|Strand videregående skole
N-4120 TAU - tlf.: +47 51 74 01 00 - fax.: +47 51 74 01 50
Strand Upper Secondary School - 1996/1997
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.
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:
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.
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.
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.
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:
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.
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 time-consuming 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.
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.
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:
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.]
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.
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.
There seems to be evidence for the following claims:
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.
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.
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.
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.
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:
Basic analytical knowledge must precede the opportunity for technical manipulations.
The value of the calculator as an aid will be dependent upon the student's knowledge of its capabilities.
When the student is proficient in the use of the calculator,
Should there be problems on the examination that must be solved without using a calculator?
Should the use of a technical manual be allowed on the examination?
Should we emphasize numerical, graphical, and analytical concepts in mathematics education?
What is central to the teaching of mathematics - now and in the future?
To what extent should (symbolic) calculators influence the course of mathematics education?
Is the use of calculators fair to students who do not or cannot buy them?
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?
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?
How actively and to what extent is it reasonable to expect that the TI-92 will be used to solve problems?
How should students be directed to use the TI-92 in the course of solving problems?
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.)
How should we introduce mathematics for the greatest profit when using the TI-92?
How can we frame intellectual challenges in science and mathematics to take maximum advantage of the TI-92 graphing calculator?
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.