The calculator and the curriculum: The case of sequences and series*

Graphics calculators have the potential to influence the curriculum in several ways, including affecting what is taught, how it is taught and learned and how it is assessed. These relationships are exemplified for the particular case of sequences and series, which frequently appear in mathematics curricula near the end of secondary school and in the early undergraduate years. Attention will focus on the ways in which these mathematical objects can be represented, viewed, manipulated and understood by students using graphics calculators. Key concepts associated with sequences and series are examined from a calculator perspective. The paper provides an analysis of the mathematics curriculum through the lens of an available technology, with a view to providing suggestions and recommendations for both curriculum development and classroom practice. The major significance of the personal technology of the graphics calculator is that it has the potential to be integrated into the mathematics curriculum, rather than be regarded as an ‘extra’ or as a ‘teaching aid’. This paper provides an analysis of the relationships between the graphics calculator and one part of the mathematics curriculum, concerned with sequences and series, with a view to understanding the significance of the technology. The paper might thus be regarded as a companion to previous papers offering similar analyses, such as Kissane (1997) for probability, Kissane (1998a) for inferential statistics, Kissane (1998b) for calculus and Kissane (2002a) for equations. To focus the analysis, it is convenient to use the structure suggested by Kissane (2002b), reflecting three different roles for technology in the curriculum. A calculator has a computational role, handling some aspects of mathematical computation previously handled in other ways. Secondly, a calculator has an experiential role, providing fresh opportunities for students to experience mathematics, and thus fresh opportunities for teachers to structure the learning programme. Finally, a calculator has an influential role, since the mathematics curriculum ought to be constructed with the available technology in mind; a curriculum constructed on the assumption that graphics calculators are routinely available might be expected to differ from a regular curriculum devoid of access to technology. Throughout the paper, we use the Casio cfx-9850GB PLUS graphics calculator to illustrate the main connections between the mathematics and the technology. This calculator is widely used in senior secondary schools and the early undergraduate year, and does not have CAS capabilities. The choice of a non-CAS calculator is deliberate: at the present time, these are more accepted by curriculum authorities and also they provide substantial pedagogical support for students and teachers. Further, an analysis of the relationships between an algebraic calculator (ie with CAS) and the curriculum can easily be constructed using the present work as a basis. Computational role Sequences are important mathematical objects, perhaps best defined as functions with domain the set of natural numbers or a subset of these. Although sequences are generally infinite structures (as the domain is infinite), in practice we are frequently interested in a finite subset. Graphics calculators are of course finite machines and thus capable only of dealing directly with finite sequences. Indeed, in the case of school mathematics, most applications of sequences and series are concerned with finite examples, which have the most plausible practical significance for students. Generating a sequence There are two essential ways in which sequences are defined, recursively and explicitly. A recursive definition specifies the relationship between successive terms of the sequence, as well as defining the starting point. An explicit definition provides a direct way of determining each term of the sequence. A sequence can be generated on a calculator in either of these ways. Consider the elementary example of the arithmetic sequence, 7, 11, 15, 19, ... . Successive terms of this sequence can be generated on a calculator by using the fundamental property that each term is four greater than the previous term, starting with a first term of 7. A graphics calculator allows this process to be automated, as shown in the screen below, in which successive terms after the second are generated by repeating the recursive command, Ans + 4, which involves only a single key press. Although this can be an efficient way of finding a particular term, it may be quite tedious (and thus error-prone) for finding terms that are not close to the first term. An explicit formula for the same sequence is given by T(n) = 7 + 4(n – 1). On a calculator, such a formula can be entered as a function and tabulated to produce successive terms efficiently, as shown below. In this case, the commands above generate the first 50 terms of the sequence, substituting the calculator function Y1 = 7 + 4(X – 1) for the sequence function T(n) = 7 + 4(n – 1). Definition as a list In order to perform computations with sequences, it is necessary to first store them in the calculator in some way, which the procedures above do not accomplish directly. The essential means of storing a sequence on a calculator is with an ordered list. In the case of the Casio cfx-9850GB PLUS, lists are restricted to 255 elements, which is more than ample for almost all secondary school purposes in practice. The screens below show some examples of how the sequence of the number of days in each month of the Gregorian calendar (in a non-leap year) can be represented on the calculator. The sequence has been entered term by term and then stored into a particular finite list (List 1). The middle screen shows the (scrollable) list, while the third screen shows the same list in a different calculator mode. A sequence of this kind has little mathematical form; indeed, it is principally a consequence of the vanity of some ancient Roman emperors. The only computational tasks likely to be of interest for an arbitrary sequence of this kind are the determination of a particular term and the sum of successive terms. In this case, the former retrieves the number of days in a particular month and the latter provides the cumulative number of days for the end of each month of the Gregorian year. The screens below show three different ways of accessing the ninth term of the stored sequence, verifying that September has 30 days, reflecting a characteristic of graphics calculators that there are frequently several ways of performing the same task. As well as specifying the terms directly, sequences can be defined recursively or explicitly, as noted earlier. Both of these are accessible with a calculator, which allows for a finite number of terms to be produced, stored and then manipulated. To illustrate these two ways of defining a sequence, consider as an example the geometric sequence with first term 5 and common ratio 2: