Compared to students in most industrialized nations, U.S. students continue to score very low on standardized mathematics tests. Though our top students do very well (they rival the best students in any nation), our average students do not demonstrate an adequate level of mathematical competence.
The following is a general synopsis of the recommendations that have emerged from the literature about reforms needed in mathematics teaching:
1. Memorization cannot be the primary method of learning mathematics.
2. Learning math is a process of construction. Children build on what they know. We cannot simply transmit information to them.
3. Algebraic and geometric concepts should be integrated throughout the curriculum.
4. Every concept ought to be presented in a variety of ways.
5. Examples and problems should be based on the real life experiences of children.
6. Cooperative learning techniques are especially helpful in learning problem-solving skills.
Focus on mental computation and estimation
The call for reform in mathematics curricula and teaching techniques is not meant to suggest that we should eliminate teaching routine computational procedures or memorization of basic math facts. Advocates of reform, including James Hiebert and Terrence Coburn, stress that routine computational procedures need to be taught. However, Hiebert and Coburn also stress that these procedures need to be taught somewhat differently in the new mathematics curriculum.
While written computation dominated elementary mathematics in the past, its importance today has diminished significantly. Hiebert points out that several reports in the last decade have recommended a de-emphasis on paper and pencil alogorithms and an increased emphasis on problem solving. Both Hiebert and Coburn believe that the teaching of computation should now emphasize mental computation and estimation over written computation.
Basic facts and procedures
In order to think and talk about mathematical ideas and operations meaningfully, students need facility with basic facts and procedures. The ability to quickly recall the 390 basic facts in addition, subtraction, multiplication and division should, according to both Hiebert and Coburn, remain a goal in the elementary curriculum. The ability to perform routine mathematical procedures, they insist, is the foundation of mental computation and estimation, and is fundamental to developing mathematical understanding and problem solving competence, the ultimate goal of learning mathematics.
Nevertheless, there is room for improvement in the way we teach computation and in the timing of the introduction of computational skills in the curriculum. Before students are asked to manipulate written symbols or learn rules for written procedures (alogorithms), they need to develop a conceptual understanding of the procedure. Some research indicates that if students are asked to memorize before developing an understanding of the concepts underlying a procedure, it becomes more difficult for them to grasp the meaning later on.
Increased use of real-life situations
For this reason, there must be increased use of manipulatives, visual aids, and oral discussion of problems based on real life situations meaningful to students. Only subsequently should the memorization of facts and procedures and written practice begin.
Hiebert stresses that mental computation and estimation are highly practical in everyday life.
With computation and estimation skills, the mental effort required to solve problems is reduced, and this enables students to concentrate on meanings and relationships. These educators believe that a renewed emphasis on mental computation will enhance understanding of math concepts and higher order thinking skills.
Coburn suggests that drills in basic facts could be made more efficient by making practice sessions shorter and more frequent. He estimates that by learning approximately 5 new facts a week, students can master all 390 by the end of 4th grade. It is important, Coburn emphasizes, not to get bogged down in too heavy a drill program.
And, again, he reminds us to spend less time on written computation, especially long, multidigit problems which do not further a child’s understanding and can be done more quickly and accurately by calculator. Reallocating math time in this way should allow a greater amount of time for concept development, mental computation, estimation and practical applications.
Increased use of visual aids
Upper elementary and middle school students’ performance on standardized tests indicates that this group, too, needs more work with manipulatives and visual aids in order to develop conceptual understanding. This is particularly true for concepts involving fractions. An understanding of the relationship of parts to a whole ought to be well established before attempting manipulation of symbols and rules for written computation with fractions.
Although most students are able to memorize rules for manipulating written fractions, many lack a conceptual understanding of these procedures and this becomes apparent on achievement tests in which they are asked to demonstrate their understanding. Generally speaking, these students also need more practice understanding the relationship between ratios (expressed as fractions), decimals and percentages.
Textbook companies and standardized test manufacturers will gradually respond to the reforms taking place in mathematics teaching. But mathematicians stres that beginning immediately, we need to make these changes in our teaching of elementary arithmetic in order to produce students who truly understand the computations they perform. Only in this way can we ensure that students will have gained the firm foundation they will need to understand higher level mathematics.
“The Role of Computation in the Changing Mathematics Curriculum” New Directions for Elementary School Mathematics 1989 Yearbook of the National Council of Teachers of Mathematics; “The Role of Routine Procedures in the Development of Mathematical Competence” Teaching and Learning Mathematics in the 1990s 1990 Yearbook of the National Council of Teachers of Mathematics pp. 31-40.
Published in ERN November/December 1990 Volume 3 Number 5