To Help Students Discover
Mathematical Relationships or Theorems,
Promote Exploration of
Many Similar Instantiations of a Class

AND IF conditions are that the learner uses computer-based procedures that are (a) interactive in real time, (b) continuously variable, (c) graphically representational and accurate, and (d) relation-preserving,

THEN use the method of promoting exploration of many similar instantiations of a class for the mathematical relation or theorem under investigation.

IF conditions are that the learner uses computer-based Dynamic Mathematics procedures which are: (a) interative in real time, (b) continuously variable, (c) graphically representational and accurate, and (d) relation-preserving AND IF the method is exploration of many similar instantiations of a class for the mathematical relation or theorem under investigation THEN the outcome will be that Dynamic Mathematics affords the discovery of previously unknown mathematical relationships or theorems .

Keywords:

Dynamic algebra
Dynamic geometry
Mathematical exploration using animations
Computer-based instruction
Mathematical discovery

Literature Review

Dynamic mathematical software "...has had a profound effect on classroom teaching wherever it has been introduced. Although not originally intended by its developers, it has also become an indispensable research tool for mathematicians and scientists." (King & Schattschneider, 1997, p. xi). There are several products available today that support this approach; some are listed by the Math Forum Corner for Interactive Geometry Software (CIGS) (http://forum.swarthmore.edu/dynamic.html) and by individual vendors. The Educational Development Center, Inc., funded by the National Science Foundation, has conducted research in this field (http://www.edc.org/LTT/DG/). The present paper will focus on two computer programs readily available from Key Curriculum Press, NuCalc^TM a, dynamic algebra program (http://www.keypress.com/product_info/nucalc.html), and the Geometer's Sketchpad^© a dynamic geometry program, for which at least one extensive bibliography exists (http://www.keypress.com/sketchpad/sketch_bib.html). In addition, a comprehensive collection of articles describing the use of dynamic geometry was published by The Mathematical Association of America (King, J., & Schattschneider, D. (Eds.), 1997) (http://www.maa.org/pubs/books/nte41.html). A survey of mathematics websites, containing dynamic mathematics references as a subset, is presented in an ERIC Digest by D. L. Haury and L. A. Milbourne (1996) (http://www.ed.gov/databases/ERIC_Digests/ed402157.html).

 

The NuCalc^TM program is provided free with Macintosh Power PC computers using Mac OS 7.5 (or later) operating systems. A dynamic book-on-disk (Erickson, 1995) that provides substantial additional documentation is available from the publisher. The software affords manipulation of algebraic symbols in equations. For instance: a variable may be moved by dragging it from one side of an equation to the other causing the equation to automatically reformulate itself. Additionally, graphical representation of equations in two and three dimensions (shapes may be rotated and transformed in real time) is possible. Users can generate families of equations, by variation of a numerical parameter inserted into a given particular equation. Other processes can also be constructed, as well. A snapshot of a 3-D graph (normally, they rotate slowly in the viewer) is shown in Figure 1.

 

 

Figure 1. NuCalc^TM depiction of the function z=xy²

 

 

 

 

 

The Geometer's Sketchpad^©, also provided by Key Curriculum Press is not free. Documentation is extensive, including a comprehensive teacher's manual (Jackiw, 1995) and other support materials (Bennett, 1999). Figures from textbooks, working models of the Pythagorean Theorem, perspective drawings, Escher-like tessellations, fractals, animated sine waves, graphs and curves are just some of the examples that may be constructed with this remarkable software. An example of an interactive process for examining the shapes of conic sections is shown in Figure 2:

 

 

Figure 2. Geometer's Sketchpad^© depiction of Conic Section Foci

 

 

 

Validity and Generalizability of the Proposed Design Principle

Based on my experiences as a high-school mathematics instructor over a period of fourteen years, I agree with this assessment (Garry, 1997):

 

There is something odd about the way we teach mathematics. We teach it as if assuming our students will themselves never have occasion to make new mathematics. We do not teach language that way...the nature of mathematics instruction is such that when a teacher assigns a theorem to prove, the student ordinarily assumes that the theorem is true and that a proof can be found. This constitutes a kind of satire on the nature of mathematical thinking and the way new mathematics is made. The central activity in the making of new mathematics lies in making and testing conjectures. (p. 55).

 

One of the objectives for making conjectures in mathematics is to provide starting points for the construction of proofs (Singh, 1997):

 

...mathematicians simply hate to make a false statement. Of course they use intuition and inspiration, but formal statements have to be absolute. Proof is what lies at the heart of math, and is what marks it out from other sciences. Other sciences have hypotheses that are tested against experimental evidence until they fail, and are overtaken by new hypotheses. In math, absolute proof is the goal, and once something is proved, it is proved forever, with no room for change. (p. x)

 

Generation of finished proofs often requires lengthy and arduous application of human ingenuity, following unanticipated paths. The difficulty of transit has been reduced by the invention of computer software according to Pollak (as cited in de Villiers, 1997, p. 21) :

 

...we find ourselves examining on the machine a collection of special cases which is too large for humans to handle by conventional means. The computer is encouraging us to practice unashamedly and in broad daylight, certain customs in which we indulge only in the privacy of our offices, and which we never admitted to students: experimentation. To a degree which never appears in the courses we teach, mathematics is an experimental science...The computer has become the main vehicle for the experimental side of mathematics. (p. 12)

 

Thus, the validity of the proposed principle is supported; the principle, like a scientific hypothesis, can only be tested by the continued expansion of the scope of its application, but never proved. General applicability of the principle is suggested by the examples of its successful use in the domain of mathematical knowledge cited in the next section of this paper. The principal constraint defining the circumstances for appropriate application of the proposed principle is the knowledge domain for which it is useful, i.e., the realm of formal mathematical proof. If the goal of new knowledge discovery is relaxed, the utility for motivating student understanding of existing mathematics by use of dynamic software is evident.

  

Examples of Application of the Design Principle

There are several examples of the use of dynamic mathematical software resulting in new discoveries. The first discussed here involved two high-school age students, David Goldenheim and Dan Litchfield, attending Phillips Exeter Academy in Exeter, New Hampshire, and their instructor Charlie Dietrich. Dietrich assigned his two geometry students with the task of partitioning a given line segment into any number of equal parts, using an Euclidean construction with compass and straightedge. The two students succeeded in inventing an entirely new way to accomplish this so-called "regular partition" by use of the Geometer's Sketchpad^©. Not only did they invent this new method (thought to be only the second approach devised in over 2,000 years), but they went on to develop another construction that creates a sequence of numbers that are the reciprocals of the Fibonacci Sequence (1,1,2,3,5,8,13,...). Their results were presented at the 74th meeting of the National Council of Teachers of Mathematics (NCTM), April 1996, in San Diego, California. A discussion of their achievements is published at the GLaD website (http://www.gfacademy.org/GLaD/). An account of their discoveries was published by the NCTM, in the Mathematics Teacher online (http://www.nctm.org/mt/1997/01/vol90-no1-euclid1.htm). Some historical connections to their discoveries were discussed by W.T.Johnston, in his article "Historical Precedents to the GLaD Construction" (http://world.std.com/~wij/glad/). A web site debunking the originality of the GLaD discovery cites an 1811 geometry text, but the conclusion drawn does not appear to be definitive. (http://jwilson.coe.uga.edu/Texts.Folder/GLaD/GLaD.Comments.html).

 

A relationship between circles and triangles known as the hemiolic crystal was discovered by Douglas R. Hofstadter (author of such books as Gödel, Escher, Bach: an Eternal Golden Braid and The Mind's I). Using the Geometer's Sketchpad^©, he constructed diagrams (dynamograms) and used the software as his verification engine. He presented insights into the intellectual processes during the course of the discovery. He described his appreciation of the power of dynamic mathematical software in these terms (Hofstadter, 1997):

 

One further key factor that mustn't be overlooked is the fortuitous existence and tremendous power of Geometer's Sketchpad^©. Somehow, this program precisely filled an inner need, a craving that I had, to be able to see my beloved special points doing their intricate, complex dances inside and outside the triangle as it changed. And my own personality welcomed a computer program to explore mathematics, and felt that it afforded visions of geometric truth, an attitude that perhaps would be a little bit less accepted by a traditional mathematician. In short, living in the 1990s and having a Macintosh and enjoying computers was also part of it. (p. 13)

