WHEN THE NATIONAL COUNCIL of Teachers of Mathematics (NCTM) published Curriculum Focal Points for Pre kindergarten through Grade 8 Mathematics: A Quest for Coherence last September, the document targeted what many feel is a need for fewer, more focused math objectives for school districts through out the United States.
Written by a committee of nine educators, including K8 teachers, the document specifies the three most significant math concepts students should learn in each grade level, given the 30 to 90 math objectives per grade that 49 of 50 states now have. The 2006 document used inputs from various sources, including mathematicians, and complements the council's widely accepted 2000 report, Principles and Standards for School Mathematics.
The federal No Child Left Behind act requires testing students in grades 3- 8 in math, and states devised their own assessment and accountability systems to demonstrate Adequate Yearly Progress in each school. But juggling scores of objectives in most states meant that teachers were stretched too thin to meet learning expectations equally well for all students. Math in the United States was therefore criticized for being "a mile wide and an inch deep," as lessons jumped from topic to topic and students failed to master the most important concepts.
American students also compare unfavorably in math with counterparts in other countries, as shown by the Trends in International Mathematics and Science Study (TIMSS), formerly called the Third International Mathematics and Science Study. In the mid-1990s, experts analyzed test results of students from countries participating in TIMSS and found that American high school seniors scored among the lowest of the participating nations. On top of this, more college students need remediation and American businesses have complained about more young workers with limited math skills. It was time for a change.
Preparing the Focal Points
In preparing the document, the authors reviewed various state curriculum requirements, met with state supervisors of math, analyzed state standards, and analyzed programs from several high performing countries on TIMSS, including Singapore, China, Japan and Korea. The Focal Points, for example, recommend that re- kindergarteners develop an understanding of whole numbers and how to compare them, that second-graders learn how to count in units and multiples of hundreds and tens, and that fourth-graders develop an understanding of multiplication, including "quick recall" of multiplication and division facts. For each grade level, the Focal Points give descriptions of concepts that the authors regard as essential, broken down by topics such as numbers and operations, basic algebra, measurement and geometry. In geometry, for example, kindergarteners should learn to identify and name shapes, such as squares and three-dimensional objects, while fifth-graders should begin to understand how to use shapes to quantify volume and to estimate.
The NCTM Focal Points are intended to help students become more focused and master key objectives so they can move seamlessly from grade to grade, especially given the high mobility of students and teachers in the United States. "The need for covering fewer topics and doing more than scratching the surface makes sense," says NCTM President Francis "Skip" Fennell. The Focal Points also "provide a vision of what should be in the foreground and what can be in the background," adds Jane Schielack, co-author of the Focal Points and professor of math and teaching, learning and culture at Texas A&M University. "It helps teachers design meaningful instruction, and students develop a deep understanding that strengthens their knowledge. It's not that districts can't do that without the Focal Points, but it helps districts that don't have the expertise and skills available."
The National Mathematics Advisory Panel
President Bush created the National Mathematics Advisory Panel in April 2006 to advise the president and the U.S. secretary of education on the best use of scientifically based research to advance the teaching and learning of math. Modeled after the National Reading Panel, the panel will examine and summarize scientific evidence related to teaching and learning math, and address topics that include:
- The critical skills and skill progressions needed to learn algebra and prepare for more advanced courses
- The role and design of standards and assessment in promoting student competence in math
- The processes by which students of various abilities and levels learn mathematics
- How training, selection, placement and professional development of teachers affect achievement of students
- Instructional practices, programs and materials that have proven effective in improving math learning
- Research needs in support of math education
Building on the Focal Points
Given the tug-of-war played out in American schools (see "Math Curriculum Eras" on p. 40), the Focal Points are considered by some to help bring back more "actual math," though Fennell and Schielack agree that they don't dictate any type of curriculum or math movement. "The Curriculum Focal Points are about important mathematics for pre-K8," Fennell says.
The Focal Points can, however, be an outline for curriculum design and instruction. "They convey knowledge and skills that are essential to educated citizens, and they provide the foundations for further mathematical learning," the document states. A curriculum built on the Focal Points also has the potential to ease learning difficulties for students who struggle with important mathematics content and to help them develop key problem solving, reasoning, and critical thinking skills. The Focal Points assume that learning math is cumulative, with work in later grades deepening what students have learned in earlier grades, without repetitious teaching. Such a comprehensive math document can prepare students for any career or professional path that they choose, the NCTM states. "The Focal Points are designed to actually provoke discussion among states and districts as they revise their own standards," Fennell says.
