Internet Exposition, Visualization and Assessment

 

Allen Klinger

University of California, Los Angeles

klinger@cs.ucla.edu

4531-C Boelter Hall UCLA1

Los Angeles CA 90095-1596 USA

 

This paper describes ways to use todayÕs technology to foster approaches to Mathematics through inquiry, a means that stresses involvement, motivation, and informality. We show how to create new materials as well as integrate past items composed by authorities. Fundamentally we reflect that the inter-net supports new kinds of tutorial exposition. Using them effectively means employing technological methods to support follow-up.

 

A second aspect of an expository style based on inter-net availability is the relative ease of integrating still or animated images, and sounds, into a new tutorial. The connectedness of knowledge is a pragmatic reality due to high computer processing speeds and communication networks. These technologies support visual and auditory information presentation. They in turn foster visualization in reasoning, analysis and design.

 

Material posted to the inter-net becomes a democratic force in this sense. Popularly available items are potentially usable for informal education. The third topic involves methods to evaluate what informal and formal learners know about topics, a necessity if information is supplied outside an organizational framework. This means measuring accomplishments. This can be done assigning highest scores to those who honestly disclose what they know.

 

1. Topics and Technology.

 

This section describes using software tools and hardware infrastructure to disseminate a personal view of Mathematics. Notation and style are substantial parts of ordinary mathematical exposition. Nevertheless the widespread use of computer network technology leads toward freedom to rethink those aspects. In fact having a sense of modern computer technology may tend one neither toward approved approaches of Mathematics nor those of Computer Science or Engineering. Instead creating inter-net materials can be for oneÕs own pleasure. In my case IÕve taken a path of presenting knowledge in a form so that others can more readily go forward.

 

The tradition of teaching difficult concepts through card games, puzzles and riddles is well established. It would be easy to list examples, even books: many volumes are collections of them. But that is an informal manner of Mathematics education. It stands in contrast to tomes on Algebra, Trigonometry, Calculus, etc. Nevertheless the recurrent failure to create positive mathematical interest in vast portions of a general population, one that is oriented toward gambling games, means that there are needs. I assert here that todayÕs technology gives means to meet those needs. This begins with a program of exposition based on the unique attributes of the inter-net. The attributes depend on such a wide array of diverse technologies that even listing their main academic sources is daunting. They certainly include Computer Science: data structures, networks, theory, programming languages; as well as contemporary computer utilities: MacromediaÕs Flash, the markup languages such as HTML, and media players such as Apple QuickTime.

 

Based on inquiry and an evolving sense of what troubles those individuals who dislike Math, I recommend fractions as one fundamental area to investigate. For many of us our understanding

 

 

of such a simple subject might make this seem a foolish suggestion. Still connecting current

views about the rigid equivalence between ratios and decimal equivalents, to subtleties of thought that arise when one considers how the ancient Egyptians would have divided nine loaves of bread among ten men, might lead to less skepticism. Even that amount surely would decline if the series and parallel forms of addition of two fractional resistors in Electronics/Electrical Engineering, say one of a half Ohm value and the other, a third Ohm, were considered. Finally that same ÒsimpleÓ addition takes on a completely different nature if baseball batting average were to be computed (for a player who bats one hit in three at bats on day one, and one in two at bats day two).

 

Two simple statements give many (and gave me) difficulties. They both show the thesis of this paper: simple things are deeply connected with Mathematical knowledge. In short form they are:

 

1. Can one divide a circular pizza pie into eight exactly equal slices with just three straight line cuts?

 

2. Can you make twenty-four from three fiveÕs, a one, using the four arithmetic operations, plus, minus, times and divide, and grouping by parentheses?

 

The first is a tricky way of leading into a more advanced concept than is usually part of elementary arithmetic (dividing) or geometry (circles, sectors). But the second points to the relative discomfort most people have with fractions.

