Indispensable Manual Calculation Skills in a CAS Environment

Indispensable manual calculation skills in a CAS environment

Wilfried Herget (Halle,

(

Halle, Germany)

Helmut Heugl (Wien,

(

Wien, Austria)

Bernhard Kutzler (Leonding,

(

Leonding, Austria)

Eberhard Lehmann (Berlin, Germany)

Abstract

Which manual calculation skills are still needed when students use graphic/symbolic calculators or computers with computer algebra systems (CAS)? What should students be able to do manually, i.e. just using paper and pencil? This text is the outcome of a two-day discussion on these questions, held by the four authors. Our answers and proposals are meant to be a challenge, aiming at sparking off a broad discussion about which permanently available manual calculation skills we still need to teach and assess.

Computer algebra systems (CAS)

Computer algebra systems (CAS) are tools which automate the execution of algebraic computations. CAS can simplify expressions, compute symbolic derivatives and integrals, plot graphs, solve equations and systems of equations, manipulate matrices, etc. In short: they automate most of the calculation skills we teach in school mathematics.

The CAS which are widely used in schools are the computer program Derive and the algebraic calculators TI-92 and TI-89. Introductions to using these tools are [Kutzler&Kokol-Voljc 2000] for Derive 5, [Kutzler 1997] for the TI-92 and [Kutzler 1998] for the TI-89.

Soon such tools will be used as a matter of course, such as we use scientific (in some countries also graphic) calculators today. Using a calculator for differentiating x³ sin²(4 x⁵) will be as common as using it to evaluate cos(1.3786) or 5.67³. The above screen images give examples of what CAS can do.

Starting point: calculator-free exam

We assume an exam comprising two parts. In the first part no modern technical tool is permitted – not even a simple scientific calculator – whereas in the second part all kinds of technology, in particular powerful calculators or computers with CAS may be used. Some countries, such as Austria, are experimenting with two-tier exams. Other countries, such as England, use two-tier exams already. We believe that two-tier exams would be a well-balanced compromise meeting both the desires of technology supporters and the reservations of those who are concerned about the use of technology in the classroom. Some fundamental thoughts about two-tier exams are contained in [Kutzler 1999].

We assume a fictitious, written, technology-free exam. We look for questions and classes of questions which we would include in such an exam.

Drawing the border line between questions to be asked in a technology-free exam and questions which would not be asked in such an exam is equivalent to listing the indispensable manual calculation skills. Therefore, the fictitious technology-free exam is a means to an end for us. Our discussion and its results are relevant far beyond the exam situation. They are fundamental for the development of mathematics education in the years to come.

After reconsidering the meaning and importance of calculation skills and restraining their role in teaching and learning, it is crucial to discuss the consequences for mathematics teaching. This will become the topic for our future discussions and work.

Three pots: -T, ?T, +T

The border line between questions to be asked in a technology-free exam and questions not to be asked in such an exam clearly depends on many parameters – including the type of school. We try to give a universally applicable answer by creating three pots, which we name –T, ?T, and +T.

• The first pot, –T (= no technology), contains those questions which we would ask in a technology-free exam. Hence these are the questions which we expect students can answer without the help of any calculator or computer.
• The calculation skills needed to answer the questions from pot –T should be mandatory from school year 8, or starting from the school year in which they are taught. The students are supposed to maintain these calculation skills throughout the remaining school years (and, hopefully, beyond school) hence teachers may assess them at any time.
• The third pot, +T (= with technology), contains questions which we would not ask in such an exam. Hence in situations in which such problems would occur, we would allow students to use powerful calculators or computers with CAS for their solution.
• The second pot, ?T, reflects our doubts, our different views, and partly also the inherent difficulties of this topic. We either were divided over the questions which ended up in this pot, or we agreed that we would not or could not put them into one of the other two pots. This pot shows how fuzzy the border line (still) is – at least for us.

Whenever feasible we outlined the spectrum and the border line of a class of questions by providing comparable examples for both –T and +T.

Higher demands during teaching and exercises

The questions we put into –T are those which we would not ask in a technology-free exam – but we would not ask them in a technology-supported exam either: