TEACHING MATHEMATICS WITH TECHNOLOGY: Using Calculators inMathematics Changes TestingAuthor(s): John G. Harvey and George W. BrightSource: The Arithmetic Teacher, Vol. 38, No. 7 (MARCH 1991), pp. 52-54Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194819 .

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TEACHING MATHEMATICS WITH TECHNOLOGY

52 ARITHMETIC TEACHER

Prepared by John G. Harvey, University of Wisconsin, Madison, WI 53706

division; and memory, percent, square root, exponent, reciprocal, and ± keys" (p. 68).

Instruction and Assessment Mathematics instruction and assessment are just different sides of a single coin. Thus, if we teach students to use calculators as tools while learning, solving problems, and applying mathematics, we should expect that those students will need to use their calculators when their learning is being assessed- * and vice versa. Therefore, as we rethink the ways that we teach mathematics so as to integrate calculators effectively, we must also rethink the kinds of test questions that we give students. Certain questions on our present tests will need to be eliminated when *

testing calculator-using students. Some of these questions will no longer be appropriate because they have become calculator dependent, that is, they test only students' abilities to manipulate their calculators and not their knowledge of mathematics. Figure 1 shows two examples of these kinds of questions. One of these questions is calculator dependent no matter what calculator is used; the other is calculator depend- ent when an algebraic-logic calculator (e.g., a scien- tific calculator or the Tl Math Explorer) is used.

Using any calculator, to get a correct answer to the first example in figure 1 one simply enters the first number (34), presses the 'x', enters the second num- ber, and, finally, presses the Q (i.e., the keystroke sequence is 34 [x] 106 '='). The second question is equally calculator dependent for students in grades 5-8 who have a calculator like that specified for these grades by the NCTM's curriculum standards, since these calculators permit students to enter the problem into the calculator exactly as given, including the pa- rentheses. The second example in figure 1 is not calculator dependent, however, when students have a calculator with arithmetic logic. Since an arithmetic- logic calculator does not include parentheses, students

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Using Calculators in Mathematics Changes Testing

matter the level at which we teach mathemat- ics, we are being asked to incorporate calcula-

tors into our instruction, to teach students both calcula- tor facility and effective ways of using calculators, and to encourage and expect those students to use calcula- tors appropriately. As early as 1 975, just three years after the introduction of Texas Instruments's Data Math calculator, the National Advisory Committee on Mathematical Education (NACOME) urged that calculators be used in mathematics instruction (NACOME 1975, 40-43). Five years later the National Council of Teachers of Mathematics recom- mended that "mathematics programs [should] take full advantage of calculators ... at all grade levels" (NCTM 1980, 1). In 1986 the NCTM's Board of Directors recommended that all students use calcula- tors to-

• concentrate on the problem-solving process rather than on the cal- culations associated with problems;

• gain access to mathematics beyond the student's level of computa- tional skills;

• explore, develop, and reinforce concepts including estimation, computation, approximation, and properties;

• experiment with mathematics ideas and discover patterns; and • perform those tedious computations that arise when working with

real data in problem-solving situations.

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1 989) expects that students will use calculators. The curriculum standards do not describe the functionality of the calculator to be used in grades K-4; but for grades 5-8, they state that students should have a calculator that includes "alge- braic logic including order of operations; computation in decimal and common fraction form; constant function for addition, subtraction, multiplication, and

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Examples of calculator-dependent questions

1. 34 x 106= 2. 12- (8 -2(4-3)) =_

using those calculators would have to pay attention to the order of operations indicated by the parentheses in the problem to compute the answer correctly. A correct sequence of keystrokes for the second example is

Some of our present test questions may be inappro- priate because the curriculum standards suggest that less attention be paid to achievement of the content tested by them. For example, the grade K~4 curricu- lum standards propose that decreased attention be paid to long division and long division without remain- ders (p. 21). Thus, to achieve these standards we will need to eliminate at least most, if not all, the test questions about long division in these grades. In grades K-4 where students are using four-function calculators, long-division- without-remainder problems could not appear on tests on which students are permitted to use their calculators, since these problems are calculator dependent; but long-division-with- remainders problems could appear if students have only simple, four-function calculators, since computing the whole-number quotient and remainder with these calculators does, I think, test students' ability to find the quotient and remainder, though the usual algo- rithm for long division is not tested. To do these problems students would first divide and get a decimal answer; then they would form the product of the divisor and the integral part of the quotient and subtract that product from the dividend to obtain the integral remainder. If, as recommended, students in grades 5-8 have calculators like the Tl Math Explorer, long-division problems could not appear on tests for which students are permitted to use calculators, since these calculators have an [INT-Η that computes both the whole-number quotient and remainder. Thus, it may be necessary, at times, to test our students in two environ- ments: with and without calculator use permitted. When we do not permit students to use their calcula- tors on tests, we should be sure that (a) the content being tested has not been taught using calculators and (b) the paper-and-pencil skills and algorithms tested

are ones that students still need to know and that they have been taught.

Calculator-based Tests "'Choosing whether or not to use a calculator when addressing a particular test question is cm important skill. Thus, not all questions on calculator-based mathematics achievement tests should require the use of a calculator* (Kenelly 1989, 47). If this conclusion is valid, then as we revise our present tests and generate new ones, we will need to make sure that our tests include, but are not completely comprised of, calculator-active questions; these tests от typically being called calculator-based tests. The problems on tests that require calculator usé are being called calculator-active items. Determining whether a test question is calculator active is a matter of judgment and may be somewhat tricky. Here is the author's definition of a calculator-active item:

A calculator-active test item is one that (a) contains data that can usefully be expbred and manipu- lated using a calculator and (b) has been designed so as most likely to require calculator use (Harvey 1989).

In grades K-8, calculator-active test questions will probably involve students in solving applications problems or searching for patterns and relationships. Here are some examples of calculator-active problems based on problems and suggestions found in the NCTM's curriculum standards.

Example I. 1 have four coins; each coin is either a penny, a nickel, a dime, or a quarter. a) If altogether the coins are worth a total of forty-one

cents, how many pennies, nickels, dimes, and quarters might I have? Is more than one answer possible?

b) What could be the maximum value of the four coins? The minimum value?

c) If I have exactly one penny and one dime, what could be the maximum value of the four coins? The minimum value?

d) What are all the possible total values of the four coins?

Example 2. Stevens School is going to have an all- school picnic. The fourteen teachers at Stevens spend $3 1 1 .50 to buy soft drinks for the picnic; they want to share equally the cost of the soft drinks. One of the teachers used a calculator to determine that each

Continued on next page

MARCH 1991 53

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teacher should pay $44.50. Is this answer correct? Explain.

Example 3, At a discount store, a six-pack of bottles of SuperGood soft drink costs $1 .29 and a six- pack of bottles of VeryFine soft drink costs $1 .33.

a) If you must buy at least three six-packs of VeryFine, how many six-packs of SuperGood and VeryFine can you buy with $15.00?

b) How many six-packs of SuperGood and six-packs of VeryFine should you buy so that the total cost of the soft drinks purchased is closest to, but not greater than, $17.50?

Example 4. Explain a rule that generates this set of numbers:

. . . , 0.015625, 0.0625, 0.25, 1,4, 16, 64, 256, 1024,4096,...

Example 5. See figure 2*

Example ó. Seventy-three students from your school are going on a field trip. The company from which you are going to charter vans for the field trip has five- passenger, eight-passenger, and twelve-passenger vans. a) If all the vans you charter must be of the same size,

how many vans will you charter and how large will each one be so that you have the smallest number of vacant seats?

b) If the charter fee for a five-passenger van is $35,

for an eight-passenger van is $50, and for a twelve- passenger van is $72 and if all the vans you charter must be of the same size, how many vans of what size will you charter so that the transportation costs of the field trip are the smallest possible?

Conclusion The examples of calculator-active items presented here will give teachers some ideas of how to generate questions of their own. As teachers discover ways of effectively integrating calculators into their own instruc- tion, it should not be difficult for them to make up problems that will accurately assess their students' learning and that will permit students to use calculators in the same ways as they have during instruction.

References

Harvey, John G. "What about Calculator-based Placement Tests?" АМАПС Review 1 1 (Fall 19891:77-81 .

Kenelry, John W. The Use of Calculators in the Standardized Testing of Mathematics. New York: College Board and Mathematical Association of America, 1989.

National Advisory Committee on Mathematical Education (NACOME). Overview and Analysis of School Mathematics Grades K-12. Washington, D.C.: Conference Board of the Mathematical Sciences, 1975.

National Council of Teachers of Mathematics. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, Va.: The Council, 1980.

National Council of Teachers of Mathematics, Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. 9

** ARITHMETIC TEACHER

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