Leaving No Child Behind Yet Allowing None Too Far Ahead: Ensuring (In)Equity in Mathematics Education Through the Science of Measurement and Instruction
by Mark W. Ellis - 2008
Background/Context: For the past century, mathematics education in the United States has been effective at producing outcomes mirroring society’s historical inequities. The enactment of the No Child Left Behind Act in 2001 was intended to address these differential educational outcomes. Given the scope of this legislation’s impact on the way in which states, districts, and schools evaluate mathematics learning and conceptualize reforms in the teaching of mathematics, it is critical to examine the possible effects this may have on how mathematical proficiency is determined and distributed.
Purpose/Objective/Research Question/Focus of Study: This inquiry raises questions about the manner in which the No Child Left Behind Act aims to improve mathematics education through an increased reliance on “objective” science. Specifically, the argument put forth here is that the policies of the No Child Left Behind Act leverage and intensify the “dividing practices” instituted in the early 20th century as a means of justifying the differential stratification of students in schools, thereby making equitable educational outcomes less likely than not. The questions guiding this inquiry are: How did these dividing practices first develop? What are the taken-for-granted assumptions under which they operate? How might technologies related to these practices, given renewed status due to the requirements of the NCLB Act, impact mathematics education?
Research Design: This inquiry takes the format of an analytic essay, drawing on both a historical perspective of efforts to improve education in the United States through a reliance on scientific methods, and an examination of recent evidence as to how the No Child Left Behind legislation’s policies are bring implemented in relation to the assessment and teaching of mathematics.
Conclusions/Recommendations: Although the intent of the No Child Left Behind legislation is to identify schools in which students are not being educated well and to compel improvement, its approach to doing so is built on a model from which long-standing disparities were constructed in the first place. The use of high-stakes standardized testing and direct instruction (DI) methods of teaching—both likely effects of the policies of the NCLB Act—reify the idea that mathematics is something to be put into students’ heads, apart from their lived experiences and daily lives. This approach to mathematics education provides a rationale for students’ (continued) stratification within an “objective” system of standardized testing and instruction. When considering reforms that aim to reduce inequities in educational outcomes, particularly in mathematics, forms of assessment and instruction must be developed and promoted that get away from the divisiveness of the traditional truth games and move toward a focus on students making sense of mathematics in ways that are meaningful, flexible, and connected to their sense of self.
The phrase scientifically based research appears 111 times in the No Child Left Behind Act. It is there with good reason. If teachers, schools and states are going to be held accountable for raising student achievement, they need the tools that will allow them to identify and utilize effective practices and programs. The only tried-and-true tool for generating cumulative advances in knowledge and practice is the scientific method. (Whitehurst, 2002)
This article aims to interrogate the possible educational and social implications of the No Child Left Behind (NCLB) Act through an examination of mathematics education as a particular arena in which its policies are played out. Since the mid-1980s, several national reports have claimed that, given the trend toward a more technological workplace, the poor performance of American students in mathematics warranted serious attention (National Commission on Excellence in Education, 1983; National Commission on Mathematics and Science Teaching for the 21st Century, 2000; National Research Council, 1989). Concerns about equity have become part of such rhetoric: The national interest is strongly bound up in the ability of Americans to compete technologically. . . . All young people, including the non-college bound, the disadvantaged, and young women, must be given the opportunity to become competent in science and mathematics (Carnegie Commission on Science, Technology, and Government, 1991, p. 15).
So it is in the interest of improving the future competitiveness of the nation that the federal government wants schools to leave no child behind,1 intending to ensure this through the monitoring of measurable objectives and the promotion of scientifically based instructional strategies (No Child Left Behind Act of 2001 [NCLB], 2002). This represents a federally mandated attempt to effect what former United States Secretary of Education Roderick Paige (2002) called a massive overhaul of the nations education system. This is to be accomplished by using the technology of scientific measurement to let research and results drive our decisions (Spellings, 2005) about what occurs in the nations classroomsor at least a particular set of the nations classrooms. But will faith in science really lead to improved mathematics learning, particularly for those historically underserved by traditional schooling practices?
Although it may seem to make good sense to base educational decision-making on the objective truth of science, it is important to recall that at the start of the previous century, a similar trust in scientific truth in the form of eugenics researcha project aimed at producing a better populace through the application of scientific means and measurementlent justification to government-sanctioned policies of social engineering affecting immigration, reproduction, and education (Baker, 2002; Duster, 2003; Gray, 1999; Selden, 2000; Tucker, 1994). This is not to claim that the use of standardized testing and promotion of scientifically based instructional practices are equivalent to the vast majority of [early 20th century] eugenics work that has been completely discredited (Dolan DNA Learning Center, 2003). Current standards of scientific practice would seem to preclude the acceptance of what then passed for evidence of genetic inequities between racial and cultural groups. The point to consider instead is how scientific eugenic endeavors to objectively classify individuals through examination and the comparison of norms and deviations masked the reality that policy decisions arising from this work were based on a narrow (and privileging) definition about what it meant to be a good person. This article offers a Foucault-inspired critique of current policies aimed at improving mathematics education in the United States and their potential impact on the nature of student learning, evaluation, and classification.
