Curriculum Improvement in Mathematics and Computer-Access for General Students

Allen Klinger, Professor

UCLA Computer Science and Engineering

<> 310 825-7695 310 825-7578 fax


This proposal develops an introductory course in computing and mathematics for students with no firm commitment to an undergraduate major. The material is multi-disciplinary and oriented to kindling interest in mathematical topics usually ignored by general students. The course involves work both in a computer laboratory and in reading and writing about historic aspects of mathematics, and the way different cultures have contributed to that field. Students orient themselves regarding computer use, and develop their exploration of mathematical knowledge. They gain familiarity with computing tools and develop their mathematics awareness and sophistication. The program improves the quality of instruction to students in both computer science and mathematics subjects by providing new materials. The materials for the course become available to the entire campus as library resources for self-study.

The three issues addressed by this proposal are: bringing about early in-depth skill at computer tools for scientifically-oriented students; acquainting students with the general power and growth of mathematics; and developing a computational studies experimental track. The first issue involves enabling students to become skilled users able to compose reports containing focussed visuals, well-written text, and appropriate mathematical models. This is a tool-oriented introduction to working with computers for a general audience: i.e., no specific-utility or application is stressed. Instead, the course demonstrates that all are able to deal with details of computer applications. The second issue addresses the need for reasoning in diverse domains, not just those of a specific course program. The activities here enhance appreciation of mathematics.

Emphasizing reading about mathematics' role in diverse cultures is helpful to general students. This is done partly by showing a non-European emphasis on mathematics: we present other cultures and societies calculating activities using history, games and puzzles. The reading coupled with activity to gain some computer familiarization strengthens their ability to succeed in a four-year college. By embedding both activities in a program to overcome mathematical inhibitions we use computers to support exploration.This perspective strengthens college-level minority students, including women, by enabling them to identify with creators of mathematics.

This proposal is to create a visually-oriented computer laboratory. The facility is to make multimedia computing accessible to undergraduate students. Computer laboratory activities expand self-study opportunities for those in the course. Students can access learning materials added to the laboratory to accomplish further growth in mathematics and computer use after they complete the course.

We will describe the program and materials to faculty at other four-year colleges and universities by: conference presentation; publication in mathematics, computer and engineering education journals; and, posting resource files on the world wide web of computers.


This proposal addresses the need for new college students to experience computer use in a meaningful context that furthers their exploration of mathematical ideas. The objective is to introduce students to a new orientation, one where quantitative models are seen as useful, enriching, and associated with pleasant activity. To do this we will create a computer laboratory to encourage experimental work with mathematical content. The computational modules will be built upon diverse sources. They will include mathematical recreations, ancient mathematics in different cultures, and contemporary computer mathematical tools: writing and evaluating expressions. Students will begin learning to use service tools in the computing laboratory: text processing, drawing figures, accessing library-data, world-wide-web browsing, and file transfer. The computational modules and associated reading material will make up the syllabus of a new introductory undergraduate course.

The purpose is to improve curriculum quality to attract students in all fields to mathematics and computer science courses. This is to be done by addressing three different issues. The first issue is enabling general students to gain a useful, focussed, tool-oriented introduction to working with computers. Second, to involve students in enjoyable activities that enhance appreciation of mathematics and stimulate further learning in the subject. Third, to provide students with opportunity to undertake meaningful and interesting computer-based experiments to develop skills. This will be done via a new visually-oriented computer laboratory designed to make multimedia computing accessible to undergraduate students. The to-be-created laboratory is based on personal computers similar to those students have, or can use. However the laboratory will include both NEXTSTEP and Windows95 operating systems. It will also have direct internet access and the UCLA Orion library information system. Also the computers will have application software ranging from writing programs such as Microsoft Word, WriteNow, OpenWrite, EquationBuilder; to drawing applications like Diagram, along with the Mathematica system, a sophisticated tool for mathematics with graphics.

The course participants will be working in two dimensions, writing mathematical expressions in EquationBuilder, composing illustrations using Diagram and other draw and paint tools, and rapidly obtaining equation graphs from Mathematica. They will also use and evaluate new creative three-dimensional solid model tools, CD-ROM materials, and multimedia presentations, some of which are described below. The proposed activities are a concrete beginning to redress widespread mathematics disinterest. They are elements in a six point program (at least the first four fall under this proposal):

1. Preparation: description of visual and mathematical computer-skills;

2. Planning: laboratory design and experiment-creation;

3. Development: preparation of visually-oriented presentations of mathematics learning units;

4. Instruction: employing teaching assistants to familiarize students with advanced computer tools;

5. Outreach: preparation of short course materials for college (university, four-year, community) and pre-college teacher training ; and,

6. Organization: matching portions of our materials to the National Council of Teachers of Mathematics curriculum standards.

All six components support curriculum improvement. The laboratory, computer experiments and mathematics learning units are steps to lessen mathematics fear and gain computer sophistication .

Preparing the Learning Curriculum

Our educational program involves an enrichment effort designed to attract general students to computer and mathematical work. To do this institutional barriers and student prejudices must be addressed. The former has partly been dealt with by pre-proposal effort: several years of teaching a similar course which lacks the laboratory features proposed here. We conclude, from past teaching that word-of-mouth, current technical development, and historical material can overcome the reluctance of specific minorities to consider taking a mathematics/computers course.

