Recalculating K-12 Math

In a downtown hotel in New Orleans in mid-January, nearly two dozen math experts from around the nation gathered to report on their progress toward recommendations that could shape the future of math instruction in U.S. schools for years to come.

The occasion was the fifth of 10 scheduled meetings of the U.S. Department of Education’s National Mathematics Advisory Panel, a 17-member panel appointed by President Bush last year. (The group also includes six unofficial members, consisting of representatives from various federal agencies.) Based on the model of the National Reading Panel, which has influenced reading instruction in the United States significantly over the last decade, the math panel is tasked with advising U.S. policy makers and educators on the effectiveness of various approaches to teaching mathematics.

It’s a topic of debate that has been heating up considerably over the last several years. In the latest Trends in International Mathematics and Science Study (TIMSS), U.S. students were ranked 15th in eighth-grade math skills, behind countries such as Australia and the Slovak Republic, while countries such as Singapore, South Korea, and Hong Kong are snagging the top rankings.

"In general, there’s widespread recognition that the U.S. is not doing well in mathematics," says Steve Ritter, chief product architect and founder of Carnegie Learning, a provider of mathematics curricula for middle and high school students. "We do, at best, average compared with other countries, and that’s below most developed countries. I don’t think anybody is satisfied."

That reality, coupled with the confusion about how math is being–and should be–taught in this country, has tossed the topic into the center of a controversy often called the Math Wars.

How contentious has this debate sometimes been? "Other than the war in Iraq, I don’t think there’s anything more controversial to bring up than math," notes one Utah school official.

Adding even greater urgency to the debate was the release in December of "Tough Choices or Tough Times," from the New Commission on the Skills of the American Workforce, indicating there are nearly 7,000 dropouts per day in U.S. schools–and math has been identified as one of the key reasons kids are dropping out, particularly when they reach the algebra level.

"One of the reasons kids fail in algebra is they haven’t mastered the skills they need–the basic fractions, decimals, percents, proportional reasoning," says Doug MacGregor, manager of instructional design for AutoSkill, a Canadian provider of math and reading software. "Without that, they can’t succeed."


The National Math Panel’s goal is to help change that. Although the group’s preliminary reports on Jan. 10-11 did not include any specific recommendations–those are expected with the panel’s final report, due out in February 2008–it was clear from these early reports that panel members aim to achieve a healthy balance between computational fluency and conceptual understanding of mathematics, an area that underpins much of the current debate.

Educators, math curriculum and software designers, and federal and state policy makers are closely watching the panel’s actions. Ritter believes the group’s ultimate recommendations will have a huge impact on the types of software programs that will become available for use in schools.

"The math panel has the potential to be very influential on what types of products we start to see come out, because it will influence the kinds of programs the government will be promoting, and publishers will respond to that," he said.

To help form its recommendations, the panel has invited various experts to testify at panel meetings, which are open to the public and announced in the Federal Register. (A schedule of these meetings also is available on the panel’s web site.)

One such expert was Melissa Kalinowski, elementary marketing director for PLATO Learning, an educational software company that provides online assessments that are tied to state standards. In her recent presentation to the panel, Kalinowski summed up the issues involved:

"There are three problem areas in elementary mathematics instruction … shallow mathematics curricula (overemphasis on rote memorization and low-level math skills), underprepared teachers (who lack understanding of how children learn math or conceptual understanding of key math concepts), and uninformative assessments (too broad to help teachers make meaningful and informed decisions in the classroom)," Kalinowski testified.

Ritter agrees that our approach to teaching mathematics is too shallow. "Our learning is very broad, but not very deep," he says. For example, "you learn about quadratic equations, but not very much about them." But there is, obviously, far more to the issue than that.

Why all the confusion over math?

The controversy surrounding how math is taught goes back at least as far as 1989, when the National Council of Teachers of Mathematics (NCTM) released its "Curriculum and Evaluation Standards." Those standards stressed the importance of conceptual understanding and problem solving.

But the standards left the impression to many that the council believed basic skills–such as fluency with addition, multiplication, subtraction, and division–were less important than learning about the process of mathematics. (For an example of the uproar this confusion has caused in some school systems, see the side story: "Looking to Singapore for success.")

In 2000, NCTM came out with the "Principles and Standards for School Mathematics," which some called a revision of the Curriculum and Evaluation Standards. The new principles seemed to emphasize more basic skills, while continuing to stress the importance of higher-level problem solving. Then, last September, NCTM released the next step in helping schools understand what and how to teach. Called the "Curriculum Focal Points," the group’s latest set of guidelines identified the most important mathematical topics from pre-kindergarten through eighth grade and offered a framework to guide states and school districts as they revise their math standards, curricula, and assessments.