The work of French philosopher-historian Michel Foucault (19261984) represents an approach to doing a history of the present that aims to better understand what is taken for granted in the discursive and nondiscursive practices of a given intellectual era or episteme:
The episteme makes it possible to grasp the set of constraints and limitations which, at a given moment, are imposed on discourse; but this limitation is not the negative limitation . . . it is what, in the positivity of discursive practice, makes possible the existence of epistemological figures and sciences. (Foucault, 1969/1972, p. 192)
Foucaults work aims to understand the meaning that is writ across the surfaces of routine practices and the conditions that make this possible.
Of particular interest to this article is Foucaults (1975/1977) explication of the concept of power/knowledge: There is no power relation without the correlative constitution of a field of knowledge, nor any knowledge that does not presuppose and constitute at the same time power relations (p. 27).
This pairing illuminates the way in which the games of truth by which knowledge is established come to be linked . . . to games and to institutions of power (Foucault, 1988, p, 16). For Foucault, power is not an object to be wielded against others, but rather a result of particular, historicized practices of knowledge making. As Foucault played with the power/knowledge concept in his analyses of madness, criminality, and sexuality, he perceived that it is within the Western episteme (the modern intellectual era) that the human subject is constituted as an object through processes of normalization and stratification. The practices through which this occursexamination and judgment do not lead to a more homogenous populous, but rather, as power becomes more anonymous and more functional, those upon whom it is exercised tend to be more strongly individualized (Foucault, 1975/1977, p. 193).
It is this aspect of Foucaults reasoning that will be brought to a critique of the view of mathematics education implied by the NCLB Act and its insistence on a particular sort of science to improve teaching and learning. Thinking about the mechanisms by which no child will be left behind in mathematics, one finds the invocation of science in order to standardize instruction, normalize measurement, and centralize decision making about who has learned how much. How might such a programintended as a means to raise the achievement of low-performing studentsactually serve to reify mathematics as inherently objective and discretely knowable, and what are the implications for how it is determined who is good at mathematics?
THE EMERGENCE OF DIVIDING PRACTICES
It is now well established that most of the scientific truths generated by the eugenics movement were far from being true, either because of poor research methods or the direct manipulation of data (Duster, 2003; Gould, 1981; Selden, 2000; Tucker, 1994). Policies that were connected with this work, including race-based immigration restrictions and forced sterilizations in the United States and Hitlers final solution, have been rightly condemned. But the patterns of thought and the practices out of which these policies emerged have not been similarly discredited and have, in fact, continued to serve their purposes quite well.
In an article outlining the roots of current discourses of special education and disability, Bernadette Baker (2002) argued that although largely discounted with respect to its immediate goals, the early 20th-century project of eugenics was both far less and far more influential than accounts of it as a discrete movement might suggest (p. 669). Baker contended that although the most overtly racist and morally offensive eugenic policies were discontinued, the dividing practices created as a means to generate data with which to demonstrate differentially distributed abilities have persisted. Rather than being manifest in discrete forms such as the anti-immigration policies of the earlier eugenic heyday, Baker argued that it is now within bureaucratic institutions, particularly in schools, that such practices are found to operate discreetly as a means to normalize and, subsequently, classify people through what are taken to be objective scientific technologies. The argument put forth here is that the policies of the NCLB Act leverage and intensify these dividing practices, making equitable educational outcomes less likely than not.
Some questions guiding this inquirys aims are: How did these dividing practices develop? What are the taken-for-granted assumptions under which they operate? How might technologies related to these practices, given renewed status because of the requirements of the NCLB Act, impact mathematics education? Following an examination of the development and employment of technologies of mental measurement in educational stratification, attention will turn toward an explication of the NCLB Act and the ways in which it may impact mathematics education. Finally, an analysis of exemplars of so-called objective and scientific approaches to school mathematics will attempt to draw connections that suggest the need for more earnest reflection about the particular path toward educational progress privileged by this legislation.
APPLYING THE SCIENCE OF EUGENICS TO EDUCATION
Looking back to the start of the 20th century, one finds a stunning example of large-scale government support of scientific research aimed at bettering the nation. The project of eugenics found enthusiastic support in the United States in the early 1900s, just after slavery had been legally ended and just as large numbers of eastern and southern European immigrants were entering the United States (Duster, 2003; Tucker, 1994). Indeed, as Carson (2004) noted, The systems of social sorting that had worked tolerably well [for some] during the antebellum period proved largely unable to cope with the powerful transformations shaping late nineteenth-century American society and culture (p. 187). As some perceived it, Anglo-American hegemony was being threatenedthreatened both from within and from without. The threat came from the illiterate foreign-born and from persons of colour (Selden, 2000, p. 241). Although started in England by Darwins statistically minded cousin, Francis Galton, the eugenics project was quickly taken up in the United States as a means of dealing with these threats to society and received ample funding from the private and public sectors.
This work produced claims of scientific truths that provided compelling scientific evidenceobjective measures claimed to be indicators of poor-quality genes and low intellectual capacity among recent immigrant populationsthat lent support to the passing of the 1924 Immigration Act, which was designed to effectively halt immigration of eastern and southern Europeans into the United States (Duster, 2003; see Brigham, 1923, and Grant, 1916, as examples of primary source documents). These same truths, spurred on by developments in mental measurement (discussed next), provided scientific justification for the spread of educational tracking and the limiting of educational opportunities for entire segments of the school-age population (Baker, 2002; Oakes, 1985; Tucker, 1994).
