Your goal as a mathematics teacher is to infuse the use of technology into your mathematics teaching.

Before beginning a discussion in this area I would like to make note that I have not started teaching yet, so any discourse will be purely hypothetical. In the current mathematics curriculum, my topic for selection is algebraic geometry (*1). Outside of my null structure (*2) approach to presenting my lessons, the main piece of technology that I will implement will be a graphing calculator.

There are certain idea's that still seem to permeate human thought. One of these idea's is that guns kill people. This is as ridiculous as the thought process that it represents. The tool is just that, a tool. It can be used for either good, or ill. It is in this light that I wish to discuss my implementation of a graphing calculator.

From the beginning, there has been some dispute over the role of computation in mathematics. Historically, computation has been used, but only to a moderate degree. "In Hilbert's time multivariate computations grew too quickly to be computed by hand" (Landau, 1999). Even for modern mathematicians, "it was as if examples were the detritus, and proof and theorem the real thing. Computation was for those who could not think abstractly-and we all knew that mathematics was abstraction" (Landau, 1999). With the rapid growth of society and complexity, the need for speed has become a pressing issue. This need has given rise to an increase in technology, some of this technology has been implemented and developed for the increase of mathematical knowledge. Computation packages such as GSP, Mathematica, Maple, Derive, as well as advanced graphing calculators have been developed by mathematical computer scientists to act as tools to facilitate learning. They offer almost instantaneous answers to complex derivations, Laplace transformations, solutions to partial differential equations, explorations in chaotic dynamics, not to mention simple algebraic operations. When talking about the subject of teaching in high-school (or at lower mathematical levels) people respond with the fear that computational skills will suffer at the expense of technology (*3). This fear is justified, but that is no excuse to take an complete alarmist position.

"To allow only students who have 'mastered' computation to use a calculator seems like an inequity of the worst kind" (Burrill, 1997).

I agree with the above quote, except that my students shall never master computational skills (*4). This is not the type of mindset that I want to cultivate in my students. The calculator shall be seen not as a privilege, or a savior, but a tool that must be utilized properly in a proper context. We never become masters of computation without the understanding to go along with it. I remember from taking classes in my undergraduate degree (*5), the great benefit that a calculator offered me. It facilitated operations and allowed for greater efficiency and mathematical (not to mention algorithmic) accuracy (*6). Even still, I could have got along without it. This is an important point that I want to make. In my high-school classes I never had use of a calculator. The brothers never saw fit to use them (*7). It would have been of great benefit when I first came to college if I had known how to use a scientific calculator before attending.

Great forces of change have been and are continuing to sweep throughout mathematics and how it is taught. Fields (no pun intended) of mathematical study long thought "the most esoteric and abstract parts of mathematics… are now used routinely in applications" (Steen, 1988). Not only in applications, "computers have also changed the way conjectures are invented and tested, the way proofs are discovered, and -in an increasing number of cases- the nature of proof itself" (Steen, 1988). Graphing calculators are mini-computers! A student can formulate a conjecture about quadratic functional coefficients, and then check their hypothesis by graphing the quadratic and noticing the effect of the conjecture. In a different way, the student may attempt to solve for the roots of a quadratic by hand, and then use the graphing calculator to check the results, by jumping instantaneously to the zeros. Research studies explored by two university mathematical educators (Kaput -Thompson, 1994) for JRME, have led them to identify three aspects for which technological implementation has a great potential for change in the experience of doing and learning mathematics: interactivity, control of learning environment, and connectivity. I would use graphing calculators to enhance my lessons, in each of the above area's. My students will be allowed to use the calculator in any way or manner that they so desired (the key is that they must learn how to use it (*8). The calculator and it's graphing functions can greatly increase the speed of my lessons (*9). It can also be used to allow for greater precision, given that their floating point is much more accurate than most human approximations. The calculator would allow me a much broader scope of the material, especially in relation to trigonometric functions. Think back to phase shifts, amplitudes, periodic oscillations and the tangent function being undefined at pi/2. Oh, how a calculator could have helped facilitate those lessons. Given enough time I plan on integrating other technologies into my lesson plans, which emphasize more hardcore mathematics (*10). During my undergraduate at Stonybrook (SUNY), we used Maple for some of the more advanced mathematics classes. It is a hope, but I dream of high-school kids one day using more advanced applications like this in my classes.