 

2. Writing and Showing.

 

The inter-net supports branching exposition. The markup language, HTML, is the fundamental enabling technology. To use HTML one needs to become acquainted with some facts from a file easily found with a search engine. Once that point has happened, it is relatively easy to present related technical topics in linked files. A link or pointer (called an anchor in HTML) is a location-finding datum. It supports transition from one file to another. This is particularly important in Mathematics exposition where students have difficulties due to unfamiliarity with related topics. To a new learner Mathematics seems like a simultaneous encounter with a large number of concepts.

 

Even more interesting, the same technical features that allow branching also support viewing images, hearing sounds, and displaying animations. Both the two questions above have answers that are better shown than written about. That is true of many concepts in Mathematics. Visuals that move people toward understanding can be very powerful. That is true even when it represents a special case, not the general instance. For instance, a not-equal-side pentagon leads to the notion of the regular shapes. It is easy to show six different representations for quantity five. This can be done using sticks, distinguishing a fifth-vertical-bar element by some means (crossing the others, augmenting the ordinary width), mimicking the shape of a hand, arranging five stones (both linearly to fourth, and as the corners of a square), and joining stones by a line in the sand to get a modern number sign. Finally, the sticks/stones and Roman/Arabic forms lead to showing how tallying takes place in different cultures. That permits showing images that can connect with people who arenÕt in the majority groups nor oriented to conventional academic curricula.

 

The visionary 1945 paper by Vannevar Bush introduced the notion of knowledge as an interconnected set. The technical notion he innovated in ÒAs We May Think,Ó is of linkages or simply links from one topic to another. That idea became the essence of computer programming where a link today is also sometimes called a pointer (see e.g., the seminal work by Knuth or the Klinger encyclopedia article listed below). Thought about what interconnected or linked ideas represent is well expressed in the related term, hyperlink or hypertext. There is a worthwhile article that expressed technical realities that now are solidly entrenched through the proliferation of the inter-net. A survey article published in 1987 by Jeff Conklin in an engineering publication, expresses the wonderful reality that is upon us today when we consider inter-net exposition. We will see that there are tools and means for making evident the connections of Mathematical material to History, Culture, Art, Music, and other subjects, without compromising its clarity or quality.

 

The truly remarkable thing about current software and computer hardware is that they are a technological means for conveying many concepts through animations. (See Software within References.) There are many differences between exposition based on the inter-net and associated software technologies and that in books and journals. For instance, the inter-net easily enables displaying a multicultural view of quantitative. The Greek culture gave rise to the area of Mathematics. Still few are aware that this knowledge included the concept of limit. A reference listed below makes it visually apparent that this too was known in the Greek world.

 

 

3. Visual Organization and Innovation

 

Although the inter-net began in the West it has great potential for good on a worldwide basis. The cost of books is prohibitive in many countries. That recognition has spawned activities like posting the text of fundamental works to the web (see e.g., Project Gutenberg and related distribution activities for ASCII or .txt format of key books). Although two figures that show how the concept of limit was known to the Greeks appear in Kasner & Newman, theyÕre now also available from my web site. However many other interesting images from other cultures that appear in AscherÕs book are not yet available. This is not through the fault of technology. Rather it is a matter of organizing students to scan in images there from Polynesian, and African cultures: the technology exists to distribute visual matter in low cost way analogous to the text distribution activities.

 

When reviewing work of Richard Feynmann, Stanislaw Ulam or Solomon Golomb, vis-a-vis their contributions to the scientific history of the twentieth century, visual thinking certainly has played an important role.  The understanding of quantum electro-dynamic (particle behavior) phenomena through Feynmann diagrams, the basic concept of the hydrogen or fusion bomb (see the appended excerpt from http://www.pbs.org/wgbh/amex/bomb/peopleevents/pandeAMEX74.html), and the key ideas supporting transmitting images from spacecraft to earth came out of Mathematical innovation by a Physicist and two Electrical Engineers based on visualization. The role visualization plays is one theme of James Adams, in his book. His work began with a Mechanical Engineering design course at Stanford University. [The nine-dot four-line problem he presents in the book generated the phrase Òthink outside the box.Ó] What inter-net exposition can do is to place the average individual in front of problems that have the effect of requiring some form of visual analysis.

