The SUOA/V circular of 15 June 1993, issued by the Norwegian Ministry of Church, Education, and Research, announced the opportunity to use graphing calculators for final examinations in mathematics. The circular contained some constraints on their use, one of which was that the calculator must not be able to perform symbolic operations.

Graphing calculators became compulsory aids for general mathematics and for teaching economics/administration in conjunction with Reform '94. At that time graphing calculators could perform only numerical calculations, but Texas Instruments announced that it was designing a new graphing calculator that would be able to perform symbolic operations, affording the opportunity for exact calculations.

The computer programs DERIVE, MathCad, Math Plus, Mathematica, and Cabri Geometry have been on the market for several years. Many of these programs can perform symbolic calculations, and they have gained considerable acceptance among mathematicians.

At Strand upper secondary school, MathCad and DERIVE had been used for several years in differential calculus, an elective subject scheduled for two hours a week in the upper class. Student response clearly indicated that the use of these programs led to enthusiasm for the subject and that they could be learned without great difficulty.

With the release of the TI-92 by Texas Instruments, where DERIVE was the basis of the calculator's symbolic operations, interest in this technological advance greatly increased. Our prior experience with DERIVE gave us high expectations for using the TI-92 in education. A small, portable calculator would offer great advantages over a stationary PC.

We knew that the Calculator-Based Laboratory (CBL) System could be used with the TI-92, and that exercises in physics and chemistry for the TI-82 and TI-83 calculators could be performed by the TI-92. We also knew that there was a great deal of help for the TI-92 calculator on the Internet, for example a program to manage the CBL system, and that it would therefore be important to construct limits for our experiment.

Conditions for a study of the use of the TI-92 calculator at Strand upper secondary school were excellent. Many students of Math 3MX had also chosen physics (3FY) and the elective differential calculus, and because the same teacher taught all three subjects, there was great enthusiasm at school when KUF/SUE approved an experiment with the TI-92 calculator. This favorable climate for testing the TI-92 raised our expectations.

We are very grateful to Texas Instruments and Ess Data, the Norwegian importer, for their contribution of sufficient numbers of TI-92 calculators for all students.

The primary goal of the study was to compare the TI-92 calculator, a graphing calculator capable of performing symbolic calculators, with graphing calculators that could perform only numerical calculations. KUF/SUE, the branch of the exam in the Ministry of Church, Education and Research in Norway, gave us an opportunity to use the TI-92 calculator for the examination in a 1997 math class, 3MX, with exercises adjusted to the capabilities of the calculator.

In addition to students at Strand upper secondary school, students in math at Sandnes upper secondary school were also to use TI-92 calculators for their 1997 examination, allowing the teachers from the two schools to cooperate on the study.

The TI-92 calculator opened the way for impressive opportunities in mathematics, and it was tempting to expand the chosen subject areas to show more mathematical connections. However, introducing more or different subjects taken by ordinary students was not relevant for the purposes of this study. Only the following questions were to be considered:

Would student motivation, calculation skills, and technical understanding increase?

To what extent would exercises have to be changed?

In physics (3FY), the TI-92 would be used with the CBL system. The CBL system interface was downloaded from the Internet, but some adjustments had to be made.

Vector analysis:
The TI-92 revealed several new ways to perform vector analysis.

Function analysis:
Treatment of different types of functions: Derivation and integration: Definite and indefinite integrals: Space and volume, rotating objects. The TI-92 was quite successful in teaching these subjects. We profited greatly when symbolic mathematical operations were performed, because the mathematical expressions could be recognized and identified in each calculation.

Calculation of probabilities:
Simulations: Statistics and testing of hypotheses. No special functions or programs were installed for these topics, since the TI-83 was superior for these sorts of calculations; however, we could rationalize the calculations in relation to the numerical calculators by defining separate functions with many variables.

Differential Equations:
It became apparent that a special module from the Internet would have to be implemented in order to benefit from working with differential equations on the TI-92. When this program was installed on the TI-92, the calculator revealed opportunities for calculations that satisfied mathematicians at the University standard.

