The Benefits of Computer Programming in Developing Algorithmic ThinkingAuthor(s): RITA L. PETOSASource: The Mathematics Teacher, Vol. 78, No. 2 (FEBRUARY 1985), pp. 128-130Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964405 .

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The Benefits of Computer

Programming in Developing

Algorithmic Thinking By RITA L. PETOSA, Adrien Black Intermediate School #25, Flushing, NY 11358

Our

school has not adopted a computer

literacy course. Instead, we have opted to direct students' interest in computers toward the study of mathematics. That is, we have infused an ongoing component of instruction in computer programming into our mathematics curriculum with interest

ing results. The focus of the instruction is on algorithmic development, which comes

about when we generalize solutions to

mathematical problems. Students move

from developing their own formulas (e.g., for finding the perimeter of a square, the sale price of an item given the rate of dis count and the list price, the complement and supplement of an angle) to creating more procedural algorithms (e.g., for find

ing prime numbers or factors of numbers or

for classifying triangles by angles given the

lengths of the sides). It seems that this emphasis on algorith

mic development has led to greater explora tion of mathematical phenomena by stu dents. The remainder of this article de scribes an example of what is happening in our mathematics classes since we began using the computer.

One student, Susan, along with her

classmates, had been discussing in their social studies class a book that has recently been greatly publicized?1984 by George Orwell. The class had discussed that the book's title was a deliberate reversal of the

digits in 1948, the year the book was pub lished. While many of the students were

conjuring up images of Big Brother, Susan made the following discovery (written in her own words).

Susan's conjecture

Take a two-digit number with the two

digits being consecutive, such as 23.

Switch the two digits around, such as 23

turned to 32. Subtract the smaller number from the larger one, 32 ? 23 = 9.

No matter what two consecutive digits you use, the difference will always be

nine.

Now choose a two-digit number, the

digits not being consecutive, such as 58.

Find the difference between the digits, 8-5 = 3. Multiply the difference by 9, 3 9 = 27. Now take 58 and turn it

around, such as 58 to 85. Find the differ

ence, 85 - 58 = 27.

The point I am trying to make is that the

product found when the difference of the two digits is multiplied by 9, is the same

as the difference of the original number

and the number when it is turned around, if the smaller number is subtracted from

the larger number.

After having Susan write her discovery in words that, as Susan said, "even her

sixth-grade brother" could understand, I

asked her to teach it to her classmates. The student were highly motivated. One student

suggested that a number like 55 was an ex

ception. Other students were quick to point out that it followed Susan's algorithm in

that

55 - 55 = 9(5 -

5).

When Susan finished her lesson, I asked the students to consider whether an analo

gous algorithm could be developed for

nondecimal-based numerals. For example, could they find a whole number, c, such that

42five-24five =

c(4-2)?

128 Mathematics Teacher

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Many students immediately guessed that c = 4 for all two-digit numerals in base five.

They were then given the task of extending Susan's discovery to systems in all other bases. Nurit's extension (also written in her own words) of Susan's conjecture follows.

Nurit's conjecture

To find the difference of a two-digit number and the number with its digits reversed, you subtract the two digits of the number and multiply their difference

by the largest one-digit number in that base. (Note: 4 ? 1=3, and 4 is the larg est one-digit number in base five.) For ex

ample :

TABLE 1

Base five

Base four

Base nine

41five 3five

-14, five x4, five

22fiv

32,

22, five

four Lfour

23fl x3f<

3four 3four

4nin.

-26n

35n

x8n

35n

Earlier in the school year, the students had been asked to code a computer program that would convert a three-digit, non decimal-base numeral to base ten, given the three digits and the base number (Elgarten and Posamentier 1984). As a result, Alec wrote a program in BASIC (table 1). Refine ments of Alec's program by his classmates followed his presentation. Also, Alec and the other students became aware that they did not know how to code a program to con vert a base-ten numeral to a nondecimal base. Our discussion of Euclid's division al

gorithm is still in progress. I then asked the students, "When, in

mathematics, does a conjecture become a

theorem?" They were familiar with Gold bach's conjecture ("Every even number

larger than 2 is equal to the sum of two

prime numbers"), which was discussed ear

lier in the school year. We discussed that we would have to show that the conjecture worked for every possible two-digit number

A Program to Convert a Three-Digit Nondecimal-Base Numeral to Base Ten

5 10 20 30 40

45

50

55

60

62

64

66

70 80 90 100

110 120 130 140 150 160 170 180 185 190 200 205 210 220 225 230 240 250 260 270 280 285 290 300 305 310 320

