The AAN Calculator: A Tool for Enhancing Instruction

The AAN Calculator: A Tool for Enhancing InstructionAuthor(s): Charles B. Vonder Embse, Frederick C. Whiteman, Suzanne K. Damarin and JamesR. C. LeitzelSource: The Arithmetic Teacher, Vol. 35, No. 7 (March 1988), pp. 12-17Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193362 .

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The AAN Calculator: A Tool for Enhancing Instruction

By Charles B. Vonder Embse, Frederick C. Whiteman, Suzanne K. Damarin, and James R. C. Leitzel

The utility of the hand-held calculator as an instructional tool is evident to the many teachers who use it. The usefulness and effectiveness of hand- held calculators in mathematics teaching is affirmed by the research litera- ture (Suydam 1982; Hembree 1986). Nonetheless no learning tool, however general its uses and purposes might be, can serve all functions for all learners. The article The Inverse Tool Principle (Belland 1985) implies that novices often should be given more sophisticated learning tools than are needed by those who have become more expert in a field.

The Approaching Algebra Numeri- cally (AAN) Calculator is a computer- based simulation of a hand-held calculator designed with novice learners in mind. Although the hand-held calculator is a useful tool for teaching numerical concepts and generalizations, the novice learner may have trouble with certain aspects of its use. In particu- lar, the AAN Calculator allows students who are new to, or uncomfort- able with, the careful recording, reading, and revising of complex quantitative expressions to enter,

Charles B. Vonder Embse is an assistant professor of mathematics at Central Michigan University. Frederick C. Whiteman is a gradu- ate student pursuing his doctoral degree in education at Ohio State University. Suzanne K. Damarin is an associate professor of education and James R. C. Leitzel is an associate professor of mathematics at Ohio State University. This team has written curriculum materials and developed and written software for the computer version of the AAN materials. Together they have a total of forty-eight years of teaching experience at the elementary, middle school, secondary, and college levels.

read, and edit such expressions as well as to evaluate them. The AAN Calculator allows several instructional approaches that the hand-held calculator does not. Errors of entry can be detected; expressions can be com- pared directly to identify causes for different values; new values can be substituted directly into expressions to investigate the effects of change.

The AAN Calculator can be used by individual students, by groups of students, or as an audiovisual aid by teachers. It can serve as an ideal environment for introducing students to the use of the calculator and for helping students clarify what's hap- pening in the "black box" called the calculator. Although the AAN Calcu- lator was developed to accompany the Approaching Algebra Numerically curricular materials (Leitzel, De- mana, and Osborne 1986), it was designed to be useful and effective for many types of instructional programs.

Description of the AAN Calculator The AAN Calculator currently runs on any Apple He or He computer and operates in the same manner as a standard hand-held calculator with algebraic logic. The four basic mathematical operations are denoted by Applesoft basic computer symbols: + (addition),- (subtraction), * (multiplication), and / (division). Other mathematical operations, such as exponentiation, square root, and logarithms, are also available with the AAN Calculator. To evaluate a mathematical expression, the user presses

either the equals key (" = ") or the return key. Any partial expression can be evaluated if it is mathematical- ly complete - that is, contains no open parentheses or incomplete operational sequences.

The AAN Calculator is an ideal tool for teaching mathematics and problem solving to students because it over- comes some of the shortcomings of the hand-held calculator. When a user keys a mathematical expression into a standard hand-held calculator, only the entered digits and some partial computational answers are seen before the answer appears. The user must remember the series of numbers, operations, and parentheses that was entered. Often in complex computa- tions, mistakes occur because incorrect keystrokes are unintentionally entered. The AAN Calculator has an expression display window that shows the mathematical expression, including numerals, operation symbols, and grouping symbols. The cor- rectness of the keying sequence can be checked before the answer is calculated. An incorrect entry of a mathematical expression can occur with the AAN Calculator, but the teacher and student can see where the mistake was made without guessing what expression was actually keyed. On a hand-held calculator, an incorrect keying sequence is not indicated as long as no improper mathematical operation is attempted.

Mathematical expressions entered on the AAN Calculator may be up to 240 characters long. The expression display window shows up to 30 characters at once and will scroll to show

12 Arithmetic Teacher



the remainder of the expression. Quick movement through long expressions is accomplished with a jump command that can position the cursor at the beginning or end of the expression. Figure 1 shows the screen for- mat of the AAN Calculator with one of the help screens visible.

If a keying sequence is entered into a hand-held calculator, changing any part of it means reentering the entire keying sequence. The AAN Calcula- tor, however, allows the user to edit a mathematical expression by inserting or deleting characters. Once the changes are made, the expression can be reevaluated immediately by pressing the return key, with the new answer appearing in the result window. The immediacy of calculation and the visible expression display make the editing features efficient.