Examination of the interests of previous general students gave rise to the background sources for the learning curriculum proposed here. We will be expanding on the outlines below in the actual work of the proposal, but include here some visuals that illustrate how we propose to incorporate two themes: 1) recreational mathematics ; and, 2) historic computational facts from diverse cultures. We expect to produce a new course that is outside of the engineering, matheamatics, physics mainstream programs, to challenge and interest students in using computers and mathematical models.

Games of chance provide a possible starting point for a course outside the traditional first year university mathematics series. Gambling and day-to-day probability-issues (weather) show that the general is familiar with random-based decisions. Recently statistical simulation of a baseball season became an interesting game to more than a million people [1]. There is similar development of widespread interest in games with roles such as Dungeons and Dragons (DD), where dice are used to add randomness. All this is a recent addition to a long history of card games (poker, bridge, pinochle, rummy) and many other kinds of play (dice, mah jong, dominoes, monopoly, checkers, chess) where there is an element of chance.

Randomness-learning in a computer curriculum will be based on card and board games. Solitaire [2]; the seed game played widely throughout Africa and in Indonesia described by Figure 1; and, five puzzles illustrated by geometric diagrams in Figure 2, introduce students to computer-mathematics by enabling in-class discussions of parity and modulo. Each item shown will be expanded into two forms, as library, and as computer-stored material (multiple image and sound files). The basis of the curriculum change proposed here is to develop and store new interactive tools for experimenting with learning about computers and mathematics.

Probability and statistics are subjects where often the mathematics is not immediately understandable. Since this is the basic tool of disease risk, voting projections, and computer simulation models (such as those that predict global warming), and quantum mechanics, some understanding is important to becoming an informed citizen. Indeed, wherever the future university graduate turns, an informed approach to the subjects of probability and statistics has clear value. Furthermore, in recent history, computer applications have used pseudorandom number generation for testing ideas since the forties. Figure 3 illustrates components of the learning curriculum proposed in probability and statistics. Student use of these items, and furthering such exploration by expanding their computer-availability, can improve understanding the underlying ideas and results of probability and statistics, and the ways that computers employ them. Widespread interest in games involving statistics contrasts sharply with the lack of broad interest in mathematics courses. Understanding many games requires some appreciation of relative-frequency probability. Figure 3 describes a series of learning activities to develop probability-understanding. The proposal envisions adding computer-based work to these text statements. Placing these ideas in an applications-oriented college course enables students to begin dealing with computers and mathematics in an overall framework.

Planning the Laboratory Experiments

Curriculum preparation involves creating computer learning experiences to illustrate readings and class discussions. That means making laboratory experiments support many objectives. For example, the probability learning activities in Figure 3 offer situations for the computer laboratory to join with the established curriculum. Alternatively, a simulated baseball season [1], a planned betting scenario involving solitaire[2], or a seed game played against a random-selection computer program, vary the standpoint and change the emphasis to noncurricular activity. The computer laboratory experiments generally, and the game /puzzle situations specifically, tend to overcome underperformance in mathematics/academic-endeavors. Thes activities use the underlying curiousity in all students. The basic structure of the experiments reinforces activity by individuals working alone. This will support individuals attaining their maximum intellectual potential in computer use and mathematical understanding. The library-housed versions of experiments will be workbooks and and illustrations. They will enable new multicultural views of mathematics and computing. The computer laboratory will be twenty-four-hour accessible to support solitary learning to the maximum extent possible.

The solids behind traditional and new (DD-game-oriented) dice: cube, tetrahedron, octahedron, dodecahedron, icosahedron, etc., lead to another topic for computer experiment. Since these and other solids create random choice in DD-like games, they clearly bridge geometry and probability. They bring up cultural icons since many are known as the Platonic solids. Still they offer an opportunity to introduce contemporary computer tools: Mathematica enables their rapid display and presentation of novel items.

Readings about these geometric solids [3] show the historic contributions of women to mathematics:

One source indicates that there were at least 28 women classified as Pythagoreans who

participated in the school. Theano, the beautiful wife and former student of Pythagoras,

became a teacher there. ... One of her treatises contained the principle of the "Golden

Mean" celebrated as a major contribution of Greek thought to the evolution of social

philosophy. The children of the marriage between Theano and Pythagoras were also

involved in the Order, and at least two daughters helped to spread the system of

thought that was developed at the school. ... Although many modern writers refer to the

Order as a "brotherhood," this idea owes more to our social lexicon than to actual

fact, for women were serious participants. Indeed, the nature of Pythagorean

philosophy required that women be a central part of the Order. (pp. 16, 17)

The realities of Egypt, particularly the fusion of older developments there with Greek culture culminate in the life of Hypatia, who "was born around A.D. 370 ... her father Theon, was a distinguished professor of mathematics at the University of Alexandria ... ; ... Hypatia was the first woman in mathematics of whom we have considerable knowledge." [3]. We will include material on the contributions of Egypt [4] and many other cultures [5] in the laboratory experiments. This will be done to reflect the interest people have in quantitative ideas. As with games, puzzles stimulate and interest all. We will use them to extend mathematical knowledge. The laboratory will have learning items derived from Figure 3, probability-oriented activities based on [1, 2]; numerical lore [6]; problems used to interest college women in mathematics [7]; collections of puzzles [8]; and diverse sources of material about solid objects.

