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2 The State of School Mathematics in the United States
Pages 31-70

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From page 31...
... The status of teachers in the society, the composition and mobility of the student population, and the extent to which external examinations determine one's life chances all constrain the ways in which mathematics is taught and learned. Countries also differ in more specific ways: parents, teachers, and students have different beliefs about the value of hard work and the importance of mathematics for one's education; whether and how students are grouped for mathematics instruction varies; mathematics textbooks are written, distributed, and used in diverse ways; and there is variation in the prevalence of tutors or special schools to coach
From page 32...
... Unique features of TIMSS included an extensive examination of textbooks and curriculum guides from many of the participating countries, a video study of eip;hth-~;rade mathematics classes in three countries, and case studies of educational policies in those three countries.
From page 33...
... In this chapter, we first take up in turn four central elements of school mathematics learning goals, instructional programs and materials, assessment, and teaching discussing the current status of each in the United States. We then examine the preparation and professional development of U.S.
From page 34...
... mathematics education since 1989 has been based on or informed by the ideas in those documents. Many school mathematics textbooks claim to be aligned with the NCTM standards, and 13 curriculum projects were funded by the National Science Foundation to produce materials for elementary, middle, or high school that embodied the ideas expressed in the standards documents.8 The NCTM standards of 1989 launched the so-called standards movement, with standards for other school subjects appearing over the following decade.9 In 1994 the reauthorization of Title I of the Elementary and Secondary Education Act furthered boosted the movement.
From page 35...
... A recent comparative analysis of mathematics assessments given to U.S. and lapanese eighth graders revealed some striking differences in the expectations held for each group, with much lower expectations in the United States.
From page 36...
... . Responding to a questionnaire in 1996, teachers of three fifths of the fourth graders and of almost three fourths of the eighth graders in the
From page 37...
... elementary and middle school mathematics curriculum has been characterized as superficial, "underachieving," and diffuse in content coverage.25 Successful countries tend to select a few critical topics for each grade and then devote enough time to developing each topic for students to master it. Rather than returning to the same topics the following year, they select new, more advanced topics and develop those in depth.
From page 38...
... Further, decorative artwork with little connection to textbook content sometimes confuses or distracts students.28 Research indicates that students can learn more mathematics than is usually offered them in the early grades, so the U.S. elementary school mathematics curriculum could be made more challenging.
From page 39...
... students can learn.34 Such a development program can be interactive, with improved learning materials yielding improved student learning that, in turn, yields improved and evenmore-ambitious learning materials. Developing teachers' capacity to acquire and use good instructional materials is also a problem.
From page 40...
... Even less attention appears to have been paid to how teachers' assessments might help improve mathematics learning. According to one analysis, "Aside from teachermade classroom tests, the integration of assessment and learning as an interacting system has been too little explored."36 As part of the 1996 NAEP mathematics assessment, teachers responded to several questions about their testing practices.37 Fourth graders were usually tested in mathematics once or twice a month, with about a third being tested once or twice a week.
From page 41...
... Currently, 19 states require that in order to graduate from high school, students must pass a mandated assessment, and several other states are phasing in such a requirement.45 In TIMSS, countries with rigorous assessments at the end of secondary education outperformed other countries at a comparable level of economic development; such assessments, however, were probably not the most important determinant of achievement levels.46 In response to calls for an end to social promotion, some states and districts have begun requiring grade-level mastery tests for promotion, typically in grades 4 and 8. Interestingly, there is some evidence to suggest that there is an almost inverse relationship between statewide testing policies and students' mathematics achievement: Among the 12 highest-scoring states in 8th grade mathematics in 1996, .
From page 42...
... Commercially published standardized mathematics achievement tests are quite variable in the topics they cover and in the proportion of these topics emphasized at each grade level.53 The tests frequently are not aligned with the teaching materials used in a district or even with the goals of the district. This misalignment further dilutes teaching efforts, as teachers must add to their long list of goals coverage of the major topics emphasized on a specific standardized test.
