ARTICLES & ANNOUNCEMENTS (CALIFORNIA FOCUS)
(1) California Elementary Education Newsletter – Focus Issue: Mathematics
Source: Terry Shorey (tshorey@cde.ca.gov) – 16 November 2001
URL:http://www.cde.ca.gov/elementary/newsletter3.pdf
* The October 2001 issue of the California Elementary Education Network newsletter addresses curriculum and instruction with a specific focus on mathematics.
Articles include the following:
= Characteristics of an Effective Elementary School Mathematics Program
= Summary of Research on Early Childhood Mathematics
= Mathematics Resources and Web Sites
= Resources for Working with Parents and Families on Mathematics
= Best (Mathematics) Practices School: Rio Calaveras Elementary School
* A study guide on Stigler and Hiebert’s book, The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom, is available at http://www.cde.ca.gov/elementary/guide.pdf
* The California Elementary Education Network (CEEN) provides resources to elementary schools and districts, organizational partners and community members to further the dialogue on standards-based achievement and support the recommendations in Elementary Makes the Grade! The CEEN web site is http://www.cde.ca.gov/elementary
(2) “Usefulness of SAT Test is Debated in California” : by Jacques Steinberg
Source:New York Times – 17 November 2001
URL: http://www.nytimes.com/2001/11/17/education/17EXAM.html
The battle for the SAT’s future in California and perhaps other parts of the nation was joined here [Santa Barbara, CA] today when the leading educator opposed to the influential college entrance exam debated its usefulness with the head of the company that oversees it.
The president of the University of California, who has proposed that his college system no longer require applicants to take the main SAT test, and the president of the College Board, which administers it, spoke out in an effort to win the hearts and minds of the system’s faculty members, who will play a crucial role in deciding whether to continue using the test.
The California system includes prestigious campuses at Berkeley and Los Angeles and is the SAT’s largest customer. The respect it gets from the College Board–and the influence it carries with other systems across the country–was evidenced by the size of the delegation the board dispatched to a research conference on testing at the university’s campus here. The College Board president, Gaston Caperton, arrived from his New York headquarters with eight top aides.
Nine months ago, the university president, Richard C. Atkinson, recommended that the California system, which is among the nation’s largest, stop requiring its applicants to take the main SAT exam. He contended that the tests had distracted students from their primary subjects and made it more difficult for many black and Hispanic applicants to get accepted into top colleges.
Since then, just one other selective public or private institution–Hamilton College in upstate New York–has said it would follow suit. But that could change if the regents who govern the California system, which has 130,000 undergraduates, adopt Dr. Atkinson’s proposal.
Before the 26-member Board of Regents can take up the president’s proposal, the plan must be approved by a series of faculty committees, which are expected to act early next year. That is why today’s forum was considered so important. The regents have said that if the president’s proposal reaches their desks, they would take it up no later than next summer. If adopted, it could take effect as soon as the fall of 2003.
In addressing 300 professors, administrators and admissions officers this morning, most of them drawn from the system’s eight undergraduate campuses, Dr. Atkinson argued that the main SAT exam, formally known as the SAT I, failed to assess what was most important to colleges.
“The SAT I sends a confusing message to students, teachers, and schools,” Dr. Atkinson said. “It says that students will be tested on material that is unrelated to what they study in their classes. It says that the grades they achieve can be devalued by a test that is not part of their school curriculum.”
Dr. Atkinson has proposed that the California system rely, temporarily, on another series of College Board exams, the SAT II’s, which are intended to measure student achievement in specific subjects, rather than general aptitude or reasoning in mathematics, vocabulary, reading and other areas. But Dr. Atkinson has proposed that those exams, too, be phased out in favor of a series of new exams that would be designed specifically to test what California students learn in high school.
Mr. Caperton, saying he regarded California as a “bellwether” that was always “a year or two ahead of everyone else,” urged the conferees to stick with both sets of SAT exams. When combined, he said, the SAT I and II provided the most accurate statistical snapshot of how a student thought and had performed in school.
“I tried to be nice when he said all those bad things about the SAT I,” Mr. Caperton, a former governor of West Virginia, said of Dr. Atkinson.
Mr. Caperton argued that the SAT I had changed drastically since its start more than 50 years ago, and he likened its evolution to that of a Chevrolet over the same period. The SAT I, he said, was unmatched in its capacity to evaluate a student’s “ability to think in words and numbers,” which he called a “very critical part of getting an education in college and doing well in college.”
Mr. Caperton delivered his remarks with some confidence, aware that his position enjoys the support of an overwhelming majority of the members of the College Board, a nonprofit consortium of 4,000 colleges and high schools that includes the California system.
Even so, Mr. Caperton drew fire from another front: Richard Ferguson, the president of ACT Inc., which produces a rival admissions exam, lobbied conferees to replace the SAT with his company’s test, which, he said, was already achievement-oriented, and “the equivalent of five SAT II’s.”
California education officials began focusing in earnest on the disparities in the SAT scores of blacks and Hispanics when compared with those of whites after the regents voted in 1995 to bar race as a consideration in admission to the public university system.
While that state policy has made it harder for some minority applicants to gain acceptance to the university system, the elimination of the SAT would probably have the effect of easing their admission.
On Thursday, the regents voted to approve a companion proposal by Dr. Atkinson to broaden the criteria by which applicants to the California system are judged.
In addition to standardized test scores and grade point averages, applicants will now be rated in a dozen other categories, including special talents, leadership skills and accomplishments in the face of “personal challenges,” including economic hardship. The model closely resembles those that dozens of highly selective private colleges have used for more than a generation.
While many faculty members who attended today’s conference said they were keeping an open mind about the SAT, they conceded that they shared Dr. Atkinson’s contentions that the tests had come to dominate students’ preparation for college and had served to draw a rough line between those students who had grown up with a rich intellectual home life, and those who had not.
Richard Watts, a professor of chemistry here who is a member of the committee that will cast the final faculty vote for or against the SAT, said he had attended all the research presentations on subjects including whether the SAT II was a better predictor than the SAT I of a freshman’s grade point average. (Dr. Atkinson’s research shows that it is; Mr. Caperton’s shows that it is not.) Dr. Watts said he was inclined to follow the lead of Dr. Atkinson, who, he said, had “usually been on target, in terms of recommending things the faculty will come to agree on.”
……………
Related article: “Debate Continues: Is SAT a Tool or Trap?” by Michelle Locke
URL: http://www.sfgate.com/cgi-bin/article.cgi?f=/news/archive/2001/11/18/state1244EST0028.DTL
(3) “Implementing Standards: The California Mathematics Textbook Debacle” by Bill Jacob
Source: Phi Delta Kappan – November, 2001
URL:http://www.pdkintl.org/kappan/k0111jac.htm
Standards and accountability are today’s education buzz words. Al Gore and George W. Bush stressed both in their October 2000 articles in the Kappan. But Kappan readers know that what happens in the classroom is more critical and that, as noted repeatedly in these pages, textbooks are a key determinant of classroom practice. This is particularly true in mathematics, where elementary teachers rarely have specialized backgrounds in the subject and where, even at the secondary level, a shortage of certified math teachers is already apparent in some districts.
Across the nation, schools are being advised to align instruction with standards. California’s accountability system links school revenue to performance on standardized tests, and a high school exit exam will be required of all graduates in 2004. So the stakes are high, and the textbooks offered to teachers are critical. In this article, I examine California’s recent “standards-aligned” mathematics textbook adoption process [with background on the development of California’s mathematics content standards and mathematics framework], which provides a lens to scrutinize the impact of high-stakes policies on classroom practice…
What Have We Lost?
Ask any California teacher who has been active in improving quality and leadership in mathematics instruction if he or she will be volunteering to serve on future committees and task forces convened by the state board. “Never again” will probably be the response. Venture to speculate on a possible reason for withdrawing from public service or allude to the “kangaroo court” of past proceedings intended to garner so-called professional input, and you will no doubt hear a resounding “Yes.” The voices of teachers have been lost in California education policy.
The idea of balance is frequently part of the rhetoric of state mathematics policy. Board spokespersons advocate balance—in theory. However, the actual text of key documents, standards, frameworks, and textbook adoption criteria exhibit an unbalanced, rigid emphasis on procedure, skill, and method. So add balance to the casualty list, and make room for flexibility as well. Districts need flexibility to define their own vision of balance in instruction and, above all, to guard against extreme policies.
