Learning with calculators: doing more with less.
Although the hand-held calculator is now almost fifty years old, it was not until around forty years ago that the first versions became available in schools in Australia. Unsurprisingly, in view of the name 'calculator', these were used mostly to undertake awkward calculations, including those of interest to scientists and engineers. Indeed, the first calculators to be used routinely in schools from the late 1970s were called 'scientific' calculators, apparently for that reason. At that stage, a calculator was described as 'scientific' when it included functions that had previously required the use of printed tables, such as those for logarithms, exponentials and trigonometric functions. Calculators offered a way for students to undertake numerical calculations efficiently and accurately. The efficiency arose from the speed of computation, compared with alternative methods, while the accuracy arose from superiorities of calculators over four-or five-figure tables in printed books.
A key purpose of this paper is to highlight the limitation of continuing to regard calculators solely as devices for handling (or even for avoiding) arithmetic. Much has changed, both in education and in technology, since the introduction of calculators, but it seems that much educational thinking about calculators continues to be stuck in the 1970s. Summarising the extensive and overwhelmingly positive research on calculators, Ronau et al (2011) noted that efforts are now needed to focus attention on effective uses of calculators in classrooms, towards which this paper is offered.
Developments in technology, education and assessment
There have been many developments in technology over the past forty years. The use of sophisticated technology has become commonplace for many Australians. This technology includes: many kinds of Internet resources; computer software, including apps, some of which are specifically for mathematics; devices such as laptops, tablets and smartphones; classroom technologies such as digital projectors, interactive whiteboards and wireless Internet. All of these have come at a cost, sometimes with equity implications, so that some schools and their students are equipped with essentially all of these resources, while others have access to very few of them.
Assessment has moved more slowly, partly for equity reasons, but mostly for security reasons, especially in high-stakes assessment, so that very few of these technological developments have been entrenched into curricula and most are prohibited from use in examinations. The exception is the hand-held calculator, versions of which are acceptable for external examination purposes throughout Australia and hence are likely to be available to essentially all students for almost all of the time. This paper explores the potential of calculators for supporting the learning of mathematics, especially in the primary, middle and early secondary years.
Developments in calculators
Calculators have also developed considerably over the past forty years. At a macro level, two major changes have included the development of graphics calculators (through the addition of a graphics screen and many mathematical capabilities to scientific calculators) and CAS calculators (through the further addition of computer algebra systems to graphics calculators). In addition, and importantly, calculators for younger students have been developed. Developments of these kinds reflect an important, but often unrecognised, change. Rather than being tools for scientists and engineers, since they were first invented calculators have been developed almost entirely as tools for mathematics education.
Changes to calculators have had educational purposes. This is especially evident in the development of more sophisticated calculators, such as graphics calculators and CAS calculators, both of which are best regarded as custom-designed technologies for mathematics education, not for professional scientists and engineers. But it is no less evident in the development of less sophisticated calculators, including those for students in the middle years and early secondary years of schooling. Amongst the developments that have clearly been designed by manufacturers to meet the needs of learners are the following:
* Consistent use of the conventional rule of order so that entering 2 + 3 x 4 into a calculator and generating the result will give 14 (instead of 20, which is still the case for many calculators and phones not designed for education).
* Improvements in user interfaces, such as multi-line displays allowing users to see what has been input into the calculator, as well as the result. Similarly, typical calculators can show earlier calculations and calculator inputs efficiently.
* The use of conventional mathematical symbolism. Perhaps the best examples of this are the representation of fractions with a horizontal vinculum and of powers in the form of superscripts.
* Mathematical functionality that matches typical school mathematics curricula.
The development of surd and fraction capabilities and the development and improvement of capabilities for statistics are the most obvious examples for scientific calculators. The development of graphing and geometric capabilities are perhaps the most obvious examples for graphics calculators.
These and other similar changes were not developed with the needs of professional scientists and engineers in mind; rather they were consciously developed to suit students learning mathematics.
In addition to functionality refinements and improvements, calculators have become significantly less expensive over recent decades. Roughly speaking, a scientific calculator for school in the mid 1970s cost around one day of average weekly male earnings; these days, a considerably improved scientific calculator for schools will cost around an hour of average weekly male earnings, a remarkable drop in forty years.
A model for calculator use in education
A key purpose of this paper is to describe and defend a model for calculator use in mathematics education. This is essentially an analytical matter, rather than an empirical one, and requires some consideration of how students learn, as well as how they are taught, as well as a study of calculator capabilities.