From page 43...
... Many states give competency tests at several grade levels, including high school exit exams. Performance on the mathematics portions of such tests has often been conlf the purpose of a test is to assess whether students have met .
From page 44...
... Many districts meanwhile have continued to use standardized comparison tests that were not necessarily aligned with their textbooks, their state goals, or their state competency tests. The combination of standardized comparison tests and state competency tests can overwhelm teachers, who have to prepare students for two kinds of highstakes tests about which they often know very little.
From page 45...
... Teachers of 93% of the fourth graders and 88% of the eighth graders in the 1996 NAEP mathematics assessment reported that they gave the topic "a lot" of instructional emphasis.59 At grade 8, algebra also received a lot of emphasis (for 57% of the students) , but that was the only other curriculum strand to receive much attention.
From page 46...
... Few students at either grade wrote reports or worked on projects more than once a week, and teachers of about two thirds said their students hardly ever did project work. For nearly half of the eighth graders and more than a third of the fourth graders, their teachers reported that almost every day they had students discuss solutions with one another, and teachers of almost all students held such discussions at least once a month.
From page 47...
... For example, in a recent survey of 85 elementary school teachers in two districts, 93% said that they were using cooperative learning, a practice in which students are grouped for instruction, are assigned roles in the group, work together on a task, are each assessed on their performance, are each held accountable for contributing to the work, and, in some versions, are taught skills for working together, promote each other's contributions, and work collectively to improve their effectiveness.63 Interviews with 21 of the teachers who had indicated they were using cooperative learning (17 of whom said they used it for mathematics) revealed that all but one had their own version of the practice, which they distinguished from the "more formal" version.
From page 48...
... Although there are some differences in the way different subjects are taught,67 the description of recitation teaching is consistent with more recent descriptions of mathematics lessons. In the mid-1970s, the National Science Foundation funded a set of studies on classroom practice, including a national survey of teaching practices68 and a series of case studies.69 After observing a number of mathematics classrooms, one researcher said: In all math classes I visited, the sequence of activities was the same.
From page 49...
... In fact, the TIMSS Video Study clearly shows that such differences are quite small compared with the substantial differences that exist between countries. Each country appears to have its own dominant style of mathematics teaching.74 In the videotaped lessons from the United States, a typical lesson begins by checking homework or engaging in a warm-up activity.
From page 50...
... lessons was independently rated as being much lower than that of the German and lapanese lessons.76 The descriptions from the TIMSS Video Study match other reports of classroom practice in mathematics. For example, a 1998 report to the California State Board of Education summarizes the conventional method of mathematics teaching in the United States, often used as the control treatment in experimental studies of new teaching approaches.77 The summary divides the conventional method into two phases.
From page 51...
... Some recent programs have attempted to bring content and pedagogy together in both teacher preparation and professional development by considering the actual mathematical work of teaching.82 Although states have long set such requirements for teachers seeking certification, some have recently begun to impose higher standards for the knowledge teachers should have to teach children at a given age or grade level, requiring teachers to take specified courses and to pass assessments of their subject matter knowledge.83 There is considerable variation across states .
From page 52...
... , impose their own standards rather than specifying courses, require a minimum number of credits in one unspecified field, or require the completion of an approved teacher education program. Thirty-seven states grant middle school certification, and the requirements fall into categories similar to those for elementary school.
From page 53...
... In the 1996 NAEP mathematics assessment, teachers of 81% of the eighth graders in the sample reported that they were certified in mathematics, and the corresponding figure for fourth graders was 32%. Those numbers were not significantly different from what teachers had reported in 1992.89 In contrast, the Council of Chief State School Officers reported in 1998 that 72% of all mathematics teachers at grades 7 and 8 in the 26 states providing data were reported as certified, 22% as not certified, and the remainder as having elementary school certification.
From page 54...
