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The NCTM Standards and Mathematics Achievement in the Elementary Grades: A Literature Review In 1989, the National Council of Teachers of Mathematics (NCTM), the largest organization of mathematics teachers in the world, published “Curriculum and Evaluation Standards for School Mathematics.” This document, which addressed both content and pedagogy, set forth a comprehensive set of national standards for mathematics teaching in the United States. These standards have been both influential and controversial (Loveless, 2001). This paper reviews the Standards’ development and subsequent revision, describes the significant debate sparked by the Standards, and summarizes research studies that purport to shed light on how the Standards affect student achievement in the elementary grades. Although research is often cited in arguments for and against the Standards, in fact there is little independent research that evaluates the actual effects of the Standards, as implemented in classrooms, on student mathematics achievement. Research evaluating the effects of the teaching methods advocated by the Standards is often cited by both proponents and critics of the Standards. Some of that research, especially more recent studies that may reflect classroom effects of the Standards, is reviewed here. Research that looks at the effects of curricula developed in compliance with the Standards, while relatively scanty, receives a greater focus. In selecting the research studies to be reviewed, the emphasis was on studies that use traditional measures of student achievement, such as standardized tests. These measures were 2 preferred because there is broad popular agreement that standardized tests measure important learning, and because of the importance of standardized test scores in American public education today. This paper concentrates on mathematics education in the elementary grades. Looking at elementary grades separately seems justified in light of the fact that most elementary teachers are not mathematics majors, and many do not have backgrounds in mathematics. Thus, as the standards are implemented in an elementary setting, they must be effective in the hands of generalists, some of whom have a weak grasp of mathematical content. This sets the elementary setting apart from most middle and high schools, where a much larger proportion of mathematics teachers have mathematics backgrounds. The History of the NCTM Standards In the decades preceding the publication of the Standards, increasing evidence suggested that American mathematics education was in dismal condition. In 1972, the first administration of the National Assessment of Educational Progress (NAEP) revealed a level of mathematics proficiency that many found shockingly low. By the time the results of the Second International Mathematics Study were published in 1987, little progress had been made. The results of United States students had declined from the First International Study in 1964, and the United States performed poorly compared to most industrial countries (Senk and Thompson, 2001). 3 The Standards were an effort to improve mathematics teaching to respond to these challenges. The Standards advocated changing mathematics instruction to focus on problem-solving, communication, reasoning and mathematical connections. In general, the Standards advocated considerably less emphasis on memorization of facts and rules, and considerably more on conceptual understanding developed through the use of manipulatives and visualization. Group and project work was encouraged, as was the use of calculators and computers (NCTM, 1989). Steve Willoughby, the President of the NCTM at the time the 1989 Standards were published, recalls that the NCTM never intended to eliminate the teaching of basic mathematical skills. However, it is clear that the Standards did not promote the learning and practice of algorithms. Statements like the following with respect to learning fractions in the 5th to 8th grade created the impression that the Standards advocated complete reliance on visualization, and were hostile to teaching mathematical algorithms: The mastery of a small number of basic facts with common fractions (for example, ¼ + ¼= ½; ¾ + ½ =1 ¼; and ½ x ½ = ¼) and with decimals (for example, 0.1 + 0.1 = 0.2 and 0.1 x 0.1 = 0.01) contributes to students’ readiness to learn estimation and on concept development and problem solving. This proficiency in the addition, subtraction, and multiplication of fractions and mixed numbers should be limited to those with simple denominators that can be visualized concretely or pictorially and are apt to occur in real-world settings; such computation promotes conceptual 4 understanding of the operations. This is not to suggest, however, that valuable instruction time should be devoted to exercises like 17/24 + 5/18 or 5 ¾ x 4 ¼, which are much harder to visualize and unlikely to occur in real life situations. Division of fractions should be approached conceptually. The NCTM Standards were widely disseminated and proved very influential (Loveless, 2001). Coincidentally perhaps, at