectively uses technology and allocates available resources to students in ways that reduce student learning. Thus, researchers could study how teachers effectively or ineffectively use resources when technological fit predicts that they would maximize or reduce students’ learning, respectively. Of course, researchers would require teacher- and student-level data to conduct such studies. As stated earlier, a possible limitation in the study of classroom diversification may be limits on access to relevant research data. Nevertheless, researchers may find that the collection of data for the study of classroom diversification is more feasible than collecting data for other lines of inquiry recently proposed by scholars of educational productivity. Monk and Rice (1999) reported that researchers have renewed efforts to improve productivity in educational systems by refocusing their investigations from linking inputs with outputs to linking costs with student outcomes using multivariate methods (e.g., Reschovsky & Imazeki, 1998) and data envelopment techniques (e.g., Duncombe, Ruggiero, & Yinger 1996). The researchers’ goal in these studies was to examine how schools organize their resources and operate various programs to achieve student outcomes. Hartman, Bolton, and Monk (2001) reviewed two approaches to school-level analysis of financial data to model the link between costs and student outcomes. The resource cost model focuses on resource consumption and uses data on the physical resources used in a school’s service delivery system. In comparison, the accounting model focuses on actual expenditures measured in dollars of the resources used by function, program, grade, or subject matter in the school. The challenge to researchers in using these models is in the collection of detailed, school-level financial information. Isaacs, Garet, Sherman, Cullen, and Phelps (1999) examined the feasibility and difficulties of collecting detailed staffing resources and expenditure data at the school level through the Schools and Staffing Survey, administered by the National Center for Education Statistics. The authors cited problems in accurately categorizing expenditures at the service level, as well as ensuring data validity, as examples of the type of difficulties related to the collection of school-level financial data. In comparison, researchers interested in the study of classroom diversification would need to gather detailed classroom-level data, but this data collection should not be a major problem. Educators and researchers already categorize students by their learning needs (e.g., bilingual, special disabilities, gifted) and attributes (e.g., ethnicity or race, economic status), and district personnel typically collect these data for reporting to state or federally funded programs (e.g., NCLB Title Programs). Researchers would need to collect detailed data about teachers, such as information on their background, training (e.g., knowledge areas, skills, and abilities), and teaching experiences. Researchers could collect these data through a survey instrument that inventories this type of teacher data. Focus groups could identify the initial inventory, and researchers could enhance the survey instrument through field testing with classroom teachers. Teachers would most likely contribute this information because the data would showcase their capacity for teaching while identifying and communicating to principals their current need for professional development or additional school resources relative to their students’ learning needs. Last, researchers would need to link teachers’ data to students’ data by teacher-student classroom assignment. Researchers could use teacher-student assignment data, typically maintained at schools and central district offices, to facilitate this process. With school officials’ permission, researchers might also access teacher-student-linked data from districts that have implemented benchmarking tests at intervals throughout the school year. In highly student data–driven states such as Texas, these include urban districts (e.g., Austin), suburban districts (e.g., Round Rock), and rural districts (e.g., Mount Pleasant). Beyond the feasibility of research design implementation or data collection, propositions from contemporary scholars support using the proposed classroom diversification framework for the study of educational productivity rather than continuing with the neoclassical educational productivity model used by researchers over the past five decades. Monk (1992) contended that the inconsistent and largely insignificant findings from educational productivity research are the result of researchers’ emphasis on school-level analyses. He argued that researchers should refocus the study of educational productivity on the classroom level. Researchers could then drive inquiry from four classroom properties emerging from educational research, properties that also resonate in the classroom diversification educational productivity model. The first classroom property is that how much a student learns depends on the identity or attributes of the classroom to which the student is assigned (Monk, 1992). This property is directly embedded in the classroom diversification concept of classroom student diversity. Classroom student diversity refers to the differences among a classroom’s students in their student attributes, unique learning needs, and modes of knowledge and skill acquisition. Classroom student diversification is the process that a classroom undergoes when a principal assigns students from these sources of classroom student diversity. Classroom student diversification divides into two types of operational strategies: related and unrelated. In related diversified classrooms, a teacher has the potential to exploit commonalities among the students to obtain economies on the basis of the sharing of teacher capacity and classroom resources. In unrelated