Anyone who knows me thinks of one word when they think of me: Chaos. This has long since been an obsession of mine. By limiting this vast subject to the mathematical parts, we are strictly talking of chaotic dynamics and stochastic differential equations. This is (as far as I know) not taught in the standard curriculum. I have been studying these area's for some time and yet have only scratched the surface of this completely amazing well of knowledge. The introductory levels of Chaos Theory (*11) are well within reach for many a laymen, and even high-school. One of the greatest ways to gain a grasp of chaos itself, is visually; this is now possible given the great array of computer applications available. Chaos theory studies underlying order and disorder of various situations. There are studies in stochastic differential equations (which are well beyond my scope as of yet) and non-stochastic dynamics. Stochastic DQ's (*12) study systems of perpetual chaos and "noise". SDQ's are a key area of probability theory. Non-Stochastic studies can offers us a grasp of the complexity of nature, without going overboard on the mathematics (for high-school at least). I have already designed one lesson plan which introduces the concept of iteration along with fractional dimensions and fractals for high-school. I am expanding and modifying this lesson plan currently, to include exact periodic orbit computations, sinks/ sources, and bifurcation's. I use Microsoft excel (for iterations on spreadsheets), graphing calculator (to do basic cobweb plotting), and GSP (for all of the in-between stuff). John Olive in his review of GSP (Olive, 1993) offers us a simple algorithm for generating a finite segment of Koch's curve using "a little ingenuity (and a lot of patience)". I have yet to use maple for some of my more advanced studies in this area, but I will get there. I will introduce the lesson which I formulated (entitled Fractus), into my teaching career when I am ready. I desire to do this to promote interest in this field which has been alluded to (and abused) by pop culture (*13). Any person with a passion can see it in the eyes of another, sometimes others pick up on that spark. If I am successful, I hope to fill the eyes of my students with a raging inferno. (*14)

1: Specifically, high-school algebra and it's geometrical equivalents (freshman-sophomore year).

2: This is detailed in part I of this essay.

3: I'm sure we all have met that kid who insisted that an answer was correct, because the calculator said so.

4: What you say? Never master? What type of teacher are you?

5: My degree was in Pure Mathematics and Applied Mathematics & Statistics (dual major).

6: We created programs utilizing slope fields to estimate anti-derivatives, Simpson's rule for integration approximations and similar topics.

7: I went to a high-school that was run by a Franciscan brotherhood. They were very traditional in their views.

8: Of course, I would be of assistance and show them how to use most of the functions for my lessons.

9: Hooking up a graphing calculator to an overhead projector can greatly facilitate the drawing of functions, not even to mention showing the students step by step hands on use.

10: Or as least as much as I can expect from high-school.

11: As I shall use this umbrella term.

12: This is a slang term among mathematicians for differential equations.

13: Need I mention Jurassic Park.

14: Final Note: Fractus was designed to be introduced at the end of the year of a standard mathematics curriculum. This is to be done of course if there is time. Alternatively, I would like to start a forum for philosophical mathematics dealing with social implications/ advancements and reality explorations (in accordance with my philosophies) in an after school club.

Burrill, G. Computation, Calculators and the "Basics", NCTM Bulletin November 1997

Kaput, J. J. Thompson, P. W. Technology in Mathematics Education Research: The First 25 Years in the JRME. JRME 1994, Vol. 25, No. 6, 676-684

Landau, S. Compute and Conjecture, commentary to Notices of the American Mathematical Society (AMS) February, 1999 Vol. 46, number 2

Olive, J. Review: The Geometer's Sketchpad Version 2.0. The Mathematics Educator Vol. 4, No. 1 Winter 1993. The University of Georgia.

Steen L. A. The Science of Patterns, Articles. Science April 29'th 1988, pg. 611-616

Steen L. A. Preface: The New Literacy taken from Why Numbers Count, quantitative literacy for tomorrows America. College Entrance Examination Board St. Olaf College. New York, 1997.

Zbiek, R. M Prospective Teachers' Use of Computing Tools to Develop and Validate Functions as Mathematical Models, JRME Online, March 1998 Vol. 29, Issue 2 p184