Experimentation using the AAN Calculator can result in discovery learning. When experimenting with different versions of a mathematical expression, the user can change one element of the expression at a time to show the effect on the final answer. For example, the expression 5 + 2*4 could be edited to (5 + 2) * 4 to show the effect of grouping with parentheses to perform the addition before the multiplication (see figs. 2[a] and 2[b]).

After a mathematical expression or partial expression is entered into the expression display window of the AAN Calculator, the user can choose to store that expression for future use or comparison. When an expression is stored in memory, the message "Expression in Memory" appears on the screen. After clearing the expression display, the stored expression can be recalled for use. If the expression display is not cleared and an expression in memory is recalled, it is inserted into the displayed expression at the cursor position. Recall of a partial expression can be used to build longer expressions for computational examples. For example, if the partial expression

" + 12" were stored, it could be recalled repeatedly to build the expression "12 + 12 + 12 + 12 + 12 + 12." This type of computation could be used to show that repeated addition is the same as multiplication. Also, an "exchange expression" op-

Fig. 1 "The AAN Calculator" with the "Edit Keys" help page displayed

Fig. 2(a) An example showing an expression that has been entered and evaluated

Fig. 2(b) The expression from figure 2(a) has been edited to show the effects of grouping using parentheses. (Note that the original result has been transferred to memory for comparison.)

eration exchanges the position of the displayed expression with the stored expression. This feature helps com- pare two expressions or work two- part problems. For example, when comparing the size of the two frac-

tions 16/75 and 61/300, the expression "16/75" could be exchanged with the expression "61/300" to show the relationship between the expression and the calculated decimal values.

The result window shows the value

March 1988 13

Edit Keys

«- Move left -» Move right I Set Insert mode D Delete a character d 1 Display start of Expression á 9 Display end of Expression Space Bar Clear Insert mode

I Expression Display | 1 Memoru 1 1 Result I

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IC) + 2)*4 I I 131 I 28]

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of the mathematical expression. It can be cleared by simply pressing the "C" key. Any time the equals (" = ") key or the return key is pressed, the result window shows the value of the mathematical expression appearing in the expression display window. If the expression being evaluated is mathe- matically incomplete or incorrect, the message "ERROR" appears in the result window for a few moments, and then the cursor moves to the beginning of the expression for editing.

The AAN Calculator also has the ability to store numerical values in memory. The numerical- value memory is similar to that on most hand-held calculators except that the value stored is visible at all times (see fig. 2[b]). This fact means that the user does not have to erase the displayed value to see what numerical value is in memory, as is necessary with many hand-held calculators. The stored value can be recalled for use in the displayed mathematical expression, or it can be used for comparison or target purposes as a visible reminder of a value being sought.

Operation of the AAN Calculator is relatively simple and can be mastered in a short time by middle school students. Help screens are built into the program to assist with the operation of the program. Any help screen can be selected and left in view as compu- tations are done. Changing from one help screen to another does not affect the currently entered expression or memory values. One choice results in no visible help screen, if that is desired. A "Quick Card" is provided with the program for easy reference to all the various program functions.

Use of the AAN Calculator The AAN Calculator has been used primarily with the course materials developed for the AAN project. How- ever, the AAN Calculator is a utility program that can be used in many instructional situations where students would benefit from the constant display of a mathematical expression and the editing features of the program. Described subsequently are

Fig.3

1 13*8+208/4+91 1 1 "ÏÏ1 I 1Ь5]

Fig. 4

ll 3*8+208/(4+9)1 1 1651 ÇZ 1201

Fig. 5

ll/2+4/7É . . I I ~51 I 1,071 426571 [

Fig. 6

|(l/2) + (4/7)1 | 1 Õ] I 1.07142Ô57]

some of the situations from the AAN Middle School Mathematics Project that take advantage of the special features of this computer program for the teaching of mathematical concepts and skills.

Order of operations

Many activities in the seventh-grade chapters on order of operations are based on regrouping elements in a mathematical expression to get different numerical results. This process is clearer when the mathematical expression can be seen and edited on

the AAN Calculator without com- pletely reentering the expression, as is necessary with a hand-held calculator. The expression shown in figure 3 can be edited so that 208 is divided by the sum of 4 and 9 (see fig. 4). The numerical result of the first computation can be stored in the visible numerical memory box. When the second result is computed, the two answers are visible side-by-side for easy comparison. The editing process also works well for showing when grouping symbols are not needed. For example, adding parentheses to the expression shown in figure 5 to get the

14 Arithmetic Teacher



expression shown in figure 6 does not change the numerical result.