Focusing on the board-approved list of mathematics textbooks for adoption, a number of programs were approved that go beyond the “drill and practice” definition. One would hope that the materials represent an acceptable middle ground, allowing for students to develop conceptual understanding, problem solving, and basic skills. But this is hardly the case. While they do not take the form of tightly scripted lessons that allow for no variation by a teacher, the materials satisfy the bare necessities of matching content to standards. Topics such as two-digit multiplication and concepts such as the meaning of division by fractions are reduced to what is known in the publishing trade as the “two-page spread.” The materials are literal translations of each standard, providing a tidy capsule lesson in which teachers are encouraged to do what they have been doing for decades: 1) explain a topic, 2) show an example from the book on the chalkboard, and 3) have the students practice similar examples. This conventional model of teaching has not provided students adequate access to algebra and more advanced topics, and it has been repeatedly cited in research studies as a major weakness of mathematics instruction in the U.S.
With the exception of one series for which current users may obtain a waiver, the 2001 adoption list excludes many successful mathematics curricula, including those identified by the U.S. Department of Education as exemplary and promising. Evidence of the effectiveness of these programs has appeared in the research literature. The studies indicate that the students in these programs make progress in skill acquisition that is comparable to that of students in other programs and that they excel in some problem-solving and conceptual areas. Unfortunately, many publishers of innovative mathematics materials declined to submit their books for the 2001 textbook adoption, after having been summarily dismissed in the 1999 emergency adoption. The bias in favor of direct instruction in the textbook criteria and on the part of the reviewers was cause for concern.
California also requires that programs align with standards on a grade-by-grade basis, and several topics were required to be introduced one year earlier than many consider to be acceptable practice. As a result, the approved list of mathematics textbooks that California districts can purchase is a limited one. The programs are designed to rush students through a long list of computational skills and procedural acrobatics, with little room for them to understand what they are doing.
These lost opportunities come at a time when teachers, mathematics educators, and mathematicians are setting a far better example elsewhere. The NCTM recently completed its Principles and Standards for School Mathematics. This volume replaces three earlier NCTM documents and was developed over a three-year period through an open and inclusive process. The team of authors represented all stakeholders, a substantial public review was included, and 15 different professional mathematics organizations provided formal input.
The value of the document is its balance, along with its delineation of the important role of skills that readers of earlier documents may have misinterpreted. It contains examples showing how problem solving can be used to develop conceptual understanding and how those understandings, in turn, can lead to acquisition and retention of basic skills. A broad base of research accompanies and supports the document. The NCTM has shown how to move forward, building upon what works. But this document has, in effect, been blacklisted in the chambers of California policy makers, adversely affecting professional development initiatives and materials adoptions. More than just a loss, what has been created is a vacuum of useful scholarship in mathematics education.
Concluding Remarks
In California, an era of devising standards and choosing curriculum came to a close in 2001 with a mathematics adoption. A new high-stakes testing system had begun in 1998, and policy makers claim that teachers have standards-aligned instructional materials with which to prepare students for the tests. Will the new era bring balanced mathematics instruction that incorporates basic skills, concepts, and problem solving, as the state board called for in 1996? Or will mathematics classes be reduced to computational drill in preparation for standardized tests?
These are not rhetorical questions. Many of the writers who contributed to the standards and framework believe that the “rigor” in the documents will lead to better reasoning and understanding in K-12 mathematics. However well intentioned, they are wrong. They failed to take into account the public’s view that “mathematics is computation,” and they failed to realize that most publishers would respond to a renewed emphasis on skills with programs that focus on drill and practice. Understanding is an afterthought.
During 1999-2001 my professional work gave me the opportunity to meet with numerous K-8 teachers, visit classes, and discuss the impact of the recent policies. The pressures to prepare students for the STAR exam consumed most of our conversations. Teachers shared with me the test-prep materials their students work through prior to the exam. Valuable time needed for conceptual development (in some cases months) is lost entirely to drill, and while some students can complete the practice pages, others are fed procedures too quickly for them to make sense. Further increasing stress levels, parents can now buy STAR prep booklets in local stores. After the spring testing season is over, many teachers report that they and their students are too emotionally exhausted for meaningful work during the rest of the year.
And the situation doesn’t promise to get much better after 2001. The state board approved the program that one of its key advisors described as not giving students “a remotely correct impression of what mathematics is about.” And the board rejected a program about which the review endorsing it said, “It contains extremely interesting and thought-provoking material.” Why? In addition to the reasons I’ve cited here, observers say that, on the day of the vote, the state board responded to pressure from the governor’s office. In other words, politics superseded quality in the choice of California mathematics programs.
