COMPUTERS (AND STUDENTS!) NEED EXPLICIT NOTATIONAuthor(s): ROBERT DUCHARMESource: The Mathematics Teacher, Vol. 71, No. 5, Computers and Calculators (MAY 1978), pp.448-451Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961296 .

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COMPUTERS (AND STUDENTS!) NEED EXPLICIT NOTATION

Here is one possible solution for students who have difficulty with the order of operations.

By ROBERT DUCHARME The University of Texas at San Antonio

San Antonio, TX 78285

Is there a difference between "computer mathematics" and mathematics of the non

computer variety? From a notational point of view the answer is a very definite yes (as some examples below will show). Thus, as

teachers, we must consider the following question: Since computers are here and their use will only increase in the future, what effect can they have on the teaching of mathematics?

This discussion will focus attention on one aspect of the computer's effect on

mathematics, namely, the notational diffi culties. That is, will the increasing use of

computers (and calculators) have any effect on the way we teach, write, and use our basic mathematics?

The central problem for most people who struggle with mathematics is one of notation. The irony is that mathematics, as we teachers often point out, is a science that reduces worldly phenomena and their

wordy descriptions to elegant symbolic code. It is the use of this coding system, with its notation and rules, that is most difficult.

For example, if we examine the notation used in school mathematics, we come to see that we continuously make use of juxtapo sitional notation. That is, we avoid the use of arithmetic operational symbols in such

simple cases as exponentiation and sub

scripted variables. Thus, we define "five cubed" as 5 X 5 X 5, and we express it as 53

using no explicit operator symbol. Here, however, is where the ambiguity

begins for the students. Soon after this new

notation is introduced, we use it in combi nation with other explicit operations, for

example, 53 X 54. What is missing here is a

useful operator symbol for "raising to a

power."

It is when we begin to juxtapose our sym bols that the average student becomes con fused. Why this conflict? Consider the

juxtapositional notation we use for sub

scripted variables. We use no operational symbol when we write 3, but we also do not mean "lower to a power of 3." The rather subtle (for new students) shift of the

digit 3 produces a very significant result. The problem is that some people "see" our

slick notation, but many others never quite understand it.

There are many other simple examples of notational problems. For example, what do we mean by 12 6 X 2? Certainly there is no problem if we express the number as (12 -s- 6) X 2 or as 12 + (6 X 2). However, the

original parenthesis-free expression, al

though valid, is often ambiguous. And any one who has ever taught algebra has seen students do this:

10

30 2f? 10 _ 2 + 3 2 + t 2

1

Normally, we point out that we are divid

ing by the number 2 + 3, and with a shift in notation this could be clearly emphasized as 30 + (2 + 3). Alternatively, it is possible to teach the fraction dividing line as a

grouping symbol equivalent to parentheses.

Enter Computer Notation Once upon a time all mathematics was

448 Mathematics Teacher

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done by hand computation. Today, how

ever, more computing is done by digital electronic computers than the sum total done by hand throughout history. How does this affect the way we write ex

pressions? To answer this, we discuss one

fact about most computer input and output devices.

Most of the data and instructions are fed to computers by punched cards made on a

keypunch machine, although a large amount is also fed on teletypes via type writerlike keyboards. In both of these situa tions there is one problem for the mathe matician. These input devices (as well as the

output devices) do not permit shifting up or down for things like exponents or sub

scripts, and they most often use only capital letters.

This is interesting, since we teach this sort of powerful symbolic notation to stu

dents, even though we are unable to "wire"

up simple computer devices to understand or interpret it. The problem for computer users, therefore, is to code basic, traditional

algebraic expressions with their heavy de

pendence on ju^tapositional notation into

simple, all-on-one-line (linear) expressions. Since algebra does not normally use an

exponential operator, we invent one for

computer use. The popular computer lan

guages use either the up arrow, j, or the double asterisk, **, to indicate ex

ponentiation. (It should be noted that the double asterisk is more popularly used, as in FORTRAN.) Thus, 52 becomes 5 ] 2 or

5**2. Pedagogically, I prefer the arrow, since it is very intuitive; the use of the double asterisk is not. Moreover, the multi

plication operator is usually the single as

terisk, *, which is too similar to the double asterisk.

Table 1 illustrates how we may translate

ordinary algebra into linear, computer-ori ented expressions. Note how much more

explicit the linear expressions are.

Although computers use regular nota tion for addition and subtraction, we see

that the multiplication operator replaces the times sign, X or dot, with a single sym bol, the *. Also, in division we n? longer

write fractions with a "top" and "bottom" but as ratios with a slash operator, /.

TABLE l

Standard Algebra Linearly Expressed

for Computer

3 + 5 4-7 6X2 or 6 ? 2

or 12+6 6

53 U

3 + 5 4-7 6*2

12/6

5|2(or5**2) T(4)

Subscripts, rather than employing a down arrow, use a more functional nota

tion, which may be viewed as an indexing scheme where we begin labeling at 1, 2, 3,

? ? . Here the preferred style is T(l), T(2), T(3), ???, rather than T1,T2,T3,

??? .(We

omit the technical distinctions here.)

Notational Conventions

The problem now is twofold. First, we

must understand how to do basic computa tions and, second, how to write them in

simple linear form. I first had occasion to see conflict in com

putation when I was discussing order of

operations in arithmetic expressions. The students were given the simple expression: 12 +- 6 X 2 = ? Much to my surprise, many

people gave the answer of one.

