by Glenda Lappan, NCTM President 1998-2000
NCTM News Bulletin, December 1998

Many different stances are taken about what standards for school mathematics should entail. Some define "standards" as statements about the mathematics to be learned that also give levels of performance expected at each grade. Others think "standards" should specify very fine-grained behavioral objectives in mathematics for each grade. "Standards" hold many other shades of meaning that focus solely on the mathematics to be learned.

Teaching Matters. NCTM is an organization of teachers of mathematics at all levels. With that perspective, NCTM views standards as statements of criteria for excellence in school mathematics programs. Notice the word program. The NCTM Principles and Standards for School Mathematics (PSSM) (the draft of the updated Standards) examines the mathematics-content goals of a program, but it also shows problems, students' work on those problems, and teaching that helps reach the content goals.

Considering teaching to be a part of important standards is entirely appropriate for an organization of teachers. Our primary goal is to ensure a high-quality mathematics education for every student. Although content is a necessary and absolutely essential ingredient, it alone does not guarantee a high-quality program. We know that teaching shapes and influences what students learn. Teaching matters.

As a result, in NCTM's Standards documents, teaching approaches are crucial in the discussions and elaborations of the Standards. This is not to say that the Council espouses only one kind of teaching. In NCTM's Professional Standards for Teaching Mathematics (1991), a basic assumption is that "teaching is a complex practice and hence not reducible to recipes or prescriptions" (p. 22). Effective teaching can look very different in different classrooms. Good teachers use different strategies at different times for different purposes.PSSM gives teaching suggestions that come from real classrooms and real teachers.

Local Control. The Council believes that provinces, states, and local districts have a responsibility to determine more-specific grade-level goals for mathematics that make sense for their own students. Therefore, NCTM's Principles and Standards document gives national vision and direction to help focus local efforts, not to preempt them. Even so, to help those who want more information as they develop their own curricula, PSSMincludes more guidance on content matters than does the earlier NCTM Curriculum and Evaluation Standards for School Mathematics.

A Closer Look at an NCTM Standard. The PSSM draft, which was released in October for public review and comment, organizes NCTM's Standards into new grade-band levels--pre-K-2, 3-5, 6-8, and 9-12. Each grade band includes 10 Standards that are the same acrossall levels but have different applications at each level. Let's look at a Standard and see how it works.

Standard 1: Number and Operation(which is the same for all grade levels): Mathematics instructional programs should foster the development of number and operation sense so that all students--

• understand numbers, ways of representing numbers, relationships among numbers, and number systems;
• understand the meaning of operations and how they relate to one another;
• use computational tools and strategies fluently and estimate appropriately.

In the grades 6-8 chapter, bullet 1 above ("understand numbers, ways of representing numbers, relationships among numbers, and number systems") is expanded on like this:

• Work flexibly with equivalent fractions, decimals, percents; compare and order these numbers efficiently and accurately, find their approximate locations on a number line; and choose appropriate and convenient forms of these numbers for solving problems;
• develop meaning for integers and be able to represent, compare, and order them;
• develop an understanding of large numbers, including the use of benchmarks to comprehend their magnitude; and recognize, understand, and appropriately use various representations for large numbers (e.g., exponential, scientific, and calculator notation);
• use number theory concepts (e.g., factors, multiples, prime factorization, relatively prime numbers) to solve problems and to understand ideas about rational numbers;
• develop an understanding of the properties of the integer and rational number systems (e.g., order, density, and additive and multiplicative inverses);
• recognize and use commonly encountered irrational numbers

PSSM then goes on to discuss what kinds of experiences students should have had with these ideas in earlier grades, and what kinds of experiences are needed to help students achieve these specific goals in grades 6 through 8. This discussion of the growth of ideas over time is an important attempt to help teachers know how what they do at their grade level fits into the big mathematics education picture for their students. This picture focuses on both teaching and learning--the inseparable components of excellent mathematics programs. That's what the NCTM Standards are about.

But please see for yourself. Be sure to get a copy of the discussion draft of Principles and Standards for School Mathematicsand let us know what you think. You can request a copy of the draft through an electronic request form on NCTM's Standards 2000 Web page ( www.nctm.org/standards/), by e-mail at  [email protected] (message must include your name, membership number, and telephone number), or by phone toll free at (888) 220-7952. The copy is free of charge for NCTM members; nonmembers will be charged a small fee for shipping and handling. Please allow at least two weeks for processing and delivery.