THE DEVELOPMENT OF MENTAL MEASUREMENT TECHNOLOGY
It is critical to understand the thinking out of which new technologies came about that would allow for the scientific management of large numbers of individuals. The idea that society could be improved through the objective measurement of mental qualities first arose from studying the application of mathematical probability theory in the natural sciences. Techniques used to estimate the orbits of planets (i.e., measurement of error and inference of the true value) were applied in the late 19th century to investigations of matters involving populations of peopleinitially physical quantities and eventually mental qualities (Hacking, 1990). British eugenicist Francis Galton (18221911) played a key role in fabricating this transition with his studies of the hereditability of physical traits.
Using the mathematical normal distribution to fit his results, Galton focused not on homogeneity as prior work had, but instead used deviations from the mean to create a stratification of the population. Focusing on the population mean not as an end, he took it to be a point from which to measure individual differences. Assuming the mean to be mediocre, it was assigned a value of zero. A person at the upper quartile boundary was labeled +1 (representing one deviation above the mean), and so on. This signaled the creation of the standard deviationan inferred scale on which to rank individuals. Galton (1892/1925) demonstrated the employment of this method in his book, Hereditary Genius. He took a mathematically manufactured scale, produced from measures of physical characteristics, as the scientific basis for inferred intellectual ability (Stigler, 1986). That Galton decreed ability to be normally distributed (Tucker, 1994, p. 42) and fit to the bell curve had profound implications for the practice of inferring relative rank from measured differences.
A new form of knowledge-making had came into existence in which judgments would be made about an individuals relative social worth and, subsequently, decisions made about the sorts of opportunities to which he or she would have access. The goal of this work as applied to education was ostensibly to determine how to better structure schooling so as to optimize students learning. However, this seemingly just rationale is based on the assumption that there are definite, meaningful, and measurable differences in mental abilities within and between groups of people. In reading the motives coolly offered by the men who developed the technologies of mental measurement, it is clear that they sought to scientifically prove, through relative statistical rankings, not only that such differences existed and mattered but also that northern and western European males represented the pinnacle of human development by virtue of their superior group rankings (see Brigham, 1923; Fisher, 1930; Galton, 1892/1925).
PUTTING A NEW TECHNOLOGY TO USE
This (mis)use of statistical inference led to what Hacking (1990) has called the displacement of human nature by the idea of normality (p. 179). A mathematical mean was transformed into something socially meaningful. Indeed, the normal distribution allowed for the development of the ultimate tool for the scientific construction and justification of educational stratification by an implied innate ability: norm-referenced standardized testing. One of the first large-scale efforts to use this new technology was led by American Louis Terman, whose Stanford-Binet Intelligence Quotient (IQ test) emerged in the 1920s (Madaus, Clarke, & OLeary, 2003; Porter, 1995). Normed with a sample of Anglo middle-class English-speaking children, it was touted as a scientific solution for what was perceived to be the problem of how to objectively identify children with an aptitude for higher learning, and those of substandard mental capacity (Tucker, 1994).
Differential ability, it was reasoned, warranted a different approach to education. In a publication prepared at the request of the National Education Association, Terman and colleagues (1922) asserted that scientific mental measurement
made it possible to classify children more accurately on the basis of native ability. Also, as a result of these findings we have come to realize the necessity of a differentiated course of study for pupils progressing along each of the so-called tracks. . . . [I]deally, provision should be made for five groups of children: the very superior, the superior, the average, the inferior, and the very inferior. We may refer to these as classes for the gifted, bright, average, slow, and special pupils. (pp. 1819)
Students, on the basis of objective statistical inference, were inscribed with labels that carried socially significant implications about their place in school and in society. They were at once depersonalized and highly individualized. That this new technology had a large impact on schooling is borne out when it is realized that by 1932, 75 percent of 150 large city school systems in the United States used group intelligence tests to track students into ability groups (Madaus et al., 2003, p. 1351).
The differential results of this large-scale scientific measurement of ability had powerful implications for school mathematics, offering what was taken to be strong evidence that mathematics was not for all. For instance, E. L. Thorndike (1923), charter member of the eugenically oriented Galton Society and president of the American Psychological Association, argued that there were excellent reasons for believing that the one in ten [fraction of children attending high school in 1890] had greater capacities for algebra and for intellectual tasks generally than the one in three of today (p. 5). Thorndike went on to present a series of tables of IQ measures and algebra passing rates, from which he proclaimed that a child whose IQ was less than 110 could not be expected to successfully learn algebra.2
It is important to note that such testing allowed those aiming to reform education to unhesitatingly label students successes or failures based on their scientific results (Blanke, 2002, p. 29). Once students were classified as scientifically (statistically) differentsubjectifiedthe need for stratified classes was easily justified. Mathematics learners in the United States remained differentially stratified in their coursework throughout the 20th century. According to data from the Third International Mathematics and Science Study (TIMSS) presented by Linn (2001), almost 75% of the eighth-grade students were in schools that offered two or more distinct types of mathematics classes (p. 33). Thus, the dividing practices that emerged within the context of eugenics-era science ensured that opportunities for higher level mathematical learning remained for most studentsand disproportionately so for non-White and low-SES studentsobjectively out of reach.
HISTORICAL PRACTICES AND RESPONSES TO THE CURRENT CRISIS IN SCHOOL MATHEMATICS
Mathematics education in the United States, as it was played out within schools (and with/in children), changed little during the 20th century (Cobb, Wood, Yackel, & McNeal, 1992; Fey, 1979; Stodolsky, 1988). Representative is Susan Stodolskys description generated from her study of mathematics teaching across 39 fifth-grade classrooms in the 1980s: Math instruction places all but the exceptional student in a position of almost total dependence on the teacher for progress through a course. In essence, the traditional math classes contain only one route to learning: teacher presentation of concepts followed by independent student practice (pp. 122123).
The persistent image of mathematics teaching and learning involved the teacher at the front of the class dispensing knowledge while students sat quietly copying notes, working on practice problems and, later, being assessed on their ability to reproduce the facts and manipulations exhibited by the teacher. This approach situates mathematics as a priori knowledge, based on objective reason alone, without taking into account the experiences students bring to mathematics or the meaning they make of what is learned . . . [and] allows students mathematics achievement to be discretely measured, quantified, and stratified (Ellis & Berry, 2005, p. 8).3 As Tate (1994) observed, traditional mathematics classrooms are structured to rank students understanding of a body of static ideas and procedures (p. 481). Access to higher levels of mathematics had historically been available only to those few deemed to possess sufficient ability.
Efforts to change this traditional approach to school mathematics that gained traction in the 1990s, characterized by the vision set forth in the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000), have started to show signs of being capable of instituting and sustaining large-scale reform (Hiebert, 2003; Schoenfeld, 2002). However, recent observational studies of mathematics teaching that have evaluated classroom practices from the perspective of NCTM-defined reform (Stigler & Hiebert, 1997; Weiss, Pasley, Smith, Banilower, & Heck, 2003) have found, amid a small proportion of high-quality lessons, that an overwhelming majority of mathematics lessons continue to focus on low-level thinking, emphasizing the acquisition and application of skills (Stigler & Hiebert, 1997) and portray[ing] mathematics . . . as [a] static bod[y] of factual knowledge and procedures (Weiss et al., p. xi).
Given this history, it is no surprise that relative outcomes on standardized measures of mathematics achievement for students have not changed much in the past century. Performance is still neatly distributed according to the historical markers of race and class (Diaz & Lord, 2005; Schoenfeld, 2002). Although not new, this achievement gap has become a focus of much attention in recent years, invoked to justify the importance of the NCLB Act. It is important to recognize that such disparate educational outcomes are now described as problematic rather than genetic. Reflecting a significant shift in the nations thinking about access to mathematical learning, calls of mathematics for all and even algebra for all are not uncommon nowadays (Moses & Cobb, 2001; NCTM, 2000; Rousseau & Tate, 2003). As Linn (2001) put it, the goal has changed from differentiated standards for a small elite and the larger masses to one of high standards for all students (p. 31).
Although the NCLB Act represents a response to this crisis, it is argued here that its strong reliance on a particular sort of scientifically driven reform will fail to break mathematics education free from the intellectual moorings to which the dividing practices of the 20th century remain tied. In fact, the NCLB Act may actually amplify the effects of the way in which children come to be identified as mathematically proficient. Given that, as mentioned previously, efforts to reform mathematics education based on principles described in the NCTM (2000) standards have recently shown promising signs of being capable of supporting a move away from traditional practices (Hiebert, 2003; Schoenfeld, 2002), it is imperative that the potential of the NCLB Act to undermine these efforts is well understood.
THE NO CHILD LEFT BEHIND ACT OF 2001: PROCLAMATIONS AND IMPLICATIONS
Signed into law by President Bush in January 2002, the No Child Left Behind Act is a legislative document, more than 600 pages, detailing what many consider to be the federal governments biggest effort ever to regulate the practice of public schooling. The legislation was developed out of an interest in closing the achievement gap that exists between minority and nonminority and disadvantaged and more advantaged students (NCLB, 2002). No specific information is given about the size of this gap, the means by which it has been measured, or how it has changed over time (see Hunter & Bartee, 2003 for a detailed explication of this). A further rationale for the NCLB Act is the concern, expressed by President Bush and the U.S. Department of Education (DOE), that schools are not producing the math excellence required for global economic leadership and homeland security (DOE, 2003). A primary goal of the NCLB Act, then, is raising the mathematics achievement of minority and disadvantaged students so that as adults, they will be able to contribute to the nations economic strength and security.
The means by which the legislation plans to effect this change is found in its repeated111 times (Whitehurst, 2002)reference to scientifically based practices. The NCLB Act requires states to develop assessments that scientifically and objectively measure students knowledge and skills in mathematics. Its focus is on the enforcement of a testing regime in which schools must assess all children annually yielding scores that can be compared across populationsthat is, standardized tests. Using the results of these tests, the NCLB Act requires growth to be measured by a complex formula that determines average yearly progress goals based on the scores of all students and scores of targeted subgroupsgiven as ethnic minorities, limited English proficient (LEP) students, and economically disadvantaged students. Schools identified as underachieving must implement scientifically based instructional programs to raise the test scores of their students. These programs are defined as ones for which research involv[ing] the application of rigorous, systematic, and objective procedures to obtain reliable and valid knowledge relevant to education (NCLB, 2002) has shown them to be effective in raising standardized test scores for the low-income and predominantly minority students classified as Title I. If the testing goals are not met, sanctions are to be brought in the form of increasingly severe punitive measures ranging from mandated district payment for private tutoring of students to the shifting of a schools governance from the local district to the state (Zhang, 2003). Because of this penalty-based approach, the testing process is a high-stakes endeavor for schools; by extension, for teachers; and, most directly, for the students taking the exams.
