The Role of Hand-Held Computer Symbolic Algebra in Mathematics Education in the Twenty-First Century:

A Call for Action!

By

Bert K. Waits and Franklin Demana

Professors Emeritus of Mathematics

The Ohio State University

Columbus Ohio

waitsb@math.ohio-state.edu

Introduction

An unparalleled opportunity exists today to deliver better mathematics education than we ever thought possible. And it can be delivered to *all* students because of the rapid expansion of inexpensive powerful hand-held computer technology with built-in computer symbolic algebra (CA) software. We fear, however, that our community is not ready due to misunderstanding, fear, and inexperience.

We believe mathematics education leaders must "hold the course on recent technological advances in education" and recognize and take advantage of what is possible *today* and will be even "more possible and practical" in the future. The Standards 2000 writing group needs to take a bold, decisive, positive position on the integrated use of computer symbolic algebra (and other computer software like dynamic geometry) in Grades 9-12 school mathematics. And they should recommend a *significant* *reduction* in the classroom time spent on learning obsolete paper and pencil symbolic manipulations.

Computer Algebra Calculators allowed on College Board Examinations

The following is from a recent College Board calculator policy announcement made in early June 1998.

The 1998-99 calculator policies for College Board math/science tests that allow or require a calculator (PSAT/NMSQT, SAT I, SAT II Math Level IC & Level IIC, AP Calculus, AP Statistics, AP Chemistry, AP Physics) will not change--therefore the Casio CFX-9970, TI-73, and TI-89 will be permitted for use on those tests since graphing calculators without QWERTY (typewriter-like) keyboards are allowed.

The Casio CFX-9970, TI-73, and TI-89 have been added to the list of approved graphing calculators for AP Calculus. The list will be updated shortly at the College Board web site: [It is now updated] http://www.collegeboard.org/ap/math/html/exam003.html

Beginning in 2000, students will be EXPECTED to bring to the AP Calculus Exams a graphing calculator with the 4 required capabilities (graph a function within an arbitrary viewing window, find the zeros of a function, compute the derivative of a function numerically, compute definite integrals numerically) BUILT INTO the calculator. Any of the graphing calculators on the current approved list will be permitted for the exams. Calculators with the 4 capabilities built in include: CASIO 9700 series, 9750 series, 9800 series, 9850 series, 9950 series, and the 9970 series; TI-82, 83, 85, 86, and 89; HP 28 series, 38G, and 48 series; and Sharp 9200 series, 9300 series, and 9600 series. Over time, the gap between first-generation graphing calculators and newer models has widened significantly. The use of a graphing calculator with the 4 capabilities built in will narrow this gap. This expectation will help ensure that all students have sufficiently powerful technology for the AP Calculus Exams." [See http://www.collegeboard.org/ap/math/html/indx001.html

Background

First we review some recent history. After a little more than two decades of numerous pioneering efforts in the U. S., we believe that we can safely say that the use of hand-held technology has forever changed the way mathematics is taught, and forever changed the way students learn mathematics. Our philosophy about using technology in mathematics teaching and learning developed through the experience gained in our work in the* Calculator and Computer Precalculus (C ^{2}PC) project* (Waits & Demana, 1994) that started in 1985. The C

Soon after our C^{2}PC project started, the Sloan Conference was held at Tulane University in January, 1986, and sparked the calculus reform movement in the U.S. (MAA, 1986). This movement was fueled by the National Science Foundation as it provided millions of dollars in grants for calculus reform shortly after the national conference "Calculus for a New Century" was held in Washington, D.C. in October, 1987 (MAA, 1987) which we attended.