the same time as the Standards were being disseminated, many states were developing their own grade level content standards as part of educational accountability schemes. The NCTM Standards therefore became the basis for many state standards. From 1989 to 1991, the National Science Foundation (NSF) issued calls for proposals to develop instructional materials based on the Standards. Ultimately, more than a dozen projects to develop new curricula and textbooks were funded. The NSF projects were field tested and implemented throughout the mid-1990’s. (Senk and Thompson, 2001). In 1999, the United States Department of Education identified ten Exemplary and Promising Mathematics Programs, many of which were standards-based, which fueled the adoptions of standards-based curricula. By the turn of the century, millions of American children were learning mathematics from materials based on the Standards (Senk & Thompson, 2001). In particular, many urban school districts adopted standards-based curricula as a requirement of participation in the Urban Schools Initiative (USI), a broad-based reform initiative that affects 22 of the nation’s largest urban areas, containing ten per cent of the nation’s public school students (Systemic Research, Inc., 2001). 5 As standards-based curricula and textbooks were introduced into schools, significant opposition to the Standards emerged. Many academic mathematicians and scientists, claiming that they had never been consulted on the Standards, reacted strongly against the idea that teaching basic skills, such as fluency with algorithms, was unimportant, and predicted a further downturn in the mathematical competency of college students. A group of 200 respected scientists and mathematicians published a full-page advertisement in the Washington Post urging then Secretary of Education Richard Riley to rescind the list of Exemplary and Promising Mathematics Programs (Senk and Thompson, 2001). At the local level, parents opposed to standards-based curricula, which they regarded as insufficiently rigorous, battled professional educators (Loveless, 2001). A national grass-roots movement, fueled by a popular website called MathematicallyCorrect.com, supported the opposition. Perhaps in response to the criticism, in the late 1990’s the NCTM began a formal process of revising the Standards. Advice was sought from the major mathematical professional societies, such as the American Mathematical Society (AMS). The AMS submitted detailed comments, suggesting that the standards should be more specific about the role of the teacher in bringing mathematical closure, the sequencing of topics, the role of algorithms in automatization, the importance of proofs, and the connection between algebra and geometry (Howe, 1999). Ultimately, significant changes were made to the Standards. The revised version, released in 2000, includes greater specificity about when students 6 should reach proficiency in certain topics, greater emphasis on building knowledge across grade levels, and a greater acknowledgement of the role of algorithms (Ferrini-Mundy, 2000). Reflecting on the revision process, Willoughby (2000) acknowledged that the 2000 revisions clarified the need for teaching both traditional skills and conceptual understanding. Despite these revisions, controversies about the Standards continue. Matters may be complicated by the fact that many textbooks, which often spark local controversies, are based on the 1989 Standards and do not reflect the changes made in 2000. However, real disputes about the correct way to teach mathematics clearly remain (Loveless, 2001). Moreover, mathematics achievement did not improve much in the years during which the controversies raged. By the late 1990’s, the results of the Third International Math and Science Study (TIMMS) confirmed that American mathematics education remains in trouble, with United States’ students below the mean in almost every mathematical area by 8th grade. In a sobering assessment of the TIMSS results, Schmidt et al (1999) identified the following features of United States mathematics education: a mile wide and an inch deep, covering too many subjects shallowly cautious, undemanding, and inconclusive texts instruction lacking in coherence highly repetitive instruction, with the same material covered year after year lack of rigor in the middle grades 7 widely differing access to educational content Thus, even as standards-based textbooks became common, and the battle over the Standards raged, little real progress in mathematics achievement occurred. Philosophical Debates About the Standards In some respects, the debate about the Standards is about values, and is therefore not susceptible to resolution by research (Hiebert, 1999). To the extent disagreement about the Standards can be characterized as a battle between “constructivist” supporters of the standards versus “behaviorist” critics, the debate represents differences in educational philosophy. Hirsch (2001), an opponent of the Standards, characterizes the debate as between the “Traditional/Classical” camp, on the one hand, and the “Romantic/Progressive“ camp, on the other. He puts himself squarely in the Classical camp. Geary (2001) rejects constructivist mathematics teaching on the grounds that humans are not inherently wired to construct mathematical understandings on their own, and that students need strong direction, organization and structure from teachers in order to master mathematics. Battista (2001) distinguishes what he calls “simplistic constructivism” from what he calls “scientific constructivism.” Citing evidence that traditional mathematics instruction does not promote understanding, Battista emphasizes the importance of sense-making to mathematics learning. However, he stresses that conceptual understanding and computational fluency both count. 