... North Carolina requires 15 credits every five years, and in Oregon, teachers must earn 24 quarter hours in their first three years of teaching.92 In an effort to encourage teachers to extend their professional development efforts, 30 states have adopted incentives for teachers certified by the National Board for Professional Teaching Standards, such as portability of certification, certification renewal, fee supports, and pay supplements.93 Standards for National Board certification are available in mathematics for teachers of students ages 11 to 15. Certification at the elementary school level is general.
From page 55...
... The main assessments reflect more contemporary educational objectives and are used to collect both national and state data, including contextual data such as teaching practices, some of which are reported earlier in this chapter.97 Except when we refer explicitly to the long-term trend assessments, the data reported here are from the main assessments. In the 1996 mathematics assessment the most recent main assessment to be thoroughly analyzed across grades 4, 8, and 12, roughly 35% of the students were below the basic level of achievement and another 45% or so were at that level, which is defined as denoting "partial mastery of knowledge and skills that are fundamental for proficient work." In the same assessment, 21% of fourth graders and 24% of eighth graders were at or above the "proficient" level, where proficiency is defined as students having "demonstrated competency over challenging subject matter" and being "well prepared for the next level of schooling." Only 2% and 4% of fourth-grade and eighth-grade students, respectively, were doing advanced work significantly "beyond proficient grade-level mastery."98 Although overall levels of achievement are low, the main NAEP assessments in the 1990s revealed significant gains.99 The gains between 1990 and 1996 have been estimated to be about one grade level.~°° According to the NAEP long-term trend, mathematics achievement improved between 1973 and 1996 at both the fourth-grade and eighth-grade levels.~°~ Performance improved even more sharply from 1973 to 1996 among black and Hispanic students.~02 Although the gap between black students and white students had narrowed through the 1980s, it widened between 1990 and 1999, especially among students of the best-educated parents.~03 This disparity repre
From page 56...
... Furthermore, the eighth graders in 1999, who compared quite well internationally in 1995 as fourth graders, were very much like the 1995 eighth graders, performing near the international average.~05 One way to quantify U.S. students' performance is in terms of the average number of points they scored on the 1995 TIMSS assessment.
From page 57...
... Performance was below the international average on geometry, measurement, and proportionality.~°8 For example, U.S. eighth graders had much weaker abilities, overall, than their counterparts in other countries to conceptualize measurement relationships, perform geometric transformations, and engage in other complex mathematical tasks.
From page 58...
... We urge, therefore, all who are attempting to improve mathematics learning in grades pre-K to 8 to reflect on the observations made in this report and to consider how they might connect and coordinate their efforts with those of others. In subsequent chapters we set forth important research, theory, and organizing principles intended to ground future efforts in fact and principled argument, to make assumptions more explicit, and to bring greater coherence to the system.
From page 59...
... 19. Council of Chief State School Officers, 1998.
From page 60...
... 42. Council of Chief State School Officers, 1998.
From page 61...
... 60. Council of Chief State School Officers, 2000, p.
From page 62...
... 92. Council of Chief State School Officers, 1998, p.
From page 63...
... (1996~. Mathematics achievement in the middle school years: IEA's Third International Mathematics and Science Study (TIMSSJ.
From page 64...
... , Research in mathematics education (pp.388-432~. Reston, VA: National Council of Teachers of Mathematics.
From page 65...
... (1995~. The National Board for Professional Teaching Standards: Making professional development "professional." In I
From page 66...
... (1997~. Mathematics achievement in the primary school years: IEA's Third International Mathematics and Science Study (TIMSSJ.
From page 67...
... U.S. mathematics teachers respond to the Third InternationalMathematics and Science Study: Grade ~ results [On-line]
From page 68...
... In Mathematical Sciences Education Board, Measuring what counts: A conceptual guide for mathematics assessment ~ p p.
From page 69...
... , Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp.
From page 70...
... (1999~. Improving mathematics education using results from NAEP and TIMSS.


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