diversified classrooms, teachers have the potential to create group learning activities that allow students of different abilities to exchange their own knowledge and share their own skills and abilities required for learning. Consequently, how principals assign teachers and students to classrooms makes a difference to student learning. The second classroom property is that individual classroom teachers can transform school-level initiatives in different ways that may result in varying educational productivity outcomes (Monk, 1992). This property is directly embedded in the classroom diversification concept of teacher capacity. A teacher’s capacity is the skills, abilities, and knowledge derived from his or her training and experiences relevant to the classroom learning enterprise. The classroom diversification definition of teacher capacity encompasses the economic concept of human capital but includes the relevance of the teacher’s capacity to students’ learning needs in the classroom. Classroom diversification makes a distinction between two types of human capital, latent and patent, in defining the relevance of the teacher’s capacity to students’ learning needs in the classroom. Classroom diversification contends that teachers in individual classrooms will transform school-level initiatives in different ways, resulting in varying educational productivity outcomes depending on the patent teacher capacity available to students in the classroom. Monk’s (1992) third classroom property is that a teacher’s experience with students can vary substantially from classroom to classroom, even when the grade level, curriculum, and student backgrounds (e.g., ethnicity, race, and ability) remain the same. His fourth classroom property is that teachers perform at different levels in different classrooms across different years. These last two properties are directly embedded in the classroom diversification concept of technological fit. When a teacher’s capacity is relevant to students’ learning needs, classroom diversification assumes that the teacher effectively uses technology and allocates available resources efficiently to students, thus maximizing student learning. When teacher capacity is not relevant or is insufficient to students’ learning needs, classroom diversification assumes that the teacher ineffectively uses technology and allocates available resources inefficiently to students, thereby reducing student learning. The relevance of the teacher’s capacity to students’ learning needs is essentially the critical factor with regard to technological fit, a concept that forms the foundation of how teachers experience their students and how well teachers affect students’ learning, classroom to classroom and year to year. In retrospect, the four emerging classroom properties may explain the inconsistencies in the educational productivity studies Hanushek (1981, 1986, 1989, 1991) found in his review of the literature. Researchers based these studies on the neoclassical educational productivity model that treats the learning process as an undifferentiated "black box," void of any human dimension, which mysteriously transforms resources into student learning outcomes. Yet, the human dimension is where the source of variability in the process of transforming resources into student learning outcomes most exists. Thus, one should expect inconsistencies across educational productivity studies where researchers assume the human dimension, in a primarily human-based production process like student learning, is excluded in how inputs relate to outputs. Stated simply, the neoclassical educational productivity model—intentional or not—excludes Monk’s (1992) four classroom properties which require researchers to consider the human dimension of the learning process itself. In comparison, classroom diversification incorporates the four classroom properties in a framework that may provide new insights into the production of educational outcomes in classrooms than other ways of representing teaching and learning. For example, some researchers have represented teaching and student learning on the basis of the relationship between actual costs of resources and student outcomes (e.g., Duncombe et al., 1996; Hartman et al., 2001; Monk & Rice, 1999; Reschovsky & Imazeki, 1998). Their goal was to investigate the questions about which resources matter, how much, and for what students. Researchers could use the classroom diversification educational productivity model to investigate these same questions, because the model does not exclude the use of the production function to relate inputs such as actual costs to student outcomes. However, researchers can also use the classroom diversification framework to investigate questions such as why teachers and students are most likely to use resources, as explained by the classroom diversification concepts of classroom student diversity, teacher capacity, and technological fit. For example, the neoclassical educational productivity model cannot address the question of why smaller class size translates into higher levels of student achievement. A simple reduction in the number of students in a classroom does not suggest a practical or theoretical explanation of the relationship between student learning and class size. Researchers have focused more recently on class-size effects as a function of student behavior (Lazear, 1999) or longitudinal student population variation (Hoxby, 2000). However, why teachers do what they do with students in smaller classes would be a more appropriate topic for class-size research to understand and replicate desired results (see Finn, Pannozzo, & Achilles, 2003). Researchers could use the classroom diversification framework to examine the effect on students’ learning on the basis of the technological fit between students’ learning needs and the teacher’s capacity in classrooms with varying student diversity groups and class sizes.
Implications for Educational Policy and Practice These insights could also help researchers resolve various contentious issues in the field of educational productivity. Such an issue is the debate among researchers about whether increased spending on education improves student learning (e.g., Biddle, 1997; Ferguson & Ladd, 1996; Hanushek, 1997, 2003). For example, Plecki (2000) noted that policy makers continue to promote educational reforms on the basis of the assumption that a relationship exists between money spent on professional development and teacher quality in the classroom and student learning. Common sense would suggest that money spent on professional development to expand a teacher’s capacity should make a make a difference to students’ learning. However, classroom diversification would contend that such capacity must be directly relevant to the students’ learning needs in the classroom. Investments in teacher development must essentially make sense within the teacher’s context and correspond to the needs and circumstances of each teacher (Darling-Hammond & McLaughlin, 1999). Classroom diversification requires that a teacher’s professional development must also result in patent capacity, not just latent capacity. Researchers have identified design principles of professional development (Hawley & Valli, 1999) and strategies to transform professional development into effective teaching in the classroom (Ball & Cohen, 1999). Policy makers’ dilemma, then, is not how to spend money on professional development to enhance teacher quality but when to spend money on professional development to expand the kind of teacher capacity that results in increased student learning. Researchers could use the classroom diversification framework to identify guidelines for when policy makers should provide professional development to a teacher, on the basis of the technological fit between students’ learning needs and the teacher’s capacity. Classroom diversification research could also guide policy in the recruitment, preparation, and induction of new teachers. Ladson-Billings (1999) stated that the resistance of prospective teachers to student diversity is a major concern among teacher educators preparing new teachers for the realities of the classroom. The result is that teacher educators are developing criteria for selecting candidates on the basis of, among other factors, candidates’ disposition to teach in diverse settings (Darling-Hammond, Berry, Haselkorn, & Fideler, 1999). Teacher educators could use the classroom diversification framework to develop criteria for prospective teacher selection into their teacher preparation program by identifying when certain teacher dispositions are more or less relevant on the basis of the technological fit between students’ learning needs and teachers’ capacity. Teacher educators should consider in the selection process prospective teachers’ adaptability to different environments if teacher preparation programs are to prepare teachers to teach students with challenging learning needs or in uncomfortable learning situations. Teacher educators could then design and embed relevant student-teaching experiences into training programs that broaden teacher candidates’ diverseness of teacher capacity required in public school classrooms and that transform teacher candidates’ latent capacity into patent capacity. Teacher educators typically assign teacher candidates to supervisory teachers during student-teaching assignments. Therefore, teacher educators need to consider two factors in implementing student-teaching experiences in student-teaching classrooms. The first is the technological fit between students’ learning needs and teacher candidates’ initial capacity. The second is the technological fit between students’ learning needs and supervisory teachers’ capacity. Teacher educators could use the classroom diversification framework to identify these factors to guide them in the design and implementation of student-teaching experiences for teacher candidates and thereby improve the likelihood of student-teaching success among teacher candidates. However, teacher candidates may require considerable teaching opportunities to acquire competence in the pedagogical domain. Teacher educators can only guide their teacher candidates on the path toward teaching expertise and provide them with the tools and dispositions to learn from their ongoing teaching experiences (Sabers et al., 2002). Therefore, professional educators need to pick up where teacher educators leave off and help beginning teachers continue their transformation of latent capacity into patent capacity through teacher induction programs. Professional educators would induct beginning teachers on the basis of student assignments and the scope of the teaching assignment and demands for new learning placed on beginning teachers, factors that can further strengthen the preparation new teachers bring to the classroom (Little, 1999). Moreover, Monk and Rice (1998) have shown in their research that this alignment between teachers’ preparation and initial teaching assignments is important to how productive novice teachers are in their initial classroom assignments. Professional educators