Stephen Weimar, director of The Math Forum, a comprehensive online resource center that supports mathematics education, says classroom practice in K8 would be different if Focal Points were used in lieu of the 2000 math standards, in that the goals of understanding and fluency in specific areas would guide instructional decisions. "Other topics can be used to support those purposes and be limited to exposure rather than mastery," Weimar says. "Students should feel that they are developing competencies and applying concepts rather than surveying topics or revisiting the same material."
Some states have already welcomed the Focal Points and are implementing them in their standards. The Florida Department of Education, for instance, is using the Focal Points as a guideline as officials write revised K8 standards, according to Cathy Schroeder, press secretary of the department. But the Focal Points are not specific enough to guide daily instruction, Schroeder says, so state officials see the Focal Points more as overarching "big ideas" for specific grade levels. Florida's new math standards are expected to be approved by the state Board of Education in June and will be implemented subsequent to professional development. The Florida Comprehensive Assessment Test (FCAT), used for NCLB accountability, will also be revised as necessary to reflect the new standards.
In Utah, the state Office of Education is revising its math standards to include the Focal Points, says elementary mathematics specialist Nicole Paulson. The Board of Education is expected to review and approve the K6 standards in June and the secondary grade level standards in August. Pending approval, the new standards will go into effect for the upcoming school year, and all 40 districts will need to comply. Utah's yearly summer professional development program for public school educators will include an instructional materials component for staff affected by the new standards. The Utah Performance Assessment tests, used for NCLB, will also be restructured to include the new standards.
In Texas, the Texas Education Agency revised the curriculum two years ago and has no plans to implement the Focal Points. George Rislov, managing director of curriculum at TEA, says the agency is "statutorily restrained from adopting a methodology" but adds that districts have authority over their own curricula. "I would say our main focus in math right now is the issue of college readiness," Rislov says. "We're trying to look at college algebra and work down from there."
For Connecticut, like Texas, local districts determine their own curricula and have control over what they teach. But the Department of Education, which is merely a guide and can provide resources, has been working with districts to "introduce and embrace" the Focal Points with some success, according to spokesman Tom Murphy. "The Focal Points are a way to introduce standards and apply them to the curriculum," Murphy says. "What they [at NCTM] say has a lot of standing, and their standards have been infused in the local curricula, so we're pretty consistent with what they are producing."
In New Jersey, the state's next update in math standards won't be until 2010. But Richard Vespucci, state math coordinator, says, "We tend to see NCTM as an authority, and we come into line with some of the practices and policies they advocate."
Similarly in California, Yvonne Evans, education program consultant, says a framework is already in place and districts need to follow those content standards. "Our standards are massive," Evans admits, and "a lot of our standards are embedded in those Focal Points." Barry Fox, assistant principal at Florence Nightingale Middle School in Los Angeles, sees the Focal Points as more of a guide for elementary teachers to recognize core concepts. California is slated to adopt new math textbooks for the upcoming school year, and the only change Fox expects is the addition of algebra readiness for eighth and ninth grades, based on the California High School Exit Exam.
In New York, Juanita Maltese, math, business and technology chairperson for the Carle Place Unified Free School District, says that a math Regents problem spurred an investigation that resulted in new pre-K12 standards in New York. The passing rate for the state's students on the 2003 math A Regents exam was 31 percent, compared to 61 percent in 2002, she says. Maltese adds that in 2005 the new core curricula for New York were issued, making sweeping changes in the elementary curriculum that in part mirrors NCTM's Focal Points. "I personally believe that once all students have been taught this curriculum from kindergarten through grade 8, then all students willbe prepared to achieve in a rigorous high school program," Maltese says. But it will take seven years to find out because that's when today's elementary students will be in high school, she says.
Since the Focal Points were published, New York State Education Department officials have discussed with Fennell and key stakeholders the next steps in building leadership in math at the state, regional and local levels, says Anne Schiano, assistant director at the Office of Curriculum, Instruction and Instructional Technology. Using the Focal Points to assist in future revisions of math curriculum, professional development and focusing classroom instruction are key priorities of state officials, she adds.