Geometry and Conic sections:
The TI-92 calculator uses Cabri Geometry as a regular program, but we did not have an opportunity to explore its possibilities.

Vernier Software's Calculator-Based Laboratory System (CBL) on the Web was very helpful.

Internet addresses:

• Vernier Software - CBL
• Texas Instruments - CBL
• Norwegian importer of CBL probes

All of the students in the experiment group also took the elective course in differential equations for two hours a week.

The program-pack ADV.92G, which can be downloaded from ftp://ftp.derive.com/pub/adv.92g, proved to be an appropriate aid.

We made studies of the general swinging equation:

where m is the body's mass, D is the spring-stiffness, and q is the damping-factor. The plot of a particular solution can look as follows:

The Norwegian National Center for Teaching Aids.

Several program and tool menus can be downloaded from the Texas Instrument Web site using the TI-92 graph link.

Some companies' home pages can serve as starting points for contact with different parts of the same trade at home and abroad. For mathematics and science teachers we recommend TeknoDidakt.

For Internet searches we recommend Kvasir.

For our evaluation of the experimental use of the TI-92 Calculator in mathematics classes, we used the educational plan and some sets of exercises designed for ordinary students and provided by the Norwegian National Council of Secondary Education.

The TI-92 calculator has inspired and motivated students of mathematics, and has been of great value for students at every level.

This can be seen, with only a few exceptions, from the results of the public written examination and assessment grades:

1. From the first examination paper:

   
                                          Test Schools   Control Schools
   
   1a) 1) Find the derived function        2p: 95, 1%     2p: 71, 0%
          of function f without using      1p:  0, 0%     1p:  9, 0%
          the calculator                   0p:  4, 9%     0p  20, 0%
          
   
       2) Find the derived function        2p: 92, 7%     2p: 52, 3%
          of function g without using      1p:  0, 0%     1p: 12, 2%
          the calculator:                  0p:  7, 3%     0p  35, 5%
          
   
   1b) Solve the integration problem       2p: 95, 1%     2p: 89, 0%
       without using the calculator:       1p:  4, 9%     1p:  2, 6%
                                           0p:  0, 0%     0p:  8, 4%
          
   
      [2p = 2 points - correct answer; 1p = partially correct answer;
       0p = incorrect or missing answer.]
       

2. Average marks on the public written examination
   

   Students from the study schools: 41 examination papers; average grade 3.90.
   Students from the control schools: 155 examination papers; average grade 3.48.

   [The Norwegian scale of marks goes from 0 to 6, with 0 (no credit given) at the bottom and 6 (highest marks) at the top.]

3. Average marks based on class work

   Test schools: Marks were calculated for the same group of pupils - 2MX and 3MX - throughout two years of school. In the ordinary course of events, the marks would have been expected to show a tendency to decline during the second year.

Several interesting tendencies became apparent when we studied the numbers in the table shown above:

1. Student papers from a 1997 3MX external examiner formed the basis for the representation. All pupils in both test groups sat for an exam in writing in 3MX, and the results from these papers were also included in the evaluated material.

2. Although the number of participants in the experiment was too small for us to draw any firm conclusions, the above survey results give us a good idea of what to expect if we begin to use the symbolic calculator for mathematics education.

3. There seems to be evidence for the following claims:
   1. Skill in calculation increases.
      
   2. Motivation for working with the subject increases.
      
   3. Problem-solving ability is stimulated.
      
   4. Professional understanding increases, making it easier to solve professional problems.

   The above claims might be discussed, but in this report we cannot supply conclusive evidence for them; nevertheless, several facts should be highlighted:
   1. There were demonstrably more correct derivation and integration solutions in the test groups than in the control group.
      
   2. The average mark achieved on the exam was considerably higher in the test groups than in the control group.
      
   3. In the test groups, the average mark for general proficiency was higher in 3MX mathematics than in 2MX mathematics.

   There can of course be other reasons for these findings. We will leave speculations to the reader.