The visible expression display and editing features of the AAN Calcula- tor are ideal for exploring the use of grouping symbols. Students are asked to edit the expression

112-5*7 + 3*29/25 + 4/2

so that the computed result is 40. Inserting and removing grouping symbols with the editing features of the AAN Calculator makes this a high- interest problem for teaching order of operations see (figs. 7[a] and 7[b]).

Exponentiation

Exponential growth is a new topic for most seventh-and eighth-grade students. This concept is presented through problems showing period-by- period growth of money, human populations, and bacteria populations. The visible expression display of the AAN Calculator helps explain this difficult concept because students can see how the growth in each period becomes exponential. Extension of the growth to more than one year can be done in two steps. First, the ex- panded yearly growth can be shown as the total at the end of the first year multiplied by the same growth rate for the second year (fig. 8). By the use of order-of-operations rules, the expression in figure 8 can be edited on the screen to become the one shown in figure 9 or, if exponents are used, the expression in figure 10. At each step of the process, the numerical result can be displayed in the result window to show that the changes made did not affect the result. This exploration could continue for more growth peri- ods. For example, the mathematical expression "(1 +.08)" could be stored in the expression memory for repeated recall to build the exponential expression "1200 * (1 +.08) * (1 +.08) * (1 +.08)." By changing the " + " sign to a "-" sign, the user can change the problem situation to explain exponential decay found in problems describing population de- crease or depreciation of value. Using the AAN Calculator in this demon- stration mode makes the abstract concepts more concrete and believable

Fig. 7(a) Original expression calculated

11 1 2-5»7+3*29/2'5+4/2l 1 I "ÏÏ1 I &2.4d1 [

Fig. 7 (b) Grouping symbols used- the result is 40.

Ь 12-C5»7)+3*(29 /(25+4)))/2ll Q] I 40]

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Fig. 8

01 I 1399.6S1 ■ ■ I ■ I ■ ■ I ■ I ■ I ■ I . ■ I ■ I I ■ ■■■■■■■■■■■■■■ I ■■.'.■.'.'■■■■■ I ■■■ ==

Fig. 9

ll200»(l + 0fiMl+.0fl)B J Ol I 1399T6Õ1

- I - ■ - ■ - ■ - ' - ' I ' ' ' ' - ■ ' ' ' - ' - ' ' ' ' - ' - ' - tr^

Fig. 10

ll200«(l+.0fi)~2l I O| I 13995ÔI [

1 . 1 . ■ . 1 . ■ . ■ - ■ ■ ■ ■ ■ .■-■-■-■-■.'■■■■■■.■■'■■■■ ■■■■■' ■'■'«'■'■'■'■' j '

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for students who traditionally think mathematics consists mainly of algo- rithms. Such a gentle break with tradi- tion is the bridge we hope to build to ease students' transition to algebra.

Exercises involving exponential growth are excellent examples of the effective use of the AAN Calculator for teaching problem solving. A straightforward problem of finding the

March 1988 15



Fig. 11

Fig. 12

Fig. 13 Answer to "6000 = 2000 (1 + .075)"" correct to eight decimal places

total amount of money on deposit after ten years at an interest rate of 7.5-percent is easily calculated. The opposite problem is not so easy. Con- sider this problem: How long will it take for money to triple in value if $2000 is invested at 7.5-percent interest? The answer to the problem is the numerical value of the exponent N in the general formula

6000 = 2000 (1 + .075)".

Traditionally, some problems at the middle school level can be solved simply by doing the inverse mathematical operation. For example, percent problems like "What is 25 percent of 80?" are solved by multiplication, whereas the problem "20 is what percent of 80?" is solved by division. However, using the inverse mathematical operation with exponential growth problems would in- volve the use of logarithms. But, with the AAN Calculator, this reverse exponential problem becomes accessi-

ble for students at this level without using logarithms. Using a guess-and- check problem-solving strategy with the AAN Calculator is a very efficient means of solving the problem. An evaluation of the expression in figure 1 1 shows the choice of ten years to be too small. Editing the 10 to 15 reveals a better estimate, shown in figure 12. Now the answer is close to $6000. The guess-and-check procedure can be continued to get an answer as close as desired, even to eight decimal places (see fig. 13). As an added feature, the target value can be displayed in the memory box for easy comparison.

Scientific notation

Scientific notation is a concept taught in both the seventh-and eighth-grade AAN materials. When learning to ex- press numbers in scientific notation on a hand-held calculator, many students become confused by the short- hand display technique. The number 3.456 x 1012 is displayed as "3.456

12," hiding the fact that 3.456 is multiplied by the twelfth power of 10. The AAN Calculator allows entry of the actual mathematical statement, 3.456 * 10 ß 12.

Converting numbers in scientific notation to decimal numbers is another opportunity for effective use of the AAN Calculator. The movement of the decimal point can be shown graph- ically. If the user enters 3.456 * 10 ß 1 and presses the return key, the result is 34.56. If the user then changes the exponent from 1 to 2 and presses the return key, the decimal point shifts one place to the right. Continuing this process shows the relationship between the power of 10 and the number of decimal places. Also, the addition of zeros as placeholders is displayed. Many more ways to use the editing features of the AAN Calculator can be used to explore multiplication and division by powers of 10 and other concepts, such as fractional and nega- tive exponents and the rules for multiplication and division of numbers involving exponents.

The concept of variable

One of the most important effects that the AAN project staff hopes to achieve from the use of the AAN Calculator in conjunction with the curriculum materials is the beginning of an understanding of the concept of variable. The use of a variable as a generalized number or an entire class of values, however, is too advanced at this level. The use of a variable as a specific but unknown number is often misleading for students because this notion promotes the idea that only one value can be chosen as a replacement for the variable. The AAN materials show students in a concrete way that many values can replace a variable in a mathematical statement and that, for each value selected, the mathematical statement can have a meaningful outcome. For example, in the problem stated previously con- cerned with finding the length of time it takes $6000 to triple (see figs. 1 1 , 12, and 13), each replacement for the value of N gives a meaningful outcome in terms of an amount of money in the account. The use of the AAN Calcula-

16 Arithmetic leacher

_2000«(l+.075)~10l I 60001 I 4122.06УЩ

l2000*(l+.075)~15l I I 60001 I 59 17.754741

|2OOO»(1+.O7 5H 5.190849471 1 I 6"ÕÕÕ1 I 6ÕÕÕ1



tor allows this multiple-replacement process to be done efficiently, not distracting the students with computational difficulties. The focus is on the concept of variable, not on computation as an end product.

Problem solving Another important effect to be achieved with the AAN materials is improved problem-solving skills. Tra- ditionally, problem-solving skills were specific to the type of problem presented. For example, the way to solve exponential problems was to use logarithms, or the way to solve algebraic substitution problems was to use fac- toring. Problem solving by the guess- and-check method is a generic strategy that will work on almost any type of problem. The computational effi- ciency gained by the use of the AAN Calculator makes this strategy a via- ble alternative to more traditional methods. The technique is simple. The student guesses the answer. The logic of the problem statement is maintained because the student is repeatedly attempting to answer the stated question. For example, when answering the question of how long it takes $2000 to triple in value at 7.5- percent interest, the student guesses ten years, twelve years, fourteen years, fifteen years, and so on. As students become more skilled at using this problem-solving strategy, the guessing process can be more aptly called "educated guessing." Students often become more attuned to the sense of the problem, the sense of the numbers they are using, and the sense of the process of the problem. The AAN Calculator is effectively used as an integral part of this problem-solving process, allowing students to do mathematics and not be held back by computational difficulties.

Conclusion The AAN Calculator is a general instructional and problem-solving tool that can be used with most of the mathematical material at the middle school level. Most mathematical activities using the hand-held calculator can be done using the AAN Calcula-

tor. The added features of visible display and editing make the AAN Cal- culator an ideal tool for exploring numerical relationships of all kinds. These features are even more crucial when instructing novice students. The AAN Calculator is a more sophisticated tool than the hand-held calculator for learning about numerical relationships. Once teachers become familiar with this computer software, many innovative uses for the AAN Calcula- tor will be discovered.

The work on which this article is based was

performed pursuant to Contract No. DPE- 8470195 of the National Science Foundation. The views and opinions expressed by the au- thors are their own and do not necessarily reflect those of the National Science Founda- tion.

The AAN Calculator disk was developed in the TABSLAB, College of Education, Ohio State University. This software will be available after 1 September 1987 under the title of The Compulator from D. C. Heath and Compa- ny, 125 Spring Street, Lexington, MA 02173.

References

Belland, John С "The Inverse Tool Principle." Educational Communication and Technology 33 (1985):51-57.

Hembree, Ray. "Research Gives Calculators a Green Light." Arithmetic Teacher 34 (Sep- tember 1986): 18-21.

Leitzel, Joan, Franklin Demana, and Alan Os- borne. Approaching Algebra Numerically. Columbus, Ohio: Ohio State University, 1986.

Suydam, Marilyn N. The Use of Calculators in Pre-College Education: Fifth Annual State- of-the-Art Review. Columbus, Ohio: Ohio State University, Calculator Information Center, 1982. W

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The AAN Calculator: A Tool for Enhancing Instruction