On the other hand, Jana Sebestik, curriculum specialist in the Office for Math, Science and Technology Education for the University of Illinois does not see any standards alignment changes in Illinois. "When you walk into a fifth-grade classroom," she says, "kids are still doing multiplication, division and fractions, and I don't see that changing."
Sample Focal Points
Grade 4 students must work to:
- Develop quick recall of multiplication facts and related division facts and have fluency with whole number multiplication.
- Come to understand decimals, including the connections between fractions and decimals.
- Understand area and determine the areas of two-dimensional shapes. The connections to the Focal Points include lessons in algebra, geometry, measurement, number and operations, and data analysis, such as continuing to use tools from third grade and solving programs by making frequency tables, bar graphs and picture graphs.
Grade 6 students must work to:
- Understand and be fluent with multiplication and division of fractions and decimals.
- Connect ratio and rate to multiplication and division.
- Write, interpret, and use mathematical expression and equations. The connections to the Focal Points include lessons in number and operations, algebra, and measurement and geometry. Problems ask students to find areas or volumes from lengths, or to find lengths from volumes or areas. Such problems extend the students' work in fifth grade on area and volume and provide a context for applying new work with equations.
Grade 8 students must work to:
- Analyze and represent linear functions and solve linear equations and systems of linear equations.
- Analyze two- and three-dimensional space and figures by using distance and angle.
- Analyze and summarize data sets. The connections to the Focal Points include lessons in algebra, geometry, data analysis, and number and operations. Students encounter some nonlinear functions they studied in seventh grade, such as inverse proportion, whose rates of change contrast with the constant rate of change of linear functions.
Math Textbook Dilemmas
Textbook publishers are not saying much when it comes to the new Focal Points. Some, like Holt, Rinehart and Winston, declined comment, while others responded cautiously.
"To comment on the NCTM Focal Points at this time gives away too much of our editorial strategy for the upcoming adoptions," says Collin Earnst, vice president of corporate communications for Houghton Mifflin, but he adds that "some publishers may incorporate those points more than others."
Cathie Dillender, marketing director for mathematics and science at Pearson Scott Foresman, says that beginning in 2008, new textbooks will be aligned to the Focal Points. The textbooks' content is now aligned with state standards "to assure success on state and national tests," she adds.
Mary Jacobsen, director of mathematics marketing for the Wright GroupMcGraw-Hill, says that Everyday Mathematics, a K6 text released as a third edition in 2006, reflects NCTM's Principles and Standards for School Mathematics and is in line with the Focal Points.
Doug Clements, co-author of SRA/ McGraw-Hill's Real Math textbook, says SRA will publish the national edition this year, with Texas and California editions in 2009. In addition, Real Math has been listed as an approved program in Georgia and New Mexico. Real Math, which uses a hierarchical structure of lesson objectives that build on each other, was launched in 2005, but SRA maintains that it meets the NCTM guidelines. A recent press release from Real Math states that each lesson for each chapter addresses specific concepts "that build understanding of the chapter focal points or big ideas."
Most educators agree that math education in the United States really won't change until the training and preparation of teachers improve. "There is increasing attention to the need for elementary and middle grades teachers to have strong content knowledge for teaching, which includes understanding the relevant and related mathematics, effective instructional practices, and how students learn," says Weimar. "The Focal Points could prompt more focus in mathematics teacher education and help new teachers achieve this depth of understanding in the core topics that they will teach."
Professor Irina Lyublinskaya, who teaches at City University of New York (CUNY), College of Staten Island, agrees that the Focal Points represent a "beginning" to a better math program. But it will take time to implement them, as the Focal Points document only teaches what to teach, not how to teach. "There are so many different approaches on how to bring kids to mastery," says Lyublinskaya, who believes students should memorize multiplication tables and facts but also understand how the numbers work.
Lyublinskaya and Stanford University math professor R. James Milgram are among some who claim that elementary school teachers and preservice teachers fear math: They don't know it well, and they don't know how to teach it well, and that impedes progress in the classroom. "To really change the way kids are taught math, you not only have to change curriculum, but you need people who understand it and implement it, and that change comes from teacher education programs," says Lyublinskaya, who also holds a doctorate degree in theoretical and mathematical physics.
"The whole teaching profession has to be brought up to a higher level," she says, adding that someone with advanced mathematics skills would likely veer from teaching in U.S. elementary and secondary schools, given low salaries and prestige as compared to private business. "If you want to attract really good teachers, you have to pay them well," Lyublinska adds.
