Strand videregående skole N4120 TAU  tlf.: +47 51 74 01 00  fax.: +47 51 74 01 50 
Strand Upper Secondary School  1996/1997
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 TI92 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 TI92 in education. A small, portable calculator would offer great advantages over a stationary PC.
We knew that the CalculatorBased Laboratory (CBL) System could be used with the TI92, and that exercises in physics and chemistry for the TI82 and TI83 calculators could be performed by the TI92. We also knew that there was a great deal of help for the TI92 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 TI92 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 TI92 calculator. This favorable climate for testing the TI92 raised our expectations.
We are very grateful to Texas Instruments and Ess Data, the Norwegian importer, for their contribution of sufficient numbers of TI92 calculators for all students.
The primary goal of the study was to compare the TI92 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 TI92 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 TI92 calculators for their 1997 examination, allowing the teachers from the two schools to cooperate on the study.
The TI92 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:
To what extent would exercises have to be changed?
In physics (3FY), the TI92 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 TI92 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 TI92 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 TI83 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 TI92. When this program was installed on the TI92, the calculator revealed opportunities for calculations that satisfied mathematicians at the University standard.
Geometry and Conic sections:
The TI92 calculator uses Cabri Geometry as a regular program, but we did not have an opportunity to explore its possibilities.
Vernier Software's CalculatorBased Laboratory System (CBL) on the Web was very helpful.
Internet addresses:
All of the students in the experiment group also took the elective course in differential equations for two hours a week.
The programpack 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 springstiffness, and q is the dampingfactor. 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 TI92 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.
Advantages Students can see the need for a basic knowledge of mathematics  in part to be able to assess the reliability of answers given by the calculator. After a short introduction to basic mathematical methods, principles can be applied to relatively complicated problems. The principles of mathematical formulas can be studied through applications that make simple variations when data are entered. Work can be focused on understanding mathematical principles rather than on timeconsuming calculations. The calculator can rapidly and accurately do 'heavy' mathematical calculations. Students will discover the need for control routines and for alternative calculations. Applications of theories will be more interesting. The calculator can generate general mathematical functions in subjects like physics through the use of collections of data (by CBL) and regression analysis. The calculator has a great capacity for work because the default menu can be supplemented by program modules from the Internet with corresponding menus. Calculator activity can be altered by replacing the default menu with the program packet ADV.92G, downloadable from the Internet. Accurate solutions to relatively complex mathematical problems, e.g. differential equations of first and second order, can be obtained. The geometry programme Cabri offers many possibilities for descriptive proofs of theorems.

Disadvantages Students have little motivation for drill and repetition of analytical calculations, which the calculator can do faster and more accurately. Students can simply become dependent on the calculator, even for relatively simple calculations. Insufficient input can have dramatic consequences for the answer. Calculator settings are crucial to the accuracy of answers obtained. The calculator may give the answer in a form that is unfamiliar to the student. The use of the calculator in upper secondary school can lead to an insufficient basic knowledge of mathematics, resulting in later difficulties in higher education if students cannot continue to use calculators. Some students may not have sufficient knowledge to use a calculator effectively. 
For our evaluation of the experimental use of the TI92 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 TI92 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:
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.]
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.]
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:
It has been a pleasure to work with the TI92 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 seconddegree 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.
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.
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 TI92, 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: