Jenchung Chuan
Department of Mathematics, National Tsing Hua University
Hsinchu, Taiwan, 300 R.O.C.
Abstract
Many important topics which appear in undergraduate mathematics and engineering textbook are omitted from the class in which the text is used, blaming the lack of time required for most students to gain a genuine understanding of these more complex subjects. Examples from integration by parts are used to demonstrate how such relatively complex topics can be included in an introductory calculus course without an unacceptably large investment of class time. The author has used one of the popular personal computer integrated software packages, Lotus Symphony, to demonstrate principles in class and to have students work certain homework problems which could not reasonably be solved by any means other than with a computer.
Spreadsheet program provides an alternative to conventional programming
languages in developing educational software. It offers the user an easy
way to create an interactive and userfriendly instructional program. It
has several builtin features such as graphic capability and data base
management capability. The program developed on the spreadsheet format
has the 'transparent black box' feature which is considered to be very
useful for educational purposes. Programming with spreadsheet program can
save a considerable amount of development time. Additional examples are
presented and several spreadsheet programming features are discussed.
Introduction
The human nervous system has the unique ability to learn through visual
images. Scientists have known for years that the learning of skills and
behavior are primarily the result of visual learning. One of the best visual
learning tools available is the spreadsheet. Lotus 123, Quattro Pro,
Excel are some of the most popular software programs based on the spreadsheets.
Spreadsheets organize our world into twodimensional boxes. Spreadsheets
are peculiarly well fitted into the human mind. Such a hypothesis is unlikely
ever to be proved or disproved, yet there is much circumstantial evidence
in its favor. Gridlike patterns of intersecting columns and rows are exceedingly
rare in nature, they are quite common in human culture. Indeed, they turn
up often enough, and in contexts unlikely enough, to suggest a human predisposition
to organize the world in twodimensional, tabular form. In what follows
we shall examine some possible applications of the spreadsheet as a mathematics
learning tool.
Table of Integrals
Browsing through the table of integrals we often come across many interesting number patterns. For example, in the expression of the indefinite integral involving exponential functions we find
òe^{ax}dx = [(e^{ax})/( a)]
òxe^{ax}dx = [(e^{ax})/( a^{2})](1 + ax)
òx^{2}e^{ax}dx = [(e^{ax})/( a^{3})](2  2ax + a^{2}x^{2})
òx^{3}e^{ax}dx = [(e^{ax})/( a^{4})](6 + 6ax  3a^{2}x^{2} + a^{3}x^{3})
òx^{4}e^{ax}dx = [(e^{ax})/( a^{5})](24  24ax + 12a^{2}x^{2}  4a^{3}x^{3} + a^{4}x^{4})
...
The coefficients
1
1 1
2 2 1
6 6 3 1
24 24 12 4 1
...
arouse curiosity: all diagonal terms equal to 1 and each of the rest in the m+1st row is obtained by multiplying m with the term above it. A moment's reflection will tell us that this is the consequence of the integration by parts formula