Preparing Math-Savvy Teachers
The National Council for Accreditation of Teacher Education, or NCATE, is on the verge of embracing elementary standards that incorporate the Focal Points to evaluate elementary teaching programs. "They will become mandatory within 18 months and voluntary immediately," says NCATE President Arthur Wise. While NCATE can't tell colleges how teachers in training should deliver lessons, Wise says, "we expect candidates to emerge with an understanding of Focal Points and how to use them."
Two years ago, CUNY started a major initiative with New York University, supported by a private foundation, to attract high performing high school students in math and science into high school teaching, Lyublinskaya says. The program provides tuition waivers, benefits, and summer work programs and internships in schools. So far, 15 students have applied for the program's 60 spots, but Lyublinskaya hopes it will grow in coming years.
In January, in a program of The Discovery Institute at the College of Staten Island, college professors started visiting K5 schools on Staten Island to help make connections between college work and elementary math lessons, says Lyublinskaya, who is also co-director of The Discovery Institute. "They observe a real classroom and discuss with the teachers about redeveloping the curriculum to make it more focused on what students need," in part based on the Focal Points, Lyublinskaya says.
Math Wars Continue
As the nation still struggles with what some call "actual math" vs. more abstract math programs, experts vary in their opinions of the Focal Points. Some say they finally shift math teaching and learning back in the right direction, with a focus, while others say they wrongly cave in to the Mathematically Correct agenda.shut up ugly face
Milgram says his crusade for so-called "actual math," which is taught in high achieving countries, came after a two-year stint in the mid-1990s at the University of Mexico when some of his smartests students lacked adequate math skills that they should have learned years before. He returned to Stanford and realized California's
standards at the time supported the reform math objectives, which are "not math," he claims. "Math is useful; it's usable," says Milgram, who played a fundamental role in the math portions of California's 1998 framework. "What these people taught was not usable. Students who are just taught this kind of material have virtually no chance of ever becoming engineers, scientists, economists or members of any other profession that uses math in an essential way."
In contrast, some say that math is more-it's about conceptual understanding, problem solving, applications and mathematical mind habits. Sol Garfunkel, executive director of the Consortium for Mathematics and Its Applications, says that reform or abstract math is most helpful for a diverse group of students in the United States. "This math is showing kids why they have to know certain objectives and why they should want to know them," Garfunkel says. "We can no longer afford to have only a small number of math-knowledgeable people."
Garfunkel, who testified before the National Math Advisory Panel last fall, also says that it's "incumbent upon us to teach math to the largest number of people, and that means different approaches early on and different from what we did in the past." He adds, "They may call it fuzzy, but we're trying to reach more kids."
The Focal Points limit what students need to learn, Garfunkel says, although NCTM is clear in stating that the three objectives for each grade should not limit what teachers teach in any given grade. "People will teach to the Focal Points and test to that," Garfunkel says, "and it will actually wind up limiting the curriculum as opposed to expanding it."
Sherry Fraser, a faculty member of the Marilyn Burns Education Associates, also testified before the national math panel, explaining that math is more than rules and procedures. Under the newest California framework, eighth-graders are required to take algebra I, even though Singapore and other high-performing countries teach an integrated curriculum in eighth grade and throughout high school. Fraser pointed out to the panel that California's standards have not improved math education. In 2006, only 23 percent of the state's students were proficient in algebra I by the end of high school, a gain of two points over four years. About 45 percent of the state's students take algebra II, and only 25 percent are proficient, a loss of four percentage points over the last four years. "Direct instruction of basic skills does not suffice," Fraser told the panel. "Moving backwards to ineffective habits does not make sense. Our children deserve more."
But Milgram says that students should be learning more, including algorithms and how to construct them. The Focal Points require that students not only learn standard algorithms of arithmetic, but why they work, Milgram says. "The best examples of algorithms are standard algorithms of multiplication, addition, subtraction," Milgram says. "If students can understand these algorithms and why they work, they can understand the mathematics that underlies a computer, so they can do a great deal more with these wonderful machines than just run Excel programs. Moreover, they will have the key foundational knowledge so that later they can look at a motor or engine and can see how it fits together and how it works."
While Milgram concedes that not every student will be an engineer or mathematician, he says studies show a correlation between a successful college education and high school math. The Focal Points are unveiling the absolute key points that students must learn, Milgram adds. But, he concedes, "it does not end the math wars."