could use the classroom diversification framework to fine-tune these alignments on the basis of the concept of technological fit between students’ learning needs and beginning teachers’ capacity. The potential for classroom diversification to guide policy and direct programs in recruiting, preparing, inducting, and continually developing teachers has implications for how teachers are certified and licensed. A measure of a teacher’s qualifications associated with knowledge of subject matter and about teaching and learning is the teacher’s license or certification. However, states generally associate the credential with the completion of a state-approved teacher preparation program at the undergraduate or graduate level (Darling-Hammond, 2000). Classroom diversification suggests an alternative vision, in which a teacher certification credential indicates a teacher’s up-to-date patent capacities by student diversity source. The certification process itself could consist of established performance-based assessments, linked to teaching standards, that delineate a prescribed set of teacher capacities relevant to students’ learning needs associated with a source of student diversity. A credential of this type, one that displays a teacher’s up-to-date available capacities, would provide principals with valuable information on which students should be assigned to teachers, or what support systems teachers may need while developing new teaching capacity in response to a new student diversity group assignment. Researchers could use the classroom diversification framework to guide development of the performance-based assessments used to construct teachers’ credentials on the basis of studies of teachers’ capacity under varied classroom student diversity assignments. Policy makers also could use classroom diversification to guide educational reform for school improvement. For example, the Southern Regional Education Board (1998) has identified five critical components essential to an accountability system for restructuring schools and districts. However, these school accountability components do not incorporate the notion of technological fit between students’ learning needs and teachers’ capacity. An accountability system based on the concept of technological fit would motivate educators to align relevant professional development with effective quality teaching that results in higher levels of student learning and performance. Essentially, such accountability systems would link student performance to that part of the schooling process over which educators have control: the development of capacity in their teacher workforce. Meanwhile, classroom diversification may benefit educators in school reform in more direct ways. In their study of teachers as mediators of reform, Olsen and Kirtman (2002) found that teachers’ implementation of pedagogical, curricular, or organizational reform is influenced by their assumptions about how students learn. A teacher’s experience with students in the classroom determines much of these learning assumptions. The fact that no clear prescriptions are available to principals for selecting teachers and assigning students suggests a need for research to better understand the consequences of these teacher-student assignment decisions to teachers’ assumptions about student learning. Using the classroom diversification framework, researchers could address this need and provide principals with guidelines for supporting systemic reform at the most fundamental level of the educational organization: the classroom.
Closing Thoughts However, classroom diversification is neither a goal nor a plan. Ideally, principals should consider in each teacher-student assignment the economies derived from the relationships among and across students in the classroom and the technological fit between the student’s learning needs and the teacher’s capacity. Thus, although there is no single strategy of classroom diversification, some strategies may work more effectively than others in affecting student learning. Classroom diversification offers an opportunity to explore these strategies and advance our understanding of the complex nature of student learning in the classroom.
Classroom student diversification measures The equations for classroom student diversification measures are as follows:
and
where, for classroom c, DTc is the total student diversification measure; GDRj is the group-related student diversification measure for student diversity source j; DRc is the total related student diversification measure; DUc is the total unrelated student diversification measure; N is the number of student diversity sources; mj is the sum of all student memberships within student diversity source j; Tm is the total sum of all student memberships across all N sources of student diversity; mj/Tm is the share of student diversity from source j of all N sources of student diversity; 1/mj is the share of student diversity arising from each student, s', in student diversity source j; ln(1 + Tm) is the weight of student diversity source j in DTc; ln(1 + mj) is the weight of student diversity source j in GDRj; and ln[(1 + Tm)/(1 + mj)] is the weight of student diversity source j in DUc.
Proof: DTc = DRc + DUc Substituting DRc and DUc as defined earlier,
Substituting GDRj as defined earlier,
But, . . .
in the DRc equation term. Therefore,
Applying the logarithmic rule of quotients on the DUc equation term yields
After the first and third term in the equation cancel each other,
Teacher capacity measures The equations for teacher capacity measures are as follows:
and
where, for teacher t, CTt is the total teacher capacity measure; GCRj is the group-related teacher capacity measure for teacher capacity source j; CRt is the total related teacher capacity measure; CUt is the total unrelated teacher capacity measure; Q is the number of teacher capacity sources; ej is the total years of experience within teacher capacity source j; Te is the total years of experience across all Q sources of teacher capacity; ej/Te is the share of teacher capacity from source j of all Q sources of teacher capacity; 1/ej is the share of teacher capacity arising from each year of experience, y', in teacher capacity source j; ln(1 + Te) is the weight of teacher capacity source j in CTt; ln(1 + ej) is the weight of teacher capacity source j in GCRj; and ln[(1 + Te)/(1 + ej)] is the weight of teacher capacity source j in CUt.
Proof: CTt = CRt + CUt Substituting CRt and CUt as defined earlier,
Substituting GCRj as defined earlier,
But, . . .
in the CRt equation term. Therefore,
Applying the logarithmic rule of quotients on the CUt equation term yields
After the first and third term in the equation cancel each other,
OMAR S. LÓPEZ is the founder and CEO of the Corporation for Public School Education K16, a private educational consulting company dedicated to improving public schools, universities, and colleges through program evaluation, educational policy research, and best-practice studies; e-mail olopez{at}cpse-k16.com. Serving clients across the United States and in Mexico, his work has resulted in educational reform at the classroom, school, district, state, and federal levels. I began in the spring of 1994 to develop the concepts presented in this article as the foundation for my doctoral dissertation. Therefore, I am deeply grateful to my committee chairperson, Dr. Deborah M. Kazal-Thresher, for her encouragement and thoughtful feedback on early drafts of my conceptual work. I also owe "mil gracias" to Sarah M. Collins for her detailed review and insightful comments on a more recent draft of this article. Last but not least, I wish to thank the anonymous reviewers for their constructive comments, which resulted in this publication. However, any errors of fact or interpretation that remain are solely my responsibility.
1 My classroom diversification perspective is from academic training in strategic management that includes the study of business diversification and corporate performance. See Datta, Rajagopalan, and Rasheed (1991); Pitts and Hopkins (1982); and Rumelt (1974) for overviews of the research in this area of strategic management.
2 A firm seeks to maximize its profits by producing a level of output at which marginal cost equals marginal revenue. Essentially, a firm expands its operation until the point at which the cost of producing one extra unit of output (marginal cost) equals the revenue received from selling that last unit of output (marginal revenue). The profit-maximizing output point can occur in any of the three stages: increasing, constant, or decreasing returns to scale. If profit maximization occurs in the increasing returns to scale stage, the firm will be operating at a technically inefficient point in that fixed inputs are underused. Therefore, a firm will always try to operate beyond this stage. If profit maximization occurs in the decreasing returns to scale stage, however, the firm is also operating at a technically inefficient point in that too much variable input is being used relative to the available fixed inputs, and therefore, variable inputs are overused. The efficiency of variable inputs and the efficiency of fixed inputs both decline throughout this stage. Therefore, maximum production efficiency must fall somewhere in the constant returns to scale stage, preferably at the boundary between the constant and decreasing returns to scale stages, at which the firm is using fixed input most efficiently and short-run output is maximized. In the neoclassical economic theory of the firm, the concept of returns to scale does permit changes in a firm’s technical capabilities as production increases in size. The result is that by the skillful application of new technologies, firms are able to expand their scale of operations without ever encountering decreasing returns to scale. However, operating in constant returns to scale does not define a firm’s profit-maximizing point in that it takes no account of prices or demand. If demand for a product is low, the profit-maximizing output could be in the stage of increasing returns to scale even though the point of optimum efficiency is in the stage of constant returns to scale (Thompson, 1981).
3 Still other resources students need to learn may manifest themselves in ways that only teachers can directly give to students, such as high learning expectations, love, reassurance, acceptance, humor, patience, inspiration, hope, persistence, empathy, creativity, and enthusiasm. Such teacher-supplied resources can create a sense of caring and belonging, a classroom environment in which student learning and growth can take place in the various domains of human development. Classroom diversification assumes that teachers also regulate these resources to students on the basis of the technological fit between students’ learning needs and the teachers’ capacity.
4 Similar diversity measures can be found in the business strategy literature. See Antoniou (1988); Hoskisson, Hitt, Johnson, and Moesel (1993); and Jacquemin and Berry (1979) for a discussion of business diversification measures. See Hoskisson and Johnson (1992), Nayyer (1992), or Palepu (1985) for examples of studies that used such measures to study business diversification and corporate performance.
5 See Panzar and Willig (1981) and Teece (1980) for more information on the concept of economies of scope.
6 A classroom strategy commonly used by teachers under conditions of diverse student ability is ability grouping, which refers to the practice of organizing a classroom of students into groups of similar ability. In theory, ability grouping should improve student achievement because its goal is to reduce the disparity in student ability levels, allowing the teacher to provide more relevant instruction within each student group of similar ability. However, Slavin (1986) contended that teachers must have the capacity to vary the level and pace of instruction according to students’ levels of readiness and learning rates for ability grouping to succeed. Teachers who differentiate and adapt instruction in their classrooms can offer students a variety of learning options designed to tap into different student readiness levels, interests, and learning profiles (Tomlinson, 1995). Classroom diversification assumes that the technological fit between students’ learning needs and the teacher’s capacity determines the extent a teacher is able to provide differentiated and adaptive instruction to students with a wide range of academic ability.
7 Classroom diversification’s focus is on the relationships across and among students on the basis of the similarities and differences of their learning needs, not on the students’ interaction derived from the teacher’s practice of cooperative learning in the classroom. My main reasons in referencing the cooperative learning strategy are twofold. The first reason is to highlight the educational competitive advantage that the teacher can derive from this instructional strategy in an unrelated student diversified classroom. The second reason is to emphasize the classroom diversification assumption that the effectiveness of such a strategy, like all learning strategies used by the teacher in the classroom, depends on the technological fit between students’ learning needs and the teacher’s capacity.
8 See Brophy and Good (1986) for a discussion that suggests that researchers should view all classroom variables as curvilinear in their effects on student achievement.
9 Classroom diversification asserts that a change in the classroom’s technological fit between students’ learning needs and the teacher’s capacity may change the classroom’s stage of operation and performance in student learning.
10 The classroom diversification notion of transforming latent human capital to patent human capital is related to situated cognitive theory, which conceives of learning as a sociocultural phenomenon rather than the action of the individual acquiring information from a decontextualized body of knowledge (Kirshner & Whitson, 1997). See Stein (1998) for an overview of situated learning in adult education.
11 I assume in this example that the teacher’s latent capacity is based on general preservice training for elementary teachers and is therefore transferable among Grades 1 through 6, but not necessarily for the teacher’s patent capacity. This capacity is based on experience teaching students in first grade. Such capacity may transfer to other grade levels in cases in which the required teacher’s capacity essential to students’ learning needs is the same. For example, a teacher’s patent capacity to teach reading to bilingual students in their native languages would mostly transfer from first grade to second grade, because the curricular standards (what students should learn) are likely to have some overlap. The same teacher’s patent capacity would not transfer if significant differences existed in the curricular standards between two grades, such as first grade and sixth grade, unless there were bilingual students in the sixth grade classroom with reading skills at or below the first grade level. Under those conditions, there is a technological fit between the teacher’s capacity and the learning needs of these reading deficient students.