We will create hands-on experiments from material we have begun assembling. Electronic mail messages in Appendix A indicate the extent to which this effort has enlisted volunteers. to a learning. Students will be asked to derive facts from the computing environment in mathematical learning activities. In addition to computer-implementation of items in Figure 3, we intend to enable many other kinds of materials in a mode supporting experiments and dialogs. Computer-presented versions of Kalah (Fig. 1); provocative puzzles (Fig. 2); and a problem in pyramid geometry whose solution is partially shown by Figure 4, will be developed. The pyramid problem statement follows.

Figure 1. Game Played in Africa and Indonesia

Pyramid problem statement.

The problem is to determine the number of faces when two pyramids are adjoined. One has a square base and four equilateral triangle sides. The second is made up of four equilateral triangles. After adjoining any triangular side of the first pyramid to one of the second pyramid, completely matching them together, and creating a single solid object, how many faces are there in the resulting solid?

Figure 2. Stimulating Mathematical Reasoning

Many other activities are part of this and have already been examined intensively by student volunteers. These items include: the game of Othello, a restricted number of exact-value coin-weighings (not by a balance), numerical mathematics involving unusual issues in arithmetic and geometry including difficult-to-evaluate limits and finding other irrationals, viewing solid shapes from a computer-movie file, composing solids with both triangular and rectangular faces. Indications of the existing work done at UCLA is in electronic mail messages and visuals presented in Appendix A.

The planning activity consists of listing an initial set of ten week-by-week tasks, and cross-referencing them to computer-familiarization and mathematics-learning characteristics. These items will be given to students with the initial course outline. The latter will have cultural and historical references to material on mathematics and computing listed below in References and Bibliography.

Developing Visually-Oriented Mathematics Presentations

Many book items will be used for initial visuals placed on the laboratory computers. Students will create similar items for a course term paper. We use materials oriented to women, e.g., [7], a book by Helen A. Merrill, who for thirty-nine years. taught mathematics at Wellesley College (a residential liberal arts college for women). Problems (90) and demonstrations there offer simple entertaining introductions to a variety of mathematical operations. Likewise, there is a rich amount of published material to open access to quantitative thinking outside the traditional framework, that is rooted in puzzles and mathematical recreations. Numerous interesting items presented first in magazine articles have been collected in books. One example is [8]; recent material by Martin Gardner (much originally appeared as Scientific American columns) is in [9]-[21]. Columns in engineering, mathematical, and science professional journals concern puzzles. Book collections of such items, e.g., [22], offer visual solutions. Figures 2 and 4 introduce problems requiring reasoning visually. In the already-presented pyramid problem [23] reasoning based on Figure 4 is helpful. (See electronic mail, Appendix A.) Students will be asked to write text solutions from computer file materials presented in the laboratory. Recitation meeting clues will assist; all reasonable solutions will be validated. For example, Figure 2 can be approached by the following hints:

a. Where are your baskets? b. What if the pizza were a cake?

c. How do you make a cocktail glass? d. What constitutes a pig-pen?

e. How much of a piece of paper do the dots occupy?

Any one puzzle may add only slightly to students' mathematical sophistication. Still within each are steps, that once taken, ease future progress in traditional mathematics courses. For instance, in Figure 2.b there are natural connections to three-dimensional Cartesian coordinate planes (above hint) and to exponentiation (two to third power). To get people to make those steps, our program uses puzzles and mathematical recreations, and it does so precisely because they are fun. As Merrill [7] put it:

Those who are trying to help our young folks to see not only that Mathematics

is a useful tool and a fine mental discipline, but that hard work may be rare good

fun, and that the subject widens out into fields of ever growing wonder and

fascination, have generally learned what a large role entertaining problems may play.

So many mathematical problems are inherently visual that there are regular columns with visual character in scientific, engineering, and mathematical journals. Some of the materials collected in [9]-[21] and all of those in [22] fit that description. Similar material is shown above in words via the pyramid problem statement.[23], and through Figure 4. (Also see electronic mail from Duke in Appendix A .) Class activities will involve creating text descriptions and visual images for puzzles and mathematical problems, for ultimate installation on laboratory computers. We have begun doing this on the following topics: Egyptian unit fractions [4]; estimation of p in different cultures [5]; evaluating a difficult-to-compute slowly-converging limit [24]; multiplication to yield a repeated digit n (product 12345679 times 9n) [22]; the 3x + 1 looping problem[20]; finding the weight of any defective coin in a set with a scale (not a balance) and a restricted number of weighings; palindromes [6]; and other mathematical issues of a special - possibly visual - character (e.g., magic squares).