The large-scale, high-stakes use of standardized testing tied to the compelled use of particular instructional practices with low achieving students is the most salient feature of the NCLB Act. As more and more tests are promulgated into classrooms and onto students, it is prudent to recall that, as was observed with an earlier generation of objective measures, the meanings of educational assessments cannot be separated from their social consequences (Wiliam, Bartholomew, & Reay, 2004, p. 43). Even though the use of traditional norm-referenced tests has declined, they are still frequently employed; in the 20052006 school year, 19 of 50 states used these as measures of mathematics achievement (Council of Chief State School Officers, 2006).4 More important, the basic principles underlying the construction of standardized tests belie their connection to the dividing practices of an earlier era. Criterion-referenced assessments, which claim to deliver scores reflecting performance against a set of criteria such as state mathematics standards (offered as an alternative to relative rankings), often share a common item pool with norm-referenced tests (Linn, Baker, & Betebenner, 2002) and are crafted following the same psychometric principles (such as validity and reliability) as norm-referenced exams (Wiliam et al.). A California Department of Education (2004) handbook explaining test construction offers some insight into this, describing how items suitable for its statewide criterion-referenced exams are developed:
The two properties most often examined are item difficulty and item discrimination. Item difficulty can be determined by discerning the percentage of students that answered an item correctly. If the p-value is .80 (80% of students answer an item correctly), the item is considered easy. If the p-value is .20, the item is considered difficult. Item discrimination refers to how effectively each item differentiates between students who know most about the content area being tested and those who know least. For example, students with the highest scores should generally get hard items correct while students with low scores generally will not. (p. 6)
Implicit within this approach is the presumption of preexisting differences that are both meaningful and measurable. As Wiliam et al. (2004) asserted, recognizing this fact of test design do[es] not mean to suggest that constructs such as mathematics are completely capricious and cannot be measured, but simply that a range of social factors intrude into the design of apparently objective assessment instruments (p. 45). The implications of using a single high-stakes test to guide the evaluation and classification of students can be severe, particularly for those who perform poorly.
Within the model favored by the NCLB Act, teachers would be not only justified but also compelled to provide instruction to lower level students that leads to increased scores. Based on historical precedent, as pressure to improve results on standardized assessments increases, it is likely that the curriculum for these students will be reshaped into one with limited content coverage and a narrow range of pedagogical techniques (Darling-Hammond & Wise 1985; Urdan & Paris 1994). This is not to say that test scores will not rise under such an approach; for many students, they will. However, such gains will be academically restraining because they are built on a model of learning that discounts learners understanding of mathematics in order to privilege their relative standing within an artificially standardized set of boundaries, thereby limiting actual opportunities for student success in higher level coursework (Tate, 1994, 2000). In a Foucaultian sense, students will come to be not only classified as proficient or not but also inscribed as sucha process that allows for depersonalized actions to be brought upon them with little regard for their personal identities, aptitudes, and interests, and for a disavowal of the social inequities that certainly are part of students lived realities. This carries strong implications for how students will come to be treated, operating under the assumption . . . that people can be known, their activities charted and quantified, their movements predicted and controlled, just as physical objects allegedly can (Code, 1995, p. 183).
EFFECTS OF NCLB-DRVIEN EDUCATIONAL IMPROVEMENT IN MATHEMATICS
The Normal is established as a principle of coercion in teaching with the introduction of a standardized education . . . the power of normalization imposes homogeneity; but it individualizes by making it possible to measure gaps, to determine levels. (Foucault, 1975/1977, p. 184)
Having developed a perspective of the intellectual roots out of which the NCLB Act emerged and having explored in theoretical terms the way in which its policies are to be applied to students experiences as learners of mathematics, manifestations of these policies in practice will be examined. The two instances that follow are offered as representative scenarios that are not unlikely to develop in multiple locations as seemingly reasonable responses to present policy mandates.
Effect One: Constructing Proficiency Through State-Level Assessments
The NCLB Act has led states to at once expand their standardized testing of student achievement, particularly in mathematics, and more narrowly define their expectations for what constitutes proficiency. Because the NCLB Act requires student achievement to be reported across a minimum of three levels (advanced, proficient, and below proficient), it is now routine that students performance on annual state exams is reported as a single digit or moniker within a three-to-five-level range. Table 1 offers examples of these for seven states (sources: California Department of Education, 2005; Colorado Department of Education, 2006; Illinois State Board of Education, n.d.; Louisiana Department of Education, 2003; New York State Education Department, 2006; North Carolina Department of Public Instruction, 2005; Oklahoma State Department of Education, n.d.).