12 Rivkin, Hanushek, and Kain (2005) found in their study on teachers, students, and academic achievement that achievement gains are related to teacher characteristics, but the effects are generally small and concentrated among younger students. For example, they found no substantial difference in teacher quality between teachers with and without master’s degrees. In comparison, the research on the differences between novice and expert teachers supports the classroom diversification notion that teaching experience links knowledge to practice. For example, researchers have found that many of the differences in the thinking and actions between novice and expert teachers can be accounted for by their pedagogical reasoning skills and cognitive schemata (Borko & Livingston, 1989; Livingston & Borko, 1990). Novices’ pedagogical reasoning skills are less developed, and their cognitive schemata are less elaborate, less interconnected, and less accessible than the pedagogical reasoning skills and cognitive schemata of expert teachers. Sabers, Cushing, and Berliner (2002) found similar differences between novice and expert teachers in the way they interpret problem solving in an active classroom in which several events are occurring at one time. These studies and others (e.g., Carter, Cushing, Sabers, Stein, & Berliner, 1988; Carter, Sabers, Cushing, Pinnegar, & Berliner, 1987; Copeland, 1987; Nelson, 1988; Peterson & Comeaux, 1987) share one common conclusion: A key to differences between novice and expert teachers is experience. As Kolodner (1983) contended, experience changes unrelated facts into expert knowledge. Likewise, classroom diversification contends that teaching experience links knowledge to practice, transforming latent human capital, which includes unrelated facts (i.e., declarative knowledge) as well as procedural knowledge, into patent human capital.
13 Pritchett and Filmer (1999) have proposed a model of educational expenditure allocation based on teachers’ utility defined by how spending on inputs affects their welfare independently of its effect on student learning. They assumed that teachers have enormous influence over the allocation of spending and such that spending is biased toward those educational inputs that also directly increase their welfare. Classroom diversification assumes that teachers have enormous influence over the allocation of their teaching effort to students in the classroom and that the expenditure of effort is also biased but based on the notion of technological fit. When technological fit in the classroom is adequate between the students’ learning needs and the teacher’s capacity, a teacher would expend more teacher effort for students. When technological fit in the classroom is not sufficient between the students’ learning needs and the teacher’s capacity, a teacher would expend less teacher effort for students. Thus, classroom diversification defines the teacher’s utility by how the expenditure of capacity in the form of effort relates to the students’ learning needs in the classroom.
14 The teacher’s amount of time devoted to a particular task essentially defines learning opportunities for students. For example, Wang (1998) examined the relationship between opportunities to learn and student learning in science. He found that opportunities-to-learn factors such as content exposure and the quality of instructional delivery were significant predictors of student performance on written and hands-on science tests. Such studies clearly show the importance of learning opportunities to student achievement, but they do not explain why one teacher over another would provide students more opportunities to learn. The classroom diversification notion of technological fit provides one possible explanation. When technological fit in the classroom is adequate between students’ learning needs and the teacher’s capacity, a teacher would provide students more learning opportunities. When technological fit in the classroom is not sufficient between the students’ learning needs and the teacher’s capacity, a teacher would provide students fewer learning opportunities. Thus, classroom diversification defines teachers’ propensity to provide students opportunities to learn in the classroom on the basis of how well the teacher’s capacity relates to the students’ learning needs.
15 See Darling-Hammond (2000) for a discussion of the curvilinear relationship between teacher subject knowledge matter and experience to student learning.
16 Classroom diversification focuses on two aspects of the classroom learning enterprise: the economies derived from the learning relationships that exist across and among students in a classroom and on the technological fit between students’ learning needs and teachers’ capacity. Researchers could extend the classroom diversification framework to include the technological fit between parents’ needs to support their children’s education and teachers’ capacity. In comparison, school diversification would center on relationships across teachers by grade level and among teachers between grade levels and on the technological fit between teachers’ capacity needs and principals’ instructional leadership capacity. School diversification could also include the technological fit between teachers’ capacity needs and the capacity of other professional staff members (e.g., counselors, special education aides, technology support staff members, and curricular specialists). Similarly, researchers could extend the concepts of the classroom diversification framework to the district, community, and school board levels. These multilevel models of diversification would form the foundation of a hierarchical diversification model.
Review of Educational Research, Vol. 77, No. 1,
28-80 (2007)
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Abstract
Classroom Diversification
SL(N), where tC(a1) and tC(a2) are subsets of total available teacher capacity, TC(A). The classroom in this situation also is generating, in the short range, increasing returns to scale in that the excess teacher capacity, tC(a2), could be exploited, if more students with learning needs for that teacher’s unused capacity were assigned to the classroom. Student learning would be maximized, but technological fit is suboptimal in both classroom circumstances. 