It is a fact that hand-held scientific calculators have *significantly* changed the high school and university mathematics curriculum in the U.S. in less than 25 years. For example, many topics that dealt with paper and pencil "computation" involving transcendental functions have been deleted. We know because we taught the material before the scientific calculator was invented and can compare with topics taught today. We estimate that at least one-fourth of the material that was typically taught in a high school "trig/functions" course or college "precalculus/college or algebra/trig" course before 1972 is simply no longer taught today in similar courses. Many sections and even some chapters in textbooks dealing with paper-and-pencil computation methods became obsolete and disappeared from the curriculum - period! Why? Because hand-held scientific calculators provided better ways to "compute" than paper-and-pencil methods. **The same thing (obsolescence) will soon happen with paper-and-pencil symbolic algebraic manipulations common today because of student use of inexpensive hand-held CA systems that now exists and soon will proliferate**.

It is important to note that *less time is now spent* on certain topics (ones made obsolete by scientific calculators) but we still "do" the same things. For example, we still "compute" the sine of 14.25 degrees but not by the time consuming method of paper-and-pencil linear interpolation. What changed was NOT the "to do’s" but the "how to" do the "to do’s." It is also equally important to note that many educators found pedagogical ways to use scientific calculators that enhanced the teaching and learning of mathematics (many references in Dunham, 1998).

Now we fast forward to the present and note that in the past ten years graphing calculators have brought the "power of (computer) visualization" to millions of students. It is a fact that graphing calculators *are* hand-held *computers* - they use computer CPU’s such as the Apple II Z80 and the Macintosh Motorola 68000. It is also a fact that graphing calculators have proven to be powerful pedagogical tools in the hands of trained teachers. There are many hundreds of examples of their use and effectiveness (Dunham, 1992; Dunham and Dick, 1994).

However, graphing calculators primarily *enhance*, not replace, what is viewed as "traditional mathematics." Graphing calculators make possible *and practical* new pedagogical approaches that facilitate representing problems numerically and graphically, but not symbolically (exact). Most graphing calculators used today do *not* manipulate symbols like computer symbolic algebra software found in popular PC computer algebra systems DERIVE, MATHEMATICA, and MAPLE.

A computer algebra system (CAS) has four broad areas of functionality.

1. Exact (rational, real, and complex) arithmetic (EA).

2. Graphing software packages for graphing functions, relations, and 3D surfaces.

3. Numerical solvers

4. Computer symbolic algebra (CA) for manipulation of (and solving) algebraic expressions (equations).

Most graphing calculators have only two of the four main CAS components. The two major components, exact arithmetic (EA) and computer symbolic algebra (CA), are not found on most graphing calculators today. Graphing calculators today use what is called a "floating point" computation engine. They display only accurate *approximations* of exact real or complex numbers. In the examples that follow we give some simple illustrations of the use of EA and CA. We use the new hand-held TI-89 CAS tool for our illustrations.

Exact Arithmetic Examples

These outputs are not possible on ordinary "graphing calculators."

EA 1. sin(15 degrees) EA 2. factor (20 factorial)

EA 3. Operations with radicals EA 4. Operations with complex numbers

EA 5. Matrix algebra computations

Computer Symbolic Algebra Examples

These operations are not possible with typical graphing calculators today.

CA 1. Expand and factor manipulations CA 2. Equation solving manipulations

CA 3. Operations with rational expressions CA 4. Trigonometric manipulations

CA 5. Differentiation manipulations

CA 6. Integration manipulations

CA 7. Matrix algebra manipulations

CA 8. Solving differential equations

Notice these examples can be done with one keystroke on the TI-89! This is indeed "awesome power" to quote one student. Soon these hand-held devices will be more popular with students than graphing calculators are today. Use of these hand-held CA tools will lead to the *unavoidable* obsolesce of what most of us have "taught" for many years (paper-and-pencil algebraic manipulation techniques). And this change will be a difficult adjustment for many teachers to make. We mathematics teachers can do the paper-and-pencil manipulations very well and we wish our students could too. It is also understandably hard for parents and legislators to understand and accept these tools because they do in single keystrokes all they recall about "mathematics."

Students will demand the use of CA because it provides a "better" tool to do the tedious algebraic manipulations common in "mathematics" today. To do otherwise is a waste of valuable teaching time and learning opportunities. We have wonderful examples of NCTM Standards based curricula, supported in development by NSF, that outline how to teach "better mathematics better." What is needed is the *additional *classroom time to devote to such innovative and sometimes changeling programs. We also need time in the classroom to explore other computer software such a CABRI dynamic geometry. Dynamic geometry should permeate every year of the high school mathematics and science curriculum (Laborde, 1998). Use of hand-held CA together with a recognition that some of what we once did is now obsolete can provide the time to spend in the classroom on more worthwhile topics!

Let's Abolish Pencil-and-Paper Arithmetic!

Professor Anthony Ralston, a distinguished mathematician retired from SUNY at Buffalo and currently at the Imperial College in London, shared with us a draft paper he has prepared titled "*Let's Abolish Pencil-and-Paper Arithmetic*." We hope he makes this paper available to the community soon as it is a must read for all mathematics educators! His ideas can easily be extended to "paper-and-pencil algebra. He writes,

Yes, I mean it. Achieving any level of proficiency in pencil-and-paper arithmetic (hereafter PPA) should no longer be a goal of elementary school mathematics. Which is not - of course - to say that teachers should not use PPA algorithms for whatever purposes they wish. It is only to say that children should not be expected to learn these algorithms and certainly they should not be tested on them (Ralston, 1998).

A Balanced Approach to Curriculum Reform

Prior to the advent of easy to use hand-held technology, we estimate about 85% of the mathematics curriculum consisted of paper-and-pencil computation. The computation involved the algebraic and analytic process of mathematics including the common symbolic manipulations of algebra and calculus (by paper-and-pencil). This type of computation usually involved very low order thinking skills, and often has been associated with the phrase "drill and kill mindless manipulations." In the pre hand-held technology curriculum, there were precious few application examples and they almost always occurred as consequences of mathematics concepts developed algebraically or analytically. Further, little or no real proof occurred in the standard courses. There is growing evidence that paper-and-pencil manipulation skill alone does not lead to better understanding of mathematical concepts. Indeed, the appropriate use of hand-held technology could provide more classroom time for the development of better understanding of mathematical concepts by eliminating the time spent on "mindless paper-and-pencil manipulations."

Figure 1 suggests important components of a modern balanced curriculum. We expect that problem solving and proof (or giving convincing arguments) will play a more important role, and paper-and-pencil computation will occupy a smaller share of a balanced modern curriculum. We do not mean to suggest that the time spent on these features should be the same. However, computation *should not* take up 85% of class time as it did in the past (and in the present in most high school classrooms).

Figure 1

Many mathematicians are becoming increasingly concerned with the perceived lack of attention to paper-and-paper skills in the new evolving mathematics reform curricula. There have been numerous articles in the "Notices of the American Mathematics Society" during 1996 and 1997 giving pro and con views about the reform effort and the use of technology. Robin Wilson (February 1997) gives a glimpse of the division among mathematicians on "reform calculus" in his article in the Chronicle of Higher Education. Speaking against reform calculus:

"This approach really shies away from anything but superficial use of skills," says Ralph L. Cohen, a professor of mathematics at Stanford University, which after seven years has decided to stop teaching the ‘reform calculus’ and to move back to something more traditional. "For students who really need to know math and use it, this wasn’t nearly sophisticated or rigorous enough."

Describing the issues involved in the debate:

"The debates are as deep as those between two different religious groups," says Ronald G. Douglas, provost at Texas A&M University, who is considered the father of the reform movement.

Speaking in favor of the reform movement:

"They (students using traditional methods) learned they could stick in a couple of key symbols, statements, and equations and put forward what were found to be acceptable solutions, even though they had no idea what was going on," says Morton Brown, a professor of mathematics at the University of Michigan.