8 The idea that conceptual learning precludes the learning and practice of basic skills is substantial source of controversy. This debate generally revolves around the memorization of algorithms, defined as procedures that are guaranteed to produce desired results (Robinson, 2000). In “Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy,” Berkeley mathematics professor Wu (1999), challenges the belief that a demand for precision and fluency in the execution of basic skills runs counter to conceptual understanding. In Wu’s words, “in math, skills and understanding are completely intertwined.” He characterizes mathematical precision and skill as the vehicles by which understanding is conveyed (p. 1). Gamoran (2001) argues that the conflict over the Standards represents a false dichotomy between rigorous content and deep understanding. Citing Stigler’s studies of how Japanese mathematics teachers teach, Gamoran maintains that the ideas in the Standards are correct, but only when implemented in the context of rigorous mathematical content. Stigler’s studies show that Japanese practices of guided exploration, focusing on student responses rather than teacher presentations, and leading students to discover the right answer work, but in a context of a longer school day, more homework, and high expectations (Stigler et al., 1999). Gamoran concludes that the methods endorsed by the Standards will also work, but that they must be used in the context of rigorous content and an insistence on deep understanding. A number of commentators explore how teacher quality influences the debate about the Standards. Wu (1999), for example, points out that when 9 proponents of the Standards criticize learning basic skills, what they may really be criticizing is poor teaching of basic skills. According to Wu, teachers who do not have the skill to guide students to understanding promote rote learning. Wu sees professional development of teachers as the key to realizing his vision of the integration of skill and understanding. Askey (2001) believes that in the United States the problem of inadequate teacher knowledge of mathematics is even worse in the context of the Standards, because the type of teaching advocated by the standards requires a deep understanding of mathematical concepts. Askey questions whether curricula developed in accordance with the Standards can adequately compensate for teachers’ lack of substantive knowledge. Studies of the Mathematics Teaching Practices Encouraged by the Standards Gamoran (2001) refers to a number of studies, including those involving Japanese teachers, which in his view demonstrate that the mathematics teaching practices encouraged by the Standards produce better student achievement than more traditional teaching practices (Hawkes, Kimmelman, and Kroeze, 1997; McKnight et al., 1987; Rohlen, 1983; Stigler et al, 1999; Westbury, 1992, 1993). Although studies such as these document the effectiveness of certain of the practices endorsed by the Standards, they often do so in specialized contexts involving highly qualified teachers, and thus may not be generalizable. Several recent, broad studies use TIMSS data to investigate the link between teaching practices and student achievement. Gales and Yan (2001) 10 used factor analysis to identify behaviorist and constructivist mathematics teaching techniques. Using Hierarchical Linear Modeling, the authors then evaluated links between the instructional practices reported in TIMSS and overall TIMSS mathematics scores. Teacher practices had mixed, and quite small, effects on scores. A regular practice of having students discuss and give the reasoning behind answers (somewhat surprisingly identified as a behaviorist practice) had a significant positive effect on scores, as did a practice of having students work on projects with no immediate correct answer (identified as constructivist). A teacher belief that mathematics is a practical and structured guide for addressing real world problems had a significant negative effect on scores. The largest effect sizes were unrelated to teaching practices. The belief that diversity in the classroom has a negative impact on student learning had a large and significant negative association with scores, and having a female teacher had a large and significant positive effect. In a similar type of study, Tomoff, Thompson, and Behness (2000) used TIMSS data to explore the relationship of four types of classroom practices to student achievement. Of the four teacher practices identified through factor analysis, hierarchical linear modeling showed that the two practices identified as constructivist, project creation and group work, had no effect on mathematics scores. One behaviorist practice, called drill and practice, had a significant negative effect on scores, while the other, called working from textbooks and workbooks, had a significant positive relationship. 