It also requires a consideration of what mathematics is--what it comprises. Indeed, the development of calculators has raised this question to some prominence, as there seems to be a widespread view (at least in the wider community, if not in the education community) that numerical calculation is the essential characteristic of mathematics. Many people seem to make the association of mathematics with arithmetic, in both directions: seeing lots of numbers is interpreted by many people as mathematics (even when that is clearly not the case, as in accounting). Conversely, mathematical activity is regularly (mis)understood as producing answers in the form of numbers.
The model described here is based on extensive work completed to support the educational use of scientific calculators by Kissane & Kemp (2013). In the following four sections, different ways in which calculators might be used are described. It is not suggested that students are restricted to using their calculator in only one of these ways at any time; in fact, frequently more than one use will be involved in a single activity.
Calculators represent mathematical objects and concepts increasingly well, at least in the sense that the representation on a screen is increasingly faithful to the representation on other media. Figure 1 shows some examples of this claim.
This is clearly beneficial to younger learners, removing previous disconnections of calculator representation and representation in textbooks or whiteboards, and eliminating the task of needing to interpret what is on the screen or juggle with syntax problems. For older learners, more sophisticated calculators similarly are increasingly likely to use conventional representations, rather than more complicated expressions, as illustrated in the calculator screens in Figure 2.
A calculator does more than merely represent mathematical objects, however. By the nature of its operation, a calculator allows for objects to be 're-presented', or presented again. As shown in Figures 1 and 2, calculators dynamically respond to inputs (shown on the first line of the display) with outputs (shown on the second line of the display). While this has routinely been dismissed as 'calculation' in the past, it is important to recognise that it is a much more important activity than that, and a source of stimulation and thus learning for thoughtful or curious users. Figure 3 shows some examples for fractions and decimals, key ideas in the middle years of schooling.
With help from teachers, students can be supported to become both more thoughtful and more curious and use such representations offered by calculators to increase their understanding about number: fractions and decimals are merely different ways of representing the same number; the same number can be represented with fractions in many different ways; divisions can be represented as fractions; and so on.
For younger students, for whom decimals and fractions are not yet comfortable ideas, integer division operations on calculators provide a different representation, as shown in Figure 4. In this case, the symbolism using R for remainder is not universal in mathematics, although the concept of division with remainder is important, and a necessary precursor to the later idea of a division producing a fraction.
In some cases, calculators routinely transform representations from one form to another, while in other cases, users need to assist this process (such as through the use of a fractions to decimals key, for example). More sophisticated transformations give rise to opportunities to learn other aspects of mathematics, as illustrated in Figure 5.
It is of central importance that when integers are represented as a product of their prime factors, only one representation is possible (which is quite different from representing a decimal as a fraction). The representation of [square root of 20] as 2[square root of 5] is sometimes referred to as 'simplifying' (although in one sense, it might not be unreasonable to think of it as 'complicating'!); such exact transformations on calculators are regrettably concealed from many younger learners by Australian examination rules, although they are routinely available to students with graphics calculators and CAS calculators. While the third screen in Figure 5 shows an example of relevance only to more sophisticated students studying complex numbers, it illustrates the same point that calculators routinely transform inputs to outputs that can be used as good starting points for students to learn about the mathematics involved.
Space precludes a fuller treatment of this matter. Users of graphics calculators will be well aware, however, of the immense power of representing functions in multiple ways, in the form of symbols, tables and graphs--the so-called 'rule of three'-- and the many learning opportunities these provide for students. While scientific calculator screens are too small to represent graphical objects, many can represent functions in two of these ways, using symbols and tables, as shown in Figure 6.
Since the 'new maths' changes in the 1960s, the idea of a function has been regarded as of central importance in school mathematics, so it is not surprising that a device to support learners might be designed to represent functions. In Figure 6, a function has been defined and tabulated and the table can be scrolled easily to overcome the limitations of screen size, allowing students to engage with the representation of a function as a set of ordered pairs. (Scientific calculators with such capabilities are not generally approved for high stakes examination purposes in Australia, however.)