Kappan readers know that the Bush Administration has launched its own education agenda calling for yearly testing of students and has proposed a new federal layer of accountability. Proponents argue that the plan will lead to higher achievement, while opponents claim it will promote a dumbed-down curriculum and encourage teaching to the tests. How is the public to decide? The California mathematics textbook adoptions provide an answer. We can expect that students will be drilled on formal skills that are aligned with the test and that high expectations for thinking and reasoning will be gone. In California, this may not have been the intent of policy designers, but it is a consequence of pushing high-stakes accountability on a system that relies on a publishing industry to produce materials that will allay the fear of failure and its consequences. What we see in California is not in the best interest of that state’s students. They need to think and reason in their mathematics classes. We can only hope that other states—and the nation—will do better.
ARTICLES & ANNOUNCEMENTS (NATIONAL FOCUS)
(1) “Curricular Controversy in the Math Wars: A Battle Without Winners” by Robert E. Reys
Source: Phi Delta Kappan – November, 2001
URL: http://www.pdkintl.org/kappan/k0111rey.htm
The continuing controversy regarding standards-based mathematics curricula developed with support from the National Science Foundation has produced a range of reactions from mathematicians, mathematics educators, parents, and other interested parties. The recent clashes in the California “Math Wars” remind us that the battle continues to consume much energy and emotion that could be used for better purposes. Differences of opinion with regard to what is important to learn and how it should be taught are nothing new in mathematics education. And at the heart of the discussion are the written curricula (i.e., the textbooks) that American students use in their mathematics classes. This is the issue I want to reflect on here…
I am involved in the National Science Foundation Show-Me Project (www.showmecenter.missouri.edu) that is dedicated to the dissemination and implementation of standards-based math curricula for middle schools… My international perspective has enabled me to see both the virtues and the dangers of some of our traditional practices. For example, the United States is the only industrialized country that does not use the metric system, and it is one of the few countries that teaches separate courses in algebra and geometry… Clearly, significant curricular change is difficult and slow. Major pedagogical change in mathematics teaching presents even more challenges…
The same factors that inhibit commercial textbook publishers from making significant changes have also made it more difficult to implement the vision portrayed in the National Council of Teachers of Mathematics (NCTM) publication Curriculum and Evaluation Standards for School Mathematics. Recognizing that publishers are reluctant to make the significant changes articulated in the NCTM standards, the National Science Foundation (NSF) supported a number of K-12 mathematics curriculum projects in an effort to produce programs that reflect the vision of the standards.. These curricula have been the subjects of much discussion in recent years…
One major concern that has surfaced in these debates has been whether these standards-based mathematics curricula are being used in schools without adequate testing regarding their impact on student learning. This is a legitimate concern…
What does it mean to be effective? …With mathematics curriculum materials, determining what is effective depends on the evidence one values. Some people place the highest priority on skill development, so any evidence of improved skill is judged positively. Others may value understanding mathematical concepts, while still others may view problem solving as most important. While these goals are not mutually exclusive, obtaining valid and reliable evidence to support them all is very difficult. And as Judith Sowder points out, once this evidence is reported, its reception is greatly influenced by what people value and will accept as evidence…
Critics have advocated “stricter controls to prevent schools from using untested programs without the informed consent of parents and students.” This claim is ironic on at least two counts. First, the traditional mathematics curriculum supported by the critics has not been tested for effectiveness, unless international assessments are used as the measure, in which case these curricula fall far short. Second, there has been unprecedented field-testing of these NSF-supported curricula over the past decade. They have been piloted, revised, field-tested in real classrooms, and revised again prior to their commercial availability. Data continue to be systematically collected, and feedback from the field is reflected in later editions. Research reporting on student achievement in a variety of grades is beginning to emerge. To criticize these curricula because of the philosophy they embody or the mathematical content of the materials is one thing. To suggest that they have not been extensively field-tested with teachers and students is blatantly untrue and irresponsible…
Calls for testing and documenting the impact of mathematics curricula will surely continue, and they should. However, the bar should be set at the same height for all publishers. If the developers of standards-based mathematics curricula are required to document the impact of those materials on student performance, then the same criterion should be applied to all companies producing mathematics textbooks for the same market…
These are exciting times in mathematics education. Despite the difficulties in designing, testing, and marketing new mathematics curricula, the need for significant improvement in student learning requires us to overcome these difficulties. All interested parties should stop trying to defend the past and work together to improve children’s mathematics education for the future.