11 Wilson, Abbott, Joireman and Stroh (2002) used Structured Equation Modeling to test a model in which parental and community involvement (called parent attributes) and staff who modeled responsible behaviors (called school attributes) were positively associated with constructivist practices, and school attributes and constructivist practices together were positively associated with student mathematics achievement on the Washington Assessment of Student Learning. According to the model, school attributes and constructivist practices together accounted for 23% of the variation in mathematics scores. None of these recent studies was published in a peer-reviewed journal. Given their mixed results, the studies shed little light on how the kinds of teaching practices encouraged by the Standards affect student achievement. Studies of Standards-Based Curricula An obvious way to evaluate the impact of the Standards on student achievement is through studies that investigate the effects of standards-based curricula. Unfortunately, no large-scale, clearly independent studies of the effects of the standards-based curricula on traditional measures of achievement were found. Project 2061 is currently evaluating how middle school mathematics curricula are impacting student learning (American Association for the Advancement of Science, 2003), and other studies are also in progress (Kramer, 2003). Most of the studies that have been reported were conducted by the curriculum developers themselves, by researchers associated with them in some 12 way, or by districts that had already made a decision to implement a particular curriculum. The ARC Center is a partnership between developers of Everyday Math, Math Trailblazers and Investigations in Number, Data & Space, three standard- based curricula, and the Consortium for Mathematics and Its Application. ARC’s Tri-State Student Achievement Study (2000) looked at the effects of these three curricula on standardized test scores in Massachusetts, Illinois and Washington State. The authors believe they identified almost all of the schools using the three curricula in each state, achieving a near census. “Reform” curricula schools were matched and compared with other, “non-reform” schools. The mean of the reform schools was at the 54th percentile of the non-reform schools, and every significant difference favored the reform schools, in every grade and in every subgroup studied. The authors conclude that this study provides strong evidence of the effectiveness of all three curricula. Among the standards based curricula, the University of Chicago’s Everyday Math program may be the most studied. Carroll (2001), conducted a longitudinal study that followed students from first to fifth grade. After four years, 171 students had been tested annually. Comparing the students’ results on items similar to those in the NEAP tests, the author found that the students kept pace with students from China and Japan, unlike typical American students. Interviews with teachers found that teachers generally liked the problem solving and reasoning emphasis of Everyday Math and its hands-on approach. They felt, however, that there was insufficient practice of certain skills, that lower 13 students struggled, and that the role of computation was not sufficiently addressed. The author’s response to these opinions was that these teachers did not understand the Everyday Math program and needed more training. In a 1992-93 study of third graders at 26 schools with 1885 students representing a mix of socioeconomic status, researchers found that the mean test scores of the Everyday Math schools were significantly higher than those of matched schools that had not implemented reform curricula (Carroll and Isaacs, 2003). Reys, Reys and Hope (1993) found that Everyday Math students scored significantly higher than comparison groups on tests of mental computation. Everyday Math students have also been found to score higher in certain geometry skills (Carroll and Isaacs, 2003). In a longitudinal study conducted by Northwestern University researchers, Everyday Math students followed from first to third grade significantly outperformed a comparison group on 55 per cent of number and operation items, 75 per cent of geometry and measurement items, and 71 per cent of data analysis items (Fusan et al., 2000). Riordan and Noyce (2001) reported that statewide standardized test scores of fourth-grade students at one Massachusetts school using Everyday Math exceeded the test scores of demographically similar students using traditional curricula. In sum, there are a significant number of studies reporting favorable results for Everyday Math. However, none of them combines researcher independence with broad, standard measures of achievement. 