While it is unhelpful to restrict thinking about calculators to their capacity to undertake numerical computation, it is important to recognise nonetheless that such a capacity is helpful for students; indeed, this was the initial attraction of calculators to mathematics education nearly forty years ago. One possible reason for the widespread confusion of computation with mathematics is that it is of practical importance, and almost everyone recognises that mathematics is 'useful' for everyday purposes. A calculator allows students to obtain numerical answers to understood problems, and is the sensible means of doing so when it is too difficult, tedious or inappropriate to use alternative methods, such as mental arithmetic, approximation or by-hand computational methods. All modern curricula, including the Australian Curriculum: Mathematics (ACARA, 2015), expect students to develop expertise with a range of computational methods, including use of a calculator.
Arithmetical computation is well-enough known to not require illustration here, beyond noting that students with a calculator have the means to undertake accurately and efficiently any practical computation that they understand. Hence practical mathematics with real measurements and real data can be undertaken, rather than be restricted to a pretend world where measurements are always in whole numbers and angles restricted to the (remarkably) few for which exact trigonometric ratios are known. Indeed, without such a computational capability, students are unlikely to have access to mathematical modelling of contexts of interest to them, or statistical analysis of realistic data from everyday sources.
It is also worth noting that computation on calculators may offer more than arithmetical convenience, however. Calculators may render some previous computational methods obsolete, or at least redefine them to be of interest mostly as historical curiosities, rather than essential components of learning. Obvious examples of this are long division and the extraction of square roots by hand. Less obvious, so contestable, examples might include Gaussian elimination and numerical integration.
Few would argue that it is not important for students to understand the logic of Gaussian elimination to solve systems of equations such as that shown in Figure 7. Yet the extent to which students should be restricted to undertaking it by hand, and need to spend the considerable time to develop fluency in what is essentially a tedious and error-prone algorithm (instead of spending time on other aspects of mathematical thinking) is much more contestable.
While faithful representation of mathematical ideas is helpful--even provocative--and a capacity for numerical calculation is essential, it is the role of the calculator as an exploratory device that offers the most promise for educational gain. As calculators are small, personal and responsive, students can use them to explore mathematical ideas and relationships for themselves in a low-risk and efficient environment. Good teachers have long known that purposeful engagement with mathematical ideas is a key element of learning. A calculator provides opportunities of that kind.
To illustrate, consider the concept of equivalent fractions, of central importance to young learners. With suitable encouragement, students can see that there are many fractions that have the same value, as shown in Figure 8. A productive task here is to challenge students to find still other fractions that are equivalent to three quarters.
In addition, students can easily check for themselves (using a fractions to decimals key) that each of these has the same decimal representation, as do other equivalent fractions, as illustrated in Figure 9. The calculator provides a fertile and responsive environment for students to explore equivalent fractions for themselves.
As another example, consider students learning about indices. A calculator allows students to explore the meaning of powers, and its consequences, especially when factors of numbers can be efficiently revealed. Figure 10 shows some examples in which a factor command has been used to decompose a result in ways that seem likely to help students make sense of what is going on (as distinct from merely calculating answers).
Exploring powers on the calculator will provide students with an opportunity to see some of the important relationships for themselves. This can be left to chance with undirected exploration, or may be provoked by suitable tasks. For example, the calculator screens shown in Figure 11 all offer opportunities for productive reflection or discussion by students, with a focus on understanding what is happening. Notice in the second and third screens that the results obtained are not what many students might expect, but potentially lead to a deeper understanding of rules for indices.
At a still later stage, students can explore powers further and in more sophisticated ways to see for themselves the generality of the relationships involved when fractional powers (much less intuitively meaningful than integral powers) are used.
In Figure 12, the calculator is being used to explore the concepts and the relationships involved. There is no suggestion here that the calculator is the ultimate preferred tool for undertaking most calculations with indices, but rather it is an intermediate tool for helping to make sense of them. At some point, students will also encounter more formal treatments of the mathematics involved (such as formal arguments for rules). In addition, they should later recognise that [7.sup.5] x [7.sup.3] = [7.sup.8], rather than use their calculator to determine such a product. The calculator is a means to an end, not the end itself.
The fourth aspect of the model for calculator use involves affirmation, concerned with users reassuring themselves in some sense about the quality of their mathematical thinking.
Sound use of a calculator should always involve the user having some sense of what to expect or some intuitions about a likely result. Good teachers have routinely encouraged students to think about results before entering them in their calculators, as a form of checking, but it is possible to be more systematic on this matter.
