Summer 2004

Talking Maths

The Calculator as ICT

Hand-held calculators for mathematics students do not supply Internet access, but do encourage constuctive communication. BARRY KISSANE gives us an overview of what makes calculators not only the most available form of technology for maths teaching at all levels, but also the most important.

DISCUSSIONS ABOUT mathematics and ICT often aim too high. In my opinion, the most important IT is the one that is most likely to be available when you need it, and for most students studying mathematics in Australia today, this is a calculator not a computer. Although students have increasingly better access to more sophisticated ICT support, curriculums in mathematics are built on assumptions of universal access to facilities, for sound social justice reasons. This is clearly the case in high-stakes examinations, and in many other educational settings, including the classrooms and homes in which school mathematics is learnt.

Calculators are distinguished from computers by the fact that they are all small and light enough to fit in a backpack or pocket, robust enough to be used outside or inside, and dedicated to doing mathematical things, so are less distracting. They are also very practicable—turn them on with a single key push and you can use them, without having to learn the latest version of the operating system, scan for viruses, look for a power point or try to remember your password. In education, properties like these are crucial.

Another distinguishing feature of school calculators is that most of them have been designed expressly for educational purposes. They are perhaps the only technology that can legitimately claim this. To see the difference, consider an example: use your calculator to calculate 2 + 3 x 4. If your calculator isn’t nearby, try your mobile phone. Even today, many office and home calculators (and all phones!) will give the (incorrect) answer of 20. Why is this incorrect? In conventional mathematics, you need to perform the multiplication before the addition to get a result of 14. It’s a universal mathematical convention that everyone needs to learn. It’s easier for students to learn things if their calculator knows how to do them, and all calculators designed for education will do this task correctly.

Of course we don’t want students using their calculators to compute 2 + 3 x 4. They should develop skills to do this mentally. But there is an issue here regarding the common misconception that mathematics and computation are the same thing. This confusion and the unfortunate term, ‘calculator’, can give the impression that the main role of a calculator in school mathematics is to calculate things. This is simply incorrect. The main purpose is to help with mathematics learning, of which calculation is only a small part. It is important that it is used well; to make sure that it fulfils these expectations. Disallowing its use will ensure that this does not happen.

Although calculation is not the main role of a calculator, it nonetheless is one of the roles. Teachers should help students choose when to calculate mentally, when to use a calculator and when to use paper and pencil. Part of this learning involves deciding when numerical results should be obtained precisely and when a good approximation is adequate. Using a calculator for calculation requires skill in estimating answers efficiently, so that judgements about their reasonableness can be made. This sort of expertise must be consciously developed by the teacher. If students are denied access to calculators, there is little prospect of this sort of competence being developed in a systematic way.

Students of different ages learn different things in mathematics, so it’s not surprising that calculators are designed accordingly. For example, primary school children with a calculator will have a first-rate opportunity to see how place value works. I have heard very young children counting, ‘… 28, 29, 20–10, 20–11, 20–12 …’ cleverly picking up part of the pattern of the number system. A calculator will never do this, but will help them to see that the pattern is more subtle, ‘… 28, 29, 30, 31 …’ Older children counting 0.1 will be heard to say, ‘0.8, 0.9, 0.10, 0.11 …’, but their calculator will help them to see the decimal place value structure of ‘… 0.8, 0.9, 1, 1.1 …’ The calculator allows students to follow their own lines of enquiry.

The calculator has the properties of the real number system embedded within it, and it is clear from research that young children can learn a lot of mathematics in an environment that allows them to take advantage of this. We sometimes still hear the maxim that children should not be permitted to use a calculator until they have learnt their tables. Instead, calculators ought to be regarded as devices that can help children learn their tables, in an analogous way to dictionaries helping them to learn to spell.

Upper primary school students need access to calculators that handle fractions as well as decimals. Teachers report that children using these calculators have new ways of understanding the links between decimals and fractions, supplementing other approaches such as those using concrete

materials. For example, a ‘fractions to decimals’ key on the TI-15 Explorer calculator helps children to see that these are alternative ways of representing numbers. For generations, many have thought of fractions as pairs of numbers—a numerator and a denominator— rather than a single number. With access to a key that links the two representations, children can learn that there is always a choice of representation and how to make sense of and intelligently use their choice in any given case. Secondary school students encounter a wider range and depth of mathematical ideas, so calculators have been developed to suit their learning needs. The best recent examples are called graphics calculators, probably to reflect their small graphics display screens. Students use them to explore many aspects of mathematics, including functions, equations, probability, matrices, derivatives and limits. Most modern Australian curriculums assume classroom access to this technology for mathematics, and (most) external examination authorities expect and permit their use in high-stakes examinations, to maintain coherence between teaching, learning and assessment and to acknowledge the importance of technology to mathematical activity at all levels these days.

Figure 1: TI-15 fraction calculator screens

Figure 2: Graphics calculator screen showing statistical data

Over the past two decades, most teachers whose students have access to this technology have come to understand that mathematics learning is enhanced by it. The best succinct summary of modern thinking on the use of a graphic calculator is the Australian Association of Mathematics Teachers’ Communiqué, arising from a recent national conference. While most Australian mathematics curricula now permit, even encourage, their use, the few that do not seem likely to change in the near future.

Figure 3: Graphics calculator screen showing CAS

Recent versions of graphics calculators have included computer algebra systems (CAS), extending their capabilities to include symbolic manipulation in algebra and calculus. As for computation and arithmetic, for too long we have confused symbolic manipulation with algebra and algebraic manipulation with calculus. Important concepts in both algebra and calculus must be learnt by students, and an environment in which some of the symbolic manipulation is routinely accessible by the calculator enhances the prospects for these ideas to be learnt. The trunk of senior school mathematics has involved algebra and calculus for generations, because of the foundation these provide for further study in quantitative fields such as science and economics. So it seems likely that we will see continued developments of the integration of technology with mathematical thinking in schools, via the use of CAS-capable graphics calculators.

So, what about the ‘C’ in ICT for mathematics, then? Educational calculators of various kinds offer a great deal of IT, but don’t offer the benefits of Internet access, responsible for adding the ‘C’ to IT. To find communication in mathematics, put a couple of students together with a calculator relevant to their level of sophistication in maths and some appropriate mathematical tasks to engage with, and you will quickly find more productive communication going on than is often the case for a student browsing the Internet.

References

AAMT (1996). Statement on the use of calculators and computers for mathematics in Australian schools, available at www.aamt.edu.au/about/policy/tech_st.pdf

AAMT (2000). Graphics calculators and school mathematics: A communiqué to the education community, available at www.aamt.edu.au/about/policy/gc-com-2.pdf

Kissane, B (1997). Growing up with a calculator, Australian Primary Mathematics Classroom, 2(4), 10-14. (wwwstaff.murdoch.edu.au/~kissane/papers/apmc.pdf)

Barry Kissane is a mathematics teacher educator in the School of Education, Murdoch University and is currently the president of the Australian Association of Mathematics Teachers.

The author owns the copyright in this article. For information related to the reuse of this work in any form please contact the publisher denise.quinn@curriculum.edu.au