COMPUTERS AND THE MATHEMATICALLY GIFTEDAuthor(s): WALTER KOETKESource: The Mathematics Teacher, Vol. 76, No. 4, Gifted Students (April 1983), pp. 270-272Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963468 .

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COMPUTERS AND THE MATHEMATICALLY GIFTED

By WALTER KOETKE Putnam/Northern Westchester BOCES

Teachers of the mathematically gifted with access to computing facilities quickly learn a very basic principle?"Never underestimate the student's ability." Some of my most enlightening experiences have occurred as my students attempt to teach me enough to understand their computer related work. And when they do this with a

patient smile, the experience is most enjoy able.

As microcomputers continue to spread throughout the schools, educators increas

ingly tend to label the near trivial as ad vanced. One finds "advanced program ming" labels on IF-THEN statements or FOR-NEXT loops in BASIC. Please don't be misled. When dealing with the gifted student, learning to program simply isn't advanced?it's fundamental. Teaching the

gifted to program is mostly a matter of

providing resources; programming itself will not be a major challenge.

Once the gifted have developed their

programming skills, however, they are pre pared to undertake applications of their skills that can be correctly labeled "ad vanced." By guiding students' interest to

applications primarily in the realm of

mathematics, teachers can observe signifi cant mathematical advances extending far

beyond the classroom. One of my favorite

images is a caricature of Albert Einstein as a child sitting in front of a microcomputer. The caption beneath the image simply reads, "What if .. ." That opportunity

must not be denied any of our students. All the problems suggested here can be

offered to students with no experience in

algebra. Grade levels have not been includ

ed, as the computer tends to make such

judgments meaningless.

1. Multiple-precision arithmetic is a

topic to which many students are attracted.

Developing an algorithm for accurately computing the sum of two integers each

containing up to 300 digits is an easy place to begin. The next step is to compute the difference of two such integers. If you are

working with young students who have not

yet used subscripts in their programs, this

problem offers a natural introduction. Sub

scripts are an example of a fourth-grade topic often mislabeled as advanced in pro gramming texts. Extending the problem to the multiplication of two large integers is the next task. Students should know how to use subroutines before dealing with

multiplication. Finally, the problem can be extended to include the division of two

larger integers. 2. Have students run each of these two

BASIC programs and explain the results.

10 FOR A = 1 TO 100 STEP .1 20 PRINT A 30 NEXT A

10 FOR A = 1 TO 100 20 IF SQR(A)<>INT(SQR(A)) THEN 40 30 PRINT A 40 NEXT A

Before suggesting this problem, try the pro grams yourself. If the results are correct, then do not use the problem. The majority of microcomputers contain an implemen tation of BASIC that will not correctly exe cute either program. Students should be aware of the potential for computational errors. They should also know what to do about such errors. Ask students to rewrite those programs so that the results expected are actually produced.

3. Is each positive integer 1 through 60 a

divisor of some integer that contains only the digits 1 and 0? For example,

1 is a divisor of 1, 2 is a divisor of 10,

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3 is a divisor of 111, 4 is a divisor of 100,

and so forth.

This one "sounds" easy but is, in fact, a rather challenging problem, especially using the computer. If you have very tal ented students, delete the l-through-60 re striction. This conjecture is true, and it has an easily understood arithmetic proof of its

validity.

It is most important that students recog nize that the computer cannot prove or dis

prove a conjecture. If thousands of cases are demonstrated as valid, we still have no assurance about those we have not tested. If we fail to find a numerator for one di

visor, we have no assurance regarding un tested numerators. Further, students must

understand that the computer is not always the best vehicle for obtaining a solution.

4. Determine all integer values of D

(D < 200) for which the decimal equivalent of the fraction 1/D terminates. After you identify the integers, try to determine a common characteristic that distinguishes those integers from all other integers less than 200.