14 The standards-based Math Trailblazers curriculum grew out of a series of quantitative, hands-on activities developed by physicist Howard Goldberg and colleagues at the University of Illinois-Chicago. Trailblazers developers evaluated the mathematics scores of third graders at eight Chicago area schools that had two years experience with the Trailblazers series. Six schools were Chicago public schools (99 per cent minority and 86 per cent low income) and two were suburban (12 per cent minority and 13 per cent low income). In the second year of implementation, most schools exceeded their own historical average scores for the third grade test, although many only slightly. On the Iowa Test of Basic Skills, given only in the six Chicago public schools in the study, five of six schools showed improvement over their historical averages. The study does not compare these gains to what may have been happening in other Chicago area schools at the time. One study of a district-developed standards-based curriculum compared the scores of students taught by ten teachers who had been specially trained in the new curriculum with the scores of all other teachers of students at the same grade level. The experimental group contained 220 students, versus 454 in the control group. Researchers administered the Iowa Test of Basic Skills to students in both groups both before and after implementation of the new curriculum. They found that the gains among the students learning the new curriculum significantly exceeded those of the control group (Adams, 1999). The results of studies of the Urban Systemic Initiative (Systemic Research, Inc., 2003) have also been cited as support for standards-based 15 curricula. While the study shows that most participating districts have made very modest gains in mathematics scores, the multi-faceted nature of the support for reform efforts makes it difficult to attribute gains to mathematics curricula. Conclusion Despite the fact that a large proportion of American students are now being taught mathematics in accordance with the NCTM Standards, there is still little reliable independent research to support the claims of its proponents. Indeed, despite all the furor, in some ways not much has changed. Almost ten years after the first publication of the Standards, the TIMSS Study painted another bleak picture of American students’ mathematics achievement (Schmidt, et al.,1999). Although the most recent NAEP results may be slightly more positive (National Center for Education Statistics, 2003), it is clear that the NCTM Standards have not been a panacea. The key to progress may lie in the mathematical knowledge of American teachers. Liping Ma (1999) pointed out that American mathematics teachers simply do not know as much mathematics as their counterparts in other countries. Ball, Lubienski and Mewborn (2001) make the case that a teacher who does not have a firm grasp of the mathematical concepts necessary to teach for understanding is unlikely to be effective in any setting. Askey (2001) may be right in pointing out that teaching the NCTM way is more challenging, and requires greater understanding, than traditional ways. That could account for the fact that in studies by curriculum developers, who generally are studying early 16 implementation situations with strong focus and high teacher support, results appear positive, while other, more general studies, are inconclusive. Ferrini-Mundy (2000), a drafter of the revised standards, agrees that when standards-based curricula are introduced into schools districts, the pedagogical features of the Standards, such as group work, projects, student discussion, and emphasis on writing, are more readily adopted than the mathematics content features. It is possible that as curricula based on the first version of the Standards, which may have leaned too far in the direction of abandoning the teaching and practice of algorithms, were introduced, teachers implemented the pedagogical aspects of the curricula without sufficient attention to the difficult work of producing deep understanding of mathematical concepts. Thus, the curricula as introduced in schools appeared to parents and other outsiders as shallow and lacking rigor. This may change as textbooks are revised. The hope is that further studies will continue to illuminate specific classroom practices that produce student mathematics achievement. Then, not only textbooks and support materials, but also teacher recruitment, selection, training, and professional development can converge around the goal of producing successful classroom practices. 17 References Adams, A.A. (1999, November). An evaluation of a district-developed NCTM standards-based elementary school mathematics curriculum. Paper presented at the Annual Meeting of the Mid-South Educational Research Association, Point Clear, AL. (ERIC Document Reproduction Service No. ED435759) American Association for the Advancement of Science (2003). Project 2061’s IERI study examines teacher practice and student learning in mathematics. 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