Early Familiarization With Sophisticated Computer Tools

We will introduce specialty aspects of word-processors: symbol font, formatting to create tables, mathematical equation and outlining features; to students beginning a four-year curriculum. We will involve them in contemporary mathematical computer-tools, in particular Mathematica because of its excellent graphic features. We will initiate early student world-wide-web (www) use. We will build on other undergraduate students' www search: see Appendix A, games; B, estimation of p.

The prior activity has located several items supporting the effort. These are basically either in text, or consist of text with modest amounts of visual elements. The items are either found in: 1) electronic mail (email) of Appendix A; 2) visuals in Appendix B; and 3) books listed in References and Bibliography. Undergraduate students using web browsers found many non-book item. Many www items are expository as are most of the book items listed, especially [25]-[30]. They contain paragraphs that are like a friend or a patient-teacher: words in a conversational tone. These items will be compiled into ten or more sets which allow student choice, and enable reading and experimentation in the laboratory on compiling student-generated text and visual materials.

We intend to enable rapid use of Mathematica, the UCLA Orion Library Information System, web browser, and other software. This will be done in a use-oriented computer-laboratory based program. The program is designed to support early learning and utilization of software, as well as growth in mathematics knowledge. Each student is to create a mathematics/computer-oriented report composed during the course, using as many as possible of the installed sophisticated computer tools. Any student who is unable to use these computer tools can still succeed in the planned course by writing a report about material in [30]-[34]. Resources for reading and reporting about mathematics and history include [3],[24] , [27]-[34]. A combination of learning some of the historic background and writing a report using computer word processors has to bring about both the results sought: some removal of mathematics inhibition and some computer familiarization.

Mathematica lets students see manifestations of many different computing and mathematical concepts rapidly. Since it interprets arithmetic commands as if it were a calculator, it is easy to start using. Whether it is used to evaluate an exponentiation, factorial, differentiation, matrix computation, or such a sophisticated computation as linear programming, it provides students with numerical response from a minimum learning effort. At the same time, it is one of a class of mathematical computer tools. Its powerful graphics enable display by computer of surfaces, shadings, and change of viewpoint. This supports visual operations we will develop once the laboratory is in place. Preliminary efforts such as the display of a tetradecahedron, a solid with fourteen rectangular and triangular faces, have taken place (see visual items in Appendix B).

Early access to Orion and to Mathematica support many skills. Orion enables ease at accurately composing a report bibliography. Mathematica opens students to the range of mathematics. Mathematica displays: a) two five-faced pyramids, Figure 4; Platonic solids; the tetradecahedron; enable student-visualizing key computer science data structures.

Extending Knowledge, Mathematics and Play

Some students will use the laboratory to extend their knowledge by developing skill with Mathematica that will serve them well in future courses in science, engineering and mathematics. This will fill a space in our curriculum, since currently, unlike the institution where Mathematica was originally developed, Caltech, few UCLA first and second year students learn to use this versatile computational tool. Similarly, some will be capable of performing a statistical experiment using random number generators on computers, calculators, or hand-simulation based on Rand's million-values book [35]. Any who wish more probability knowledge will be guided to the text used for a general-undergraduate student mathematics course in probability, [36].

The evolving nature of the activities we seek to open up to general students by the curriculum change, is, of course, open to their own innovation. Many of the materials will make it clear that mathematics and computer science are activities that grow through individuals' thought and effort. For example, we will use [37] where a chapter (pp. 242-253) presents mathematical aspects of palindromes as an example. A short version of palindromes based on [6] or [31], p. 146, will be on laboratory computers or in class presentation materials. This will be a lead-in for students who wish to extend their knowledge by reading [37], and possibly using its extended introduction for a report.

The laboratory gives students the opportunity to investigate recent contributions. For example, through [38], p. 123, they can explore the calendar concept of perverse months, due to Prof. Kirby Baker of the UCLA Mathematics Department. They can look at a blind-chess-game called Kriegspiel [39]; although it is discussed there, and originally in a puzzle-column on which the book is based, the name means war-game and its play was a part of post World War II history at the neighboring research institution, the Rand Corporatio. Other puzzles in [39] may stump most, but a few are good candidates for to be installed on the laboratory computers, particularly polychrome, a map coloring problem which uses the countries of South America, and the rack concerning locating billiard balls, both for their strong visual nature.

A puzzle-column of [40] extends palindromes to equations and asks a question that opens another line of play. What equations are still valid when reversed left to right? Its example, 473 is 43 times 11, but 374 is 11 times 34, offers a way to start students on laboratory exploration that is at once mathematical, inquisitive, potentially rich in solutions, amenable to computer experimentation, and not oriented to a traditional curriculum.

Learning from Games and Puzzles

Learning from Games

Learning from Games

Learning from Games

Learning from Games

ComputerLaboratory Learning Curriculum in Probability and Statistics

(relative frequency, geometric probability)

Probability concerns systematic ways to handle chance. Chance occurs in situations like dice-playing. Systematic probability-study began with the question: When tossing a pair of ordinary dice, how many tosses should the player expect to make before rolling double 6? Two mathematicians, Pascal and Fermat, solved the problem and showed where a third person had gone wrong (and why professional gamblers were right). One way to start considers throwing a one six-sided die and predicting the chance of throwing a three. Experience shows three occurs about once in six tosses, when the die is thrown many times. Computers easily conduct equivalent experiments. where many is sixty-thousand.

