Chapter 8 Standard 11

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

CHAPTER 8: THE TEN CONTENT STANDARDS

STANDARD 11: NUMERICAL OPERATIONS

All students will develop their understanding of numerical operations through experiences which enable them to construct, explain, select, and apply various methods of computation including mental math, estimation, and the use of calculators, with a reduced role for pencil-and-paper techniques.

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 pencil-and-paper computational procedures. Most people defined proficiency as a combination of speed and accuracy with the standard algorithms. Now, though, the societal reality is that adults who need to perform calculations quickly and accurately have electronic tools that are both more accurate and more efficient than any human being. It is time to re-examine the reasons to teach pencil-and-paper computational algorithms to children and to revise the curriculum in light of that re-examination.

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, they should be able to work quickly and easily with changes in order of magnitude, and they should be able to be satisfied with the results of an estimation when an exact answer is unnecessary. Mental mathematics skills, too, play a more important role 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 these types of computations to do them mentally rather than looking for either a calculator or pencil and paper. And, probably of greatest importance, a students knowledge of the meanings and uses of the various arithmetic operations is still an essential concern. Even with the best of computing devices, it is still the human who must decide which operations need to be done and in what order to answer the question at hand. The construction of solutions to lifes everyday problems, and to societys larger ones, will require students to be thoroughly familiar with the mathematical operations and processes that are available.

The major shift in this area of the curriculum, then, is one away from drill and practice of pencil-and-paper 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 pencil-and-paper 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 pencil-and-paper 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 pencil-and-paper algorithms, or algorithms devised by the children themselves, may well be better choices than the standard algorithms, which were built mostly for speed. The excessive use of drill, necessary in the past to develop reflexive automaticity, is no longer necessary and should play a much smaller role in todays curriculum.

For procedures involving even 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 pencil-and-paper 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 there is to do so for any other component. An enlightened mathematics program for todays children will empower them to use all of todays tools rather than require them to meet yesterdays expectations.