Mathematically Correct Presents
In our view the NCTM standards present a vague, somewhat grandiose, readily misinterpreted view of what American children should learn in mathematics. Moreover, the view fails to meet what we would consider to be the meaning of "standards." Standards should involve a progression of accomplishments or competencies that are to be demonstrated at defined times in the child's schooling. The NCTM standards give no indication (beyond four-year intervals) of the sequence with which the content is to be presented and are not helpful to the classroom teacher in designing lessons that meet the standards.
The NCTM standards list goals with which no one would be likely to disagree. Of course we want children to value mathematics, to be mathematics problem solvers, to be confident of their ability, and to be able to reason and communicate mathematically. Certainly students must develop a number sense, have concepts of whole number operations, and the other kinds of skills and knowledge indicated under NCTM's curriculum standards. But the published standards do not integrate these two important components: the general attitudes and mathematical skills. Another important aspect is how realistic it is for teachers to implement these vague ideas in their everyday teaching. In many cases, schools elect to emphasize students' general attitudes and fail to provide them with opportunities to learn fundamental mathematics skills. As a result, the new NCTM approach of teaching math is often criticized as being fuzzy and susceptible to multiple interpretations.
The goal in establishing standards is to set up something that can be implemented. How is the regular classroom teacher expected to be able to implement these standards? It is interesting that although a high percentage of American teachers indicate they are aware of the standards, many are unable to provide explicit examples of how they have done to put the standards into practice. In many cases, the demonstration of their teaching to the standards is very confusing to the students. The problem is in the manner in which the standards are organized and written as well as their lack of sufficient clarity to enable their implementation.
The NCTM standards include the following types of goals and problems. "In grades 5-8," for example, "the mathematics curriculum should include numerous and varied experiences with problem solving as a method of inquiry and application so that students can use problem-solving approaches to investigate and understand mathematical content...", etc. Problems at different grade levels include examples such as the following: "I have some pennies, nickels, and dimes in my pocket. I put three of the coins in my hand. How much money do you think I have in my hand?" or "Students are given a carefully drawn picture of a roller-coaster track. The challenge is to sketch a graph (with no numbers) to represent the speed of the roller coaster versus its position on the track." How are these to influence the everyday conduct of mathematics classes? How does one integrate mathematical computation with such problems? Where does one go after presenting these examples? They may have some use for a very well prepared teachers and a very bright group of students, but they are likely to be of little help to the majority of teachers, teaching the majority of our children.
A graduate student just came by and we read him some examples from the NCTM volume. "Why those aren't standards," he said with some indignation. We agree. Compare the NCTM standards with the Japanese curriculum guidelines where provision of developmental and systematic cues for instruction are a main focus. Here are examples from the second grade:
Objectives: To help pupils deepen their understanding of the concept of number and notation. Furthermore, to help them understand the cases in which such calculations as addition, subtraction, and multiplication are applied, and to enable them to perform basic calculations.
To help pupils understand gradually the meaning of quantities such as length and volume, and their measurement, and to enable them to measure length and volume.
To help pupils to understand gradually the concept of fundamental geometrical figures and space.
And then the contents offer very specific guidelines for the mathematics skills expected of children at that grade level. For example:
For Number and Calculation.
To help pupils understand the concept of number and how numbers are represented, and to help them develop their abilities to use numbers.
To count objects by re-arranging them into groups of the same size or by classifying them. To classify and arrange simple matters and express them by using numbers. To know the way to represent numbers by the decimal numeration system, and the size and order of numbers up to 4-digit numbers. to help pupils deepen their understanding of addition and subtraction, and develop their abilities to use them. And so forth.
Japanese and Chinese teachers do not permit the everyday use of calculators or computers in mathematics classes because they want the students to understand the concepts and operations necessary for the solution of problems. Early reliance on these aids does not provide the necessary experience in manipulating concrete objects, an experience which helps children learn the basics of mathematics. Only at the high school level, after students have developed a clear understanding of mathematical concepts are East Asian students given the opportunity to use a calculator as a tool in solving mathematics problems.
Professor Harold W. Stevenson
Department of Psychology, University of Michigan
Co-author (with James Stigler)
The Learning Gap: Why Our Schools Are Failing
and What We Can Learn from Japanese and Chinese Education, (1992)
Director, Ethnographic Case Studies Project
Third International Mathematics and Science Study (TIMSS)
[Note: This passage from Dr. Stevenson developed in the context of email messages concerning the NCTM Standards]