1. Conduct an actual and a computer experiment to study:

a. Coin toss. Take a penny and throw it in the air ... and do this one hundred times recording the number of heads that come up when it lands. Write out the probability of heads that you find based on the results. Give another way to gauge the probability of heads.

b. Card draw. Choose a card from a deck of fifty-two and record it. Replace the card in the middle of the deck. Shuffle. Repeat the draw and other steps one hundred times recording the number of King, Queen and Jack (royal) cards that you observe. Write out the probability of seeing a royal card that you find based on the results. Give another way to gauge the probability of drawing a royal card.

c. Toss dice. Take two dice. Roll them and record their sum. Repeat this one hundred times. Write out the probability of getting a total of seven that you find based on the results. Give another way to gauge the probability of a total of seven.

2. Compute In Roulette. In one form of the game there is a wheel with 38 sections. The numbers on the sections are 0, 00, 1, 2, 3, and so forth up to 36. A game consists of rolling a ball until it stops in a section, selecting its number. If 0 and 00 are not odd nor even, find the probability of an even number. Evaluate the probability by a computer experiment.

3. Obtain and view video and workbook [34] (reference 4). Write a report on cited workbook pages .

4. Report on prediction using encyclopedia articles on probability, chance, random, and average.

5. Read about randomness on [21, p. 122]. Write a report discussing six improbable but extremely probable events you define. [The cited page concerns of the behavior of p from the 710,100-th decimal digit on, where seven 3's in a row appear.]

Figure 3. A Sample of Computer Laboratory Activities for Probability Learning

Developing Visually-Oriented Mathematics Learning Resources

Mathematica Illustrates that a Tetrahedron Fills Gap Between Two Square-Based Pyramids

Figure 4. Pyramid Geometry

References and Bibliography

[1] Waggoner, Glen, Editor, Rotisserie League Baseball, New York: Little, Brown and Company, 1994, ISBN 0-316-91712-5

[2] Morehead, Albert H., Mott-Smith, Geoffrey, The Complete Book of Solitaire and Patience Games, New York: Bantam Books, David McKay Company, Inc., 1949, 1979, ISBN 0-553-13373-X.

[3] Osen, Lynn M., Women in Mathematics, Cambridge MA: The MIT Press, 1974, ISBN 0-262-15014-X.

[4] Gillings, Richard J., Mathematics in the Time of the Pharaohs, Cambridge, MA: The MIT Press, 1972; NY: Dover Publications, Inc., 1982, ISBN 0-486-24315-X.

[5] Joseph, George Gheverghese, The Crest of the Peacock - Non-European Roots of Mathematics, London UK: I.B. Tauris & Co. Ltd Publishers, 1991.

[6] Friend, J. Newton, Numbers:Fun & Facts, New York: Scribner, 1954.

[7] Merrill, Helen A, Mathematical Excursions, New York: Dover Publications, Inc., 1933, 1957.

[8] Frohlichstein, Jack, Mathematical Fun, Games and Puzzles, New York: Dover Publications, 1962.

[9] Gardner, Martin, Riddles of the Sphinx, Washington D.C.: Mathematical Association of America. 1987.

[10] Klarner, David A., editor, The Mathematical Gardner, Boston MA: Prindle, Weber & Schmidt; Belmont CA: Wadsworth International, 1981.

[11] Loyd, Samuel, Mathematical Puzzles, Selected and Edited by Martin Gardner, New York, Dover Publications, 1959-60.

[12] Gardner, Martin, Time Travel and Other Mathematical Bewilderments, New York:

W. H. Freeman, 1988.

[13] Gardner, Martin, New Mathematical Diversions from Scientific American, New York: Simon and Schuster, 1966.

[14] Gardner, Martin, The Scientific American Book of Mathematical Puzzles & Diversions, New York: Simon and Schuster, 1959-61.

[15] Gardner, Martin, Sixth Book of Mathematical Games from Scientific American, San Francisco CA: W.H. Freeman, 1971.

[16] Gardner, Martin, Knotted Doughnuts and Other Mathematics Entertainments, New York: W.H.

Freeman and Company, 1986.

[17] Gardner, Martin, The Unexpected Hanging and Other Mathematical Diversions, ChicagoIllinois: The University of Chicago Press, 1969, 1991, ISBN 0-226-28256-2.

[18] Gardner, Martin, Mathematical Carnival, New York: Alfred A. Knopf, Inc., 1965, 1966, 1967, 1968, 1975, ISBN 0-394-49406-7.

[19] Gardner, Martin, Gardner's Whys & Wherefores, Chicago IL: Univ. Chicago Press, 1989.

[20] Gardner, Martin, Wheels, Life and Other Mathematical Amusements, New York: W.H. Freeman and Company, 1983, ISBN 0-7167-1588-0.

[21] Gardner, Martin, aha! Gotcha: Paradoxes to puzzle and delight, New York: W. H. Freeman and Company, 1975, 1982.

[22] Nelsen, Roger B., Proofs Without Words: Exercises in Visual Thinking, Washington, D.C. : The Mathematical Association of America,1993.