In our opinion much of the debate revolves around misconceptions on both sides about the goals of the reform effort. In our opinion, it is not the goal of the reform effort to abandon algebraic or analytic techniques. Yet teachers sometimes give this impression or mistakenly believe that this is true in their zealous advocacy for the use of technology. For example, the* Curriculum and Evaluation Standards for School Mathematics* (NCTM, 1989) clearly states that certain algebraic techniques should receive decreased attention in the curriculum. Some teachers wrongly infer that if less attention is good, then no attention is better. A careful reading of the Standards shows that this was never the intention of the authors.

Some paper-and-pencil skills are and will continue to be an important part of the curriculum. However, the role of paper-and-pencil computation will change dramatically in the future because of hand-held technology. As noted earlier, CA provides a "better mouse trap" for much computation. We must recognize and exploit this fact.

Our new challenge is to think about computation differently. Each paper-and-pencil algorithm should be analyzed to see if the procedure contributes any understanding to the process. If not, it should be removed and performed with technology. For example, there is probably widespread agreement that the square root algorithm and finding trigonometric and logarithmic values from a table by interpolation are obsolete. The concept of interpolation is not obsolete, as it is an important idea in mathematics. Using interpolation to find values of trigonometric and logarithmic functions from a table is obsolete. Hand-held computer symbolic algebra will soon make many of the paper-and-pencil factoring algorithms obsolete but *not the process* of factoring (which is a key concept in the fundamental theorem of algebra). The same will be true for many of the paper-and-pencil symbolic procedures typically taught today.

We believe computation should be done in one of the following three ways today and in the future. (By computation we mean those manipulative procedures associated with paper-and-pencil arithmetic, algebra, and calculus.)

1. Mental computation

2. Paper-and-pencil computation

3. Computation done with technology

Our challenge is to decide when a given computation method is appropriate. We believe that some computations will be judged to be mental or paper-and-pencil computation in one course (or section of a course), but then should be done with technology in subsequent courses (or section of the course). This pedagogical technique is called the *white-box black-box* principle. For example, partial fraction decomposition in calculus is a "black-box" procedure best done with CA technology. But integration of functions in beginning calculus is *first* a white-box procedure (using paper and pencil). That is, we allow the use of some algebraic, non-calculus, "black-box" CA procedures while not allowing any "black-box" CA integration procedures (until the skill or concept is learned). Professor Bruno Buchberger, Research Institute for Symbolic Computation in Linz, Austria, first introduced the white-box black-box principle. This excellent CA pedagogical approach is outlined in detail along with other good examples in the book by Heugl, Klinger, and Lechner (Addison Wesley, 1996).

We also need to analyze paper-and-pencil procedures to see if technology can add understanding about the underlying concepts. The following examples illustrate that full use of CAS technology (including CA) can deepen student understanding about mathematics concepts.

CA Pedagogical Example 1. Exploring the local and global behavior of rational functions.

We begin by using a CA tool to apply the division algorithm to a series of rational functions. The first goal is to sketch (paper-and-pencil) the graph of the rational functions by examination of the two more elementary functions (whose graphical behavior is assumed to be known) that make up the rational function. Then computer graphing supports the paper-and-pencil sketches. Next a series of computer graphing "zoom-out" explorations lead to the student discovering that for x large, the rational function behaves "exactly" like a simple quadratic. Finally, the student is led to describing the local and global behavior in the general case of .

We illustrate this exploration with one typical example as shown in the figures below.

a. Apply the TI-89 CA *"propFrac("* command
to find quotient and remainder for the rational function .

b. Use previous knowledge to *sketch with paper and pencil* the two simpler functions y1 = 16/(x - 2) and y2 = x^{2} - 8x - 17 on the same axes.

c. Compare the paper and pencil sketch with a computer / graphing
calculator graph of
in the window [10, 20] by [-100, 100].

