Who is tired of hearing the Standards repeat the statement "appropriate use of calculators at all levels as a legitimate and important tool for learning and doing mathematics" (Discussion Draft, 1998, p. 43) without further explanation or direction? I know I am. The following is a discussion for the specific use of the graphing calculator for middle school and high school mathematics. Arguments for both sides of the issue are presented on should we or should we not use graphing calculators as a regular classroom tool. My arguments on their use and place in the Standards is presented and supported.
        Graphing calculators have been on the market for about ten years. They are becoming an integral part of mathematics instruction and a necessary classroom item. This phenomenon generates several concerns. Some of these concerns are as follows. How do graphing calculators affect curriculum and instruction? How do they affect students' learning? How reasonable is it for every middle school and/or high school student to have a graphing calculator? How do graphing calculators alter testing and other forms of assessment? This paper will explore these issues and present varying viewpoints.
        The National Council of Teachers of Mathematics and its Curriculum and Evaluation Standards (1989) had a large impact on curriculum. When talking about curriculum we look at these Standards to guide our educational goals. One Standard to focus on with graphing calculators is Communication. How well can students communicate mathematically with the use of technology? NCTM dictate and assume that every student will be able to use appropriate technology at all times. This assumption and realization have drastically changed the direction of curriculum in mathematics. Waits and Demana (1996) have noted that in the past pencil and paper algebraic manipulations were necessary procedures needed to solve mathematical problems, but now with calculators readily available we can solve more complex problems applicable to real-world situations. They believe the community can no longer ignore how these students use technology based symbolic algebra in a mathematics curriculum. It is time to redefine "basic skills."
        Many curriculum projects have taken on the challenge of explicitly defining and using graphing calculators in mathematics instruction. "The emergence of powerful technologies for numeric and graphic mathematical calculation suggests changes in the traditional focus of school curricula on procedural skills in arithmetic and algebra." (Schoen, Fey, Hirsch, & Coxford, 1998, p. 3.) The overwhelming need for appropriate use of technology is outlined in the Core - Plus Mathematics Project (CPMP). These authors have developed a curriculum integrated with the use of the graphing calculator; complete with instructions for teachers and students. As found in the Contemporary Mathematics In Context Sampler by CPMP on page 4 (Features and Benefits) is the type of calculator expected of all students to have and use. Graphics Calculator: Numerical, graphics, and programming/link capabilities of graphics calculators are assured and capitalized on, as appropriate, throughout the curriculum.

    *Allows all students access to important and useful mathematics.

    *Enables teachers and students to concentrate on big mathematical ideas.
 

    *Fosters the representation of mathematical concepts in multiple ways: numeric, symbolic, and
      graphical.
 

    *Allows the investigation of rich applied problems that involve large amounts of data.
 

    *Promotes the development of versatile ways to solve mathematical problems.

        This sampler further develops, explains, and provides activities for students and teachers to explore mathematical situations with the use of graphing calculators as a necessary tool. This a good project with high definition but is not as widely recognized as NCTM Standards. The nation will be looking for what is acceptable and necessary on the next NCTM Standards document that will be printed. This kind of definition for technology and graphing calculators is eminent.
         This document will launch the goals and criteria for what students will be taught for the next ten years or more to come. The draft document as written is behind the times. The new Standards document needs to be ahead of the times so that teachers, schools, and students have something to work towards. The National Council of Teachers of Mathematics should be concerned with what should be taught and how it should be taught as a STANDARD of teachers and schools to compare themselves with. NCTM should not use the logistics of can these standards be reached (i.e. cost of calculators, availability of calculators, etc.) The logistics is the responsibility of schools and school districts. With out clear definition of tools necessary to reach the standards, the standards will not be reached.
           Some are concerned that calculators will inhibit a student's understanding of mathematics, and we can hear the echo of the "crutch" premise now related to algebraic manipulations and curve sketching. Demana and Waits (1990) respond that we should include the paper and pencil tasks necessary to understand the mathematics in the curriculum but stop spending large amounts of time teaching obsolete pencil and paper tasks/manipulations and allow the use of technology to help students understand mathematics even better. Shultz (1991) supports this notion when he says "calculators can be used to support concept building in Algebra so that the language through which most of the mathematics is communicated need not be a foreign language to students"(p. 37). He believes technology helps to visualize mathematical relationships. Educators from both sides of the argument will have to become curriculum reformers together.
        Since curriculum is changing, so is instruction. NCTM advocates the use of calculators at all levels of mathematics instruction: starting with four function calculators in elementary school, scientific calculators for middle school, and graphing calculators in high school. Highly proficient skill in arithmetic and algebraic manipulation has been rendered practically obsolete due to the explosion of technology in this decade (Demana & Waits, 1990). According to Demana and Waits (1990), graphing calculators are most likely to have an immediate impact on curriculum for grades 9-12. From the article, "The Role of Technology in Teaching Mathematics"(Demana & Waits, 1990), several questions form the teacher's point of view are proposed and then commented on by Demana and Waits.

