Descriptive Statement

Numerical operations are an essential part of the mathematics curriculum. Students must be able to select and apply various computational methods, including mental math, estimation, paper-and-pencil techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, and other kinds of numbers. With calculators that perform these operations quickly and accurately, however, the instructional emphasis now should be on understanding the meanings and uses of the operations, and on estimation and mental skills, rather than solely on developing paper-and-pencil skills.

Meaning and Importance

The wide availability of computing and calculating technology has given us the opportunity to significantly reconceive the role of computation and numerical operations in our school mathematics programs. Up until this point in our history, the mathematics program has called for the expenditure of tremendous amounts of time in helping children to develop proficiency with paper-and-pencil computational procedures. Most people defined proficiency as a combination of speed and accuracy with the standard algorithms. Now, however, adults who need to perform calculations quickly and accurately have electronic tools that are more accurate and more efficient than any human being. It is time to re-examine the reasons to teach paper-and-pencil computational algorithms to children and to revise the curriculum in light of that re-examination. Mental mathematics, however, should continue to be stressed; students should be able to carry out simple computations without resort to either paper-and-pencil or calculators. Fourth-graders must know the basic facts of the multiplication table, and seventh-graders must be able to evaluate in their heads simple fractions, such as What's two-thirds of 5 tablespoons?

K-12 Development and Emphases

At the same time that technology has made the traditional focus on paper-and-pencil skills less important, it has also presented us with a situation where some numerical operations, skills, and concepts are much more important than they have ever been. Estimation skills, for example, are critically important if one is to be a competent user of calculating technology. People must know the range in which the answer to a given problem should lie before doing any calculation, they must be able to assess the reasonableness of the results of a string of computations, and they should be able to be satisfied with the results of an estimation when an exact answer is unnecessary. They should also be able to work quickly and easily with changes in order of magnitude, using powers of ten and their multiples. Mental mathematics skills also play a more importantrole in a highly technological world. Simple two-digit computations or operations that involve powers of ten should be performed mentally by a mathematically literate adult. Students should have enough confidence in their ability with such computations to do them mentally rather than using either a calculator or paper and pencil. Most importantly, a student's knowledge of the meanings and uses of the various arithmetic operations is essential. Even with the best of computing devices, it is still the human who must decide which operations need to be performed and in what order to answer the question at hand. The construction of solutions to life's everyday problems, and to society's larger ones, will require students to be thoroughly familiar with when and how the mathematical operations are used.

The major shift in this area of the curriculum, then, is one away from drill and practice of paper-and-pencil procedures and toward real-world applications of operations, wise choices of appropriate computational strategies, and integration of the numerical operations with other components of the mathematics curriculum. So what is the role of paper-and-pencil computation in a mathematics program for the year 2000? Should children be able to perform any calculations by hand? Are those procedures worth any time in the school day? Of course they should and of course they are.

Most simple paper-and-pencil procedures should still be taught and one-digit basic facts should still be committed to memory. We want students to be proficient with two- and three-digit addition and subtraction and with multiplication and division involving two-digit factors or divisors, but there should be changes both in the way we teach those processes and in where we go from there. The focus on the learning of those procedures should be on understanding the procedures themselves and on the development of accuracy. There is no longer any need to concentrate on the development of speed. To serve the needs of understanding and accuracy, non-traditional paper-and-pencil algorithms, or algorithms devised by the children themselves, may well be better choices than the standard algorithms. The extensive use of drill in multi-digit operations, necessary in the past to enable people to perform calculations rapidly and automatically, is no longer necessary and should play a much smaller role in today's curriculum.

For procedures involving larger numbers, or numbers with a greater number of digits, the intent ought to be to bring students to the point where they understand a paper-and-pencil procedure well enough to be able to extend it to as many places as needed, but certainly not to develop an old-fashioned kind of proficiency with such problems. In almost every instance where the student is confronted with such numbers in school, technology should be available to aid in the computation, and students should understand how to use it effectively. Calculators are the tools that real people in the real world use when they have to deal with similar situations and they should not be withheld from students in an effort to further an unreasonable and antiquated educational goal.

In summary, numerical operations continue to be a critical piece of the school mathematics curriculum and, indeed, a very important part of mathematics. But, there is perhaps a greater need for us to rethink our approach here than to do so for any other component. An enlightened mathematics program for today's children will empower them to use all of today's tools rather than require them to meet yesterday's expectations.

NOTE: Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.