Explain why it seems that the graph of the rational function *is* y1 for x "near 2" and y2 for x "away from 2."

d. Graph the original rational function together with y2 in the viewing window given in part c. Explain why y2 is considered a "parabolic asymptote." Further explain how this is a generalization of the "slant asymptote" concept.

e. "Zoom out" (we use window [-100, 100] by [-10000,10000]) and compare the graph of the rational function with y = x^{2}.

f. Explain what you observe and can conclude from part d. Explain the meaning of "global behavior."

g. Repeat a. - e. for other rational functions of the same type.

g. Generalize using .

CA Pedagogical Example 2. Validating CA solutions.

Consider the "exercise" of evaluating the definite integral given in Example CA 6. We use the TI-89 "*integrate*" command to do the computation.

What "is" the exact answer ?
Clearly it can be found quickly by using a CA tool. But *what is it*? How
do we know it is correct? Rather than asking students to do the tedious "paper-and-pencil"
manipulations (no real understanding is involved) to find the "exact"
answer, a much better series of questions can now be asked.

a. How do you know this definite integral exists?

b. Describe a "problem" for which the integral is the "answer."

c. *Estimate* the answer without using * CA* and
compare your estimate with the CA solution. This solution is easy and involves
a computer or graphing calculator graph in the interval [0, /3]
by [0, 1] together with the observation that the area is "about" the
same as that under the line shown

To conclude that the CA answer *must *be near 0.25 requires real understanding of calculus concepts (not low order manipulative skills).

The above two examples illustrate that the real pedagogical power of computer algebra is from combining its use* with* paper and pencil, *and* (computer/graphing calculator) graphical, *and* (computer/graphing calculator) numerical representations.

The scientific calculator of the next century

By the end of this year (1998) there will be several relatively inexpensive hand-held CAS products available to students at local retail and discount stores. The new TI-89 from Texas Instruments and the Casio fx-9970G are two such products. There will undoubtedly be more from other companies. They will cost $100 to $150.00, which is *less* than the cost of early graphing calculators in equivalent 1998 dollars! And they are far, far more powerful. They are truly like having DERIVE or MATHEMATICA like software in your pocket!

There is almost no integrated use of CA in U.S. high school mathematics today. However, many colleges and universities are using PC based CA in calculus, linear algebra, and differential equation courses today. And the number is growing (See Solow, 1994). There are many projects and programs in other countries that are developing programs using CA in high school mathematics. They are listed in the reference section by country. We can learn from their experiences.

Some cautions are in order when using CA in the teaching and learning of mathematics.

CA solutions are subject to error and interpretation. Consider the simple definite integral . One new hand-held CA tool (the Casio fx-9970G) produces the answer ln(2) while another (the TI-89) gives the solution as "undef" as shown . Only the TI solution is correct since this improper integral diverges. Obviously some understanding of the concept of a Riemann integral must exist in order to interpret CA answers.

Another example: we teach in elementary calculus that differentiation and integration are almost "inverse" operations as illustrated here

However, when we try this using a CA tool with a bit more complicated example, we find something that will give us pause as shown.

What is going on? Why? Teachers will have to develop a new set of skills to deal with CA effectively in the classroom. It is our responsibility to provide those skills.

Importance of Mental Algebraic Skills

We believe that mental skills will become even more important than they are today, because students need to interpret the reasonableness of results obtained with technology. We also believe that some paper-and-pencil algebra manipulation skill is still important, particularly when new topics are introduced. Our challenge is to determine how much paper and pencil is necessary before moving on to full utilization of CA.