Having obtained that much mathematical insight the enthusiastic learner should be eager to list the coefficients in a spreadsheet program. The procedures are:
1. List the row numbers 1,2,...,10 on the first column (Column A).
2. Enter 1 at cell B1.
3. At cell B2 type in the formula $a1*b1.
4. Copy the formula from cell B2 to B2..B10
5. Copy the formulas from cell B1..J10 to C2
Would you believe the spreadsheet could lend itself well to the creation of the table of integrals?
We have just applied a simple recursive technique to generate number patterns. We may now proceed with slightly bolder attempt: let's generate a number pattern that corresponds to a double recursion procedure.
From the table of integrals we see the following listing of integrals involving cos x and sin x:
òcos x dx = sin x
òx cos x dx = cos x + x sin x
òx^{2}cos x dx = 2x cos x + (2 + x^{2}) sin x
òx^{3}cos x dx = (6 + 3x^{2}) cos x + (6x + x^{3}) sin x
òx^{4}cos x dx = (24x + 4x^{3}) cos x + (24  12 x^{2} + x^{4}) sin x
òx^{5}cos x dx = (120  60x^{2} + 5x^{4}) cos x
+ (120x  20x^{3} + x^{5}) sin x
òx^{6}cos x dx = (720x120x^{3}+6x^{5}) cos x + (720+360x^{2}30x^{4}+x^{6}) sin x
............
òsin x dx = cos x
òx sin x dx = x cos x + sin x
òx^{2}sin x dx = (2  x^{2}) cos x + 2x sin x
òx^{3}sin x dx = (6x  x^{3}) cos x + (6 + 3x^{2}) sin x
òx^{4}sin x dx = (24 + 12x^{2}  x^{4}) cos x + (24x + 4x^{3})sin x
òx^{5}sin x dx = (120x + 20x^{3}  x^{5}) cos x
+ (120  60x^{2} + 5x^{4}) sin x
òx^{6}cos x dx = (720360x^{2}+30x^{4}x^{6}) cos x + (720x120x^{3}+6x^{5}) sin x
............
These formulas are the consequences of the integration by parts formula


Therefore the corresponding table of coefficients
can be generated when the spreadsheet is appropriately arranged. This table consists of blocks of coefficients separated by spaces. Each block consists of four rows. The first row of the mth block represents the coefficients of òx^{m}cos x dx that correspond to the factor of cos x and the second row those of sin x. Similarly the third and the fourth row of the mth block correspond to the coefficients of òx^{m}sin x dx that belong to cos x and sin x respectively. At the beginning we fill in 1 and 1 in block 0 to reflect the integrals

Next, we fill in four cells in block 1 dictated by the above two recursive formula with m = 1. As in our earlier spreadsheet procedure, the work is completed when we copy the formulas ``downwards" then ``diagonally".
Number Theory
Examining the similarity among the modular multiplication tables offers the learner an opportunity to formulate and to discover the fundamental principles of number theory. The appeal for using the spreadsheet program to generate these modular multiplication tables is immediate.
Note that, for a prime integer p, each row of the multiplication table for Z_{p} contains all the elements of Z_{p}. From this we see that if 0 < b < p, then a ® ax is a permutation of Z_{p} and the diaphantine equation

has a unique solution. A slight modification of the worksheet yields a new table of 0's and 1's showing which cells in the upper half of the original table belonging to the ``second half" of Z_{p}.
It can be seen that those columns with even sums correspond precisely to those numbers in Z_{p} having square roots. This fact has led us to the discovery of Gauss' Lemma, the most crucial step in the proof of the quadratic reciprocity theorem: a number A is a quadratic residue modulo an odd prime p if and only if an even number of terms
A, 2A, 3A,···,(p1)A/2
belongs to the second half mod p. As a further illustration, we consider the table of geometric progressions in Z_{p} created with the spreadsheet.
It is almost impossible to miss the appearance of the identical terms 1 in the (p1)st rows in all these tables. What this suggests is none other than Fermat's Theorem:
If a is any integer not divisible by a prime number p, then

In the work of scientist, formulating the problem may be the better part of discovery, the solution often needs less insight and originality than the formulation. Thus, letting the students have a share in the formulation, the teacher not only motivate them to work harder, but also teach them a desirable attitude of mind.
Fibonacci Sequence in a Multiplicative System
Leonardo da Pisa was one of the leading mathematicians of the Middle Ages. He is famous today because of a sequence of numbers
1,2,2,3,5,8,13,21,34,...
that resulted from one obscure problem in his book Liber Abaci. This sequence was not given any real significance until the 19th century when the French mathematician Edouard Lucas coined the term ``Fibonacci numbers" and became intrigued with it, its properties, and the areas in which it appears. From the abstract point of view, the notion of a Fibonacci sequence could be considered in algebraic systems which are more general than the additive semigroup structure of the integers. It is possible to extend the definition of the Fibonacci sequence in an arbitrary multiplicative system. For example, let the ``multiplication table" on the set M = {1,2,3} be given by

