SELF-EXPLANATORY SYMBOLIC COMPUTATION FOR MATH EDUCATION

F. Lichtenberger Inst i tu t fur Mathematik

Johannes Kepler Universitat A-4040 LINZ/AUSTRIA

In this position paper we wil l t ry to state some new ideas concerning the design of software systems for symbolic computation which are to be used in math educa- t ion. Up to the late seventies computer algebra systems were running only on big mainframe computers. The muMATH-80 system was the f i r s t to be used on microcom- puters running under the CP/M operating system. Since the f i r s t hand-held com- puters that can run CP/M are on the market, i t seems obvious that in a few years we wil l have computer algebra systems on cheap hand-held or even pocket com- puters. The use of such computer algebra systems wi l l probably have more influence on high school math than the appearence of electronic pocket calcula- tors did have. I f we now take a closer look at muMATH, one can see that most topics of mathematics that are relevant in high school, can de treated by the system. Some parts that had not been covered in the muMATH-80 version are covered by the muMATH-83 version, l ike for example vector algebra and simple d i f ferent ia l equations. This shows the direction where things probably wi l l go. Sooner or later all topics of high school math or even undergraduate math wi l l be covered, at least in principle, by systems running on microcomputers. Further developments in the hardware f ie ld wil l accelerate this process, e.g. i f the standard size for RAM storage in micros and hand-held computers increases from now 64 KB to, say, 256 KB.

These perspectives could lead to the opinion, that we only have to wait for one of the next versions of, say, muMATH, having then the ideal computer algebra system to be used in education. In fact, such a system would be a very helpful tool in conventional math education. But we should go a step further and try to build a system that could serve as a real didactical device. This means that students should not only be able to use such a system for doing the "mechanical" parts of those topics in mathematics that they just have comprehended, but the system should help them to understand new topics as well. We wi l l give now some suggestions how the design of such a system should look l ike.

AS a f i r s t point, as much as possible of the differences between the way existing computer algebra systems are working and the way students learn to do their computations should be eliminated, i.e. the computer should do all com- putations in (nearly) the same way as the student is used to do them. We, there- fore, take a rather pragmatic position: Look through all the math books used in high schools and write a system that covers all the topics found in these books. The system should ask for user interactions in cases solutions can not be found mechanically and perform all those computations that can be mechanized. Interactive and fu l l y mechanized computations should be done in a way that is as close as possible to the way the student is used to work when using pencil and paper. In the ideal case, the output produced by the system should be in such a form that i t could be cut out and pasted into one's copy-book. Consequently, this means that in a computer algebra system for education the standard problems

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sometimes must be solved by other algorithms than they are solved in existing systems. As an extreme example, for the computation of the GCD not only the Euclidean algorithm should be available, i f pupils have learned to calculate the GCD by examining the factorization of the two numbers.

As a second point, in addition to having as much work as possible done "automatical ly" by the computer, the student should also have the choice to control al l computations step by step. For example, in symbolic integration he should also be able to say something l ike "Try the subst i tut ion method with the fol lowing subst i tut ion . . . " and to find out then i f the proposed subst i tut ion has s impl i f ied or solved the problem. In a more or less d idact ica l ly feasible way, this feature is already in some of the exist ing systems, e.g. using the data type "equation", subst i tut ion commands, etc.

A th i rd important feature of such a system should be the poss ib i l i t y of getting a l l detai ls of a computation also a poster ior i . That means, that in the example of symbolic integrat ion one f i r s t asks the system for the solution of the problem and then would l ike to ask the system which method i t had used for calculat ing the results. The system should then answer in a way l ike "I have used the subst i tut ion method with the following subst i tut ion . . . " , thus "explaining" the process of f inding the solution.

I f we have a closer look at th is subsystem, we see that th is "explainer" is a crucial point in making the system a good didactical device. I t should be able to break the protocol of the computation of a solution into pieces of arb i t rary size, the extreme case being to give no intermediate results at al l on the one hand, or to give a total trace of al l of the executed routines on the other hand. Choosing the r ight size of these intermediate steps, which strongly depends on psychological facts and personal taste, surely is a d i f f i c u l t task and wi l l probably require some empirical studies. The system should provide the poss ib i l i t y of asking for further explanation i f one did not understand a par- t i c u l a r step of the computation, i .e. the system should divide this step into smaller pieces.

We also would l ike to remark that the problem of adapting the algorithms used by the system in such a way that they can easily be understood by the students appears in automatic theorem proving as well as in computer algebra. Many proofs or at least parts of the proofs that students have to perform can be done automatically. A system for automatical theorem proving should be able to "explain" those parts of the proofs by sp l i t t i ng them into comprehensible parts, as we claimed i t for computer algebra systems. Since nobody uses the resolution method when proving theorems by hand at school, the system has to use another algorithm, for example some system of "natural deduction".

F inal ly we want to remark that our suggestions probably w i l l make a system for symbolic computation less e f f i c ien t and powerful than exist ing or future systems not bu i l t for the special purpose of education. We believe this does not matter, because those systems only must be able to solve small problems, which in the extreme case, could be solved by a student by hand as well.

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