Table 1. Performance Levels for Selected State Mathematics Assessments
The assessments designed ostensibly to ensure that no child is left behind are used to generate labels for students that neatly (re)create a hierarchically organized system eerily similar to that offered by Terman et al. (1922) early last century as a means of classifying students on the basis of IQ score (theirs was a five-part continuum from gifted to special). This is a stark indicator that present policies remain tethered to the same patterns of thought from which the dividing practices such as norm-referenced IQ testing emerged. As with the way in which IQ scores were used to objectively stratify students on the basis of race and class, providing inequitable access to educational opportunities, there is a concern that rather than being diagnostic in purpose, offering useful feedback about students strengths and weaknesses, the assessments will become prognostic, taken as a signal of students potential for success. In effect, it is the student, not her or his performance on an assessment, that is marked.
In the past few years, several studies have documented educational practices indicative of this effect. Evidence is found of low-performing students being perpetually held in nontested grades and encourageddirectly or indirectlyto leave the schooling system (Haney, 2000; Nichols & Berliner, 2005; Orfield, Losen, Wald, & Swanson, 2004), while students on the statistical cusp of being proficient receive some measure of help (Booher-Jennings, 2005; Diamond & Spillane, 2004). This demonstrates how extreme reliance on objective scientific measures can lead not to some utopian world in which every child is brought ahead, but rather to a system in which many children are left behind in good conscience by educators and administrators because they are unworthy of efforts to educatethey are simply unsatisfactory.
Effect Two: Instructional Practices For Those Left Behind
In considering the sort of instructional practices likely to be scientifically proved to raise test scores of low-achieving students to a state-defined proficient level, it is important to understand what such proficiency requires. Although this is a matter of much discussion and debate, United States Secretary of Education Margaret Spellings has signaled her support for aligning proficiency on state tests with the basic level of achievement on the National Assessment of Educational Progress (NAEP). In discussing with reporters the discrepancies between many states reports of student proficiency in mathematics and states performance on the NAEP mathematics exam, Secretary Spellings encouraged the comparison of state test results with the percentage of students performing at the basic [italics added] level on the federal [NAEP] test (Dillon, 2005; see Finn & Ravitch, 2006, for further commentary about this apparent shift). The significance of this is dramatic, for basic denotes the ability to use algorithms to calculate answers to routine problems, whereas proficient signifies evidence of conceptual understanding and the ability to problem solve (National Assessment Governing Board, 2004; see fourth-grade achievement level descriptions). Thus, proficiency may come to be defined as basic skills rather than a richer multidimensional construct of mathematical proficiency offered by the National Research Council (2001) and embedded in the design of the NAEP mathematics assessments (National Assessment Governing Board).
If the goal for low-achieving students is narrowly defined as improved performance on tests of basic computational skills, as opposed to more complex learning outcomes, the sort of instructional practices found to deliver such results will be correspondingly narrowly defined (Kennedy, 1999). With this in mind, direct instruction (DI), champion of the Project Follow Through study that tracked the achievement of thousands of Title I students within several instructional programs over a period of years, is an approach that has been deemed by some to be one of the only instructional programs for mathematics with scientifically based research proving its effectiveness in improving students computational skills (American Institutes for Research, 1999; Baltimore City Public School System [BCPSS], 2003; Bereiter & Kurland, 1981). According to a March 2003 report issued by BCPSS detailing the impact of DI programs in 13 of their underachieving Title I schools, Its major goal is to improve the fundamental education of children from economically disadvantaged backgrounds in order to increase options later in their life (p. 2). In gauging the success of DI, the BCPSS report stated that although DI student achievement on nationally normed exams was no better than that of students in the control group, the DI cohort scores were significantly higher on the Maryland Functional Mathematics Test (p. 41), a test of basic computational skills. Furthermore, in a high-profile speech about what is known to work in teaching mathematics, the director of the Institute for Educational Science, Grover Whitehurst, proclaimed, We know that direct instruction can help students learn computational skills and understand math principles. As a corollary, we know that children don't have to discover math principles on their own or work with authentic open-ended problems in order to understand mathematical concepts (Whitehurst, 2003).
Given years of scientifically based research documenting success with Title I students development of computational skills and strong support from high-profile education officials such as Grover Whitehurst, the DI program appears likely to meet the standards for instructional practice set by the NCLB Act.5 As such, it is important to understand how the DI method works. In their book, Theory of Instruction: Principles and Applications, Engelmann and Carnine (1991), two of the primary developers of DI, offer this insight into the theory behind the programs tightly scripted and timed lessons: Faultless communication by the teacher leads to effective understanding by students. A faultless presentation rules out the possibility that the learners inability to respond appropriately to the presentation, or to generalize in the predicted way, is caused by a flawed communication rather than by learner characteristics (p. 3). This objective approach, with its faultless presentation of content, removes all claims of subjectivity from the teacher. The mathematics content is delivered efficiently to students so that they have the best opportunity to learn. This in turn allows for an accurate assessment of learner characteristics and the determination of their true level of achievement. This objective information about students is then used to form homogenous groups according to students skill levels. . . . The instructional pace is set accordingly for each learning group (BCPSS, 2003, p. 9). This is the epitome of the application of an objective scientific approach to education.