A sequence {a_{n}} on M is called Fibonacci if

Since this artificial algebraic system enjoys none of the nice properties commonly found in the real number system, almost nothing from the traditional theory of Fibonacci sequence could be adopted. We shall proceed to illustrate how interesting mathematical problems centered around this new notion could be motivated by the spreadsheet display.
It is seen that whenever the ``rule" of multiplication is changed, a different set of Fibonacci sequence appears. But there is one phenomenon common to all these Fibonacci sequences: they all become periodic after the first few terms. Can this fact be explained by any mathematical reasoning? Indeed, this phenomenon is nothing but a consequence of the Pigeonhole Principle. (Among the first 10 consecutive pairs of the sequence

some two pairs must have the same product, by looking at the multiplication table.) Having established this spreadsheetmotivated fact rigorously, it is natural to follow up with the search for multiplicative structures on M that yield Fibonacci sequences of the longest period. Prior to finding a good algorithm for doing so, we must find a environment suited for testing various hypothesis. The interactive inputoutput nature of the spreadsheet makes such examination feasible.
Pursuing along the same path, one may wish to analyze the behavior of Fibonacci sequences in a finite permutation group. The law of composition of two permutations can be viewed as a particular instance of tablelookup. For instance, the composition of permutations

is the permutation

here s·t(1) equals 2 since in the table

2 is located on column 4 ( = t(1)), row 1. Thus, to repeat the composition for successive permutations, we merely have to enter one formula involving the spreadsheet function @INDEX and followed by the copy command. Tabulating Fibonacci sequences this way results in the conclusion that all sequence are periodic without exception.
Again a logical explanation is sought. (This is due to the cancellation
law.) As in the previous case, it is tempting to ask such questions as:
what are the possible periods occurring in a given group? Is there an algorithm
determining the period from the initial data? In what way do the Fibonacci
sequences depend on one another? The intuitive basic approach to all such
problems relies on the raw calculation performed with the spreadsheet.
Powers in Z_{2}[x]/(x^{4}+x+1)
Let Z_{2}[x] be the ring of all polynomials with coefficients in Z_{2}. For a(x) and b(x) in Z_{2}[x], we say that

(a(x) and b(x) are congruent modulo x^{4}+x+1) if x^{4}+x+1 divides a(x)b(x). Write [f(x)]_{x4+x+1} for the congruent class mod x^{4}+x+1: two polynomials belong to the same congruent class mod x^{4}+x+1 if they are congruent modulo x^{4}+x+1. The addition and multiplication of congruent classes are given by


These operations make the set of congruence classes into a ring which is denoted by Z_{2}[x]/(x^{4}+x+1). To investigate the algebraic structure of this ring, the spreadsheet may be used to compute a, a^{2}, a^{3},···, in which a = [x]_{x4+x+1}.
From this table we see that a^{15}
is the same as 1, so the nonezero elements form a multiplicative cyclic
group with a as its generator. The ring Z_{2}[x]/(x^{4}+x+1)
therefore has the structure of a field. It can be shown that all finite
fields are similarly constructed. All of a sudden spreadsheet has been
converted into a computational tool for finite fields!
Conclusion
Spreadsheet program provides an alternative to conventional programming languages in developing educational software. It offers the user an easy way to create an interactive and userfriendly instructional program. It has several builtin features such as graphic capability and data base management capability which are not included in this discussion. The program developed on the spreadsheet format has the 'transparent black box' feature which is considered to be very useful for educational purposes. Programming with spreadsheet program can save a considerable amount of development time. We have seen that the spreadsheet program serves as a versatile tool for mathematics learning. Examining the current curriculum we find that many important topics which appear in undergraduate mathematics and engineering textbook are omitted from the class in which the text is used, blaming the lack of time required for most students to gain a genuine understanding of these more complex subjects. We do suggest the instructors look into the potential power of the spreadsheet as a learning tool, and to explore its many intuitive and straightforward applications.