Barry Kissane

Although advanced scientific calculators of recent years include many features intended for educational use, they are still frequently misunderstood as mostly devices for numerical computation. Following a model for the educational use of calculators, this article describes and illustrates several ways in which the use of a calculator facility to construct tables of values can have educational value in the secondary school. Examples include the study of linear and quadratic functions, algebraic equivalence, equations, sequences, series, limits, convergence, differentiation and integration. The article concludes by observing that the educational value of calculators derives from the experiences they offer students, not merely from their capacity to generate numerical answers.

Graphics calculators have been available to students in secondary school in some countries now for more than thirty years, although of course their capabilities have been developed in various ways to support the school curriculum over that time. The most frequent use of these devices seems to be concerned with the representation of functions, including in particular their graphical representation, which was an important component of a previous paper in this magazine (Kissane, 2016). However, the success of graphics calculators is due in no small part to their use for a much wider range of mathematical capabilities. In this article, the focus is on their potential to help students to learn about chance phenomena, which are generally addressed in schools through the study of probability. The history of probability in secondary schools is relatively short and generally unfortunate. Unlike many other parts of the secondary school curriculum, such as algebra, geometry, trigonometry and calculus, probability has been studied in schools only recently, and was relatively rare in most countries as little as fifty years ago. One part of the reason for this is likely to be that probability is a relatively recent inclusion in mathematics itself, dating from around the sixteenth century (Hacking, 1975). Until quite recently, much of the probability work in schools has been excessively formal, with a focus on the algebra of probabilities, but with less attention paid to the nature of everyday random phenomena. Yet in recent times, probabilities have become more evident and explicit in our daily world, a good example of which is weather forecasting, now regularly accessed by many people on their smartphones.

While technology is a prominent feature of modern mathematics education, frequently the technologies of interest require substantial investment of resources. Less sophisticated technologies - in particular, calculators - are frequently overlooked and often not even regarded as part of a school’s ICT activities, even though they are widely available to students and accepted for use in examinations. (Kissane, 2016). In previous Conference Editions of Reflections, I suggested some ways in which a calculator might support students learning about some aspects of Number (Kissane, 2017b) and Measurement (Kissane 2018), drawing on the model proposed by Kissane and Kemp (2014) for using calculators in education developed. In addition to computation, other aspects of calculator use proposed include representation of mathematical objects and concepts, exploration of mathematics and affirmation of student thinking. As well as in the original paper, these four aspects are described and illustrated elsewhere (including Kissane (2017a), so will not be elaborated further here. In this paper, a further example of these ideas is offered and briefly discussed, concerned with Statistics. A number of aspects of statistics are addressed, including learning about descriptive statistics, understanding the nature of various statistical measures and the Normal probability distribution and using bivariate statistics to analyse real data. While a scientific calculator of course eases the computational burden on students, and is virtually indispensable if real data are to be encountered by students, its major contribution might be to provide an environment in which important statistical concepts can be readily and productively explored.

Calculus is fundamentally concerned with understanding and measuring change, which is why it has proved to be such a useful tool for more than three hundred years and has frequently been studied at the end of secondary school. The concept of a derivative is critical to the study of calculus, and is concerned with how functions are changing. In this article, we will outline how a modern graphics calculator can be used to explore this idea. We will use a particular graphics calculator, the CASIO fx-CG 20, which does not have computer algebra capabilities. Too frequently, students focus on the symbolic manipulation aspects of calculus, which are appropriate for a later and more general treatment of ideas, but not as helpful for an introduction. Of course, similar explorations can be undertaken with other graphics calculators and with various kinds of computer software.