Staff Development

Successful technology based curricular reform requires two important ingredients. First, you need to provide teachers with technology based materials (for example, Demana, Waits, Clemens, & Foley, 1997; Finney, Demana, Waits, & Kennedy, 1999). These two books (in their first editions) are considered to be the first textbooks to fully integrate graphing calculators in precalculus and calculus. Now most secondary and collegiate algebra, precalculus, and calculus textbooks incorporate graphing calculator technology. Second, teachers need to be trained in the appropriate use of technology to enhance the teaching and learning of mathematics. We found that we needed to develop a massive professional development program for teachers that is now called the T^{3} (Teachers Teaching with Technology) program (Demana & Waits, 1997). See http://www.math.ohio-state.edu/~waitsb/ for more information about T^{3}.

The key to implementing the technology-based approach to teaching mathematics we use is to provide teachers with intensive start-up training and regular follow-up activities. We cannot expect teachers to make fundamental change in their teaching without a good deal of help and ongoing support. This may seem like a hopeless task, but our work shows that it is indeed possible. Providing such training in the future to our huge teacher corps on how to effectively integrate CA in the high school curriculum should be a major focus of all mathematics educators.

The Lesson of History

In most high schools today a majority of the classroom time in mathematics classes and student time on homework is spent on *soon to be obsolete* paper-and-pencil algebra manipulation skills. CA tools, particularly inexpensive, easy to use, hand-held tools like the TI-89 will drive many of these topics from the curriculum. This is the LESSON OF HISTORY! These CA tools will be the scientific calculators of the new century! This conclusion is unavoidable.

We also believe that talented mathematics educators will find creative ways (many not yet thought of) to use CA to enhance the understanding of mathematics. Remember that "new pedagogical approaches" were created after the introduction of the scientific calculator. The same thing will happen with hand-held CA.

History tells us that we can and should teach the same *content* topics, but we should expect the methods we will use "to do" or "to apply" the topics will change (and likely be much faster). For example, some reformers have said it is no longer necessary or desirable to teach factoring. We believe they are wrong. The mathematical topic of factoring *is a major and important topic*. It MUST remain in the curriculum. However, in the past factoring was a mental or tedious paper-and-pencil exercise that often hid the really beautiful underlying mathematics. Recall using the "rational zeros theorem" to factor 2x^{3} - 5x^{2} - 9x + 18? What a painful experience for students - and it took a good deal of time to do just one example! With CA this polynomial can be factored instantly. What is important and was often lost in the fog of tedious computations was recognizing what the factors can tell us about the behavior of the expression. The *concept* topic of factoring *is* important! Integrating CA into the curriculum means the same topics can be taught in less time so more time can be devoted to new mathematics, better mathematics, understanding, proof, problem solving and so forth.

Summary

We know there will need to be textbooks that fully integrate CA and this will take time - it is just now beginning to happen with graphing calculators. We know there will be tremendous need for staff development. Change is difficult for all of us. We need to recognize that it is hard to teach "understanding" and even harder to assess it meaningfully. Much research in needed. However, that should not stop the need for a bold decisive positive position now on CA in school mathematics.

We know today there is a backlash against "math reform." We know that some view "technology" as something to avoid. We need to communicate clearly that technology is not "mathematics" but simply provides better tools TO DO some mathematics manipulations. What is mathematics? It is important to consider this question from the view of a parent - a taxpayer - a legislator. They likely experienced mathematics in the "glory" days of endless paper-and-pencil exercises and drill. Their view of mathematics is one of practicing (to mastery) rote paper-and-pencil procedures - to solve, to factor, to simplify, etc. Even some of our colleagues think such frequent, tedious "practice" is necessary. It is no wonder our critics are skeptical.

We believe what is needed today and in the future is a school and university mathematics curriculum that takes advantage of computer technology to assist students in gaining mathematical understanding, in becoming powerful and thoughtful "thinkers," communicators, and problem solvers. We seek a *balanced approach* to the use of technology.

Some of us in mathematics education have an appreciation of the deeper and richer understanding of mathematics that is possible when technology is used effectively. The great challenge for the Standards 2000 writing team is to make clear to the "public" that such good mathematics is both possible and desirable.

