More than 15 years after its publication of influential national standards in mathematics, a leading professional organization has unveiled new, more focused guidelines that describe the crucial skills and content students should master in that subject in elementary and middle school.
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The National Council of Teachers of Mathematics last week released “Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics,” a document that supporters hope will encourage the polyglot factions of state and local school officials, textbook publishers, and teachers to set clearer, more common goals for math learning.
While the report is being published by the NCTM, it was reviewed by numerous math experts from across the country, some of whom have strongly disagreed with the organization’s past positions on essential skills. The new document reflects an attempt to overcome those conflicts and focus on a number of crucial, agreed-upon concepts.
“Focal Points” offers grade-by-grade advice for what students should be taught in various areas of math. Here is how students should progress, for selected grades, in the area of number and operations:
Pre-K Develop an understanding of whole numbers and how to count and compare them.
Kindergarten Use numbers to solve quantitative problems, count numbers in a set, and create a set within a given number of objects.
2nd Grade Learn how to count in units and multiples of hundreds, tens, and ones; understand multidigit numbers in terms of placevalue, and how to compare and order numbers.
4th Grade Develop understanding of multiplication, including “quick recall” of multiplication and division facts; select correct methods to make mental estimations and calculations.
6th Grade Know the meanings of fractions, multiplication, and division; understand relationships between decimals and fractions, and how to multiply and divide them, using multistep problems.
8th Grade Use linear functions, linear equations, and their understanding of the slope of a line to solve problems; understand verbal and graphical representations of functions; describe how the slope of a line and the y-intercept appear in different verbal, graphical, and algebraic representations.
SOURCE: National Council of Teachers of Mathematics
“I would hope that this has a large impact, because I believe it gets it right,” said R. James Milgram, a Stanford University mathematics professor and a critic of the math organization’s previously issued national standards. He was one of 14 individuals who provided an outside, formal review of the document. “I would like to hope that this represents a new era of cooperation,” he added. “I hope that what this represents is an end to the math wars.”
The NCTM’s publication of voluntary national standards in 1989 served as a guidebook for many states’ drafting of academic standards in math—or expectations for what students should know in that subject. Yet the content of those state standards, which typically serve as blueprints for state tests, varies enormously. Some of the states’ documents are packed with nearly 100 expectations per grade level, NCTM officials say. Supporters of the “Focal Points” report say it can help state and local officials pare down those goals.
“States and school districts are looking for guidance,” NCTM President Francis M. “Skip” Fennell said in an interview. The message behind the report is that “this is the blueprint,” he said. “We start here.”
Curriculum Document
“Focal Points” is not meant to replace the previous standards, Mr. Fennell said, but rather serve as a next step offering more focused help in the earlier grades. The new document, in fact, includes links showing where the previous standards are covered in it.
The guidance is being released at a time when improving math and science education is receiving significant attention at the federal level, where elected officials and senior appointees see a need to produce more highly skilled workers and gird the United States against rising foreign economic competition. Corporate leaders are lobbying Congress to take steps to improve the quality of teaching in those subjects and encourage more students to consider math- and science-related professions.
Some policymakers could be especially receptive to the new NCTM guidance. Earlier this year, President Bush established the National Mathematics Advisory Panel, a 17-member expert body charged with producing recommendations by next year for improving math teaching and learning. Mr. Fennell serves on that panel. The group heard a presentation on the Focal Points at its meeting last week in Boston.
The 41-page “Focal Points” focuses solely on curriculum, not on teaching strategies or tools to help students learn, such as technology.
For each grade level, it offers paragraph-long descriptions of concepts the authors regard as essential, broken out by topics such as numbers and operations, basic algebra, measurement, and geometry. Those expectations become more demanding with each grade.
For instance, in geometry, “Focal Points” says that kindergartners should learn to identify and name basic shapes, such as squares and triangles, as well as three-dimensional objects. In 1st grade, children should begin to recognize those shapes from different perspectives, describe their similarities and differences, and develop an understanding of concepts such as symmetry. In 5th grade, students should begin to understand how to use shapes to quantify volume and make estimations, on the way to learning even more advanced skills, the document says.
Isolating What Matters
Officials of the NCTM, a 100,000-member organization with headquarters in Reston, Va., say they hope the document will help schools and states as they struggle to raise student test scores in grades 3-8, under the mandates of the federal No Child Left Behind Act. The law requires that students in those grades be tested annually in math and reading. The need for guidance, the officials say, is especially acute at the early grade levels, where many teachers lack certification and training in math and rely heavily on textbooks.