    What can I do with a graphing calculator?

    Graphing calculators are inexpensive powerful pocket computers that are easy to program, have excellent statistical features, includes a graphics interface, perform matrix algebra and various other algebra manipulations. "The graphing calculator is a vehicle for implementing many important aspects of the Standards for all students"(Demana & Waits, 1990, p. 28). Because of this last statement it is so important for the new Standards document to include specifics on graphing calculators since their use is so essential to many other aspects of the Standards.

    What equipment does my school need? Is it Is it affordable?

    The Standards say one demonstration computer at $2000, however, programmable graphing calculators for overhead projectors are available for about $250. Also, about eight classrooms can be equipped with calculators for the price of one computer and projection device. As each day passes graphing calculators become more affordable. They cost little more than textbook themselves.

    Will technology change what and how I teach?

    Yes, technology will dramatically change what and how mathematics is taught. Graphing calculators are encouraged to be used with algebraic manipulations, zeros, factors of polynomials, and graphs all previously done by pencil and paper. Graphing calculators have many other uses. These were just a few. With this technology we can exploit the power of visualization. Emphasis that used to be placed on exact answers, which are rarely needed, has subsided in favor of approximate answers which are more realistic in real-world situations. Dick (1992) adds to this notion by expressing that the graphing calculator removes the constraints with which teachers and textbooks relied on artificial nice examples and exercises. Yet again, how the graphing calculator ties into another aspect of the Standards.

    How do I justify the use of technology?

    "Technology is here to stay"(Demana & Waits, 1990, p. 29). Today's workplace is dependent on technology and the sooner students are familiar with technology the better. To ensure success, students must become proficient with technology and effectively reason with mathematics. Dick (1992) supports Demana and Waits with the statement that calculators are a great machine but it is of little use unless the student knows what to do to solve the problem. The inclusion of specifics on the graphing calculator in the new Standards document will be one of the most important justifications for teachers, schools, parents, students, and revised text books.

    How do I learn about using technology?

    Teachers first need access to technology. Then they will be able to attend workshops which are offered at local, state, regional, and national levels. Schools or districts could organize a technology support group to share ideas and minimize frustration.

    How can I get started using technology in my classroom?

    Encourage students to purchase graphing calculators of their own. Demonstrate on the overhead calculator things that can be done. Consult the Standards for examples. This is exactly why the Standards need to include examples in the new text. Have your school order texts that incorporate technology. This is also an opportunity for the Standards to direct the public towards good text book and or curriculum projects that align themselves with the Standards.

    What if my students know more about the technology than I do?

    Encourage students to explore the calculator and share with the class what they have found. Borba (1995) also advocates student exploration of the calculator. He claims this leads to a better understanding of mathematics for everyone involved. Borba continues, students are then motivated to confront more complex problems. Allowing teachers to say "I don't know," gives the student a feeling of control in their learning. As time goes by and teachers become more familiar with the technology, both students and teachers will have proficient calculator skills which will aid both students and teachers in solving mathematical situations throughout their lives.

    What pitfalls should I be aware of when using technology?

    What you see on the display screen is not always accurate, be aware of behavior hidden from view. This has to do with proficiency of calculator use and with time and practice students will overcome this.

    Is the role of Algebra and Calculus in mathematics changing?

    It changes the focus of instruction from algebraic manipulation to the understanding of algebra and calculus as languages that allow for modeling problems. Since graphing calculators have had such an impact on the changes in curriculum and instruction, lets take a look at their affect on students and their learning.
        There is one question directly pertain to students' learning from the article, "The Role of Technology in Teaching Mathematics"(Demana & Waits, 1990). Will the Algebra skills of students using technology be hurt? Quite the contrary, students using graphing calculators become good problem solvers and gain a deeper understanding of algebraic concepts and procedures. The Standards believe this also which is why the Standards needs to set limitations and specify what is appropriate. Now teachers have the freedom to determine and interpret appropriate. With out guidance, definition, and experience appropriate is an ambiguous term. Teachers may feel that their use of technology is appropriate but have no basis of comparison.
        Technology is rapidly improving and we must prepare students for lifelong learning and train them to be good problem solvers able to employ technological advances as they develop. Dick (1992) adds his findings of technology's affect on students' skills:

    a. Concentration on the problem-solving process,

    b. Gain access to mathematics beyond the students' level of computational skills,

    c. Explore, develop, and reinforce concepts including estimation, computation, approximation,
        and properties

    d. Experiment with mathematical ideas and discover patterns,

    e. Perform those tedious computations that arise when working with real data in problem-solving
        situations,

    f.. Remember to cover the necessary basic skills to understand the mathematics being used with
        thecalculator.