 

Professor Hofstadter submitted his conjecture to H. S. M. Coxeter, a noted British geometer, to verify its originality. He produced a set of computer files to accompany his article, in the folder hofstad-mac, from Swarthmore's "Downloadable Computer Files" (http://forum.swarthmore.edu/dynamic/geometry_turned_on/download/index.html). Some of the files provide interactive scenarios which help explicate the remarkable insights into his discovery processes. All must be opened with The Geometer's Sketchpad^© (TGS). Part of the geometrical concepts in Hofstadter's article can be found at the "JavaSketchpad^© DR3 Gallery" (http://www.keypress.com/sketchpad/java_gsp/gallery.html). Look for the interactive file 9 Point Circle. This file does not require installation of TGS, it only requires a Java-capable browser.

 

Relation of the Design Principle to More General Principles

The proposed principle is related to the following perception principles listed in Winn (1993):

 

"1.4c. The configuration of parts into perceptual units, arising from the presence of an emergent property, is influenced by the physical proximity of the parts to each other, in time as well as in space" (p. 63).

 

The emergence of properties is sought during the exploratory phase in the development of a mathematical proof. Dynamic mathematical programs afford the arrangement of the elements needed for the proof in easily modifiable configurations. These programs also offer a variety of similar cases by moving elements within a configuration.

 

"5.1. By and large, the structural conventions of diagrams, charts, and graphs may be used in metaphorical ways in order to make abstract ideas more concrete and easier to grasp" (p. 97).

 

In some sense, diagrams are metaphors representing the constructions of mathematics. They originate in the human mind and have no physical existence outside the mind. Mathematics itself is a kind of metaphor for nature. We do not know why the language of mathematics is so effective for the description and prediction of natural phenomena, but science demonstrates that it works.

 

"5. 5. The relative position of elements to each other in charts and diagrams is important in determining the nature of the perceived relationship among them" (p.101).

 

This has been demonstrated by the examples described in the previous section. Manipulation of diagrams by use of dynamic software is not just important, but may be absolutely necessary to perception of unexpected relationships.

 

Relation of the Design Principle to Psychological Theories

This section will identify some psychological elements peculiar to dynamic software use and provide pointers to the more general psychological knowledge base. A potential psychological effect of the use of dynamic mathematical tools is that the explorer, being so impressed with the revelations appearing during investigation with the software, may be unmotivated to construct the corresponding proof. Michael de Villiers (1997) offers arguments to counter this suggestion:

Although I have often achieved confidence in the general validity of the conjecture by seeing its truth displayed while objects undergo continuous transformation across the screen ..., this provides no personally satisfactory explanation of why it may be true. It merely confirms that it is true, and even though the consideration of more and more examples may increase my confidence even more, it gives no psychological satisfactory sense of illumination. There is no insight or understanding into how it is the consequence of other familiar results. It has been my experience that the more comvimced I become, the more motivated I also become to find out why it is true. (p. 22)

Another psychological dimension motivating proof, again suggested by de Villiers (1997) is that:

Mathematicians know that proving something is an intellectual challenge that can be compared to the physical challenge of completing an arduous marathon or triathlon. In this sense, proof serves the function of self-realization and fulfillment. Proof is a testing ground for the intellectual stamina and ingenuity of the mathematician. To paraphrase Mallory's famous comment on his reason for climbing Mount Everest: "we prove our results because they're there." (p. 23)

 

Besides the idiosyncratic psychological attributes of the use of the proposed principle listed above, the obvious connection to recent theories is to constructivism, in both the literal and figurative sense. An annotated bibliography of references for the use of Constructivism in Mathematics, from the RUMEC Conference on Research in Mathematics Education (Central Michigan University, Sept. 5 - 8, 1996) can be found at the MAA website: (http://www.maa.org/t_and_l/sampler/construct.html). Using the keywords (Constructivism) AND (Mathematics) AND (Education) yielded 392 documents at the ERIC site: (http://ericir.syr.edu/Eric/). The role of "Constructivism in Visual Message Design And Learning" (Anglin, Towers, & Levie, 1996, p. 757) cites E. H. Gombrich, who claims that the meaning of a picture depends on what the viewer constructs when observing it.  

Author's Opinion of the Design Principle

My education included degrees in Physics, and Professional Certificates in Computer Science and Electrical Engineering (Telecommunications). My work experience has included Secondary Education instruction in Physics and Mathematics, and participation in space flight projects. From a personal perspective, learning involves visualization and the ability to deal with spatial relationships. During the teaching phase of my career, I was involved in the development of a mathematics tutorial project, when personal computers were just becoming available. It seemed to me at the time that graphics on the computer would play a significant role for educational applications, but the implements available at the time were extremely primitive: character-mapped graphics rather than bitmapped, for example. Processor clock rates and RAM capacity precluded the real-time dynamic graphics only recently available to the private computer user. The advent of powerful dynamic graphics tools like NuCalc^TM and The Geometer's Sketchpad^©, with the facility they offer for mathematical discovery, are simply amazing to me. The kind of power formerly available to military/industrial research institutions, now on the market and accessible to both pre-college and university students, is in my opinion one of the most incredible tools for the opening of mathematical knowledge. Without the existence of this class of software, the application of the proposed principle would be difficult.

  

Related Mathematical Software Sources

Other examples of mathematical software assistants, such as MATLAB (http://www.mathworks.com/index.shtml), Mathematica (http://www.wolfram.com), and Maple (http://www.maplesoft.com/cybermath/share/r_geom.html), while extremely powerful, require the user to learn syntactical rules worthy of a complete computer language, and do not lend themselves easily to use by the mathematical novice. Thus they are not nearly as suitable for the application of the proposed principle, but might provide avenues to the professional researcher in mathematics. Sites like The Geometry Center: Center for the Computation and Visualization of Geometric Structures (http://www.geom.umn.edu) provide links to more sophisticated approaches of representing mathematics graphically.

 

References

Anglin, G. J., Towers, R. L, & Levie, W. H. (1996). Visual message design and learning: The role of static and dynamic illustrations. In D. H. Jonassen (Ed.). Handbook of research for educational communications and technology. (pp. 755 - 794). New York: Simon & Schuster Macmillan.

Bennett, D. (1999). Exploring geometry with the geometer's sketchpad. Berkeley, CA: Key Curriculum Press.

de Villiers, M. (1997). The role of proof in investigative, computer-based geometry: Some personal reflections. In J. King, & D. Schattschneider, (Eds.). Geometry turned on!: Dynamic software in learning, teaching, and research. (MAA Notes 41). (pp. 15-24). Washington, D.C.: The Mathematical Association Of America.

Engel, C. E. (1950).A history of mountaineering in the alps. London: George Allen And Unwin LTD.

Erickson, T., & Avitzur, R. (1995). Introducing dynamic algebra with NuCalc^TM. Berkeley, CA: Key Curriculum Press.

Gary, T. (1997). Geometer's sketchpad in the classroom. In J. King, & D. Schattschneider, (Eds.). Geometry turned on!: Dynamic software in learning, teaching, and research. (MAA Notes 41). (pp. 55-62). Washington, D.C.: The Mathematical Association Of America.

Hofstadter, D. R. (1997). Discovery and dissection of a geometric gem. In J. King, & D. Schattschneider, (Eds.). Geometry turned on!: Dynamic software in learning, teaching, and research. (MAA Notes 41). (pp. 3-14). Washington, D.C.: The Mathematical Association Of America.

Jackiw, N. (1995). The geometer's sketchpad: User guide and reference manual. Berkeley, CA: Key Curriculum Press.

King, J., & Schattschneider, D. (Eds.). (1997). Geometry turned on!: Dynamic software in learning, teaching, and research. (MAA Notes 41). Washington, D.C.: The Mathematical Association Of America.

Pollak, H. O. (1984). The effects of technology on the mathematics curriculum. In A. I. Olivier, (Ed.). The Australian experience: Impressions of ICME 5. Centrahil, Australia: AMESA.

Singh, S. (1997). Fermat's enigma: The epic quest to solve the world's greatest mathematical problem. New York: Walker.

Winn, W. (1993). Perception principles. In M. Fleming & W. H. Levie (Eds.), Instructional message design: Principles from the behavioral and cognitive sciences (4th ed.). (pp. 55-126). Englewood Cliffs, NJ: Educational Technology Publications.