The current curricular changes in mathematics education articulated in the recently released Principles and Standards for School Mathematics provide an opportunity for rich and substantive dialogue. We can all hope that mathematicians will use this document “as a tool for focused, constructive efforts to improve pre-K-12 mathematics education.” If they do so, we can look forward to valuable discussions about ways to improve the mathematics education of our children. Opportunities to reflect on what we do and don’t know and to explore new options are bound to emerge. And, in the process, mathematicians, mathematics educators, classroom teachers, and school administrators will be able to pool their collective wisdom and energies to develop mathematics programs that will help all students learn relevant and challenging mathematics throughout their school years. This is an important task that requires a united effort if we are to be successful, and nothing short of success is acceptable.
(2) “Standards-Based Mathematics Curriculum Materials: A Phrase in Search of a Definition” by Paul R. Trafton, Barbara J. Reys, and Deanna G. Wasman
Source: Phi Delta Kappan – November 2001
URL: http://www.pdkintl.org/kappan/k0111rey.htm
… In this article we describe six central characteristics of “standards-based” mathematics curriculum materials and discuss how these characteristics are related and how each one sets the stage for and supports the others.
= Standards-Based Materials Are Comprehensive:
A primary concern in all curriculum reform is the inclusion of knowledge, understandings, processes, and skills that constitute competency in a field. Most long-time observers of U.S. mathematics programs have been struck by their narrow focus on skills and procedures…Thus the first characteristic of standards-based materials is a focus on core mathematics for all students. This mathematical core must address the general literacy goals of school mathematics, the mathematics that is considered central to preparing America’s youth for the work force, as well as the mathematics necessary to provide a foundation for continued, advanced study of the subject. Within this framework, mathematics includes such familiar strands as arithmetic, algebra, geometry, statistics, and probability. It also addresses broad themes, such as dimension, quantity, uncertainty, shape, and change…
In standards-based materials, the importance and interconnectedness of understanding and skills are recognized. A balanced approach to knowledge and skills can help students acquire the mathematics they need for solving problems and also establish a foundation for later study of more sophisticated mathematics. Although career paths may differ in terms of mathematical expertise required, all students need a core of mathematical knowledge (content, processes, understandings, and skills) if they are to keep career options open and find satisfying and productive jobs or occupations.
= Standards-Based Materials Are Coherent
Coherence refers to the presentation of mathematics so that the core ideas of the subject are highlighted and cause students to see it as an integrated whole. If students are to think mathematically and use mathematics as a tool for solving problems, coherence is crucial, and establishing connections among the big ideas of mathematics fosters coherence. When carefully established, these big ideas provide anchoring for the important pieces of mathematics — terminology, definitions, notation, and skills — that emerge from them. Standards-based materials promote coherence through an initial focus on big ideas, with an emphasis on connections and links to related mathematical ideas and applications…
= Standards-Based Materials Develop Ideas in Depth
The Third International Mathematics and Science Study (TIMSS) describes textbooks used in the U.S. in recent years as being “a mile wide and an inch deep.” This comment refers to the large number of topics covered at each grade and to the shallow development of each. It is not unusual to see a superficial presentation of important topics, such as symmetry and similarity, repeated at nearly the same shallow level from grade to grade with the primary focus on rules and techniques rather than on mathematics. By contrast, standards-based instructional materials recognize that there are important ideas in mathematics that must be developed at varying levels of depth as students mature mathematically…
There is a natural connection between coherence and developing mathematics in an in-depth way. When the key ideas of the discipline are presented coherently and instruction promotes coherence in learning, then there is a greater inclination for teachers and students to pursue ideas in greater depth and for discussions to focus on fundamental notions.
= Standards-Based Materials Promote Sense-Making
One repeatedly hears adults remark, “Math never made sense to me.” These individuals then identify some point in their schooling when they no longer understood the mathematics they were learning or why they were learning it. Unfortunately, they never made it past the symbols, definitions, rules, and formulas of mathematics. When given the opportunity to make sense of a mathematical idea, they are surprised — : and delighted—: by the experience…
One of the most consistent findings of research over the past 50 years is that understanding increases the ability to learn, remember, and use mathematics. As James Hiebert and his colleagues note, “Understanding is crucial because things learned with understanding can be used flexibly, adapted to new situations, and used to learn new things. Things learned with understanding are the most useful things to know in a changing and unpredictable world.” They also argue that developing a deep, connected understanding of mathematics (i.e., making sense of mathematics) promotes the learning of skills. What’s more, learning with understanding helps students develop confidence in their mathematical abilities.
