a.  Title

Mathematical Exposition and Visual Knowledge

 

b.  Topic Area

Other Areas of Mathematics (Possibly Mathematics Education)

 

c.  Presentation Format

Complete Research Paper

 

d. Author

Allen Klinger

e.  Department, Affiliation

Henry Samueli School of Engineering and Applied Science

Computer Science Department

University of California Los Angeles

 

f.  Mailing Address

4532-K Boelter Hall

UCLA

 Los Angeles, CA 90095-1596

U.S.A.

 

g.  E-mail Address

klinger@cs.ucla.edu

h.  Phone Number         

310 825-7695

i.  Fax Number

310 794-5057

j.  Corresponding Author

Allen Klinger


1. Introduction

 

A well-established mathematical structure with relationships, names, and notations, exists and is a core educational subject. Yet there is a pervasive and general sense that the structure is Òtoo hardÓ or generally inaccessible to the ordinary person. This leads to an unfortunate partition of people into the scientifically attuned (a presumably small elite) and the averse. Existence of large numbers of mathematically averse individuals comes with a high social cost. People avoid studies that would qualify them for higher-paying employment. Others believe that a slot machine is a worthwhile gamble and spend funds that they really need.

 

This paper concerns learning and knowledge in mathematical areas. The purpose of this last phrase is to begin to distinguish thinking from application. The idea is that success in the field calls upon a set of skills. Those skills share subtle and complex issues involving description Ð visual knowledge. Experience with real problems may fail to enable independent thinking because of traditional discipline boundaries.

 

One becomes discomforted when a puzzle is posed in words if its key idea is outside the realm where training takes place (for people succeeding in a traditional mathematical, applied science, or engineering curriculum). Fractions and ratios illustrate this. A curriculum conveys training in some useful tools. Two there are: decimal equivalents of whole number ratios, and fraction arithmetic skills. Neither kind of training has any value when considering the following three ordinary ideas all of which pertain to different mathematical areas.

 

1.    Military time: Twenty hours as the equivalent of ten p.m.

2.    Baseball arithmetic: One half plus a third as two fifths.

3.    Price: A herring and a half is a penny and a half; Six herrings costs what?

 

All three may seem ordinary through non-school experience, but they relate to abstract mathematical notions (modulo arithmetic, algebra).

 

This paper contains a series of questions that communicate ideas, parts of the mathematical knowledge. The questions are generally stated in words, ordinary language, not using special terminology. They are available here to form the body of a different kind of introduction to mathematics, one that lessens the anxiety2 and opens the subject to otherwise able and different3 individuals. We below note contributors to mathematicsÕ from the ÒdifferentÓ group. Two statements by Albert Einstein convey some of the rational for this paper.

 

The creative principle [of science] resides in mathematics (1933).

 

Do not worry about your difficulties in mathematics; I can assure you that mine are still greater.

 

Quotations like the first statement appear frequently: physicists saw that mathematics gives ways to close in on underlying realities. But the second statement has a quixotic appeal. It combines two issues:  Einstein was not a good student in school; and, his work wrestled with difficult concepts, things beyond the ordinary and accepted body of mathematical knowledge.

 

This paper focuses on the role of ideas in mathematics, and how to convey that both specific concepts and the overall body of mathematical knowledge are a common heritage of all of humanity.

 

2. Conventional Mathematics as Preparation.

 

The truest test of learning involves more than finding a way to apply some body of knowledge. Situations where one must develop an answer by synthesizing many kinds of information clearly call for more than simply matching what is asked, to some things that have been already learned. Conventional mathematics curricula convey skill at issues that are sometimes useful, such as long division and converting a fraction given as a ratio of two integers to decimal form. Nevertheless, many are confused by rate times time equal distance problems, find trigonometry baffling, or simply choose to stop participating in learning things offered under the heading of algebra. Unconventional problems such as the following three questions make the above concrete (a productive direction for thought Ð a hint - follows each; answers are appended):

 

a. WhatÕs half of eleven?

                                    Roman numeral representation; horizontal cut.

 

b. Can you put nine apples in four baskets so that there are an odd number in each?

                                    In signifies inclusion; nest baskets.

 

c. Is it possible to divide a circular pizza pie into eight exactly equal segments with three straight line cuts?

                                    Group cut items; relate numbers three and eight. 

 

Question a concerns representation or notation. (It also puts forward a word statement far different than rate times time equals distance.)  All three questions are completely stated in language: they make no use of special mathematical symbols.

 

Another problem is like b in its use of both four and nine. Many know Òconnect nine-dots with four straight lines where the pen/pencil does not leave the paper.Ó The problem and several innovative solutions appear in AdamsÕ book4.  That problem led to the common linguistic maxim: Think outside the box!

 

Arithmetic, geometry, and algebra are three parts of the mathematics curriculum. They describe some major areas of mathematical knowledge. Is the material taught there adequate? Does it make best use of learning time? (Consider the availability of digital computers and inter-net sources?) Does it prepare students to think when confronted with questions like a, b, and c?

 

The next question puts the above squarely before us because it involves arithmetic, challenges, and often causes discomfort.

 

d. Can you combine the four digits, 1, 5, 5, 5 using only the four arithmetic operations, plus, minus, multiply and divide (and parentheses) to form the quantity 24?

                                    This is second grade math.

                                   

People have difficulty with d. This isnÕt a question for which people tend to immediately write down an answer. This supports considering what the mathematics curriculum actually does. It is possible that beginning from the earliest school grades, the need to communicate special approaches, terms, and notations, has driven away the underlying sources of mathematics.

 

The origin of mathematical knowledge is explored in DunhamÕs book (mathematics), AscherÕs cross-cultural works (ethno-mathematics), and also via calendar issues CampbellÕs examination of myth (anthropology). The start to mathematical knowledge had a greater focus on the visual, things that can be seen.

 

3. Visualization

 

Those outside the mathematical tradition, like Ramanujan, Fermat, and Gardner, show that involvement in questions is one way that it grows. The questions may seem odd. For example, Gardner displays3 answers to the problem of composing numbers using only four fours, arithmetic operations, adjacency/concatenation, and square roots. The answers appear up to twenty É except for nineteen Ð for example, 15 = (44/4) + 4; and 13 = (44/4) + Ã4.

 

Though it seems analytical, there is visual challenge in finding a four-fours form of numbers from:

 

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20}

 

Likewise, the following fact known to the early Greeks challenged me for several years until I saw it as a visual issue:

 

e. The cube of every positive integer is a sum of consecutive odd integers.

                                             What is a cube?

 

The understanding of statement e depends on making a visual interpretation of the language, both as the hint suggests and since we need to see that consecutive or adjacent odds are things like three and five. Under this thereÕs a fundamental mathematical idea. The linguistic statement tends to obscure the importance of the word odd: other than the elements {the, of, is, a}, it is the smallest word in the sentence. In fact by distinguishing two concepts, odd and even, progress can come from considering the first six words of statement e.

 

Another small but significant word in statement e is sum. Sums are written either in a line as 6 + 7 + 203 + 59 + 834 + 4002 + 314 or as a vertical list. The list aids our senses to enable success with executing an addition algorithm (specifically carrying numbers from say the integersÕ to the tensÕ place). But even if one moves to seeing a cube, it may be much harder to come up with the consecutive odd integers that represent seven-cubed 73 = 343, three-hundred forty-three, than to rewrite and add the above seven numbers. This is especially so because we arenÕt trained to deal with the opposite of sums, taking a quantity and partitioning it into constituents.

 

Both question c and statement e deal with visual issues involving partitions. Some of the earliest human writing does so as well. In ancient Egypt partition methods ensured equality of amount, and also the perception of equality. (TodayÕs methods divide nine loaves of bread among ten men giving out nine large portions and one set of nine one-tenth slices. The older approach would begin with nine two-thirds-sized portions, reserving two one-thirds for the tenth person. The remainder would come from tenths of one-third loaves, some as six-thirtieths or one-fifth, the others as one-thirtieths.)

 

Partitioning connects to mathematical knowledge two ways. As a general subject one place is in combinatorics, but for statement e, the best idea source is through two- and three-dimensional (Cartesian) coordinate systems. For question c the partitioning for a solution comes from repetition. But the existing organization of mathematical knowledge doesnÕt lead to recognizing repeated partitioning (if halving, division by powers of two: halves of halves, the second power of two, 22 = 4).

 

4. Extension Ð From Investigation to Maturation

 

The word extending indicates moving beyond an initial mathematical idea, as in the case of perceiving that since 23 = 8, three divisions of the original pizza folded over or arranged so all pieces are cut each time, solves c. To some extent extending, as in this example, moves from some field of mathematics to another one (arithmetic to geometry in this case). But another way to see what happens is to stick with ideas instead of fields.

 

This problem statement appeared in Tau Beta PiÕs Bent journalÕs Brain Ticklers column:

 

f. We are interested in finding a sequence of 2n + 1 consecutive positive integers, such that the sum of the squares of the first n + 1 integers equals the sum of the squares of the last n integers. The simplest such sequence is 32+ 42 = 52.

                                             What relates the numbers to one another?

 

The first thing one sees is consecutive numbers. All are squared, the results are summed in groups, one sum comes from the lower n + 1, the other from the upper n. A solution can be algebraically derived, or based on ideas/concepts Ð come from an investigation.

 

A careful look at the equation 32+ 42 = 52 enables responding to the hint by stating these key facts (the appendix shows combining these facts to yield a useful result):

 

- There is a biggest number in the list of those squared that has n + 1 entities.

- The remaining numbers, those less than that biggest number, are smaller by an integer, as in the example where the difference is one.

- The n numbers in the list of those squared on the right of the equals sign are bigger than the biggest number left of the equals sign by an integer, as in the example where the difference is one.

 

Anyone interested in f is likely to seek out the next few examples. Once done, even if it is only to obtain the immediate subsequent equation, it is possible to hazard a guess, the same thing that is implied by the use of n in the statement of f.  That guess (conjecture) is that there are infinitely many equations like 32+ 42 = 52. Notice that this moves the issue beyond the ordinary (É  interested in finding a sequence of 2n + 1 consecutive positive integers É [emphasis added]). What the guess does is take the question into a bigger area: specifically, no matter how big the number n there will always be a sequence of 2n + 1 consecutive positive integers É.

 

The above conjecture is (mathematically) a better question than f. A word expression at end of the last paragraphÕs final sentence states the central issue: the property of equations that look like 32+ 42 = 52 occurs in infinitely many cases. Most people do not know this: in April of 2004 I was in that group.  [IÕd sought help about equalities of this type by sending out electronic mail with subject line ÒPythagorean Magic.Ó] When something is true for more than one case, even if it is only true for two, it may be generally true. Experience led to knowing that, and thus to the extension and conjecture.

 

Experience gave rise to current mathematical knowledge. The knowledge stems from achievements of unnamed artisans and workers. Some people from ancient times, many Greek and Egyptian, are associated with specifics. But there are others less known and unknown, who contributed important steps. Whatever the origin, and whichever names are used to label ideas, only the ideas (and their accessibility) are the important matter.

 

4. Conclusion

 

Many simple ideas and concepts form the core of mathematical knowledge. The concepts include whether a number is even or odd. They involve comparing two numbers and seeing which is less, and which more (or greater).

 

Like the presumed difficult nature of mathematics, it is not magic to resolve questions like a through f. The world is filled with facts that people have observed, learned from, and described in language and notation known as mathematics. People in that field know some things from experience. The statement that best describes possessing significant amounts of that knowledge is maturity. One gains mathematical maturity as one does in life, by extending questions, investigating where they lead, and retaining a body of experience.

 

Today the inter-net enables disseminating visual and auditory material on a worldwide basis. It is easier than ever to show an image, say something to explain it, and communicate to people far away about mathematical knowledge.

 

 


References

 

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

 

Ascher, Marcia, Ethnomathematics, A Multicultural View of Mathematical Ideas, Pacific Grove, CA: Brooks/Cole Publishing Company, 1991.

 

Ascher, Marcia, Mathematics Elsewhere, Princeton NJ: Princeton University Press, 2002.

 

Campbell, Joseph, The Masks of God: Oriental Mythology, NY: Viking Penguin, 1962-91.

 

Dunham, William, Journey Through Genius: The Great Theorems of Mathematics, NY: John Wiley & Sons, Inc., 1990.

 

Einstein, Albert, Ideas and Opinions, 274, June 10, 1933,

 

Gardner, Martin, The Numerology of Dr. Matrix, NY: Simon and Schuster, 1967.

 

Lakoff, George and Nunez, Rafael, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, NY: Basic Books, 2000.

 

National Council of Teachers of Mathematics, Principles and Standards for School Mathematics.

 

Tobias, Sheila, Overcoming Math Anxiety, NY: W. W. Norton, 1978 and 1994, revised.

 

Tobias, Sheila, TheyÕre Not Dumb, TheyÕre Different: Stalking the Second Tier, Tucson AZ: Research Corporation, 1990.

 

Appendix

        

a. Horizontally cut through XI yielding VI in the upper half. The concept is number representation.

 

b. It can be done if one nests at least two baskets. Suppose two are nested. With one apple in the top one and two in the bottom, itÕs possible to have an odd number in all four. (One in the top, three total in the bottom basket; three in each of the other two.) The concept used is hierarchy, something basic to using digital computers.                    

 

c. It is possible to divide a circular pizza pie into eight exactly equal segments with three straight line cuts. Just make two cuts to create exactly equal quarters, and line them up with points at the middle of the other slices arcs. Then cut a last line. As indicated by the discussion, this can be recognized from 23 = 8. So the concept is exponential notation and its connection to partitions. For those familiar with Cartesian coordinates, four quadrants in the x-y plane correspond to eight octants in x-y-z 3-space.

 

d. To combine the four digits, 1, 5, 5, 5 via only the four arithmetic operations, plus, minus, multiply and divide (and parentheses) to form the quantity 24, one must allow oneself to work with fractions. The only meaningful fraction to get to 24 is 1/5. It is a sign of how early we learn to be uncomfortable with the mathematical knowledge that people tend to resist the fact that they know things like 5[5 Ð (1/5)].

 

e.  Two questions must be answered to represent a cube of any positive integer. 1) How many elements are in the sum of consecutive odd integers? 2) Since the Greeks knew mathematics as geometry, what is the visual idea of a cube? The first questionÕs answer comes from the second one. It is easiest to look at two examples: 23 and 33. Of course the geometric version of each is more meaningful, but immediately we need solids that are 2x2x2 and 3x3x3. In language, they look, respectively, like: eight cubes, four above another four; and, three layers stacked each involving nine cubes arranged in 3x3 fashion. For 23 take one cube from the top layer and adjoin it to the bottom one, getting 8 = 3 + 5. For 33 the middle layer is already odd. Take two cubes from the top layer and adjoin them to the bottom getting 27 = 7 + 9 +11. But every number is either even or odd5. If even, do as in 2x2x2 except that for 4x4x4, 6x6x6, É , the number of consecutive odd integers must be the same as the number of layers (4, 6, É ). For cubed even integers, subtract 1, 3, É from layers above the plane with equal numbers of layers above and below it. Add those same numbers to the layers below. This yields things like 43 = 64 = 13 + 15 + 17 + 19. For cubed odd integers, subtract 2, 4, É above the middle plane, and add them below it. This yields things like 53 = 125 = 21 + 23 + 25 + 27 + 29.

 

f. Separate a sequence of 2n + 1 consecutive positive integers into three parts: a) the largest number in the lower n + 1 entities which we call a: b) the least n (remainder of the lower n + 1); c) the n greatest values. Each element of b) is less than a by an integer, beginning with 1, and proceeding in steps of 1, since these are consecutive. Each element of c) is greater than a by an integer, beginning with 1, and proceeding in steps of 1, since these are consecutive. Since every element of all three sets is to be squared, and since b) and c) elements are identical except for the sign between their a) and integer values, after squaring items of form a, (a Ð i) and (a + i), and rearranging, we get the expression a2 = 4a · i, where summation is from 1 to the largest n needed to construct the smallest element of b) (or the largest element of c) from a. That yields a as always 4 times the sum of the integers, namely 4, 12, 24, 40, 60, É

 

Internet Resources

 

National Council of Teachers of Mathematics

 

http://illuminations.nctm.org/

http://standardstrial.nctm.org/document/index.htm

 

Allen Klinger, UCLA Computer Science Students, Diverse Sources.

 

http://www.cs.ucla.edu/~klinger/nmath/pyth.html color animation shows triangle squares (on legs, hypotenuse); motivates Pythagorean theorem.

 

http://www.cs.ucla.edu/~klinger/nmath/Factorial.html two-color animation of factorial example; two different speakers say the word factorial

 

http://www.cs.ucla.edu/~klinger/size.html factorial, exponents É

 

http://www.cs.ucla.edu/~klinger/triaL1.jpg half of eleven

 

http://www.cs.ucla.edu/~klinger/trial3.jpg baseball arithmetic

 

http://www.cs.ucla.edu/~klinger/fraction1.jpg baseball/resistor arithmetic

 

http://www.cs.ucla.edu/~klinger/trial4.jpg apples in baskets

 

http://www.cs.ucla.edu/~klinger/pizza1.jpg pizza pie

 

http://www.cs.ucla.edu/~klinger/radiia.jpg Greeks, limits (Kasner/Newman)

 

http://www.cs.ucla.edu/~klinger/tet1w.gif (Mathematica) display,14-faced object

 

pointers to diverse mathematical resources

http://www.cs.ucla.edu/~klinger/nmath/index2.html including

http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html links, curves

http://www.gap-system.org/~gap/ from University St. Andrews, Scotland.

http://www.nist.gov/dads/ Dictionary Algorithms and Data Structures

http://mathworld.wolfram.com/ World of Mathematics

Footnotes

 

1 Lakoff and Nunez, op. cit., takes a linguistic approach to math. They show that the arguments leading to infinitesimals and derivatives require going outside pure reasoning into realms of faith.

2 Tobias, op. cit., introduced the term Math Anxiety.

3 Tobias, op. cit., pointed out social factors make math and science courses, difficult for people who are ÒDifferentÓ.

4 Adams, op. cit., stresses visual reasoning.

5 Odd and even is a key concept.