An early form of affirmation, still sometimes seen in printed material, is for students to complete calculations mentally or by hand and to then check their efforts with a calculator. When the focus is on developing manual expertise with computation, this might be a defensible, although low-level, activity. Context is important here: while it would not be appropriate to use a calculator as an alternative to learning or checking recall of tables (such as 7 x 8), it may be a sensible use in developing and practicing mental or approximate methods of computation (to handle tasks like 7.24 x 6.1).
There are more sophisticated uses of verification, however. Some calculators have a Verify mode, explicitly designed for students to check their thinking in some sense. Figure 13 shows some examples of this, in which a response of 'True' or 'False' is shown for the given inputs.
In these examples, the user is checking their own thinking, not relying on the calculator to generate the original responses.
In the first example, the calculator is being used to refine approximate mental computation skills. In the other two examples, the key idea is that of an identity, one meaning of an equals sign. In a class of students, instructed to give different values to the variables involved, the critical idea of a variable is involved. (Indeed, if some students assign a value of zero to either A or B, the exceptional case of equality will be produced, and the expression declared true, for an interesting discussion point).
This idea of affirmation might also be invoked more generally, even when a Verily mode is not available. On a graphics calculator, identities can be powerfully explored by comparing tables and graphs for functions thought to be the same. This is more difficult on a scientific calculator, but the example in Figure 14 offers a possible mechanism (using a calculator with a capacity to tabulate a pair of functions simultaneously, a nice feature available on all graphics calculators, but generally not available on calculators for Australian schools).
These examples are not meant to suggest that the calculator's purpose is to undertake such computations, and nor is the calculator sufficient to establish the mathematical results involved. Rather, the calculator is being used to affirm students' thinking--or to disconfirm it, of course--and thus help to provide a means of making sense of the mathematics involved. Ultimately, a formal argument is needed to establish an identity; the calculator is used to motivate the need for such an argument.
Calculators and curriculum in Years 6 to 10
The few references to technology made by the Australian Curriculum: Mathematics (ACARA, 2015) seem to suggest that digital technologies are regarded as computational devices, through the repetition of the phrase, "with and without digital technologies" in content descriptions. Yet a careful reading (by the author) of the number substrands for Years 6 to 10A, suggests that learning related to 37 of the 38 content descriptions can be supported and enhanced through the careful use of a modern scientific calculator, many more than are flagged in the official documentation. It seems opportune for teachers and students to take advantage of these potentials, using tools that are widely available to students.
Significant changes to calculators since the 1970s have rendered them as purpose-built devices to support the learning of mathematics, in addition to undertaking computations.
The model described in this paper is offered to clarify the precise ways in which that educational promise might be fulfilled, through exploiting the calculator's capabilities for representation, computation, exploration and affirmation.
Australian Curriculum Assessment and Reporting Authority (ACARA) (2015) Australian Curriculum: Mathematics F-10. Retrieved 12 Jan 2015 from http://www.australiancurriculum.edu.au/mathematics/
Kissane, B. & Kemp, M. (2013) Learning Mathematics with ES PLUS Series Scientific Calculator. Tokyo: CASIO, Inc. Retrieved 6 Jan 2015 from http://edu.casio.com/education/activity/
Ronau, R., Rakes, C., Bush, S., Driskell, S., Niess, M. & Pugalee, D. (2011). NCTM Research Brief: Using calculators for learning and teaching mathematics. Retrieved 29 Jul 2012 from http://www.nctm.org/news/content. aspx?id=31192
Caption: Figure 1: Developments in representing fractions, powers and radicals on scientific calculators.
Caption: Figure 2: Calculator representation of more sophisticated mathematical expressions.
Caption: Figure 3: Re-presenting fractions, decimals and division.
Caption: Figure 4: Representing division of whole numbers with remainders.
Caption: Figure 5: Further examples of re-presentation on calculators.
Caption: Figure 6: Representing a function symbolically and numerically.
Caption: Figure 7: Solving simultaneously 2x + 3y = 7 and 5x - y = 2 on a calculator.
Caption: Figure 8: Some fractions that are equivalent to three quarters.
Caption: Figure 9: Equivalent fractions have the same decimal representation.
Caption: Figure 10: Exploring and understanding powers.
Caption: Figure 11: Further explorations of integral powers.
Caption: Figure 12: Exploring fractional indices.
Caption: Figure 13: Using Verify mode to check thinking.
Caption: Figure 14: Exploring the identity: sin 2x = 2 sinx cosx.
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|Publication:||Australian Mathematics Teacher|
|Date:||Mar 22, 2017|
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