 

Some would protest that this is Òtoo complicated,Ó or that Òproblems form the widespread aversion to Mathematics,Ó The immediate counter to that is the popularity of gambling: Las Vegas, government-sponsored lotteries, and casino activities. Games with cards, boards, and other formats, are seen as entertainments, not drudgery. Architects ranging from Escher to others known more from their residential or public building designs, or their theoretical contributions as in the case of Christopher Alexander, are of widespread interest and their designs (see SchindlerÕs house number 175), images, and books sought after.

 

Both architects and engineers work in the real world: they apply knowledge to make things.

The two fields value models if useable/useful, but are ready to discard them when they are seen to be unhelpful. Schindler discarded the notion of isolated characters for numbers. He hung a single metallic sign based on small segments joining the offset two horizontal bars to represent three numbers. The standard he applied is usefulness. That same process is clearly involved when calculating the addition of two fractions, either in sports statistics [Òtotal successesÓ divided by Òtotal trialsÓ] or the familiar to Electrical Engineers formula for the parallel combination of two resistances (see appendix). So considering these two means and the standard method for fraction addition, there are really three valid ways.

 

4. Assessing Learning Accomplished

 

For more than sixty years there has been a remarkably successful methodology for determining exactly what individuals know. The triangle in the following figure, and associated numerical weights derived and presented in the FIE paper show one version of that approach. Here there is a visual presentation of thirteen responses to three alternative completions of a statement. The non-vertex points represent partial knowledge. The triangle mid-point signifies Ònothing knownÓ. [Emir Shuford posted to the web many papers concerning this technology. James Bruno first communicated using discrete points with an alphabetic labeling, and essentially the relative weights or value of answers at all thirteen of the triangle points discussed in an appendix of the following reference http://www.cs.ucla.edu/~klinger/csd_report.html.]

 

The paper http://www.cs.ucla.edu/~klinger/csd_report.html gives detail about using the thirteen-response system (see figure or http://www.cs.ucla.edu/~klinger/tri.jpg) in a UCLA graduate course: it has been used in first year, and senior courses as well. The following three questions illustrate how statements can be composed that require students to examine their knowledge or work out a problem solution (in order to do better than guess an answer). The five shown below examined basic ability in Mathematics. To see the overall potential of these assessment tools consider new web-expository material [http://www.cs.ucla.edu/~klinger/size.html] introducing exponentiation and factorial. Three examples follow showing how I use three possible completions to a given statement, in a way that rewards only those who truly know a mathematical topic.

 

1. When two quantities, say a and b, are compared, we write a is greater than b (or a is more than b) as: a > b. Which statement is true?

(a) 1/103 > 1/10-3                                 (b) 103 > 1/10-3                                    (c) 103 > 10-3

 

2. For the given pair of equations which is true about the variables x and y ?

2x + 3y = 2;                             6x + y = 12

(a) x and y have the same sign. (b) x and y are whole numbers.             (c) x > y.

 

3. In four births, which is most probable:

a) Two male, two female.        b) Three of one sex; one of the other. c) All four the same sex.

 

5. Opportunities

 

Today very powerful computer software tools exist that implement mathematical notation, knowledge, and procedures (APL, Mathematica, Maple, etc.). Notwithstanding their availability, the tendency is for there to be fewer individuals seeking out Mathematics as a source of basic knowledge. Beginning with a traditional educational infrastructure to disseminate a personal view of Mathematics. Notation and others can more readily go forward. In spite of the existence of list-serve, newsgroup, and electronic mail dissemination of Mathematics, today in the United States Computer Science graduating seniors are hesitant to express fundamental notions in their field to their peers. They avoid showing interest in coding, modulo, and many other simple computational issues. Nevertheless they are outstanding in their general ability to absorb Mathematical ideas. The general population has been shut out more thoroughly from access to Mathematics.