Work on the Curriculum Focal Points included analysis of all 49 states that have preK8 curriculum frameworks, as well as frameworks from Singapore, Japan, China, and Korea, according to NCTM. The Focal Points identified three key areas of learning per grade level. For example, multiplication of whole numbers is a focal point at the fourth-grade level, and division of whole numbers is a focal point at the fifth-grade level.

Following the publication of the Focal Points, however, the press began to challenge the NCTM’s position on what students should be learning, saying the council had reversed its earlier position on the need for students to learn higher-level problem-solving skills. The Wall Street Journal, for example, referred to the Focal Points as a "remarkable reversal" by the organization.

In response, NCTM President Francis Fennell sent a letter to the editor, saying, "Contrary to the impression left in your article, learning the basics is certainly not ‘new marching orders’ from the NCTM, which has always considered the basic computation facts and related work with operations to be important … As stated in NCTM’s 1989 and 2000 standards, conceptual understanding and problem solving are absolutely fundamental to learning mathematics."

Whether the Focal Points were a reversal, a revision, or simply the next step in offering teachers of mathematics an understanding of what students need to know in order to fundamentally understand math, people in the mathematics field consider them to be a much more balanced approach, says Ritter.

"What NCTM did, correctly and very well, was to define goals, which were not to produce human calculators but to develop people who could understand more complex subjects," he says. Where the NCTM went wrong earlier was that, in talking about methods, the group "deemphasized the practice of basic skills."

The National Math Panel contains several people who were critics of the NCTM’s approach, believing it to be too focused on the application of mathematics and not enough on basic skills, according to Ritter. However, in its preliminary progress reports, the panel claims that its members believe the NCTM is on "sound footing" with its new Focal Points.

Problem areas in teaching math

The confusion surrounding what, exactly, should be taught to students is made worse by the absence of a single, national math curriculum. While states have adopted their own curricula, many of these are extremely broad, so teachers end up teaching "a mile wide and an inch deep," as the saying goes.

The mathematics world is beginning to recognize that students need to learn both the basics and the conceptual skills. Students must understand where the concepts come from and what they mean, so if they forget a formula, they can perhaps figure out the problem anyway, says Mary Ann Stine, director of curriculum and instruction for the Everett School District in Washington. And that’s a particular challenge, Stine notes, because teachers often are not taught, themselves, how to teach math.

"There’s a difference between someone who’s a mathematician and someone who’s a math teacher. The skills are different," Stine says. She adds, "We don’t expect people with a third-grade reading level to teach reading, but we do that in math a lot."

Another problem, according to Stine, is that teachers aren’t taught how explain to parents why the math they are teaching might look different from how parents learned in the past. "They do not understand how to explain to parents why it’s different and how it will help kids in the future, and they end up saying things that shoot us all in the foot," she says.

Teachers, many experts say, need better training in how to teach the concepts. "One thing that’s really hard about math instruction is that being conceptual is really demanding," says Angus Mairs, senior manager of High School Transformation for the Chicago Public Schools. "Leading with the conceptual is difficult, because a teacher’s own math learning must [encompass] those concepts. [Teachers] may gravitate toward the skills, because that’s what they were taught."

The advent of technology, too, is an important development in math instruction–not only because it can help teachers teach math, but because it has changed what students need to learn.

Statistics, for example, has taken on a larger role in elementary education. "In life, in the news, you always see charts and graphs and displays of data, and people need to be able to understand those data," says Stine. "Because of the data we can crunch using computers, we use [statistics] more, and it’s much more important in our lives."


Stine notes that schools are beginning to teach probability and statistics in the elementary grades, having students use tools such as surveys and then create charts, graphs, and statistical displays to demonstrate what they know about the information they’ve gathered.

But again, teachers must be taught how to teach statistics at an elementary-school level, and for the most part, they are not getting that learning.

Until now, these challenges have contributed to the country’s failure to improve on its mathematics knowledge as quickly as other countries, experts say. Students who fell behind either failed or were expected to make up the class in summer school or the following year. Now, with the increased accountability ushered in under No Child Left Behind and the calls to strengthen the competitiveness of American students, schools are finding it’s no longer acceptable to allow kids to fail.

Addressing the problems

One approach to addressing the problems faced by math teachers is "Response to Intervention," or RTI, a phrase that has recently gained traction in education. The idea originally was developed to identify and help learning-disabled students, but it since has been recognized as a successful series of practices that can keep kids from failing in all areas of education.

"The idea behind Response to Intervention is you want to detect when a student is on the path to failure, so you can change the path," says Ritter. The first key in RTI is formative assessment: evaluating students, not to see whether they’re passing or failing, but for the purpose of driving instruction.

This is an area where technology can be invaluable.

"The advantage of technology is that you can track those needs," says Ritter. For example, Carnegie Learning’s Cognitive Tutor and other software programs monitor students as they work on problems, identifying areas where their skills need development.

The next step in RTI is differentiated instruction. Again, software programs can help. In a typical math class, a teacher might give all students the same 25 problems to work on. "If you were having problems with negative numbers and I was having problems with fractions, we’d get the same question," says Ritter.

With Cognitive Tutor, each child might still do 25 problems, but the problems would be different, tailored for each child. "It’s not that you tell the computer you need help with fractions," Ritter explains. Rather, the program sees where a student is struggling and offers targeted, individualized questions to help that student. (For a look at how Chicago school officials are using Cognitive Tutor to help reinvent instruction in the city’s high schools, see the side story: "How Chicago schools are transforming math instruction with technology.")

Technology solutions also can help illustrate the concepts behind math problems–the area where many teachers need the most help.

Chicago’s Mairs uses this example: Imagine a teacher who wants to teach students how to create a graph based on the acceleration of an object. Using a laptop and a projector, the teacher can project the image of a person on a skateboard ramp, along with a graph of what the skateboard’s movement looks like. The teacher then can explore with the students what will happen if the skateboard speeds up, slows down, or changes direction. Many computer programs allow students to learn using examples they encounter in their own lives. And, Stine adds, "the artificial intelligence can automatically identify the strengths and weaknesses of kids, so they don’t keep working on the stuff they already know."

Algebra readiness

This brings us back to what the report, "Tough Choices or Tough Times," identified as one of the greatest barriers to a high school diploma: algebra readiness.

"Teachers are so tied to their curriculum–they have to get through things–that they don’t always have time to go back and find the gaps in their students’ knowledge," says AutoSkill’s MacGregor. "You can offer students great instruction in quadratic equations, but if they don’t have a strong understanding of integers, it’s not going to help them."

Teachers have to find these gaps, and teach to fill the gaps, before students reach algebra. That, MacGregor says, is one strength of his company’s Academy of Math software. "By giving teachers the knowledge of where their students are breaking down, [teachers] can come up with lesson plans that would meet those needs," he says.

Ritter agrees that a lack of basic skills is what leads to students failing algebra. He says Carnegie Learning has developed a program, called Bridge to Algebra, to address this very need. Through a review of middle school math, Bridge to Algebra presents a broad spectrum of topics that the system will customize based on the individual needs of students. "When it learns that a student understands statistical topics but not fractions and conversions, it will teach them the latter," Ritter says.

The Los Angeles Unified School District is in the middle of a three-year contract with Carnegie Learning to implement the company’s Algebra I and Bridge to Algebra programs. Scott Svec, a math teacher at the city’s Columbus Middle School, teaches sixth, seventh, and eighth graders who are highly diverse in their nationalities, math skill levels, and English-speaking abilities. The recent implementation of Bridge to Algebra has improved his students’ performance and given them the confidence to work both independently and collaboratively, he says.

"One girl who is in my class for the second year was a big behavior problem. Now, with this program, she works independently–and she has reached Level 9 in ten weeks with very, very limited English ability," Svec says. "Many kids ask to work on this program at lunch, which blows my mind. My [students with] behavior problems have turned into my better students. [Bridge to Algebra] is the best thing that has happened to these kids and me since I started here."

‘…Teaching [educators] to fish’

It is success stories like Svec’s that school software developers hope National Math Panel members will consider carefully when drafting their recommendations.

But as the math community watches the panel’s progress with great interest, its biggest concern is that the panel–and any subsequent federal grants that are made on the basis of its recommendations–will favor specific software programs or instructional approaches over those with equally strong track records.

That’s a criticism that has been leveled at the federal Reading First program, which distributes about $1 billion a year in grants based on the recommendations of the National Reading Panel. In a strongly-worded rebuke of Reading First last year, the Government Accountability Office essentially confirmed this charge, saying the Education Department showed favoritism in awarding certain grants.