The following transcript and accompanying narrative of an episode from a DI classroom illustrate the reality for students at City Springs Elementary School, one of 13 Title I schools in Baltimore using DI mathematics in 2001:
The dog has 99 fleas and wants to get rid of 70 fleas, the teacher says. How many fleas did the dog have?
NINETY-NINE FLEAS! the students shout.
Would you add or subtract?
They then take out their lined workbooks and write the problem down, separating the parts of the problem with thick lines:
NINE MINUS ZERO IS NINE! NINE MINUS SEVEN IS TWO!
Read the whole problem!
NINETY-NINE MINUS SEVENTY EQUALS 29!
The students then move on to lesson 21.
The students chant their answers, in a tone somewhere between a recitation in church and soldiers counting cadence. They do this 100 times a day, five days a week.
Student behavior is also tightly regulated. When a student is sitting at a desk, he isnt just sitting, hes in the listening-learning position or LLPfeet firmly on the floor, back square in the seat, arms crossed on the desk. . . . Students are tested, as many as two or three times a week. This enables the good students to move forward more rapidly, but also enables the school to determine who the poor students are who need more work . . . the worst students are drilled again and again until they get it right. (Wooster, 2001, p. 41)
Pedagogically, a picture develops of the teacher leading the class, the clear authority and arbiter of truth. Students are under control and ready to perform on cue, their responses timed to keep pace with the teachers inquiries. The material is organized a priori into what have been determined to be logically sequenced bits of mathematical knowledge. Frequent objective testing serves not only as a means of coercion for students to internalize these bits of knowledge but also as a device through which to inscribe worth. Those who excel are rewarded by being allowed to move ahead of the others, earning them the label, and accompanying status, of good students. Those who fail to do sothe worst studentsare sanctioned, subjected to repeated drill until they get it right.
The DI approach, one that is well positioned to meet the definition of scientific evidence demanded by the NCLB Act, reifies the idea that mathematics is disembodiedsomething out there to be put into students heads, apart from their lived experiences and daily lives. This serves to both reinforce mathematics as a source of immutable truth and provide a rationale for students (continued) stratification within an objective system of standardized testing and faultless instruction. Learning is at once depersonalized, disconnected from students lives, and subjectifying, serving as a means with/in which students are scientifically assigned to their true level.
A CRUEL IRONY: THE NEW EUGENICS
And so it is that already marginalized students are (re)inscribed with labels taken to be objective evaluations of their mathematical ability. Though some may make it ahead of their peers on standardized measures of achievement, as Title I students, they retain their inscription within societys other labels of race and gender and class. Their relative within-group success may or may not provide entry into advanced levels of mathematics that require more than the ability to carry out simple computations (Knapp & Shields, 1990; Tate, 2000). These higher levels of coursework, too, are guarded by tests that by design create a divided distribution of achievement and remain disproportionately middle class and White (and taken-as-White). Indeed, realizing the (almost) futility of such an effort, Baker (2002) sharply observed that the new eugenics operates through examinations and observations in more insidious if unintentional ways. The analyses of exam results that point to which populational groups fail or succeed at what do not simply tell us of the existence of racism or sexism or classism (p. 694).
This is further exacerbated when achievement is individualized to the level of each student, upon whom the judgment of worth is delivered and appropriate actions taken. It is clear that the odds are stacked against the students whom the NCLB Act intends to help, with its reliance on traditional methods of constructing scientific truths about student achievement.
In startling and devastating insight, Baker (2002) realized that it is not easy to escape the effects of the ingrained dynamics of power/knowledge that constitute our own subjectivities and that, as such, these same dynamics inhere in the well-intended evaluation efforts and classificatory practices (p. 696) that are now used as a means to effect change. In some sense, no matter the intervention, it will always be the children of privilege who are found to be proficient.6 As Nel Noddings (1998) suggested, if the knowledge associated with privilege is just that . . . then members of the dominant group are likely to shift the locus of power to something else (p. 114) once the less privileged have achieved a measure of parity. The practices of standardized testing, with their reliance on measures of statistical difference to produce valid scores, assure that this will be the casethere must always be someone left behind.
Expressing frustration at this cruel irony, Baker (2002) charged that those who claim to care about the education of all students have not, will not, do not want to, or do not know how to give up the act of classifying, sorting, and hierarchizing human beings, reduced in the end to ability levels or test scores (p. 96). Given its use of the eugenic-era technologies of measurement and stratification tied to a view of mathematics as absolute truth, the implementation of the No Child Left Behind Act seems likely to create another legacy of legitimizing inequality in the name of social improvement.
RESISTING TRADITIONAL TRUTH GAMES: REDEFINING REFORM IN MATHEMATICS EDUCATION
The point to be made is not that all testing leads to the unfair stratification of people. Rather, it is that an awareness of the historical intellectual roots of the dividing practices from which efforts to standardize education emergedevidenced by their development in conjunction with a societal desire to stratify students inequitablymust lead to the more cautious and critical use of these technologies in education. Although the intent of the NCLB Act is to identify schools in which students are not being educated well and to compel improvement, its approach to doing so is built on a model from which long-standing disparities were constructed in the first place. Particularly in mathematics, forms of assessment and instruction must be developed and promoted that get away from the divisiveness of the traditional truth games.
