The Effect of the NCTM Standards On Student Achievement by zoz11082

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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
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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).
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       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
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       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).
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         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
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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
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             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.
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      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
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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)
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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.
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       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
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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.
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       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
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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.
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