Indispensable Manual Calculation Skills in a CAS Environment.
Wilfried Herget, Helmut Heugl, Bernhard Kutzler, Eberhard Lehmann · The Knowledge Bank (The Ohio State University) · 2000
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. (Author) Reproductions supplied by EDRS are the best that can be made from the original document. Indispensable Manual Calculation Skills in a CAS Environment Wilfried Herget (Halle, Germany) Helmut Heugl (Wien, Austria) Bernhard Kutzler (Leonding, Austria) Eberhard Lehmann (Berlin, Germany) 1 PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS BEEN GRANTED BY TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) This document has been reproduced as eived from the person or organization originating it. Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this document do not necessarily represent official OERI position or policy. 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. 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. F2Y Algebra F3Y Cal c Other F5 PrgmI0 FS, Clean Up f(x3sin(x))ax (6.. x3)cos(x)+ 3-(x2-2)sin(x) MAIN RAD AUTO FUNC 1/30 16E1 F2, Algebra F3Y Ca lc pi+ Other FS Prgrd 0 Ffi Clear a-z... solve(x2 + x 1 = 0, x) + i) x 2 or x2 + -1=0 x) MAIN RAD AUTO FUNC 1/30 The CAS which are widely used in schools are the computer program Derive and the algebraic calculators TI92 and TI-89. Introductions to using these tools are [Kutzler K the practical application of science to industry and other fields; the total technical means and skills available to a particular human society; the terminology of an art or science. However, the use of the word became ingrained in the academic literature about the use of calculators and computers for teaching. Therefore and because we want to evade confusion we use it here as well. 2 BEST COPY AVAILABLE S.ity. 1 unn R / R 711(1/1 HergeVIleugYKutzler/Lehmann: Indispensable Manual Calculation Skills in a CAS Environment 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: These questions appear sensible only in the context of appropriate problems, but not as isolated questions. Their best use could be to test how well a student can operate a calculator. The questions we put into -T describe long-term manual skills. In order to reach this goal it certainly would make sense to let the students practice with more demanding examples at some stage. To some extent it could make sense to let the students practice some of the examples from +T even without technology.