a. Title
Mathematical
Exposition and Visual Knowledge
b. Topic Area
c. Presentation Format
d. Author
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
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:
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.
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.
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.
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
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.