REM ALEC'S PROGRAM PRINT "THE 7-356 THEORY" PRINT "HERE'S THE THEORY" PRINT "_" PRINT "FIRST YOU TAKE A TWO DIGIT

NUMBER AND" PRINT "SUBTRACT THE SMALLER DIGIT

FROM THE LARGER ONE." PRINT "THEN YOU MULTIPLY THE

DIFFERENCE BY" PRINT "A NUMBER ONE LESS THAN THE

BASE NUMBER." PRINT "FINALLY, IF YOU TAKE THE

ORIGINAL NUMBER" PRINT "AND SWITCH AROUND THE DIGITS

AND THEN SUBTRACT" PRINT "THE SMALLER NUMBER FROM THE

LARGER ONE," PRINT "YOU WILL GET THE SAME ANSWER

AS WHEN YOU MULTIPLIED." PRINT PRINT "LET'S TRY IT." PRINT "_" PRINT "ENTER A TWO DIGIT NUMERAL IN

ANY BASE." PRINT "ONE DIGIT AT A TIME PLEASE." INPUT A, PRINT "ENTER THE BASE NUMBER PLEASE.' INPUT IF A>B THEN GOTO 180 IF B>A THEN GOTO 200 IF A = B THEN GOTO 200

C = A-B PRINT A;"-";B;"

= ";C GOTO 210 C = B-A PRINT B;"-";A;"

= ";C R = N-1 D = C*R PRINT C;"X";R;"

= ";D Z = A*N + B X = B*N + A IF Z>X THEN GOTO 280 IF X>Z THEN GOTO 300 IF X = Z THEN GOTO 300

Y = Z-X PRINT Z;"-";X;"

= ";Y

GOTO 310 Q = X-Z PRINT X;"-";Z;"

= ";Q PRINT "AND THAT'S HOW IT WORKS!" END

in every possible base. We agreed that even

with the assistance of Alec's program, this

undertaking would be impossible. We then

discussed the difference between inductive

reasoning, which we were trying to apply to

this case, and deductive reasoning. I men

February 1985 129

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tioned that algebra, that is, generalized arithmetic, might be helpful. I did go

through the following algebraic derivation for base ten only. The students were able to follow but not reproduce it, since, being seventh graders, they had not yet been

taught the algebraic manipulations that were necessary.

Algebraic justification

Let t be the tens digit of a two-digit number, and let u be the units digit of a two-digit number. Then,

10*+ lu -It - lOu

9t- 9u = 9(t -

u).

We had discussed the distributive property of multiplication over subtraction earlier in the school year.

Later I asked the students to explore three-digit numbers. Some students might conclude that for any base 6 and any three

digit number represented by

b2x + by + z,

the absolute value of the difference of the number and the number with its digits re versed is always equal to

(b2-l)\x-z\ or

(? + l)(?-l)|*-z|.

Students' exposure to computer program ming all year enabled them to develop an

algorithmic approach to use as a basis for

solving the problem.

REFERENCE

Elgarten, Gerald H., and Alfred S. Posamentier. Using Computers. Melilo Park, Calif. : Addison-Wesley Pub

lishing Co., 1984. m

DEPT. OF MATH. & COMPUTER SCIENCE UNIV. OF NEBRASKA AT OMAHA, OMAHA, NE 68182

Math. Center Director. Tenure Track Asst. Prof, position. Duties include administering the Pre-Calculus instruction in the center, some classroom teaching, and research. Qualifications: Ph.D. in Math, or Ed.D. with strong math, background. Previous administrative- experience is a plus. Send a resume, transcripts, and 3 letters of recommendation to Charles Downey by March 20, 1985. UNO is an equal opportunity/affirmative action employer.

Activities Swap and Share Session The Editorial Panel of the Mathematics Teacher will offer a special "swap and share" session at the annual

meeting in San Antonio. The session is designed es

pecially for middle and junior high school teachers, who should bring twenty-one copies of one activity that has been successfully used in the classroom. The teacher can then swap a copy of this activity for one of another teacher. Bring twenty-one copies of one ac

tivity and leave with twenty different activities (the MT will keep one copy for possible publication). Please refer to the program for the specific time and room of the session, and come prepared !

Authors Needed for "From the File" The Editorial Panel of the Arithmetic Teacher is en

couraging teachers to submit material for students in

grades K-8 that can fit on a 4 in.-by-6 in. index card in the format of "From the File." See recent issues of the

journal for examples. These items are intended to serve as a teachers' exchange of tips on bulletin-board

ideas, teaching aids, projects, games, and interesting problems, Three copies of each item should be sent to the Arithmetic Teacher, 1906 Association Drive, Reston, VA 22091.

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130- -Mathematics Teacher

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