Calculators and Constructivism

Calculators and ConstructivismAuthor(s): Grayson H. Wheatley, Douglas H. Clements and Michael T. BattistaSource: The Arithmetic Teacher, Vol. 38, No. 2 (OCTOBER 1990), pp. 22-23Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194690 .

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RESEARCH INTO PRACTICE

Calculators and Constructivism

conflicting views have emerged about the place of cal-

culators in elementary school mathematics. Some teachers and many par- ents believe that the use of calculators will undermine mastery of the "ba- sics" and thus should not be used, at least until students "know their facts" and are proficient with paper- and-pencil computations. Others suggest that in today's society, facility with calculators is essential. The Na- tional Council of Teachers of Mathe- matics has, for many years now, held the position that calculators should be used at all grade levels.

From a constructivist perspective (von Glasersfeld, in press), a calculator can aid mathematics learning when it -

1 . permits meaning to be the focus of attention;

2. creates problematic situations; 3. facilitates problem solving; 4. allows the learner to consider more

complex tasks; and 5. lends motivation and boosts confi-

dence.

The use of a calculator can present potential mathematics learning oppor- tunities by creating problematic situations. Consider the following two-per-

Prepared by Grayson H. Wheatley Florida State University Tallahassee, FL 32306 Douglas H. Clements State University of New York at Buffalo Buffalo, NY 14260 Edited by Michael T. Battista Kent State University Kent, OH 44242

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son activity. (Use nonscientific, algebraic-logic calculators; check to make certain this activity will work.) Guess my number Pat chooses 59 as a mystery number and enters [59] 0 [59] [=]. The calculator display shows "1." She hands the calculator to Wanda and asks, "Can you find my number? You may enter your guess and push the equals key. When you get "one" you have found my mystery number."

Wanda replied, "I'll guess forty." |40| F] [0.67796611

At this point, Wanda has little idea what all these digits mean. Her "problem" is giving meaning to the display of digits to achieve her goal. She next tries 90.

[90] R [1.52542371 To Wanda, this looks like a bigger number. She follows this trial with the following entries.

7Õ|F1 [1.1864406 Щ [g [1.0169491 5Ü| R [0.8474576 56] R 10.9491525 59]ВШ

Through this game, Wanda was able to give meaning to strings of digits showing on the calculator's display. The two numbers on each side of the "dot" took on special significance while the other digits were ignored. Successive experiences playing this game would deepen Wanda's concept of decimals. (This activity works because the calculator is storing the op- eration + 59 and performs that oper- ation on each of Wanda's entries.)

Jenny, a third-grade student, re- sponded to the task shown in figure 1 by adding a number to 85 and 63 until she found one that gave 244 as a sum. For her this task was not one for which the subtraction key on a calculator could be used. She entered the addends in the order they appeared on the pages from left to right, 85 (her number) and 63. In just a few minutes she was able to determine the number that gave the required sum. The calculator was useful in freeing Jenny from computational rules and allowed her to focus on numerical relationships. In fact, Jenny did not have a reliable method for determining differ- ences with three-digit numerals.

Patrick, another student in the same

ARITHMETIC TEACHER



class, first tried to solve this problem without a calculator but soon asked, "Is there any way the calculator can do 148 plus how much will give 244?" In asking this question, he was taking a first step in relating a how-many- more question to a subtraction task. When he learned how to use the subtraction key to achieve his goal, Patrick was enthusiastic about this newfound capability and proceeded to try other subtractions. For each of these students, the calculator created a problematic situation and led to meaningful mathematical activity.

Considerable time is required for students to construct action schémas that include calculator use (Wheatley and Wheatley 1982). Students have been observed doing long division by paper and pencil while a calculator was lying near their hand. This situation is analogous to a carpenter's put- ting a power saw in the toolbox but continuing to use a familiar hand saw. With calculator availability and op- portunities to use a calculator, students will slowly build patterns of action that include reaching for a calculator. Activities such as the "range game" facilitate this transi- tion.

The "range game" The range game (Reys et al. 1979) is a versatile calculator activity that can be used at any grade level. It has proved to be a powerful experience in building number sense. The task shown in figure 2 is to find a number that when multiplied by 75 will give a result in the range from 400 to 625. Students are instructed to enter 75 in their calculator and choose a number they think will put them in the range. Ricardo chose 10 and found it was too large. He then tried 6, arriving at 75 x 6 = 450. But other numbers besides 6 will work. Students are encouraged to find all numbers (thinking whole numbers) that will satisfy the condition. The question "What is the smallest number that works?" can generate an interesting discussion. As students begin trying decimals such as 5.4, getting 405, and then 5.35 getting 401.25, they have an opportunity to construct meaning for decimals and develop

OCTOBER 1990

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number sense in the way the term is used in the NCTM's Curriculum and Evaluation Standards for School Mathematics {Standards) (1989).

Because calculators display quotients in decimal form, students may be productively perturbed when using a calculator. What does 3.456138 mean? In solving the problem "One bus can carry 34 students. How many buses will be required to carry 489 students?" Julie divided 489 by 34 on a calculator, looked at the display, saw a string of numbers with a dot, and exclaimed, "Goodness!" She quickly put the calculator down and did the division using the standard paper-and-pencil procedure. Being unfa- miliar with a calculator, she expected the display to show her the type of information she would have attained if paper-and-pencil methods were used. She was unprepared for a decimal form and did not give it numerical meaning. A calculator fosters rich op- portunities to reason mathematically, in this example to reflect on quotients expressed in decimal notation. The calculator activities "guess my number" and "range game" described in the foregoing offer just such opportu- nities.

In the 1990s, the view of mathematics as a fixed set of procedures to be mastered by practice is giving way to a different conceptualization, that is, mathematics as the activity of con- structing patterns and relationships. Calculators can play an important role in students' construction of mathematical relationships. However, we must think of their use in a quite different instructional setting and with activities markedly different from ex- plain-and-practice lessons or even so-

called discovery teaching where the teacher directs students to a principle to be learned. Students must be free to construct meaning in ways that make sense to them. Shifting attention from carrying out tedious paper-and- pencil computations to solving prob- lems and building relationships results in conceptually powerful learning. When used in a learning environment compatible with constructivism (Cobb et al. 1988) as illustrated in this article, calculators can facilitate students' construction of number patterns that form the basis of mathematical rea- soning by allowing many complex computations to be performed quickly and accurately while attention re- mains focused on meaning. The activities described herein suggest how calculators can be used to achieve these goals. References

Cobb, Paul, Erna Yackel, Terry Wood, and Grayson Wheatley. "Research into Practice: Creating a Problem-solving Atmosphere.11 Arithmetic Teacher 36 (September 1988):46- 47.

National Council of Teachers of Mathematics, Commission on Standards for School Mathe- matics. Curriculum and Evaluation Stan- dards for School Mathematics. Reston, Va.: The Council, 1989.

Reys, Robert, Barbara Bestgen, Terrence Coburn, Harold Schoen, Richard Shumway, Charlotte Wheatley, Grayson Wheatley, and Arthur White. Keystrokes: Multiplication and Division. Palo Alto, Calif.: Creative Pub- lications, 1979.

von Glasersfeld, Ernst. "An Exposition of Con- structivism: Why Some Like It Radical." In Constructivist Views on the Teaching and Learning of Mathematics. JRME Mono- graph. Reston, Va.: National Council of Teachers of Mathematics, forthcoming.

Wheatley, Grayson, and Charlotte Wheatley. Calculator Use and Problem Solving Strate- gies of Grade Six Pupils. Final Report. Wash- ington, D.C.: National Science Foundation, 1982. V

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Calculators and Constructivism