What might be an alternative method of reforming mathematics education that avoids truth games that unfairly divide students? Donna Haraway (1991), thinking about how to reform the discourse of objective scientific truth without denying it has any value whatsoever, offered a practical approach to this problem. She argued not for complete relativism but for partial, locatable, critical knowledges sustaining the possibility of webs of connection that lead to a re-visioning of what may count for rational knowledge from standpoints that acknowledge positioning from somewhere, rather than being hidden behind claims of an objective universality (p. 178).
Thinking about what this might mean for mathematics education, there must be an awareness of the dichotomous and divisive truth game that has been played. For all students to have opportunities to more meaningfully constitute their (mathematical) selves, a conscious move must be made away from traditional practices tied to the idea of an objective absolute truth about mathematics and student achievement that can be accurately measured (albeit artificially constructed). One principle to adopt immediately was suggested by education scholar Linda Darling-Hammond (2006): We must use assessment mechanisms that evaluate standards over time and that can reveal progress, rather than use tests that are continually renormed to reproduce the normal curve and that eliminate items when too many students have come to know the answers to them (p. 17).
This would cast assessment as a process that is more informative than evaluative and ensure that standards (as reflected in tested items) do not remain perpetually out of reach. In an article arguing for the alignment of assessment with a learning culture Lorrie Shepard (2000) stressed the need to change the social meaning of evaluation [such that] students and teachers look to assessment as a source of insight and help instead of an occasion for meting out rewards and punishments (p. 10). In such a model, assessment is ongoing, transparent as to its criteria and feedback, and used to inform teachers and students about their progress toward robust learning standards. This must be an explicit effort on the part of teachers, an act of resistance against the pernicious effects of proficiency testing on their curriculum (p. 10). A good place to start is with investing in the development and implementation of formative assessment practices that are known to be effective in supporting students meaningful learning of mathematics (e.g., Black, Harrison, Lee, Marshall, & Wiliam, 2002; Gallagher, 2004).
Following Haraways (1991) thinking one step further, reforms must be aimed at opening spaces within which multiple knowledges are recognized, a move that would view mathematics not as disconnected external facts and procedures but as meaningfully connected sets of ideas about which students could develop personal understandings. This would require a conscious move away from measuring learning against fixed, hierarchical models of individual development and toward the consideration of the dialogical and dialectical processes by which students and teacher interactively (re)construct and negotiate mathematical understandings within the situated local ecologies of classrooms and communities. Attention would be focused on students making sense of mathematics in ways that are meaningful, flexible, and connected to their sense of self (Kumashiro, 2001).
In assessing learning outcomes within such a model, credence would be given to outcomes such as students application of mathematical thinking to problems relevant to their lives and communities, students interpretation of the validity of mathematical solutions, and students confidence in their own mathematical abilitiessomething called for by the National Research Councils model of mathematical proficiency, which encompasses five interwoven strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (National Research Council, 2001). Moving in this directiontoward a more complex model of both mathematical knowledge and student achievementwould necessarily recognize the privileging and hierarchical game of truth through which present inequities are (re)inscribed within students, its power thereby dissipated and spread among many other measures of students proficiency.
Whether this approach to reforming school mathematics can really work is less and less a matter of speculation because examples of success exist (e.g., Boaler, 1997, 2006; Campbell, 1996; Knapp at al., 1995). Notable among these is the Algebra Project created by Robert Moses, the civil rights-era founder of the Student Non-violent Coordinating Committee (Moses & Cobb, 2001). This program for middle school mathematics students in underserved urban and rural communities is built on a curricular model grounded in starting the learning process with a common physical, hands-on experience in which all students are engaged and from which their mathematical understanding develops. Attention is given to students sense-making, both personal and collective, through activities of exploration and communication. Learning is taken to be connected to students sense of self, not simply a matter of internalizing a disconnected body of knowledge.
The matter of greater uncertainty is whether such reform efforts that resist traditional notions of accountability will be allowed to disrupt the legacy of the dividing practices in mathematics education that privilege the few at the expense of so many. Are we prepared for more diverse groups of students coming to be successful in learning mathematics as personally and socially meaningful? Or will we continue to hide behind the veil of objectivity that justifies leaving many children behind so that some may be labeled as being ahead? The present course, set by the well-intentioned policies of the NCLB Act and interpretations of federal education officials, appears to steer strongly toward the latter of these possibilities.
1 Note that the directive to leave no child behind is a trademark of the Childrens Defense Fund and has been so since long before the creation of current federal government initiative bearing a similar name.
2 This claim is made despite the low correlation (r = 0.19) between IQ scores and algebra grades (Thorndike, pp. 3536).
3 This is not to say there had not been instances of reform that have had small-scale positive impacts on school mathematics teaching and learning. See Ellis and Berry (2005) for a discussion of some of these.
4 Interestingly, in recent years, the use of augmented norm-referenced assessments, which have had items added to better reflect state content standards, has been prompted by the United States Department of Education as cost effective (Corwin, 2003) and touted by testing companies as a way to satisfy NCLB assessment requirements and to obtain useful norm-referenced data about their student populations (Zucker, 2004, p. 8).
5 See Hake (2005) for a critical discussion of the research about direct instruction methods in science teaching.
6 Indeed, the Washington Post recently reported about area suburban districts in which advanced is kind of proficient for us (school principal, quoted in deVise, 2006).
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