[23] Davis, Robert B., Learning Mathematics:The Cognitive Science Approach to Mathematics Education, Norwood, NJ: Ablex Pub. Corp., 1984.

[24] Kasner, Edward, and Newman, James, Mathematics and the Imagination, New York: Simon and Schuster, 1940, 1963; ISBN 1-55615-104-7 Redmond, WA: Tempus Books of Microsoft Press, 1989.

[25] Adams, James L., Conceptual Blockbusting, San Francisco CA: W. H. Freeman, 1974.

[26] Stevenson, Harold W. and Stigler, James, W., The Learning Gap - Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education, New York : Summit Books, 1992.

[27] Beckmann, Petr, A History of Pi, Boulder, CO : Golem Press, 1982.

[28] Davis, Philip J., The Thread, A Mathematical Yarn, New York: Harcourt Brace Javanovich, Publishers, 1983, 1989.

[29] Humez, Alexander, Humez, Nicholas, and Maguire, Joseph, Zero to Lazy Eight, New York: Touchstone, 1993.

[30] Kline, Morris, Mathematics for the Nonmathematician, New York: Dover, 1967.

[31] Pappas, Theoni, The Joy of Mathematics, San Carlos CA: Wide World Publishing/Tetra, 1986, 1987, 1989, ISBN: 0-933174-65-9.

[32] Pappas, Theoni, More Joy of Mathematics, San Carlos CA:Wide World Publishing/Tetra,1991,

ISBN: 0-933174-73-X.

[33] Dunham, William, Journey Through Genius: The Great Theorems of Mathematics,

New York: John Wiley & Sons, Inc., 1990.

[34] (Video) The Story of p, Pasadena, CA 91125: Project MATHEMATICS!, California Institute of Technology; Program Guide and Workbook (to accompany the videotape).

[35] The Rand Corporation, A Million Random Digits With 100,000 Normal Deviates, New York: Free Press, 1955.

[36] Brown, Robert and Brown, Brenda W., Essentials of Finite Mathematics, New York: Ardsley House, Publishers, Inc., 1990, pp. 222-227.

[37] Gardner, Martin, Mathematical Circus, New York: Alfred A. Knopf, Inc., 1979.

[38] Gardner, Martin, The Magic Numbers of Dr. Matrix, Buffalo, New York: Prometheus Books, 1985.

[39] Silverman, David L., Your Move, New York: McGraw-Hill, 1971.

[39] uSilverman, David L., Your Move, New York: McGraw-Hill, 1971.

[40] The Bent of Tau Beta Pi, Spring 1995.

Appendix A

1. Electronic Mail Descriptions of Activities and Issues for Learning Materials

Text on Adjoining Pyramids W. Duke

Text on Coin Weighing W. Duke, T. A. Brown


Date: Wed, 22 May 96 12:49:31 PDT

To: <>

Subject: The Pyramid Problem

Professor Klinger,

Here is a geometry problem I came across in the book "Learning Mathematics" by Robert B. Davis, published in 1984 by Ablex Publishing Corporation, New Jersey. After you try it you may want to include it in one of your math merit badges.

The following problem appeared in the 1980 PSAT and the answer believed to be correct by the Educational Testing Service experts was, in fact, incorrect.

Construct two pyramids with every edge of equal length, L. The first pyramid has a square base and four equilateral triangle side. The second pyramid has an equilateral triangle base and three more equilateral triangle sides. Glue any triangular side of the first pyramid to any triangular side of the second pyramid, completely matching the two glued triangles together, and creating a single solid object. How many faces does the resulting solid have?

You can stop reading here and try to determine the correct answer without help.

The Educational Testing Service and nearly all of the student test takers selected the incorrect answer of 7 based on the first pyramid having 5 faces and the second pyramid having 4 faces for a total of 9 with 2 faces disappearing when glued together.

If your answer was also 7 and incorrect, you should stop reading here and try the problem again.

If you have convinced yourself that the correct answer is 5 you may proceed.

One student correctly answered that there were only 5 faces remaining because in two separate cases what had been two distinct faces became a single face after gluing. Why did that student clearly see the correct answer and the experts did not?

So now comes the $64,000 question: Can you describe a simple representation of this problem that demonstrates that the correct answer of 5 faces is immediately obvious?

Definitely stop here and try to construct such a representation without reading the following answer.

If you have given up trying, you may proceed and visualize the following representation. Being limited without the aid of a blackboard, I will do my best to draw you a mental picture using only e.mail text.

Put the four-sided pyramid out of sight. Place the square base of the five-sided pyramid flat on the table. Construct another identical five-sided pyramid and place its square base also on the table. While keeping both of their square bases flat on the table slide the two five-sided pyramids together such that one edge of one pyramid's square base completely coincides with an edge of the other pyramid's square base.

Draw a picture at this point if you have trouble visualizing the two identical pyramids side by side. It is obvious that the two square bases lie in the same plane, the plane of the table. It is also obvious that two of the four remaining triangular faces of the first five-sided pyramid lie in the same plane as the two corresponding faces of the other five-sided pyramid. A closer inspection confirms that the remaining two corresponding faces of each pyramid lie in parallel but separated planes.

In each of the two five-sided pyramids construct an altitude line which is a vertical line connecting the top or apex of the pyramid to the center of the square base. Observe that a straight line drawn on the table from the center of the square base of one pyramid to the center of the square base of the other pyramid has a length identical to all of the same length edges of the pyramids.

Get ready! Draw a straight line through the air connecting the tops of the two pyramids. You should be comfortable that the length of this line is the same as that of the line connecting the centers of the square bases and therefore the same length as all of the edges of the pyramids. The four lines you have drawn square center to square center, top to top, and the two vertical altitude lines form a rectangle.

Take a deep breath and go find the four-sided pyramid you set aside earlier. Set it down in the valley formed between the two five-sided pyramids already sitting on the table. Are you convinced that the four-sided pyramid with its four equilateral triangular sides exactly fits between the two five-sided pyramids with its exposed top edge coincident with the top to top line you constructed?

Now it is obvious that the two hidden faces of the four-sided pyramid coincide with the two now hidden triangular faces of the five-sided pyramids. So must the two remaining exposed faces of the four-sided pyramid lie in the same planes as the exposed coplanar faces of the five-sided pyramids to either side.

So put a drop of glue between the four-sided pyramid and one of the five-sided pyramids and toss the loose five-sided pyramid into the trash.

The resulting solid sitting on the table has only 5 exposed faces.

Steven Young published this solution, known for obvious reasons as the "pup tent" solution, in his article, "The Mental Representation of Geometrical Knowledge" in the Summer 1982 Journal of Mathematical Behavior.

Good luck,



Date: Thu, 30 May 1996 05:12:51 -0700


Subject: My "computer software" proposal stuff...

The associated computer lab will help achieve the main goal of the course: formalized individual exploring of computer applications and mathematics. The ten week quarter would be composed of several "learning units". These units are activities that are either done in the computer lab or individually researched by the student (here the student is shown the excellent research potential of modern Internet search engines).

The computing lab would be a tool to focus the non-science student to overcome any uneasy feelings of computers and math. Class progress may be measured several ways, for instance, one could have the student complete established "learning units" in the computer lab and turn in interim reports. Lab exercises would be mental exercises from different areas of mathematics and computer science presented in the form of games and puzzles (see appendix for samples). The actual graphics and coding would have to be done by dedicated staff for the class (see "Required software for staff"). At the end of each learning unit, or set of coherent learning units, the student will report on the activity ... ... visual analysis of presented problems and clear exposition are complementary to the course ...

Through the use of the computer lab, non-science students not only gain experience with software well know by initiated science majors, but also remove self-inflicted barriers to math and computers. Barriers are formed for a variety of reasons ... the computer lab is to provide a non-pressured, yet structured environment to stretch the students mind and foster individual interest ... By using modern computer resources we remove the barriers (to) computer technology and mathematics.

From: "Christine A. Tai" <>


Cc: Team <>,,,

Subject: Platonic Solids

Professor Klinger,

We have complete a draft of the QuickTime movie with the platonic solids according to your ideas. For this draft, we have chosen the tetrahedron. The movie is finished in functionality (i.e. the user can rotate the solid interactively using Apple QuickTime Virtual Reality software and view the solid from all horizontal sides). It can use some refining, however, in the appearance of the figure (in terms of resolution and details). The movie can also be made so that the figure can be rotated and viewed from all angles (not just horizontal) but a rig would be required for this. Nevertheless, the existing draft can give you a general idea of how we can utilize the Apple QuickTime technology to further education in the mathematic and/or related fields.


Christine Tai <>, Judy Lee <>

Pavel Vesely <>

Wei Michael Sun <>

From: "Christine A. Tai" <>


Cc: Team <>,,,

Subject: Platonic Solids

Professor Klinger,

We have complete a draft of the QuickTime movie with the platonic solids according to your ideas. For this draft, we have chosen the tetrahedron. The movie is finished in functionality (i.e. the user can rotate the solid interactively using Apple QuickTime Virtual Reality software and view the solid from all horizontal sides). It can use some refining, however, in the appearance of the figure (in terms of resolution and details). The movie can also be made so that the figure can be rotated and viewed from all angles (not just horizontal) but a rig would be required for this. Nevertheless, the existing draft can give you a general idea of how we can utilize the Apple QuickTime technology to further education in the mathematic and/or related fields.


Christine Tai <>

Judy Lee <>

Pavel Vesely <>

Wei Michael Sun <>

From: Date: Fri, 31 May 1996 02:47:28 -0700


Subject: Polyhedra with "dropped" faces...

Hi Dr. Klinger,

I have just found the necessary commands ...for making Platonic solids appear with any number of dropped faces ... commands:



Show[Graphics3D[Drop[First[%], (1)]]]

... should produce a cube with a top missing ... using the 3d viewpoint selector we can rotate the figure any way we want in 3 space. I've also tried it with Icosahedrons. ... (picture inside a hollow

Platonic solid is now realizable in Mathematica).