But what do we mean by “understanding”? Is it simply reproducing a definition or remembering the steps of a procedure? In recent years, cognitive science has provided new insights into the nature of understanding. Rather than view learning as collecting and stacking pieces of knowledge in our heads, it is more productive to view understanding as a web of interconnected ideas…
Students’ ability to make sense of mathematics in the classroom depends on the expertise of the teacher to select good tasks, engage students in thoughtful reflection, and create a classroom environment that supports reflection and communication. Standards-based curricula have an important role in providing guidance to the teacher and posing problems and questions in the text materials that encourage reflection.
= Standards-Based Materials Engage Students:
Standards-based materials engage students physically and intellectually through problems and tasks. By engagement we mean more than hands-on tasks, and the purpose goes far beyond “making math fun.” The tasks are carefully selected to draw students into the study of mathematics by directing and focusing their thinking on important mathematics. The emphasis is on intellectual engagement. Problems and tasks that raise students’ curiosity are posed. When a task is intriguing and poses a challenge, students are more likely to pursue a solution and explanation… In standards-based curricula, the use of contexts, problems, projects, and other tasks to engage students and to connect mathematical ideas provides a platform for learning and allows for the development of a belief that math is not only important but also interesting. Student engagement is further supported by a coherent, in-depth treatment of mathematics in lessons that emphasize making sense.
= Standards-Based Materials Motivate Learning
The sixth characteristic of standards-based instructional materials addresses a long-standing issue in school mathematics: linking mathematics with its applications. While it has been common for traditional materials to include a few “application problems” in exercise sets or even an entire lesson on a particular application, instructional materials have not substantively incorporated applications as part of their core. And most students fail to see the practical relevance or usefulness of the mathematics they learn…
The use of applications to contextualize mathematical study is an important characteristic of standards-based materials. Identifying good applications that reveal the underlying mathematics without overshadowing it is a challenge. However, when designed well, such materials promote coherence and sense-making. They also stimulate student interest and engagement and the development of a healthy, accurate view of mathematics as a useful discipline.
= Implications for Teaching
The development of curricular materials based on the six characteristics presented here is crucial to improving opportunities for students to learn important mathematics. Training new and current teachers has also been identified as a particularly important factor… For teachers to be able to change their role and the nature of their classroom environment, administrators, supervisors, and parents must expect, encourage, support, and reward the kind of teaching described in this set of standards.
Our experience suggests that the changes that teachers must adopt are neither trivial nor quickly attained and will require ongoing support over an extended period. Teachers need the opportunity to work through new instructional materials, to confront issues associated with new teaching strategies, and to increase their own knowledge of mathematical content…
Summary
Reaching the goal of developing mathematical power for all students requires the creation of a curriculum and an environment that are both very different from much of current practice. This means that significant change must occur at many levels if the important and necessary goals of mathematics education reform are to be met.
Student learning has to go beyond the learning of specific concepts and skills. It needs to include attention to the development of a mathematical disposition that causes students to be confident in using mathematics, flexible in exploring mathematical ideas, perseverant in working on mathematical tasks, interested and inventive in doing mathematics, inclined to reflect on their thinking and to value mathematics and its uses and to appreciate its role in our culture.
Instructional materials have a particularly important role in making these changes happen, for they affect the mathematics students encounter and how they encounter it, the processes students use, the way teachers teach, and what is assessed. They are also important because of their central place in American education. As Deborah Ball and David Cohen have noted, “Unlike frameworks, objectives, assessments, and other mechanisms that seek to guide curriculum, instructional materials are concrete and daily. They are the stuff of lessons and units, of what teachers and students do.”
We have discussed six central characteristics of instructional programs that represent efforts to capture the vision and essential elements of the NCTM Standards documents, and we have shown ways in which they collectively support greater student learning in mathematics. We believe that unless teachers and students have access to such materials, we will fail to achieve our common goal of high levels of mathematical literacy for all students.