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DOI: 10.3102/003465430298571
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 |
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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. 