In written testimony given to the math panel in November, Mark Schneiderman, director of education policy for the Software & Information Industry Association, spoke for many when he said:

"We would urge that the math panel’s findings and recommendations focus toward broad principles and practices of effective math instruction and curriculum, rather than toward identifying specific interventions, programs, and products. To the degree you do look at specific interventions, we encourage you to look foremost at the underlying reasons for effectiveness in a manner that can more broadly inform educators as well as developers. We believe your most appropriate role and greatest impact will come not from giving educators the fish–i.e., a list of interventions and products–but by sharing with them the knowledge you gained and teaching them to fish themselves. Only then can we keep up with ever-changing instructional programs and practices in the face of evolving research and technologies."

By Jennifer Nastu, a freelance writer in Colorado who writes frequently on technology and education.


National Mathematics Advisory Panel

National Council of Teachers of Mathematics (NCTM)

NCTM’s Curriculum Focal Points

Shaping the future of math instruction

Last May, Education Secretary Margaret Spellings announced the 17 expert panelists, and six "ex-officio" members, chosen to advise educators and policy makers on the best use of scientifically based research to advance the teaching and learning of mathematics. "To keep America competitive in the 21st century, we must improve the way we teach math, and we must give more students the chance to take advanced math and science courses in high school," Spellings said.

The National Math Advisory Panel, chaired by Larry Faulkner, president of the Houston Endowment and President Emeritus of the University of Texas at Austin, released its first progress reports on Jan. 11. However, because the panel’s research is not complete and its final report is not scheduled for release until next February, these preliminary reports did not include any recommendations.

The panel consists of four working groups: Conceptual Knowledge and Skills, Learning Processes, Instructional Practices, and Teachers. Each group presented its own report.

Conceptual Knowledge and Skills Task Group

This group aims to define the key math concepts and skills that should be taught at various grade levels. Its progress report included a topical list of the concepts and skills needed for pre-kindergarten through eighth grade math, derived from an analysis of state curriculum standards and the math curricula for Japan, Korea, Flemish-Belgium, Singapore, and Chinese Tapei, among other sources.

The group has pinpointed the key concepts and skills that students should learn regarding numbers and operations, algebra, geometry and measurement, and data analysis and probability. In putting together its list, the group appears to have sought a proper balance between computational fluency and conceptual understanding.

For example, under numbers and operations are 11 distinct skills, including "develop automaticity of multiplication facts and related division facts and fluency with the multiplication and division of whole numbers," as well as "define ratio and rate in terms of multiplication and division" and "develop an understanding of and apply proportionality, including similarity."

The group also is looking at having some algebra learned before high school: "Write, interpret, and use mathematical expressions and equations" and "analyze and represent linear functions and solve linear equations and systems of linear equations" appear in the list.

"With the following topical list for levels preK-8, it is important to provide the instructional time and emphasis needed for students to develop concepts, solve problems, and compute," the group notes.

It also says the panel’s final recommendations might articulate grade-by-grade expectations–and, if so, the National Council of Teachers of Mathematics’ Curriculum Focal Points "will be seriously considered."

In addition, the Conceptual Knowledge and Skills Task Group’s report outlined the key algebra skills and concepts that students should learn, ranging from symbols and expressions to quadratic relations and functions–but it noted that, while there is general agreement about the scope and sequence of skills in pre-kindergarten through eighth grade, no such agreement exists yet regarding algebra skills.

Learning Processes Task Group

This group aims to identify the key principles of learning and cognition and apply them to the study of mathematics, to help shape and inform the work of the other groups.

In its work so far, the group has explored how children learn, why they "choke" during testing, individual and group differences in how kids learn, and how brain science and learning math work together.

In examining the available research on how kids learn, the group has identified what skills children already bring to school–there is an inborn sense of quantity that each child possesses, but it’s not sufficient by itself to begin school-based learning–and has looked at whole-number learning. Its next steps are to examine how students learn fractions, estimation, algebra, and geometry, and to draw explicit links with the work of the other task groups.

The Learning Processes group’s work so far demonstrates why a healthy balance between computational fluency and conceptual understanding is so necessary.

Studies show that practice leads to automatic retrieval of declarative information or execution of procedures, the group’s report says–but conceptual knowledge is what promotes the transfer of skills to new situations.

Still, this automatic retrieval of math facts is necessary to help students overcome the anxiety often associated with testing. "Choking" on tests occurs when competency-related thoughts–that is, the high stakes involved in the exam–intrude on students’ working memory, according to the group’s preliminary report. Its solution: "automaticity" of test-related content.