1. Professional experiences

   It has been a pleasure to work with the TI-92 in Math 3MX. Math subjects obviously become more understandable when using a symbolic calculator. The best use of the calculator came from studying how different changes in input were important for the output.

   We could concentrate our attention completely on studies of professional phenomena instead of using a lot of time for analytical calculations.

   We continually had to repeat the procedure for the analytical calculations - and they are still a part of the foundation of mathematics - but sometimes long and elaborate analytical calculations seemed like a distraction from the process of learning.

   For the physics lessons, working on the physical formulas was especially relevant, but the calculator wasn't really much help in physics.

   The optional subject differential calculations became considerably more understandable when the program package ADV was downloaded from the Internet. The students especially profited from programs that solved general first- and second-degree equations, finding the direction graphs and then one particular solution.

   Students found that it wasn't so easy to solve differential equations in the analytical, manual way, and therefore the calculator freed the subject from a heavy burden.

2. Educational and methodological experiences

   Many new challenges had to be met, but in an open dialogue with students, we finally got the procedure on a constructive path. Although it was often tempting for the teacher to demonstrate the latest technical finding on the calculator, best effects were seen when students found the technical possibilities themselves and demonstrated these to the rest of the group. Since most of the calculators were equipped with a switch for transferring the graphical calculator image to an overhead, there were always pupils who wanted to show possibilities that they themselves had found. After a time, some pupils reached very high levels of competence in using the calculator.

   It proved to be relatively easy to motivate students to study at home when a problem was to be solved on the calculator. The learning outcome was excellent.

3. Experiences with technology

   It was interesting to see how some students, especially some of the girls, changed their attitude toward the calculator as a technical instrument during the year. The first weeks were characterized by an almost negative relation to the machine's possibilities, but after a while we saw a noticeable change in attitude, one that had a positive effect on the whole group. It actually seemed as if something had to mature before the realities could be taken seriously; then it became clear that frequent use of the calculator was a necessary condition for its use as a technical support in particular situations. Most students became competent users of the calculator during the year.

   All of the necessary downloading from the Internet and transferring between machines proved to be more difficult than we had anticipated. Since the calculator memory is organized in folders, it was very exciting each time to see where the saved files would end up. The program packages on the Internet, however, included detailed descriptions for both downloading and use, so all problems were solved.

As we evaluate the advantages and drawbacks of using a symbolic calculator for mathematics education in the Norwegian high school, it is important to consider the situation from the point of view of the student. The fact is that the symbolic calculator is an indispensable aid for students who already understand the basics of analytical mathematics manipulations and who can evaluate answers found using calculators. It is not so clear, however, that students who do not have the necessary understanding of the basic axioms of mathematics can simplify and rationalize their way to mathematical understandings with the help of a technological aid. This point must be given great importance in evaluating the results of any study.

In individual cases, proving that manual calculations were in agreement with answers obtained using a calculator appeared to be an almost insoluble problem for students. It could be problematic to prove the correspondence between manually calculated answers to specific exercises in derivation and integration and the corresponding answers supplied by the calculator. Thus the calculator raised doubts, and an absence of analytical knowledge was clearly demonstrated to be a drawback.

On the other hand, during recent decades mathematics education has been overly concerned with doing exercises over and over, and in many cases mathematical understanding has suffered as a result. Now, access to a technological aid that can be combined with a basic analytical education in mathematics will certainly make an important difference in mathematics education in the schools.

The question whether symbolic calculators should be allowed in mathematics education is an emotionally charged discussion. The arguments cover the whole spectrum, from the skeptical - keeping math clean - to the enthusiastic - strengthening useful mathematics. It seems important to direct significant resources toward activities that will inform those entitled to offer their opinions. In the next round the situation will naturally be discussed before a decision is made.

