Speaking Out for Calculators
l was interested in Patricia Hollingsworth's comments on Kumon Mathematics (A Reformer's 'Retrogression': Speaking Out for Kumon Mathematics," Commentary, Nov. 27, 1991), especially since her view of the use of calculators seemed to preclude the need for drill and practice or for memorization.
It is certainly true that the mathematics reform movement suggests placing less emphasis on paper-and-pencil computation using the standard algorithms (long division, long multiplication) and more emphasis on reasoning (when should we divide, multiply, add, or subtract?). But this does not mean that we no longer expect students to memorize their basic number facts.
In fact, if we want students to feel comfortable with numbers, they need to have control of a certain body of facts, the very minimum of which would be the addition, subtraction, multiplication, and division facts up through, say, the 10's or 12's. Keep in mind, though, that having properly learned them, students realize that the subtraction facts are derived from the addition facts, the division facts derived from the multiplication facts.
What the new thinking on learning is saying is that we need to help students see why the facts work (using manipulatives), and why we need to know them (relating them to real-world problems). Finally, as students use facts to do more sophisticated computation, we need to encourage their involvement in the process (for example, having students construct their own algorithms).
So, where do calculators come into the picture? First, calculators can be used to help students construct their own knowledge by allowing them to explore--within the construct of a mathematical lessen or on their own. For example, by using the constant feature on a simple four-function calculator, students can "skip count" by say, 3's:
This experience with repeated addition helps them better understand the multiplication fact: 7 x 3= 21. Calculators can also be used to perform computations that would be possible to do by paper and pencil, but whose completion would have little purpose other than the process itself. For example, I have given the following problem to various audiences:
For many, it takes several minutes to work out the problem, and I invariably get a distribution of at least 10 different answers in a group of 30 people. Why not just use a calculator?
Yet, using such an example, we can see that we still want students to know that the answer should be about 6000 x 7000, or 42,000,000. This is done mentally and requires use of a basic fact, an understanding of place value, a feel for estimation, and an understanding of multiplying by powers of 10 (10's, 100's, and 1000's).
To learn this number sense does take experience-- experience that may come in the form of drill and practice, students talking and writing about their mathematical observations or conjectures, and students being tested (assessed) on their understanding. As Ms. Hollingsworth noted in her essay, mathematics is sequential in that skills and terminology learned at one level are applied or extended at the next level. Math learning does not always come easy, and it may require much more time for some students than it does for others. This is not unlike learning in other subjects (for example, music or speaking a foreign language). Calculator use should be a part of this learning process.
Recently, I worked with a group of high school, junior- level girls in a special summer project called Miss, for Mathematics Intensive Summer Session. The girls were selected because they were college bound, were doing well in other courses such as English and social studies, but were failing or doing poorly in math. Some had failed Algebra II, some had done poorly in Algebra I and Geometry but had not yet taken Algebra II. The purpose of MISS was to see if we could turn them around with a month-long seminar experience and prepare them to succeed in Algebra II the following school year, when they all would be seniors.
The students studied math from 8 in the morning to 3 in the afternoon on the campus of California State University at Fullerton during the month of July. They were free to use a sophisticated graphing calculator at any time and were taught to use the calculator's powers functions as they were needed in the course.
Now, as Patricia Hollingsworth pointed out, girls are especially vulnerable to dropping out of math, and research has shown that they tend to give way to boys when using technology like computers. But in the MISS project, we found that the biggest boost to girls' confidence came from their ability to master this sophisticated calculator and use its power to help them understand the math. What is needed are model lessons to help teachers get a sense of this benefit and learn the proper classroom use of such technology.
We as educators need to use calculators in a way that enhances mathematical learning. Calculators will not replace the need for number sense and for learning basic mathematical facts, from multiplication to the definition of a trigonometric function. But calculators can be used to help students explore and analyze mathematical relationships, compute, and graph functions. This power exists and we cannot ignore its potential as a teaching tool.
David L. Pagni is a professor of mathematics at California State University, Fullerton. He is co-director of the Calculators and Mathematics Project, Los Angeles (CAMP -LA), a National Science Foundation-supported project that has developed model mathematics lessons in grades K-8 that make use of the calculator.