IN 1995, IN A BACKLASH TO THE reform math approach in schools, parents in southern California created the Mathematically Correct organization. These parents were dissatisfied with what they term on the group's Web site "newnew math," particularly College Preparatory Mathematics, an integrated secondary school mathematics program.
Mathematically Correct has become the most influential organization challenging the NCTM agenda. "The advocates of the new, fuzzy math have practiced their rhetoric well," the Web site states. "They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas."
The California Board of Education agreed with critics of the NCTM standards and scheduled a rewrite of the framework, as there was public pressure, particularly from the Mathematically Correct group, to improve the teaching of reading and math in schools. Meanwhile, most other state governments have adopted math standards that are closely aligned with NCTM's standards. And the NSF created various programs to encourage use of those standards, including Everyday Mathematics for K6, Math Trailblazers for K5, and Connected Mathematics for grades 6-8.
Math Product Implications
Product developers will have to decide whether or not to adopt the NCTM Focal Points. Here are initial reports from the field.
- Apex Learning www.apexlearning.com
Apex Learning offers Web-based courses containing all the instructional activities, content and assessment for mastery of math, says David Dwyer, chief academic officer. "Our courses are NCTM and state standards-based, meaning they are composed of original curriculum created specifically to cover the math standards," he says. "As the standards change, an enormous advantage we have is that we can easily modify our content to keep pace with changing standards." Initial analysis shows that the Math Fundamentals curriculum is well aligned to the Focal Points and no immediate changes are required, Dwyer adds. With its new pre calculus course, Apex will analyze the Focal Points and begin adjustments to prepare for the course's release in the fall of 2008.
- CompassLearning www.compasslearning.com
Ann Henson, vice president of curriculum and instruction for CompassLearning, says the Focal Points are a point of reference to be used "to prioritize where we will add more depth to existing concepts." The product is designed to provide a strong foundation of critical math skills to ensure success by focusing on key skills but in a context that supports a deep understanding of math concepts, she says. "There are skills for which the student must develop fluency (addition, subtraction, multiplication and division facts, and rounding), but this has to be done in a manner which enables the student to make connections with the concept, such as place value, and the actual algorithms that go along with math operations." CompassLearning will also add content for grades 7 and 8, with more items to use for formative assessments.
- HELP Math www.helpprogram.net
Barbara Freeman, chief operating office for Digital Directions International, says that when the Focal Points were published, they seemed to be modeled after the HELP Math program. The company has added even more modeling and visualization to give students a richer visual experience, she says. HELP Math, which was developed by Digital Directions International and is distributed by Discovery Education, is a multimedia, supplementary Web-based program designed to remove language barriers from the learning of math skills and content. HELP Math targets English Language Learners with a focus on Spanish-speaking middle school students.
- Hotmath www.hotmath.com
Hotmath.com supports teachers, parents and students using major math textooks. It explains the odd-numbered homework problems for students. Chuck Grant, president of Hotmath.com, says that the Web site explains the problems from textbooks in the style of each textbook. "We await the new textbooks and publishers' actions in creating new homework problems," he says.
According to Barclay Burns, founder of Learning.com, the Focal Points created the curriculum map for their new product Aha!Math, which is aligned to the Focal Points and state math standards. Teachers can search by Focal Point or state standard. Aha!Math uses a community of learners model, simulating the classroom environment where students learn the language of math and can use multilayered games and simulations with feedback. Burns believes the Focal Points create a balance between conceptual math understanding and conceptual math fluency. Animations make it possible to use conceptual morphing to show computation. Learning.com has developed a fluid process of listening and developing and can update and respond to customer needs.
- McGraw-Hill's SRA RealMath www.sraonline.com
McGraw-Hill's SRA Real Math co-author Doug Clements says that because Real Math is a research-based program, and the Focal Points are guided by that research, "consistency was expected." Real Math is the online, interactive counterpart for the SRA texts.
- Qwizdom www.qwizdom.com
Qwizdom is a Focal Points-based product set for a fall release that will offer formative assessments and a supplemental math curriculum. "The Focal Points will map Qwizdom's curriculum," says Patrick Leonard, vice president of curriculum and business development. "The Focal Points make it easy for teachers who are new or not experts at teaching math." Qwizdom also offers the curriculum- standards-aligned and gradespecific program ReadySet Math, which is already based on the Focal Points. Leonard says that online products can do what texts can't, and that is to quickly customize for every state, district, and student need. In many cases, digital changes can be made in an hour. -Ken Royal
Math Curriculum Eras
The math curriculum has evolved through multiple influences.