Many ideas have been put forth for how to test the use of the symbolic calculator in school, and many people clearly believe in this technological aid. In our study, where we chose to adjust the original 3MX examination tasks for the use of the TI-92, only minor adjustments were necessary and it was shown that no significant change occurred on the exam when students were given access to a symbolic calculator. Our experience this year has indicated that significant questions still remain to be addressed regarding the use of the symbolic calculator in teaching:

1. The value of algebra and of basic analytical knowledge must be evaluated independent of the technological options.

2. Basic analytical knowledge must precede the opportunity for technical manipulations.

3. The value of the calculator as an aid will be dependent upon the student's knowledge of its capabilities.

4. When the student is proficient in the use of the calculator,
   1. motivation and technical understanding will increase,
   2. tasks will be more descriptive,
   3. parameters will play a larger part, and
   4. tasks will generally be more complex.

1. Is it useful to employ the TI-92 calculator in mathematics education but then forbid it on the examination?

2. Should there be problems on the examination that must be solved without using a calculator?

3. Should the use of a technical manual be allowed on the examination?

4. Should we emphasize numerical, graphical, and analytical concepts in mathematics education?

5. What is central to the teaching of mathematics - now and in the future?

6. To what extent should (symbolic) calculators influence the course of mathematics education?

7. Is the use of calculators fair to students who do not or cannot buy them?

8. How should teachers and others entitled to offer their opinions be informed and trained to create a constructive debate on this topic before it is decided whether the symbolic calculator should be allowed as an aid on the examination?

1. How much insight and knowledge about the TI-92 calculator are reasonable to expect for an exam in Math 3MX? Most students become highly competent in its use, but what can be expected? Are there fields currently of great interest? Fields currently of no interest?

2. How should problems be formulated so that the solution will show current technical insight and competence? Should several ways of solving problems (for instance graphic, numerical, analytic) be required, without the use of a graphing calculator? How should the problems be formulated?

3. How actively and to what extent is it reasonable to expect that the TI-92 will be used to solve problems?

4. How should students be directed to use the TI-92 in the course of solving problems?

5. What knowledge will be important and central in mathematics when the TI-92 is available as a technological aid? (numerical calculation of approximation values to square roots, algebraic calculations, techniques for calculating integrals and differentials, etc.)

6. How should we introduce mathematics for the greatest profit when using the TI-92?

7. How can we frame intellectual challenges in science and mathematics to take maximum advantage of the TI-92 graphing calculator?

1. Technical use
   • The TI-92 is practical and intelligent, with functional menus. The fact that menus are strategically placed and that the same function exists in several places makes it easier to use.
   • Ordinary use is reasonably uncomplicated, but the calculator also has functions for higher mathematics that are not for beginners.
   • Programs can be difficult to access, to find again on the calculator, and to use.

2. How do you experience the TI-92 as an aid in Math 3MX?
   • We can focus on understanding mathematics because calculations become easier.
   • Mathematical theories and formulas can be explored efficiently. By making small changes in parameter values and pushing a key, we are able to see how the changes affect the result.
   • With large manual calculations it helps to have access to the answer.

3. Has math become easier to understand when using the TI-92?
   • When using the TI-92, possibilities related to theories and formulas can be explored by making small changes in input values in order to study their effect on output values.
   • The analytical calculation process is important, but when using TI-92 we get to see the material from different angles.
   • When the traditional and the technical solutions can be presented and discussed in parallel, we obtain the maximum foundation for mathematical understanding.

4. How should the TI-92 be used in mathematics training?
   • In some fields learning to use the TI-92 can be presented as extra work when both technical and traditional methods must be learned.
   • Competence gained will depend on the regular and active use of the calculator.

5. How good are the manuals?
   • It is a great advantage that the manuals are in Norwegian and that they are available in two versions, an easy version for beginners and a fuller version for those who have gone further in their use of the calculator.

6. General comments
   • It is easy to become dependent on the TI-92.

   • When the TI-92 is available, it is hard to be motivated to work through manual and more cumbersome methods of solutions.

   • The calculator is great in mathematics. It is brilliant for mathematical calculations and accounts.

   • The calculator's effects are only positive: I experienced the opportunity to use the TI-92 in 3MX as a great inspiration for the work.

   • How will more advanced science studies (science and mathematics) turn out without the use of the TI-92?

   • There should be no interest at present in abolishing the use of the calculator.