Selected published papers and books about pedagogical computer algebra and computer algebra systems use in specific countries (by country)

US

Demana, F., Waits, B. K., Clemens, S. R., & Foley, G. D. (1997). *Precalculus, A Graphing Approach*, Fourth Edition. Menlo Park, CA. Addison Wesley

Finney, R. L., Demana, F., Waits, B. K. and Kennedy, D. (1999). *Calculus: Graphical, Numerical, Algebraic.* Menlo Park, CA. Scott Foresman Addison Wesley Publishing Co.

Glynn, J. (1994) *Exploring Math from Algebra to Calculus with Derive, A Mathematical Assistant*. MathWare, Urbana IL

Heid, M.K., (1988) Resequencing skills and concepts in applied calculus using the scomputer as a tool, *Journal of Research in Mathematics Education* 19.1, 3-21.

Heid, M.K., Sheets, C. and Matra, M.A. (1990) Computer-enhanced Algebra: new Roes and Challenges for Teachers and Students, in T. Cooney and C. Hirsch (eds.), *Teaching and Learning Mathematics in the 1990’s*. NCTM 1990 Yearbook. NCTM: Reston, Va.

Heid, K.K. *et al*. (1998) *Concepts in Algebra: A Technological Approach*, Everyday Learning Corporation - P.O. Box 812960, Chicago, Illinois 60681

Keller, B. (1994) Symbol Sense and its Development in Two CAS Environments. in L. Lum (ed.) *Proceedings of the Sixth annual International Conference on Technology in Collegiate Mathematics.* Reading, Mass. Addision-Wesley.

MCTM SIMMS (1998).* Integrated Mathematics , A Modeling Approach using Technology (six levels*). Simon & Schuster Custom Publishing 2055 South Gessner, Suite 200 *Houston, TX 77063*

Israel

Nurit Zehavi, "New Teaching Practices using a CAS," (1996) in __The Role of Technology in the Mathematics Classroom__. M. C. Borba, *et al*, editors. Proceedings of Working Group 16, ICME-8, Seville, Spain. UNESP, Sao Paulo, Brazil

France

Artique, M., M. Defouad, M. Duperier, G. Juge, J. B. LaGrange. (1998) *Integration de Calculatricies Complexes dans L’enseignement des Mathematiques au Lycee*. [300 page France CAS research report.] University of Paris, France.

Hirlimann, Anne (editor) (1993) *Enseignement des Mathematiques et Logiciels de Calcul Formel - Derive un outil à intégrer*. Ministére de l'Education Nationale, Paris, France

LAGRANGE, J-B., (1986) "Using a Computer Algebra System in the Mathematics Classroom" in __The Role of Technology in the Mathematics Classroom__. M. C. Borba, et all, editors. Proceeding of Working Group 16, ICME-8, Seville, Spain, 1996. UNESP, Sao Paulo, Brazil

Austria

Aspetsberger, K. and Kutzler, B. (1989), Using a computer algebra system at an Austrian high school, *Proceedings of the Sixth International Conference on Technology and Education. * Linz, Austria.

Heugl, H., Walter Klinger, Josef Lechner: Mathematikunterricht mit Computeralgebra-Systemen (1996) (*Ein Didaktisches Lehrbuch mit Erfahrungen aus dem österreichischen Derive -Projekt*). Addison-Wesley, Bonn.

Schneider, Edith and Walter Peschek. (1998) L’utilization de la TI-89 dans L’enseignement des Mathematiques en Handelschademien" Research paper presented at the Montpellier, France CAS conference May, 1998. University of Klagenfurt, Austria.

UK

Math Association (1995), Calculators and CAS and their use in mathematics examinations, in the *Mathematical Gazette*, Vol. 79, No. 484.

M. Taylor, (1996) ‘The Effect of CAS on A-level Mathematics’, Hugh Neill in the Proceedings of *Mathematics for 16 to 19 year olds*. SCAA, England

Flower J & Oldknow A. (eds). (1996) *Symbolic Manipulation by Computers and Calculators*, UK Mathematical Association, Leicester, UK.

