Indispensable Manual Calculation Skills in a CAS Environment

Herget/Heugl/Kutzler/Lehmann: Indispensable Manual Calculation Skills in a CAS Environment

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.

Other Important Skills and Abilities

It goes without saying that other important skills and abilities exist in addition to calculation skills. In a CAS teaching

and learning environment many of those skills and abilities will keep their importance. Several will become more

important. In any case, they are indispensable also (for details see [Heugl 1999]). Examples of such abilities are:-

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Herget/Heugl/Kutzler/Lehmann: Indispensable Manual Calculation Skills in a CAS Environment

• finding expressions

• recognizing structures

• testing

• visualizing

• using technology properly

• documenting calculations or problem solutions properly. 2

x

The ability to visualize allows a person to make a “proper sweep of the hand” to sketch a graph of, for example, or

sin( x ) .

Among all the skills and abilities teachers are supposed to teach in math classes, calculation skills have played and will

play an important role. We teach them not only for their own sake (if we did, their relevance would be severely

challenged by the availability of powerful calculators and computers), but to some extent because they are

prerequisites for the attainment of “higher” abilities such as the above mentioned. Therefore the above mentioned and

other abilities play a decisive role when judging the importance of calculation skills, hence they were part of our

discussions. This is partly documented by some of the annotations we give.

Mathematics Education Will Not Become Simpler!

We do not believe that mathematics education will become simpler – the contrary is true. The suggested lower level of

manual skills reflects our believe that CAS will become standard tools for mathematics teaching and learning. It also

reflects what we believe is our realistic approach as to what we want students to know throughout their school career

and beyond. A consequence of the new tools is that mathematics becomes more useable and probably more demanding

– but definitely not simpler. After the very unfortunate discussion about “7 years of teaching mathematics is enough”

in the German and Austrian press some years ago we definitely do not want to create a similar debate about “trivial

symbol manipulation is enough.” Most important for us is the distinction between the goals “perform an operation” (to

some extent this can be delegated to a calculator) and “choose a strategy” (this cannot be done by the calculator.)

It goes without saying that the following exposition has an impact on many aspects of teaching mathematics: the

teaching methods, training methods, homework, curricula, the topics we teach, what teachers need to know, etc. We

broached these issues but did not elaborate them. Therefore we do not mention them here.

Our Goal: Permanently Available Minimal Calculation Skills

We want to spark off a long overdue discussion about the mathematical, methodological, and administrative

consequences of using CAS and other mathematics software for teaching and learning mathematics.

This text is meant to be challenging, maybe even provocative. Let us face the challenges of the new tools and let us

take the necessary steps! In particular this demands the willingness to say goodbye to familiar things if we see the

necessity for it.

Questions and Classes of Questions

For this article we restrict ourselves to questions for which one could use powerful calculators or computers with CAS.

Arithmetic – long term minimal competence

?T +T (with technology)

–T (no technology)

⋅ ⋅

01 compute 3 40 compute 3.2987 4.1298

02 compute 81 approximate 80 to ... digits

03 estimate 80 simplify 80

04 ⋅ 3

calculate 11 11

05 factor 15 factor 30

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Herget/Heugl/Kutzler/Lehmann: Indispensable Manual Calculation Skills in a CAS Environment

The example 80 (and its variants –T03, +T02, and +T03) demonstrates how important and decisive the formulation

of a question is for putting it into a certain pot. The less important the manual calculation skill becomes, the more

important becomes the appropriate formulation in order to clarify the objective of the question. This becomes even

clearer with some of the questions in the next sections. We agreed that the importance of the teaching goal

“estimation” goes far beyond the given example (–T03). It is so important that we need to reach it without technology

– although it may be useful to use a calculator as a pedagogical tool, for example when testing the quality of an

estimation, when computing the error, or when demonstrating the purpose of estimations.

To avoid a misunderstanding we repeat what we said earlier: The questions from pot +T are questions we would not

ask in a technology-free exam. We would not ask these questions in a technology-supported exam either, because these

questions appear useless as such, their best use might be to test how well a student can operate a calculator.

The questions from +T just require the skill of evaluating an expressions which typically comes from a more

complicated problem. In the long term this should be delegated to a calculator. We need to make sure that the students

understand what these expressions mean. But for testing such an understanding, we need different types of questions.

Nevertheless – this is another reminder – it certainly could make sense to use questions from +T in both technology-

free and technology-supported “training units.” This could be needed in order to make the questions which we put in

Basically our proposals obey the following rule: elementary calculations (such as the factoring of an integer with only

two factors, e.g. 15) are an indispensable skill (therefore these questions belong to –T ), whereas calculations requiring

a repeated application of elementary calculations (such as the factoring of an integer with three or more factors, e.g.

30) may be delegated to a calculator.

Fractions– long term minimal competence

?T +T (with technology)

T (no technology)

01 2 2 4

10 ⋅

simplify 7 :

simplify 2 5 6

5

02 2 3 2

10 100 x y

simplify simplify

5 5

10 10 xy

03 1

simplify 2 : 2

04 2

simplify 1

2

05 5 a

simplify 5

06 a ⋅

simplify 5

5

07 2

2 x a b

⋅ ⋅

simplify simplify

x y b 3 ac

08 2 x

2

simplify 3 x : 3

5 y

09 a a a

− − +

simplify 2 a simplify 2 a

3 3 7

10 a a

+

simplify 3 7

11 5 2

−

simplify x x Page 4 of 8 / June 1, 2000

Herget/Heugl/Kutzler/Lehmann: Indispensable Manual Calculation Skills in a CAS Environment

12 2 5 2 x

−

− simplify

simplify x y x 5

100 =

–T01: Here we want students to see the obvious calculation 4 . This is not trivial!

25

–T02: Expressions like this are needed in physics. 1

–T03: A corresponding alternative question (of higher value) would be: “Why is 2 : equal to 4?” This involves the

2

ability to recognise structures. +

a c ad bc

+ =

We deliberately would not test if the rule was learned by heart. We consider this a back ground goal

b d bd

– by which we mean a goal which does not need to be tested explicitly in a written exam. Learning such a rule by heart

only leads to students who stolidly apply it for adding two fractions – instead of using the most often more appropriate

a c ac

⋅ =

approach of computing the least common multiple of the denominators. Equally, is only a background