Since there is not a well-defined set of characteristics to be determined, this prob lem is open-ended in nature. Students' de

scriptions of characteristics are secondary to the mathematical explorations they pursue in an effort to determine the charac teristics. Students develop the ability to

make, test, and then evaluate mathematical

conjectures with such problems. Open ended problems can often lead the gifted into a variety of unexpected but quite valu able sidetracks.

5. A sequence of numbers is determined

by the following rule: "Each number after the first is the sum of the squares of the

digits in the preceding number." For exam

ple, beginning with 23 the sequence is

23, 13, 10, 1.

2*~+3* l*~+72 1* ( 2

Beginning with 12, the sequence is

12, 5, 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58.

One conjecture based on these examples is that for all positive integers this rule will

always produce a sequence that contains a 1 or a 58. Verify that the conjecture is true for each of the integers 1 through 1000 or

determine an integer in this interval for which the conjecture is not true.

6. Over two hundred years ago, Christian Goldbach stated that every even number greater than or equal to 6 can be

expressed as the sum of two prime num

bers. He also stated that every odd number

greater than or equal to 9 can be expressed as the sum of three primes. Write a program that can produce a

table containing each of the even integers 6

through 1000 and two primes whose sum

produces each of the integers. Goldbach's

statement, known as Goldbach's conjec ture, has never been proved or disproved. It continues to be one of the many un

solved problems of mathematics.

Simulations have become very important in many areas of society, and understand

ing their applications and limitations is

important. The last two examples are simu lation challenges. For each problem, the student should write a program to simulate the situation and supply the data necessary to answer the question.

7. Imagine a carnival booth that offers the following game:

A regular deck of fifty-two playing cards is shuffled. Two cards are then drawn from the deck. If either one or both of the cards is a diamond, you win one dollar. If neither card is a diamond, you lose one dollar.

Sounds like a pretty good way to get rich, doesn't it? If you play this game 1000 times

(you didn't really want to see the rest of the carnival anyway), how much money can

you expect to win?

8. Suppose you are a baseball manager in the unhappy situation of having an

entire team with individual batting

April 1983 271

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averages of .220. To make matters worse, when a player gets a hit there is only a

l-in-10 chance that the hit is a double rather than a single. What is the average number of runs your team will score in a

nine-inning game? (Assume that a single will advance each runner one base, whereas a double advances each runner two bases.)

BIBLIOGRAPHY

Ahl, David H. Computers in Mathematics: A Source book of Ideas. Morristown, N.J.: Creative Com

puting, 1979.

Ball, W. W. Rouse. Mathematical Recreations and

Essays, rev. ed. London: Macmillan & Co., 1940.

Creative Computing, published monthly by Ahl Com

puting, Morristown, N.J.

Dwyer, T., and M. Cutchfield. You Just Bought a Per sonal What?. Hancock, N.H.: Byte Publications, 1980.

Gardner, Martin. Aha! Insight. New York: Scientific

American/W. H. Freeman & Co., 1978.

-. Mathematical Carnival. New York: Alfred

Knopf, 1975.

Kemeny, J., and T. Kurtz. BASIC Programming, 3d ed. New York: John Wiley & Sons, 1980.

Schuk, Fred. Master Book of Mathematical Rec reations. New York: Dover Publications, 1968.

AN ACCOUNT OF A MATHEMATICIAN'S EDUCATION

( Continued from page 228)

often learn from one another as well as from the distinguished faculty. It is essen tial for a mathematics student to be point ed in a fruitful direction by a good advisor. For my advisor, I was fortunate to have had a teacher of advanced mathematics without peer, Elias Stern. His influence on

my mathematical development was deci sive.

After graduating from Princeton in 1969, I remained at Princeton one more year as an instructor, then moved to the University of Chicago. Chicago is a major center for

my field of mathematics, and I learned a

great deal from my colleagues. My four

years in Chicago were mathematically very active for me. Since 1974 I have been back at Princeton. I continue to work at my re

search, and I have never stopped enjoying it.

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272 Mathematics Teacher

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