 

Fundamentally today most economic opportunity goes to those who know how to work effectively with information technology. Mathematical thought is at the heart of that subject. Inter-net exposition has the ability to break through the habit of not wanting to think about Mathematics. Visual means are at the center of the communication difficulties that cause general individuals to resist continued learning. Using the inter-net to show visuals or to lead people through inter-related ideas by using links is a means for educating the many to work with modern technology. Assessing what has been learned by informal review of web material can be made a step toward qualification for admission to study about a future in one of a variety of technical careers.

 

References

 

Bush, Vannevar, ÒAs We May Think,Ó Atlantic Monthly, 176-1, July 1945, 101-108. Available from http://www.theatlantic.com/unbound/flashbks/computer/bushf.htm

 

Conklin, Jeff, ÒHypertext: An Introduction and Survey,Ó IEEE Computer, 20-9 (1987), 17-41.

 

Golomb, Solomon W., Polyominoes [with more than 190 diagrams by Warren Lushbaugh, NY: Charles Scribner's Sons, 1965], Second Edition, Princeton NJ: Princeton Univ. Press, 1994.

 

Ulam, S. M., Adventures of a Mathematician, Berkeley CA: Univ. of Calif. Press, 1991.

 

Klinger, Allen, Human Computer Interactive Systems, NY: Plenum Press, 1991.

 

Alexander, Christopher, et. al.,  A Pattern Language: Towns, Buildings, Construction,

Oxford Univ. Press, 1977.

 

Klinger, A., and Salingaros, N., "A Pattern Measure," Environment and Planning B: Planning and Design 2000, 27-4, July 2000, 537-547.

 

Klinger, Allen, "Training and Thinking," The Tau Beta Pi Bulletin, LXXV-3, March 2002, 3-5.

 

Klinger, Allen, ÒExperimental Validation of Learning Accomplishment,Ó Proc. Frontiers in Education, 1997.

 

Klinger, Allen, Finelli, Cynthia J.  and Budny, Dan D., ÒImproving the Classroom Environment,Ó Proc. Frontiers in Education, 1999.

 

Finelli, Cynthia J., Klinger, Allen, and Budney, Dan D., ÒStrategies for Improving the Classroom Environment,Ó ASEE J. of Engineeering Education (invited), Oct. 2001, 90-4, 491-498.


Appendix A

 

Software Macromedia, Dreamweaver (web files), Flash (animations); Adobe, Photoshop (image-handling), Illustrator (drawing).

 

Web Files

Stanislaw Ulam

Corin Anderson http://www.the4cs.com/~corin/motm/stan_ulam.html

Public Broadcasting System (PBS)

http://www.pbs.org/wgbh/amex/bomb/peopleevents/pandeAMEX74.html

 

Allen Klinger

http://www.cs.ucla.edu/~klinger                                     Basic Starting Point

http://www.cs.ucla.edu/~klinger/csd_report.html           Assessment Paper Technical Report

http://www.cs.ucla.edu/~klinger/size.html                      Expository: Factorial, Exponentiation

http://www.cs.ucla.edu/~klinger/blocks.html                  Expository: Stimulate Thinking

http://www.cs.ucla.edu/~klinger/math.html                   Bibliographical

http://www.cs.ucla.edu/~klinger/pizza1.jpg                    Problem 1 Solutions.*

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image005.gif

                                                                                    Greek Visual Showing Limit**

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image001.gif         

Non-equal-side Five-edged Shape.