Instructional Practices Task Group

This group is looking at what research says about the effectiveness of various instructional approaches to mathematics. It has focused in particular on teacher-centered (direct) versus student-centered ("inquiry-based") instruction, as well as the importance of "real-world" problems and approaches.

According to the group, the benefits of real-world instruction–which uses problems and examples grounded in practical applications of math–include greater student motivation before lessons and engagement during lessons. Supporters of real-world approaches also say they have the potential to raise achievement and lead to greater long-term retention. But there are dangers in this approach as well, the group’s report suggests.

In its 2005 "State of State Math Standards" report, a review of the math standards of all 50 states, the Fordham Foundation described "excessive emphases" on "real-world problems," the group says. The foundation reportedly warned, "Excessive emphasis on the ‘real world’ leads to tedious exercises in measuring playgrounds and taking census data, under headings like ‘geometry’ and ‘statistics,’ in place of teaching mathematics."

Some of the topics the Instructional Practices group will be examining more closely are the ideal sequencing of tasks–that is, are real-world tasks more appropriate at the beginning of a lesson, where they can boost students’ motivation, or at the end of the lesson, where students can apply what they’ve learned?–and the appropriate amount of time that math teachers should spend on real-world tasks.

Teachers Task Group

This group has four areas of focus: teachers’ knowledge of mathematics, teacher education and professional development programs, elementary math specialists, and recruitment and retention of math teachers.

In this first area of focus, the group aims to develop "a firmer understanding of the mathematical knowledge needed for teaching." The answer to this question, the group says, is "crucial" to its understanding of the other focus areas.

In these other three areas, the group seeks to determine which approaches might be most effective. For example, what kinds of teacher education programs have been shown to be effective? Do particular designs or curricula make a difference in teachers’ instructional skills, or their students’ achievement?

Implications of the panel’s work

"The buzz in the air is that the recommendations of the math panel will emphasize more basic skills," says Steve Ritter, chief product architect and founder of Carnegie Learning. He adds, "My expectation is that the report will have a reasonable balance between understanding the need for students to become fluent in their basic mathematics skills and the need for application of mathematics and problem solving."

One of the concrete things Ritter imagines will happen as a result of the panel’s work is that the ratio of basic, "naked" math to application problems offered in textbooks and practice sets will start to change–with more application problems, and fewer basic problems, being included.

On the other hand, Mary Ann Stine, director of curriculum and instruction for the Everett School District in Washington, is not certain how much influence the math panel’s recommendations really will have, particularly when it comes to curriculum.

"Some schools don’t get any federal money, so I suppose if you don’t get federal dollars, you wouldn’t have to pay any attention to the math panel at all," she said. "If [federal officials] want people to pay attention to the math panel, they’ll have to tie dollar amounts" to implementing the group’s recommendations.

Who’s who on the National Math Panel

In addition to Faulkner, panelists include:

• Deborah Ball, Dean, University of Michigan School of Education;

• Camilla Benbow, Dean of Education and Human Development, Vanderbilt University’s Peabody College;

• A. Wade Boykin, Professor and Director of the Developmental Psychology Graduate Program, Howard University Department of Psychology;

• Francis Fennell, Professor of Education, McDaniel College, and National Council of Teachers of Mathematics President;

• David Geary, Curators’ Professor, University of Missouri at Columbia’s Department of Psychology;

• Russell Gersten, Executive Director, Instructional Research Group; Professor Emeritus, University of Oregon College for Education;

• Nancy Ichinaga, former Principal, Bennett-Kew Elementary School, Inglewood, Calif.;

• Tom Loveless, Director, Brown Center on Education Policy and Senior Fellow in Governance Studies, The Brookings Institution;

• Liping Ma, Senior Scholar for the Advancement of Teaching, Carnegie Foundation;

• Valerie Reyna, Professor of Human Development and Psychology, Cornell University; "Wilfried Schmid, Professor of Mathematics, Harvard University;

• Robert Siegler, Teresa Heinz Professor of Cognitive Psychology, Carnegie Mellon University;

• Jim Simons, President, Renaissance Technologies Corp.; former Chairman of the Mathematics Department, State University of New York at Stony Brook;

• Sandra Stotsky, independent researcher and consultant in education; former Senior Associate Commissioner, Massachusetts Department of Education;

• Vern Williams, Math Teacher, Longfellow Middle School, Fairfax, Va.; and

• Hung-His Wu, Professor of Mathematics, University of California at Berkeley.

–Jennifer Nastu

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