Three years after calling for a reordering of elementary and middle school math curricula, the nation’s largest group of math teachers is urging a new approach to high school instruction, one that aims to build students’ ability to choose and apply the most effective problem-solving techniques, in the classroom and in life.
Cultivating those skills will make math more useful, and more meaningful, to students, the National Council of Teachers of Mathematics argues in a document scheduled for release this week.
“Focus in High School Mathematics: Reasoning and Sense Making” is a follow-up to the NCTM’s 2006 document, “Curriculum Focal Points,” which offered grade-by-grade content standards in math for prekindergarten through 8th grade. “Focal Points” won general praise in math circles, even from some of the NCTM’s strongest critics.
The high school document has both a different purpose and a different structure. It is not a suggested set of content standards, but rather a framework that attempts to show how skills that the NCTM considers essential—reasoning and sense-making—can be promoted across high school math.
While the new guidelines say that understanding math content and procedures is important, they also argue that students need to learn to apply that knowledge in different situations—a skill that proves essential in everyday situations and in the workforce.
“Reasoning and sense-making are at the heart of mathematics from early childhood through adulthood,” NCTM President Henry S. Kepner Jr. says in an introduction to the document. Cultivating those skills, he writes, “will prepare students for higher learning, the workplace, and productive citizenship.”
NCTM officials also argue that those abilities will help produce more students who are more interested in, and capable of, going into math- and science-centered occupations, a major concern of American policymakers.
“We keep teaching that learning to carry out complicated procedures is what math’s about,” said W. Gary Martin, a professor of mathematics education at Auburn University, in Alabama, who chaired the committee that wrote the document. “To me, the real question is, can [students] do anything with it?”
Something in Common
The 100,000-member math teachers’ group, based in Reston, Va., is releasing the document at a time when policymakers at the federal and state levels are pushing for more consistency in what students are taught.
Forty-eight states have agreed to take part in a venture called “Common Core,” designed to produce common standards in language arts and math. It is being led by two organizations that work with states, the Council of Chief State School Officers and the National Governors Association. The NCTM is one of several organizations that have offered comments on a draft of the math section, which is organized by content area and focuses on preparation for college and the workforce. (“New Standards Draft Offers More Details,” Sept. 30, 2009.)
NCTM officials provided a prepublication copy of the high school report to the Common Core authors, said Jason Zimba, a professor of mathematics and physics at Bennington College, in Vermont. The two documents touch on many of the same main ideas, he said.
“They ought to be reinforcing,” said Mr. Zimba, who is a member of the Common Core math working group. “We’ve been trying to be on the same page the whole time.”
Both documents, for instance, emphasize mathematical practices, or students’ ability to adapt math strategies to solve new problems, although they describe those skills in somewhat different language.
New Meaning in Math?
The NCTM’s high school guidelines explain how reasoning and sense-making can be applied in different areas of math. They also offer numerous examples of how such applications might play out in the classroom, presented through model dialogues between teachers and students covering math lessons.
Focusing on reasoning will not saddle teachers with additional burdens, but rather produce more engaged students, the NCTM authors argue.
“Currently, many students have difficulty because they find mathematics meaningless,” they write. “Without the connections that reasoning and sense-making provide, a seemingly endless cycle of reteaching may result.”
The NCTM did not organize “Focus in High School Mathematics” by grade, partly because American students take different classes, such as introductory and advanced algebra, at so many different points in high school, Mr. Kepner explained in an interview.
While “Curriculum Focal Points” was aimed mostly at state officials, curriculum developers, and publishers, the high school document is mostly targeted to “instructional leaders,” such as teachers, curriculum specialists, and administrators, in addition to state officials and the publishing industry, Mr. Martin said.
The NCTM published voluntary academic-standards documents in 1989 and 2000, which have shaped curriculum, instruction, and textbooks nationwide. Critics have said those standards paid too little attention to crucial math content and strayed from standard problem-solving approaches.
With the publication of “Focal Points” three years ago, the NCTM won over some detractors, who said the document offered clear guidance to schools on elementary and middle school math.
More Content Wanted
Early reaction to the high school document was mixed.
Albert Cuoco, who directs the Center for Mathematics Education at the Education Development Center, in Newton, Mass., called the NCTM publication a “breath of fresh air.”
Mr. Cuoco said other standards documents and resources, such as a 2008 report by the White House-commissioned National Mathematics Advisory Panel formed during the second Bush administration, have sought to present “clear and unambiguous” lists of essential math topics. What’s missing for many teachers is “the glue that holds them together,” or a sense of how those topics relate to one another and why they are important, said Mr. Cuoco, whose center works on math curriculum, teacher training, and policy.
The NCTM’s new document and others like it that explain in plain language why topics matter represent “a new genre in framework-writing,” Mr. Cuoco said.
Mr. Cuoco, who reviewed a draft of the document, was less enamored with its use of hypothetical examples designed to demonstrate how reasoning works in math class, many of which describe a teacher interacting with “student A” or “student B.” Mr. Cuoco said he would have preferred more direct testimonies provided by actual teachers. “It didn’t ring true to me,” he said.
Vern Williams, who served on the national math panel and has been critical of the NCTM, said he did not want to make extensive comments on the document until he had read it in more detail. But he said the language in the publication reminded him of the NCTM’s 1989 standards, which he believes promoted a weak approach to math instruction, as opposed to “Focal Points,” which he regarded as much more content-rich and useful.
“It would be nice if the content was described more,” said Mr. Williams, a math educator at Longfellow Middle School in Fairfax County, Va., who has won a number of awards for his teaching. “It’s too vague.”
Wilfried Schmid, another math panelist and a math professor at Harvard University who also was critical of the NCTM’s 1989 and 2000 standards, said he agreed with the organization that nurturing math reasoning skill is important, and that many students become overly reliant on procedures.
But he said the new document places too much emphasis on “abstract” skills, rather than on math content and math “as a sequential subject.” There needs to be more of a balance between content knowledge and reasoning and problem-solving ability, he said.
“They are diagnosing a problem, and I largely agree with the diagnosis,” Mr. Schmid said, but the high school document “would lead to a worse situation.”
Mr. Kepner countered that the NCTM was trying to avoid repeating or adding to the long list of math content topics promoted in individual states. It instead wanted to “go further” in giving that content greater meaning for teachers and students, he said.
Other Resources Coming
In an effort to make “Focus in High School Mathematics” useful to the public, the NCTM is also publishing short, separate resources that translate the document for teachers, administrators, policymakers, and families.
The organization also plans to publish additional “topic books” to explain how reasoning and sense-making can be cultivated in different areas of math, such as in algebra and geometry, in greater detail than the current document does.
“This is only the first step,” said Mr. Martin of Auburn University. Overall, NCTM officials hope the document will help high school students see the relevance of math, and improve their performance, which on national tests has been mostly stagnant for decades.
At times, high school math instruction “seems impervious to change,” Mr. Martin said. “It’s just a Herculean task. ... We can’t just keep going down the same path. We keep repeating the same mistakes, year after year.”