From: Date: Fri, 31 May 1996 03:37


Subject: Infinite Product Draft

There is no analytic, closed form expression for this infinite product:



| |

| | Cos(Pi/n)

| |


Surprising? Maybe not at first glance, but if we consider the limit of the above expression we find that it converges very, very slowly. A closed form would give us essentially unlimited precision if we knew it in terms of other transendental numbers and a finite number of arithematic operations (*,/,sqrt,+,-). If this were an engineering application, a high degree of precision would usually be necessary. But, because of it's slow convergence, we must find faster series representations for higher degrees of accuracy. For N in the millions we only get a few digits of accuracy ... First, let me present the geometry that ... makes it an interesting problem. ... The limit above arises from ... :

... take a unit circle and inscribe an equilateral triangle in it, then inscribe another circle inside the triangle, then inscribe a square inside the new circle, then inscribe another circle inside the square, then ... inscribe a pentagon and so on (every time the number of sides of the polygon inside the new circle increases by one). The radius ... converges to a limit: given by the infinite product above.

Using a computer it is easy to show that the limit's value is 0.1149....

Is there any way to find an analytic solution: with irrational numbers, roots, or perhaps ... gamma functions (... convert ... cos(pi/n) to its exponential form). ...

... references to publications for the infinite product problem ( Product[Cos[Pi/n], {n, 3, infinity}] ):

Math and the Imagination, 1942, page 311 *0*

Mathematical Spectrum 26:4, 1993/1994 pages 110-118 *1*

C.J. Bouwkamp, An infinite product, Ned. Akad. Wet. 68(1965), 40-46 *2*

(Bouwkamp) Indagationes Mathematicae vol. 27, 1965, part 1, pp 40-46 *3*

Reference *2* and *3* may be the same paper. Both (*1* and *2*, *3*) authors say that a closed form probably doesn't exist (Bouwkamp on the front page, Mathematical Spectrum on page 117).

Mathematical Spectrum goes about taking the logarithms and using power series for cos x and log (1-x) and even uses Bernoulli numbers to find a form of the series that converges quickly. The Bouwkamp article does about the same (numerical approximations).

Here's some Mathematica code I found in sci.math newsgroup (using



Re: HELP!!! Infinite Product!!! Return-Path: bruck@paci

Organization University of Southern Californi Date Thu, 15 Feb 1996 19:59:18 -0700

Newsgroups sci.math Message-ID <>

In article <bruck-1502961848200001 ...>, (Ronald Bruck) wrote:

:In article <4g0ko6$>, (Kevin Johnson) wrote: ::Xander Kenobi ( wrote: ...(what's the value of)

::> inf. approx 0.114942 (?) i let this go a lot farther,

::> PI cos(pi/n) but.. began to question.. cos() function ..

::> n=3 I don't see any obvious expression to get this

:A suggestion for numerical computation: use the series for log cos x,


: log cos x = -x^2/2 - x^4/12 - x^6/45 - (17*x^8)/2520 - (31*x^10)/14175 - ...


:and sum this for x = pi/n for n = 3 to infinity. You get something involving the zeta function of even values Then exponentiate at the end. Hmmm. But the series must diverge when x = pi.

Yes, that's the ticket. In Mathematica:

logcos = Normal[Series[Log[Cos[x]],{x,0,60}]];

from100 = logcos/. x^n_ -> Pi^n(Zeta[n] - Sum[1/k^n,{k,1,100}]);

from100dec = N[from100,100];

val = Product[N[Cos[Pi/n],100],{n,3,100}] * Exp[from100dec]

The result is 0.114942044853296200701040157469598742830795337200863516844023396518966 \01282535305117794

The idea is to compute, say, \prod_{n=101}^\infty cos(\pi/n) by taking the log. The first line of Mathematica gives the series ... through degree 60 (the Normal strips off the O() term and converts logcos into a polynomial). ... substitute x = Pi/n from n = 101 to infinity and sum. Mathematica knows ... Zeta[n] for even n in terms of Pi and the Bernoulli numbers, so from100 is an exact expression, no numerics involved. The next line converts it to 100-place decimal. The final line exponentiates from100dec to ... product from101 to infinity, and then multiplies by the 3..100 terms.

You can probably do it with a lot smaller numbers than 60 and 100. The answer stays the same if you increase the 60 or the 100. ... took about 49 seconds on a Power Macintosh 9500/132. ... I still prefer the interpretation that it's the average value of

cos(Pi * (1 \pm 1/2 \pm 1/3 \pm 1/4 \pm 1/5 ...)), \pm = plus-or-minus.

--Ron Bruck ...



As you can see, the algorithm given is very efficient and accurate.

For any engineering purposes, having a quick algorithm with a large degree of accuracy is sufficient. But I still wonder if a closed form is possible...along that branch, I have found the Euler-MacLaurin formula:

n/ n/

n / /

--- | |

\ f(k) = 1/2(f(0) + f(n)) + |f(x) dx + | f'(x) p(x) dx

/ | |

--- /0 /0

k=0 / /

p(x) = x - [x] - 1/2 = Sum[ -1/Pi Sin[2 Pi k x] / k , {k,1,Infinity}]

(Apply) the above for f(k)= log cos (Pi/(k+3)) and the problem is solved!

Mounitra Chatterji