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image002.jpg

Sticks, Stones, Six Five Representations#

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image004.jpg

Tally In Three Cultures#

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image006.gif

Stanislaw Ulam^

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image008.jpg

SchindlerÕs 175 Number Sign

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image010.jpg

Fraction Addition and Electric Circuits

http://www.cs.ucla.edu/~klinger//Internet_Exposition_6_21__files/image011.gif

Fourteen-Faced Solid                           

http://www.cs.ucla.edu/~klinger/tri.jpg                          Thirteen Responses, Three Statements@

 

Appendix B

 

Thermonuclear Weapon [Source: PBS in References]

 

President Truman announced in January 1950 that the United States was about to embark on an all-out effort to develop a hydrogen bomb, Ulam began calculating whether physicist Edward Teller's design É would work. É

 

Ultimately, Ulam and a fellow mathematician Cornelius Everett concluded that Teller's model É would never work. ... But a year later Ulam accidentally came up with a new scheme that would prove to be a breakthrough, and he reluctantly took it to Teller. Teller recognized that, though there were problems with the idea, Ulam had hit on the solution. Together the two men converted it into a design for the superbomb that everyone at Los Alamos immediately recognized would work.

 

Their plan was to place an atom bomb inside a heavy shell that would also contain a capsule of hydrogen fuel. When the atom bomb exploded, in the fraction of a second before the whole assembly blew itself apart, the shell would confine the radiation from the atomic blast long enough to heat and compress the hydrogen fuel, setting off a fusion reaction. ... Ulam's idea was the use of material surrounding the fuel capsule that would magnify the energy of the radiation.

 

Decades later physicist Hans Bethe wrote: "The new concept was to me, who had been rather closely associated with the program, about as surprising as the discovery of fission had been to physicists in 1939 ÉÓ [Comment: instructors can lead pupils toward the penultimate paragraph image, and the importance of visualization by using this and other web-materials.]

 

Electric Resistance                   OhmÕs Law is                         E  =  I * R       (1)      

Series -             R  =  R1 + R2                                                                               (1a)        

                        Parallel -          Apply (1) noting that the same electric potential E goes across the total resistance R and also each of the two parts, R1 and R2. Since the current I divides, it equals I1 + I2, implying::           E  = I * R  =  (I1 + I2) * R  =  I1 * R1                (2)

                        E  = I * R  =  (I1 + I2) * R  =  I2 * R2                (3)

From algebraic manipulation of the rightmost equalities of (2) and (3) aided by substituting I for (I1 + I2), and replacing I2 by (I - I1), one finds a solution of the two equations for R or 1/R:             R     =  1/[(1/ R1) + (1/ R2)]                 (4)

                                    1/R  =  (1/ R1) + (1/ R2)                        (5)

The following steps lead to these results.

(2) ->                           R  =  [I1 * R1]/[ I1 + I2]                                     (2Õ)

(3) ->                           R  =  [I2 * R2]/[ I1 + I2]  =  [(I - I1)/I] * R2            (3Õ)

Substituting (2Õ) in (3Õ) and solution of the result for R yields (4) and (5).

Multiplying (4) by R1* R2 yields the parallel resistance formula:

                                    R  =  [R1 * R2]/[R1 + R2]/                                  (1b)

Expressions (1a) and (1b) are well known to Electrical Engineers and Physicists (as S. Ulam, S. Golomb, and R. Feynmann). They extend the ordinary notion of fraction addition, just as the addition of numerators divided by sum of denominators does in sports statistics and graphics.

 

Appendix C

 

Visuals support the text exposition of these three themes. Seven appear as supplements below. They are:

image001.gif                Figure 1. Five-Sided Shape

image002.jpg               Figure 2. Six Representations of Quantity Five

image004.jpg               Figure 3. Three Representations of Tallying the First Five Quantities

image005.gif                Figure 4. Nested Regular Polygons With Inscribed Circles

image010.jpg               Figure 5  Sports and Electric Resistance Fraction Addition

image011.gif                Figure 6. Fourteen Faces and Thirteen Points on Three Planes

image012.jpg               Figure 7. Thirteen Responses to Three Statement Completions

 

Creation of the visual items was often through some assistance of others. The image * was drawn by Gavin Wu. The image ** is scanned from Kasner & Newman. The # images were significantly improved through the efforts of Yu-Chian Tseng. The ^ drawing of Ulam is from his autobiographyÕs cover courtesy of Corin Anderson, and attributed to Zygmund Menkes. The information in @ is the original notion of James Bruno.

First Draft