With such technology available, students need a fundamental understanding of mathematical concepts and processes which can then be applied along with critical thinking and reasoning skills in mathematics. According to Demana and Waits (1990), "Technology empowers students to solve difficult problems."(p. 27)
        Another advantage that this technology has shown us is the misconception that only bright students are able to do interesting mathematics. Technology has reduced the time we devote to paper and pencil drill, leaving more time for exploring and investigating mathematics with application to real-life situations. Nicolas Miller is a perfect example. He was a tenth grader taking Algebra for the first time. He has a learning disability which slowed down his learning. After reading the article, "From an E to an F in First -Year Algebra With the Help of a Graphing Calculator" (Bethell & Miller, 1998), one can really see the importance of such technology in the classroom and its benefits to students of all abilities. This article is the result of an interview with Nicolas Miller, a math student, and his teacher Sandra Bethell. The use of the graphing calculator allowed Miller to get past his frustrating disability and explore mathematics with a clear understanding of the processes they were studying. In fact, Miller was so capable with the calculator he was able to help his classmates.
        Calculators stimulate interest, understanding, and the desire to solve complex problems and find exact answers (Embse & Engebretsen, 1996). Stick (1997) believes graphing calculators have helped students' retention and performance. He did a study with two calculus classes, one with calculators the other without calculators. At the end of the study he found that students taught with calculators understood calculus better, were more interested, and were able to see calculus as applied in the world. Students commented, "we could see what was happening and learning became fun"(Stick, 1997, p. 360). Stick concluded that by giving a graphical representation first, helped make the transition to analytic methods much easier and more understandable.
        Coxford and Hirsch (1996) argued that calculators have helped educators believe that all students can learn mathematics and that students must learn more. This pair proposed and tested a new curriculum. They used a heterogeneous group and a four-phase system where graphing calculators and cooperative learning were central to the curriculum. Phase one started with a class discussion and launching of a situation. In Phase two the context was set and students worked in groups of four and set to explore the mathematical features of the situation. Then in Phase three, the groups organize their thinking about the big mathematical ideas, processes, and algorithms. Lastly, Phase four followed with students sharing and summarizing the activity and then applied their knowledge to other similar situations. This study found heterogeneous grouping helped all students to clarify their understanding of mathematics and provided support for struggling students. "The extensive use of the graphics calculator as a tool for learning and doing mathematics helps students whose limited computational abilities previously prevented them from advancing in the study of important mathematics" (Coxford & Hirsch, 1996, p. 25).
        No one believes that bringing a set of graphing calculators into the classroom will have a magical effect on students. However, studies have shown there is a difference. Dunham and Dick (1994) looked at a study where content, instruction, and testing were identical; the only difference was the presence of graphing calculators for half the group. The students with graphing calculators displayed better graphical understanding, the ability to link equations with their graphs, could read and interpret graphical information, were able to obtain more information from a graph, were successful at finding algebraic representations of a graph, and understand global features of functions as a whole. Now that we know the effect of graphing calculators in curriculum, instruction, and learning; the next question is how feasible is it for students to have one.
        This is an important issue. It will be even more significant if the Standards includes them in the new document. There are many solutions that school districts can implement. Dick (1992) points out that access to graphing calculators is rapidly being rendered moot. "Affordable hand-held calculators with the capabilities to graph functions and relations, manipulate symbolic expressions including symbolic differentiation and integration, compute with matrices and vectors, and perform high-precision numerical integration and root-finding of functions will provide the reality of mathematics classrooms where every student has tools rarely available on mainframe computers 20 years ago" (Dick, 1992, p.1). More specifically, students do have access to a graphing calculator. Most classrooms that I know of have calculators for students to use in class. The problem comes in when students have homework. Virginia has addressed that obstacle by purchasing enough calculators so that teachers issue a calculator along with the textbook for students to use all year at anytime. This requires a large amount of money up front. Much of Maryland has only been able to provide classroom sets to mathematics teachers.
        I received thirty TI 83's to use in my classroom last year along with every teacher in Charles County. Many other counties in Maryland have been using graphing calculators for years. Maryland and many other States are implementing a high school assessment test necessary for graduation. Maryland is designing end - of - course high school assessments with the following guidelines: "Use of current technology is expected throughout the test. The mathematics tests will require a graphing calculator. Students should have access to the graphing calculator that they regularly use during instruction. Therefore, no constraints or limitations should be placed on the type of graphing calculator used on the test. Questions should be written so that students who have graphing calculators that have more capabilities do not have and unfair advantage over those who have more limited calculators. At a minimum, calculators must have the capability to do the following: table functions, point plotting, linear fit, solutions to systems of equations, statistics (mean, median, mode, interquartile range), maxima and minima, trigonometric function values, and matrices. (The College Board & Educational Testing Service, 1997, p. C-23) The report further recommends that 10% of the questions require the graphing calculator and that 40% of the questions where a calculator may be useful or necessary. It is assumed that students know how to use the graphing calculator. Soon, I believe, every mathematics student will carry a graphing calculator just as they carry a textbook.
        Since graphing calculators have so drastically changed curriculum and instruction, it has also affected testing and assessment. Harvey (1992) categorizes tests into three parts: 1. technology-inactive - where no opportunity to use the technology exists, 2. technology-neutral - problems easily solved without technology, and 3. technology-active - use of technology is essential or greatly assists the completion of the problem. First, teachers need to examine their current tests to see how best to align them with content and processes which support technology, reasoning and open-ended items. The Standards requirethat appropriate use of technology be incorporated into the curriculum (Thompson, Beckmann, & Senk 1997). Rewriting tests completely is time consuming and requires immense commitment. Thompson et al. suggest that a few test items on each test be modified to meet new goals. They included some examples where technology can be applied:

