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Technology and Teaching Maths
RAY PECK explores the technological changes that have taken place in mathematics education and answers the question ‘Do I have to be good with technology to be a good maths teacher?’.
PERHAPS THE SINGLE biggest change in the teaching of mathematics in recent years has been the growth of the learning technologies applicable for the study of mathematics.
However, while some teachers have embraced technology, others have been less enthusiastic. Does this make the former a ‘better’ teacher of mathematics than the latter?
It’s an old story, but for me as a student, my maths technologies were 4-figure tables and a slide rule. In 1969, the Stats department at Melbourne University had a calculating machine. To multiply 123 by 37, you entered 123 and turned the handle 37 times!
The first real technology challenge for me as a beginning secondary maths teacher in the early 70s was mastering the new scientific calculator. Reverse-polish was not a sex position but rather an order of operation used by some scientific calculators! With sine, cosine and tangent buttons the 4-figure tables became redundant. In addition to learning how to use the new calculators myself, it also became necessary to understand how my students were using them. A good teacher of mathematics must demonstrate ‘calculator faux pas’. Modelling the capacity of an ordinary calculator to ‘get it wrong’ they highlight these variations and use them to deepen mathematical understanding.
There was now more need for me to re-emphasise the importance of estimation and checking calculator results for reasonableness.
Research shows that a calculator is a valuable problem-solving tool and that, if used effectively, can create wonderful opportunities for students to explore aspects of number and tackle ‘real-world’ problems where often the numbers are large or ‘messy’. A frequent student response to the question asked in the Victorian Middle Years Numeracy project, ‘What makes maths easier or better for you?’ was ‘Use of calculators, materials diagrams, pictures and/or games’ (Siemon, 2001, p 48).
In the early 80s, computer technology began to have a major impact on mathematics classrooms. Seymour Papert’s book, Mindstorms, was bedside reading and there were BBC machines running LOGO and driving ‘turtles’ everywhere. Many maths educators found themselves learning to program in LOGO and there were opportunities to explore relationships in geometry on a computer, create beautiful symmetrical patterns and save the work in code. The computer language BASIC enabled some maths educators to introduce their students to simulation. Many MCTP (and now Maths 300) activities are accompanied by software originally written in BASIC. My students and I had fun simulating the workings of a blood bank and the game of cricket.
With the advent of cheap PCs onto the market, there was an explosion of available technology, from the humble spreadsheet (wasn’t that just for accountants?) through to geometrical and graphing software that suddenly needed evaluation and mastery. Could I really produce a perfect parabola, label it and cut and paste it into a worksheet or examination? You mean I can now animate a point on a circle and show that the angle subtended at the centre is always double that on the circle? But how?
In the early 90s, hand-held graphic calculators, with the capacity to graph functions, perform sophisticated statistical analyses, manipulate matrices, run and store programs, and link to PCs could be used in year 12 examinations. Real data (temperature, radiation, etc) from experiments could be captured from data-loggers, enabling opportunities for mathematical modelling and analysis. Social data from the Internet could be downloaded. Students could pass programs (and games!) from one calculator to another or write programs for their hand-held calculators. To create a program to perform mathematical routines, a deep and generalised understanding of the underlying mathematical principles is needed. What’s more, I could now project my calculator screen and demonstrate mathematical concepts and procedures in an interactive way. Surely this was the ultimate.
Well actually, no. The latest technology, set to revolutionise mathematics classrooms in Australia and the world, is Computer Algebraic Systems (CAS). CAS machines enable the user to perform algebra and calculus. Exact solutions to equations (such as √2) can be obtained. While available for some time, hand-held CAS is only now finding its way into Australian year 12 examinations. In Victoria, for example, a CAS trial is in its fourth year and in 2006, all year 12 mathematics students will be able to use them in exams. Interestingly two of the three Victorian year 12 maths subjects are also likely to have one exam that is ‘technology free’. Some university mathematics courses in Victoria still prohibit the use of even basic calculators in exams!
One reported benefit of CAS use in algebra and calculus lessons is the increased opportunity for the ‘Explore, Discover, Conjecture, Confirm’ style of teaching. The teachers involved in the CAS trial in Victoria reported
The unrestricted use of CAS has led to dramatic changes in pedagogy and assessment in each of the three schools. Traditionally, the focus of senior mathematics courses has been on how to carry out mathematical procedures, such as solving equations and differentiating expressions. CAS has made it possible to increase the emphasis on the understanding of concepts and on helping students to decide when and why it might be appropriate to apply a particular procedure. Unrestricted access to CAS has challenged us, as educators, to start inventing new paradigms for the teaching and learning of senior mathematics.
(Garner et al, 2003)
Back to the original question: Do I need to be good with technology to be a good maths teacher? While research into the effective teaching of mathematics does not specifically highlight teacher expertise with technology, it’s hard to imagine an effective teacher of mathematics in this day and age who is not good with technology. The Australian Association of Mathematics Teachers (AAMT) Standards for Excellence in the Teaching of Mathematics in Australian Schools (visit www.aamt.edu.au/standards/) make reference to ‘technologies’ in three of the ten Standards.
1.3 They are aware of a range of effective strategies and techniques for: promoting enjoyment of learning and positive attitudes to mathematics; utilising information and communication technologies.
2.2 The professional development they undertake enables them to develop informed views about relevant current trends (including teaching and learning resources, technologies).
3.2 A variety of appropriate teaching strategies are incorporated in the intended learning experiences, enhanced by available technologies and other resources.
In the Standards, the emphasis is on knowledge of mathematics, students and learning, and professional attributes and practice. Knowledge of current technology is assumed. It’s how the technology is used in the classroom to enhance the learning and enjoyment of mathematics that is seen to be important.
Keeping up to date with developments in technology is sometimes hard work but mostly fun and rewarding. All teachers need to be learners. To know one’s subject well and master new technology and work out how to use it to enhance learning is incredibly demanding. In a more industrialised workplace it would be a clear case for a productivity claim!
AAMT (2002). Standards for Excellence in Teaching Mathematics in Australian Schools, Australian Association of Mathematics Teachers, Adelaide.
Garner, S, McNamara, A & Moya, F (2003). ‘CAS: The safe approach’, in Clarke, B et al (eds) Making Mathematicians, Mathematical Association of Victoria, Melbourne.
Siemon, D (2001). Middle Years Numeracy Research Project: 5-–9, RMIT University.
Steen, L (1999). ‘Twenty questions about mathematical reasoning’, in Stiff, L (ed) Developing Mathematical Reasoning in Grades K–12, National Council of Teachers of Mathematics, Reston, VA.
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