On further investigation, it was indicated

by some people that they were practicing the "My Dear Aunt Sally" rule of multi

plying before dividing before adding before

subtracting. A few students even felt that the problem was undefined, since it ap peared ambiguous as to which operation should be performed first. And, indeed, it is

ambiguous! Moreover, the result is arbi

trarily defined. The whole incident provided an unex

pected opportunity to discuss the arbitrary way we define our calculational procedures. Of course, for computer-oriented people, 12-1-6X2^ 1 simply because they say it is

not! We assume that we teach order of

operations to our students; yet when they first use computers (and some calculators),

May 1978 449

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they find out that the programmed hier

archy of the machine is not the same as

their own.

Certainly a case can be made for avoid

ing such ambiguous expressions as a - - b X c by the simple use of parentheses. Here

again we see that by using certain explicit symbols, we can arrange our priorities. Thus we can write (12 -r 6) X 2 or 12 + (6 X 2), depending on which result we desire.

Computers are essentially programmed to scan arithmetic expressions and perform them in a simple (but arbitrarily pre scribed) manner. Thus, the rules for a par enthesis-free expression are these:

Pass #7: The expression is scanned for all

exponential computations, and these are

performed from left to right, Pass #2: The second scan performs all

multiplications and divisions as they ap

pear, from left to right. Pass ?3: The final pass calculates all ad

ditions and subtractions as they appear, from left to right.

For example, the expression

12-^6X2 + 33 + 3- 2+ 15 + 6 + 3 + 42 X 5 - 3

is processed in the order given by the cir cled numbers:

Pass #7; 12 v6X2 + 33 + 3

-2 + 15+ 6^-3 + 42 X 5 - 3

Q) O @ Pass #2: 12 -=- 6 X 2 + 33 + 3 - 2

@ ? + 15 + 6^3+16X5-3

? ? Pass #3: 4+11-2+15

+ 2 + 80 - 3 = 107

These rules are maintained even when we use parentheses to affect the order of oper ations. The only modification is that the first pass will now compute the parenthet

ical expressions from left to right, from in nermost out. (And within a parenthetical expression, the normal order holds?left

to-right exponentiation, multiplication and

division, and addition and subtraction.) The next three passes then do exponents, multiplications and divisions, and addi tions and subtractions.

So we see that computers do their arith metic in a standard but arbitrary way. And we also see that parentheses can be very

helpful in making clear the order of oper ations. As mentioned earlier, many stu dents would avoid such cancellation mis takes as

16 Mi 2 + 3 Z+ 3 4

1 if they used linear computer notation,

which forces the division by the sum 2 + 3 and not just by 2; that is, we would write

16/(2 + 3). There is just no opportunity with computer notation to develop the can

celling habit. Of course some occasions do require the

use of parentheses. For example, in using the formula for the area of a trapezoid with bases bx and b2 and height A, we have

Area = \{bx + b2)'h.

In linear computer notation we may write

this in several ways:

(1) AREA = 1/2 * (Bl + B2) *

(2) AREA = 0.5 * (Bl + B2) *

(3) AREA = (Bl + B2) * H/2 (4) AREA = Bl + B2 * H/2

Although formula 4 will not produce the correct result, formulas 1, 2, and 3 are valid. And we see that they each need pa rentheses. Formula 2 is somewhat more ef ficient than formula 1, since it uses one less

computation. (You might say that we did the division 1/2 so as to save the computer one step.) However, formula 3 also uses

just three computations, and we do not per form the division ourselves.

Thus, we see that the use of parentheses not only may enhance the readability of an arithmetic expression but also may be re

450 Mathematics Teacher

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quired. Certainly their liberal use should be

encouraged for the sake of explicit clarity.

Implications for the Future

Since the use of computers and calcu lators in the mathematical sciences will in

crease, a need arises for the efficient, ex

plicit coding of basic algebra. Our common

juxtapositional notation cannot be read

into, nor printed by, computers or calcu lators. Hence there is a need for a com

plete and consistent set of explicit operators that will alleviate the very common pitfalls of order of operations, incorrect can

cellations, and simple misunderstandings of such basics as exponents and subscripts.

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BILOXI MEETING 16-18 November 1978

Come to the beautiful

Mississippi Gulf Coast

Come to Biloxi for an NCTM name-of site meeting, 16-18 November 1978. Hosts for the meeting will be the

Mississippi Council of Teachers of

Mathematics; headquarters will be the Biloxi Hilton.

This meeting will emphasize the basic skill areas endorsed by the National Council of Supervisors of Mathematics. Saturday morning has been set aside for a special "Metric Hours" time, during which all sections

and workshops will be devoted to

metric matters. The program will

feature nationally known speakers who will inform, stimulate, and chal

lenge you. These include Shirley Hill,

Stanley Bezuszka, Virginia Newell, John Egsgard, Vernon Hood, Zalman Usiskin, Thomas Hill, and Ross

Taylor. Outstanding section speakers and workshop leaders will provide information and materials to enrich

your teaching. On Friday evening the special at

traction will be a Seafood Jamboree featuring the delicacies of the Gulf Coast.

Make plans now to see old friends, attend informative meetings, and enjoy the Mississippi Gulf Coast in Novem ber. See the September 1978 NCTM Newsletter for additional information and a housing reservation form.

May 1978 451

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