Question 1: What is the volume of a box with dimensions 5cm by 10cm by 3 cm?

Question 1 revised: Does a box with a volume of 150 cubic inches exist whose dimensions are the maximum allowable by the U.S. postal service? Justify your answer.
 

Question 2: Let f(x) = - 3x² + 2x
        a) What is the maximum value of f?

        b) On what interval(s) is f increasing?

Question 2 revised: f(x) = - 3x² + 2x

        a) What is the maximum value of f, to the nearest tenth.

        b) On what interval(s), to the nearest tenth, on which f is increasing.

        This is a good start for improving test assessment and more information is given in this article by Thompson, Beckman, and Senk. The three authors suggests a change of ten percent of test items to shift the emphasis and direction of the test to using graphing calculators. Such examples are often included in the Standards documents. Very few examples exist for the graphing calculator. Such examples should be included in the standards document for Standards 1, 2, 4, 5, 6, and 8. The incorporation of graphing technology would enhance students mathematical understanding for all six of these standards.
        We must also consider other forms of assessment, such as: a multi-step, complex problem solving, or reasoning process to assess. Just allowing the use of graphing calculators is not enough, such an essential tool to instruction must show in technology-active assessments. Harvey (1992) emphasizes that "students value what is graded." So we need to grade students' ability to use the calculator with mathematics. The problem is that very few materials and resources incorporating technology are published. It is difficult to find assessment items for the graphing calculator. Also, few guidelines are established for developing good assessments. One thing we do need to consider is the more complex and involved the task the fewer items we should include on an assessment (Thompson et al., 1997). In addition, more time may be needed for students to complete the tasks. To conclude, Senk (1992) suggests using various types of assessment which include electronic devices with the use of a graphing calculator, items presented on paper with the use of a graphing calculator, and paper and pencil items without the use of a graphing calculator. He feels this will give a balance and complete portrayal of what a student's abilities are.
        When I began teaching, I was of the philosophy "brain as muscle" and thought students should be able to perform algebraic manipulations by hand without any assistance. Since then I have attended various seminars, training sessions, and classes. My idea of graphing calculators in the classroom has evolved and I do feel they are essential and an important part of a mathematics education. The Standards need to evolve also. Not much has changed in the Standards description of technology use from the 1989 document and the Draft 1998 document. However much in the way of price, availability, capability and use of such technology has changed greatly. The Standards needs to express more clearly the expectations for technology and the graphing calculator in the technology standard. NCTM need to stop dancing around the issue and using colorful language and make a stand that is clear!
        The curriculum standards that I found most applicable to this issue are; reasoning, communication (involving technology), interdisciplinary, patterns, and problem solving. The assessment standards that relate to graphing calculators are equity, mathematics, learning, and coherence. The equity standard concerns with the availability of graphing calculators, but as time goes by and calculators become affordable this concern will diminish. Mathematics, students need to know and be able to do mathematics and calculators can only help. As for the learning standard, we need to advance students learning and let that direct the teacher's instructional decisions. Last is the coherence standard, ties everything together, making sure the use of graphing calculators is aligned with curriculum, instruction, and assessment.
        Many aspects of graphing calculators have been considered. They are prevalent in our society and it is to the benefit of the students for them to be educated in their use. New assessments must be developed to check student understanding and teacher instruction. Graphing calculators allow students to see multiple representations and extend their current mathematical knowledge. Much of what I have read fits with the Standards and what they wish to accomplish. Stick, said it well with "Both the instructor and students have to share excitement about the technology to make it work."(1997, p. 360)
        "Technology is vital to the study of mathematics. While today that technology takes the form of computers and graphing calculators, it is essential that technologies continue to reflect current standards. With the change in technologies, the mathematical processes change." (Maryland State Department of Education, 1996, p. 1) The National Council of Teachers of Mathematics desperately needs to address this in the technology standard. Without a national goal for mathematics education where does that leave our students? I hope to see more than just the statement "appropriate use of calculators at all levels as a legitimate and important tool for learning and doing mathematics" in the final Principles and Standards for School Mathematics.
 