Critics of how math is taught in the United States have long derided what they call the “mile wide, inch deep” approach to the subject, in which teachers bounce from topic to topic, without encouraging mastery of the most important concepts. That wrongheaded strategy, they say, stems partly from the hodgepodge of expectations set by different states and schools.
• Jane F. Schielack, professor of mathematics, and of teaching, learning, and culture, Texas A&M University (chairwoman of writing team)
• Sybilla Beckman, professor of mathematics, University of Georgia
• Randall I. Charles, professor emeritus of mathematics, San Jose State University
• Douglas H. Clements, associate dean for educational technology, State University of New York at Buffalo
• Paula B. Duckett, retired, District of Columbia public schools
• Francis M. “Skip” Fennell, professor of mathematics, McDaniel College, NCTM president
• Sharon L. Lewandowski, mathematics support teacher, Bryant Woods Elementary School, Columbia, Md.
• Emma Trevino, senior program coordinator, Charles A. Dana Center, University of Texas at Austin
• Rose Mary Zbiek, associate professor of mathematics education, Pennsylvania State University
Publishers, in an attempt to meet those competing demands, attempt to pack too many concepts into textbooks, and math teachers lose sight of which lessons are most important, the critics say.
“Why is a 3rd grade student lugging around a 738-page textbook?” Mr. Fennell said. “You can’t tell me every one of those 738 pages is equally important.”
Looking beyond middle school, the NCTM is forming a task force to address the core math skills of high school students, Mr. Fennell said. Because students’ high school course schedules by grade vary greatly in different schools and states, that undertaking could focus more specifically on what math content should be covered in classes with titles such as Algebra 1 or Algebra 2, he said.
To date, the most widely cited NCTM document among educators was “Curriculum and Evaluation Standards for School Mathematics,” a set of voluntary national standards, first published in 1989 and updated in 2000. Those standards, originally published at 258 pages, set detailed expectations for what students should know in K-12.
But the group’s standards also have drawn steady criticism from some parents, teachers, and university mathematicians, who say the group’s guidance placed too little emphasis on students’ mastery of basic arithmetic and memorization of basic number facts, such as the automatic answer to easy multiplication and division, in favor of less concrete conceptual skills.
That misguided approach, detractors say, has been copied by many states in their standards and worked its way into classrooms. Defenders of the standards, however, reject those charges, saying that K-12 math lessons have long been too reliant on drill and rote learning and fail to build students’ ability to reason and solve problems in different contexts.
In recent years, advocates on both sides have said they see those debates as counterproductive. Any math teacher or mathematician, they acknowledge, knows that students need a combination of basic skills and conceptual understanding.
What Kind of Recall?
Several observers agree there is evidence of a cease-fire in the “math wars” in recent years. The easing of tensions was apparent, they say, in the publication last year of a paper titled “Reaching for Common Ground in K-12 Mathematics Education.” In it, a number of scholars, including Mr. Milgram of Stanford University and Richard J. Schaar, a former president of the business unit for Texas Instruments, identified several agreed-upon math skills students should master.
“We always realized that the number of math topics that matter is relatively small” in the early grades, Mr. Milgram said. The strength of “Focal Points” is to “isolate out the topics … that are important,” he said. “They are not trivialized.”
Throughout its new document, the NCTM seems to address squarely the skills that critics have accused the organization of neglecting. For instance, the document says that 4th graders should develop “quick recall of the basic multiplication facts and related divisions facts” in studying numbers, operations, and algebra. Second graders, it says, should be able to use “basic addition facts and related subtraction facts.”
Mr. Fennell disputed the suggestion that “Focal Points” represents a shift in the NCTM’s approach. He said his organization has always recognized the importance of building students’ ability to memorize certain basic math facts and procedures. “If that wasn’t clear before, [we’re] saying that now,” Mr. Fennell said.
But he also recalled that authors of the new report engaged in heated debates over the exact terminology used in describing various math skills, weighing phrases such as “immediate recall” of number facts and “instant recall” before settling on “quick recall” in one section.
Cathy L. Seeley, the immediate past president of the NCTM, said the report could prompt some states to retool their math standards, even if those changes came about more slowly than in the 1990s, when many states were only beginning to craft NCTM-inspired documents. “State standards are dynamic,” she said, “they’re not fixed.”
Ms. Seeley, who helped initiate the writing of “Focal Points,” does not believe the document will squelch all debates about how math should be taught, or what topics belong at various grade levels. But she predicted the new document could at least narrow that discussion.
“Perhaps we needed these 15-plus years of discussion to understand more clearly what everyone was talking about,” she said. “It has always struck me, over and over again, that we really agree on more than we disagree.”
