a. Title

Mathematical, Visual, and Linguistic Knowledge

b. Topic Area

Other Areas of Arts and Humanities

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 “too hard” 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, and to highlight the positive and negative aspects of mathematics’ use of symbolic representation. The idea is that success in the field calls upon a set of skills. Those skills share, not the application of a formal structure involving terse notations, but something more subtle and complex. This work notes that social acceptance, description, and language all play roles.

There is a visual aspect to knowledge, and to reasoning in mathematics. That is even true when we stick to the linguistic or notational issues. Here is an example. In calculus a lengthy derivation involving increments¹ leads to fixing notations, such things as D_xf(x) or if y = f(x), just D_xy, or the commonly used older forms df(x)/dx and dy/dx. But a realistic functional word-portrait alternative exists in the following short paragraph.

Calculus enables working with rates-of-change. A car has position, velocity (speed, direction), and acceleration (rate-of-change of velocity). When it brakes, it decelerates (negative acceleration). Knowing how the velocity changes over time (the acceleration) makes it possible to figure out all kinds of things. Many scientific facts can be expressed in terms of how some property (pressure, charge are examples) relates to a rate-of-change of some other thing. (Change involves differences. Differential calculus describes this.)

People pursuing scientific or technology studies focus for a few years on a series of problems that may be solved using calculus. They learn the body of knowledge, notation, and are trained to fit both to real world questions. In some ways this is like a London cab driver learning the routes and streets – in aggregate called the knowledge. However, as we shall see below from areas as diverse as division or partitioning, hierarchy, and numerical decomposition, it is not enough to acquire knowledge and notation. Even a large series of problems applying them may fall short in terms of educating individuals: making it possible for them to think on their own.

The difficulty in thinking can result in two responses. The first is blaming one’s own inadequate mathematical ability. The second turns the source of failure outwards and lays blame on the discipline. We show below that neither accurately states what occurs. In the following there are many problems stated that have caused great discomfort to individuals well qualified in the sciences. The source of the discomfort is neither their inadequacies, nor those of mathematics as a structure, but rather the experience of being thrust into the use of an unfamiliar language.

One becomes discomforted when a different dialect or pattern of intonation replaces the familiar. The same sort of thing occurs when a puzzle is posed in words if its key idea is outside the realm where training takes place for people absorbing a traditional mathematical 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 mathematics’ from the “different” 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 – a hint - follows each; answers are appended):

a. What’s 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 “connect nine-dots with four straight lines where the pen/pencil does not leave the paper.” The problem and several innovative solutions appear in Adams’ 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 isn’t 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 sky has to be one stimulus to mathematical knowledge. As evidence there is an initial focus on the moon in Dunham’s book (mathematics), navigation and other issues in Ascher’s cross-cultural works (ethno-mathematics), and calendar issues there and in Campbell’s examination of myth (anthropology). The land and oceans, planting crops and traveling far, also started the mathematical knowledge. All these things require that de-emphasis on symbols and techniques, and more focus on the visual, what can be seen.

3. Visualization

Some people really like to consider quantitative questions: I know a dentist who does. An editor who influenced many people mathematically, Martin Gardner, and two people famous as mathematicians also fit the description. Fermat was a 17^th century lawyer and government official in Toulouse, France. Ramanujan lived just thirty-three years, dying in the 20^th century, having worked in his own unique way perceiving things, and not providing traditional proofs.

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 … except for nineteen – for example, 15 = (44/4) + 4; and 13 = (44/4) + √4.

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 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 there’s 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 integers’ to the tens’ 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 aren’t 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. (Today’s 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. The systems are in 6^th/7^th/8^th grade materials made available through the inter-net by associates of the National Council of Teachers of Mathematics. For question c the partitioning for a solution comes from repetition. But the existing organization of mathematical knowledge doesn’t lead to recognizing repeated partitioning (if halving, division by powers of two: halves of halves, the second power of two, 2² = 4).

4. Extension – 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 Pi’s Bent journal’s 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 – 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 (… interested in finding a sequence of 2n + 1 consecutive positive integers … [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 ….

The above conjecture is (mathematically) a better question than f. A word expression at end of the last paragraph’s 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. [I’d sought help about equalities of this type by sending out electronic mail with subject line “Pythagorean Magic.”] 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, They’re Not Dumb, They’re 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, it’s 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 – (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 question’s 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, … , the number of consecutive odd integers must be the same as the number of layers (4, 6, … ). For cubed even integers, subtract 1, 3, … 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, … 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 – 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, …

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 …

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 “Different”.

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

5 Odd and even is a key concept.