J. Monaghan and T. Etchells (eds.) (1993)* Computer Algebra Systems in the Classroom*. University of Leeds Press. Leeds, UK.

Germany

Bärbel Barzel (ed.): (1995) *Derive Days Düsseldorf - Tagungsband***. **Landesmedienzentrum, Rheinland-Pfalz/Germany.

Spain

Miguel de Guzmán, Baldomero Rubio: (1993) análisis Matemático 3 - sucesiones y series de funciones, números complejos, Derive, aplicaciones._Ediciones Pirámide, Madrid Spain.

General publications about pedagogical use of computer algebra and CAS

Berry, J. S., Manfred Kronfellner, Bernhard Kutzler, John Monaghan* *(1996*) Computer Algebra in Mathematics Education*. Proceedings of Derive CAS Research Conference, July 31-Aug 3, 1996, Honolulu, USA. Chartwell-Bratt, Bromley, UK

Berry, J. S., E. Graham, and A.J.P. Watkins*. *(1997)* Learning Mathematics through the TI-92*. Lund, Sweden Chartwell-Bratt Publishing Ltd

Böhm J. (Editor) (1992) *Teaching Mathematics through Derive*.

Proceedings of Krems 1992 Conference, April 27-30, 1992, Krems, Austria.

Chartwell-Bratt, Bromley UK

Bright, G. W., Waxman, H. C., & Williams, S. E. (Eds) (1994), *Impact of Calculators on Mathematics Instruction*. Lanham, Maryland, University Press of America.

Bright, G. W., & Williams, S. E. (1994). Research and evaluation: Creating a complete picture of the impact of calculator use on mathematics teaching and learning. In L. Lum (Ed.), *Proceedings of the Sixth International Conference on Technology in Collegiate Mathematics*. Reading, MA: Addison-Wesley.

Comstock, M., & Demana, F. (1987). The calculator is a problem-solving concept developer. *Arithmetic Teacher, 34*(6), 48-51. Reston VA. NCTM

Demana, F., & Leitzel, J.R. (1988). Establishing fundamental concepts through numerical problem solving. In A. F. Coxford & A. P. Shulte (Eds.), *The ideas of algebra, K-12: 1988 yearbook* (pp.61-88). Reston VA: National Council of Teachers of Mathematics..

Demana, F., & Waits, B. K. (1997). A Zero-Based Technology Enhanced Mathematics Curriculum for Secondary Mathematics. In A. Ralston & H. Burkhardt (Eds.), *Proceedings of WG 11, ICME 8*. England.

Demana, F., Schoen, H., and B. Waits. (1992) "Graphing in the K-12 Curriculum: The Impact of the Graphing Calculator," chapter in *Integrating Research on the Graphical Representation of Functions*. NCRME University of Wisconsin Monograph, T. Romberg, E. Fennema, & T. Carpenter, Editors. Madison, WI.

Demana, F. and B. Waits. (1994) "Graphing Calculator Intensive Calculus: A First Step in Calculus Reform for All Students" in the Proceedings of the Preparing for a New Calculus Conference. Anita Solow (Editor) MAA Note. MAA, Washington D.C.

Demana F., J. Harvey, and B. Waits. (1995) "The Influence of Technology in Changing the Nature and Role of Algebra and in Revolutionizing the Way it is Taught," in the Proceedings of ICME-7, Quebec, Canada. The Journal of Mathematics Behavior, volume 14. Ablex Publishing Corp. Norwood, NJ.

Dunham, P. (1992). Teaching with Graphing Calculators: A Survey of Research on Graphing Technology. In L. Lum (Ed.), *Proceedings of the Fourth Conference on Technology in Collegiate Mathematics*, 89-101. Reading, MA, Addison Wesley.

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