 

Bethell, S. C., & Miller, N. B. (1998, February). From an E to an A in first-year Algebra with
    the help of a graphing calculator. Mathematics Teacher, 91 (2), 118-119.

Borba, M. C. (1995, July-August). Teaching mathematics: Computers in the classroom.
    Clearing House, 68 (6), 333-334.

The College Board & Educational Testing Service. (1997, August) Designing end - of -
    course high school assessments. A final report to the Maryland State Department of Education.
    (1), c23-g4.

Core - Plus Mathematics Project. (1999). Contemporary mathematics in context: A unified
    approach sampler. Everyday Learning Corporation. 2-99.

Coxford, A. F., & Hirsch, C. R. (1996, May). A common core of math for all. Educational
    Leadership, 53, 22-25.

Demana, F., & Waits, B. K. (1996, December). A computer for all students-revisited.
    Mathematics Teacher, 89 (9), 712-714.

Dick, T. P. (1992, January). Symbolic-Graphical calculators: Teaching tools for mathematics.
    School Science and Mathematics, 92, 1-5.

Dunham, P.H., & Dick, T. P. (1994, September). Connecting research to teaching: Research
    on graphing calculators. Mathematics Teacher, 87 (6), 440-445.

Embse, C. V., & Engebretsen, A. (1996, December). A mathematical look at a free throw
    using technology. Mathematics Teacher, 89 (9), 774-779.

Harvey, J. G. (1992). Mathematics testing with calculators: Ransoming the hostages. In T. A.
    Romberg (Ed.), Mathematics Assessment and Evaluation: Imperatives for Mathematics
    Educators.  139-168.

Hirsch, C. R. (1995, November). Teaching sensible mathematics in sense-making ways with
    the CPMP. Mathematics Teacher, 88 (8), 694-700.

Maryland School Performance Program. (1996, September). High school core learning goals.
    Maryland State Department of Education. 1-36.

National Council of Teachers of Mathematics. (1998, October). Principles and standards for
    school mathematics: Discussion draft. 9-335.

Pollak, H. O. (1997). Solving problems in the real world. Teachers College, Colombia
    University. 91-105.

Senk, S. L. (1992). Assessing students' learning in courses using graphics tools: A preliminary
    research agenda. In T. A. Romberg (Ed.), Mathematics Assessment and Evaluation:
    Imperatives for Mathematics Educators. 128-138

Schoen, H. L., Fey, J. T., Hirsch, C.R., & Coxford, A. F. (1998, September). Issues and
    options in the math wars. Phi Delta Kappan. 1-20.

Shultz, J. E. (1991, May). Implementing the Standards: Teaching informal Algebra. Arithmetic
    Teacher, 38, 33-37.

Stick, M. E. (1997, May). Calculus reform and graphing calculators: A university view.
    Mathematics Teacher, 90 (5), 356-360.

Thompson, D.R. & Beckmann, C. E. & Senk, S. L. (1997, January). Improving classroom
    tests as a means of improving assessment. Mathematics Teacher, 90 (1), 58-64.

Waits, B.K., & Demana, F. (1990, January). Implementing the Standards: The role of
    technology in teaching mathematics. Mathematics Teacher, 83 (1), 27-31.