MATH EDUCATION IN AMERICAN SCHOOLS HAS SEEN AN EBB and flow over the past century, shifting from what some refer to as the "actual math" of basic skills and rote memorization, to math that is more abstract and emphasizes logical explanations. And the socalled math wars continue.
- In 1915, William Heard Kilpatrick, one of the nation's most influential education leaders, recommended in his report The Problem of Mathematics in Secondary Education that traditional high school math be only for a select few. But mathematicians objected, as documented in "A Brief History of American K-12 Mathematics in Education in the 20th Century" by David Klein, and NCTM was created in 1920. NCTM's 1923 report justified the study of math for its intrinsic value and proposed new math curricula for schools. But the Kilpatrick report held greater influence, Klein states, and math education during the 1930s infused math in other subjects.
- In the 1940s, the "Life Adjustment Movement" emerged, and programs focused on practical consumer math problems rather than algebra, geometry or trigonometry. But such programs were criticized for lack of academic content, and by the late 1940s developments in radar and atomic energy underscored the importance of math.
- In 1957, the launching of the Russian satellite Sputnik highlighted the low quality of math and science in American schools and the "New Math" era was born, during which mathematicians introduced curricula that focused on abstractions. A year after Sputnik, Congress passed the National Defense Education Act to advance education in math and science, and the American Mathematical Society set up the School Mathematics Study Group to develop new K12 curricula, including calculus in high school. In the early-1970s these structured programs gave way to the "Open Education" era that emphasized self-directed student investigation and experimentation.
- In 1980, NCTM published An Agenda for Action, which described the shape that K12 math programs should take, outlined 10 recommendations and focused on the fundamental principle that students must learn how to solve problems. The 1983 report A Nation at Risk, by the National Commission of Excellence in Education, also addressed an array of problems in U.S. public education.
- In 1989, NCTM published Curriculum and Evaluation Standards for School Mathematicians, expanding a vision for teaching and learning in grades K4, 5-8 and 9-12. It was widely accepted and endorsed by the National Science Foundation, the American Mathematical Society, the Mathematical Association of America and the Council of Scientific Society Presidents. Basic skills and general math principles were to be learned through "real world" problems, with decreased attention on "complex paper and pencil computations" "long division" and "rote memorization of rules." The standards reinforced the general themes of progressive education dating back to the 1920s by advocating student-centered, discovery learning.
- In the 1990s, the math wars heated up again when math textbooks with "radically diminished content and a dearth of basic skills" were published that failed to develop fundamental arithmetic and algebra skills, Klein states.
- In 2000, NCTM'S Principles and Standards for School Mathematics elaborated on the 1989 standards for grades pre-K2, 3-5, 6-8 and 9-12. Although they were accepted by various organizations, critics claimed the standards were still short on mastery of basic arithmetic and memorization of number facts. Those who defended the standards claimed that K12 math lessons had historically failed to build students' ability to solve problems in different contexts.
- In January 2006, President Bush called for 70,000 high school teachers to be trained for Advanced Placement courses in math and science, 30,000 math and science professionals to enter teaching, and early help to be provided for students who struggle with math so that they would have better chances in getting high-wage jobs. "If we ensure that America's children succeed in life," Bush stated, "they will ensure that America succeeds in the world." -Angela Pascopella
The Future of Math Education
Like others, Milgram worries about the future of math education in American schools, though he says it's heading in the right direction given the Focal Points. The majority of people graduating with advanced degrees in science and engineering from U.S. universities appear to be foreign-born, Milgram says, and many are returning to their native countries to live and work.
As a NASA Advisory Council member, Milgram realizes the nation is in need of competent mathematicians and scientists who are also U.S citizens. Schools are not producing enough to uphold the space program. "We are not producing the first-line people that a group like NASA needs," he says. "We have to learn from our competitors, particularly from what they do better than we do, and training citizens well in mathematics is something that many of them do far better than we do. The Focal Points do this. They apply the lessons learned from studying what these high achieving countries do and why what they do works, to create a series of ecommendations for our own mathematics instruction that will go a very long way toward helping us remain competitive."