The humble calculator, now both significantly ubiquitous and laughably convenient, is a technological advance that should have changed the face of mathematics education. Lagrange (2006), wrote:
‘Mathematics education has thus to reconsider the study of a domain, taking the obsolescence of traditional techniques into account, and to conceive new techniques as components of new praxeologies for this domain. Thinking of new techniques linked to the use of computer tools and of their possible epistemic value is not easy because mathematical culture is implicitly linked with paper-and-pencil techniques and is not accustomed to the idea that other tools can support conceptualization. However, this is indeed possible.’
Note the future tense here: this had still not happened in 2006 (and, arguably, some eleven years later). While it is true that the introduction of the calculator has “perturbed the primary maths curriculum” (Pimm & Johnston-Wilder, 2004) and undeniably affected assessment practices in the UK, its relationship with curriculum and pedagogy has been inconsistent and fickle; in short, use of calculators seems to be a particularly political issue, with recurrent narratives around ‘21st century learning’ versus ‘back to basic rote learning’ taking prevalence by turns. ‘It has been suggested that technology, per se, does not improve student learning. It is the curriculum in which it is embedded, and the accompanying pedagogy, which may determine the ultimate effectiveness of technology implementation in mathematics classrooms’ (Highfield & Goodwin, 2008). Grayson (2013) found that not even half of countries surveyed in Trends in International Mathematics and Science Study – a ... More 2007 had policies or statements about the use of calculators in Grade 4 (upper primary) curricula. Why?
‘Controversy still surrounds the integration of calculators in the teaching and learning of mathematics even though their use has been consistently advocated since the late 1970s’ (Sparrow et al, 2011). It would appear that much of the research is now decades old, suggesting this may be a fruitful area of focus for current mathematics education researchers. But what does that body of research say?
During the 1980s … a battle of wills [began] between educators across the world who saw these machines as either handy little classroom-elves, or devilish little desk-goblins.
The history of the calculator
First, a short history lesson. The first pocket-sized calculators that we might recognise were designed and created in the 1960s and became popular in the 1970s, giving us some 45 years’ worth of research to draw on when considering their place in the mathematics classroom. Scientific calculators – with the ability to handle trigonometric, exponential and logarithmic functions and scientific notation – have also been around since the 1970s, with graphical calculators bringing in the ability to visualise graphs and manipulate tables of variables in the 1980s and becoming more popular in the 1990s.
During the 1980s some states in the US began to require calculator use in exams; others banned them – and thus began a battle of wills between educators across the world who saw these machines as either handy little classroom-elves, or devilish little desk-goblins, as interlopers or allies to the cause of promoting good mathematical understanding. Others have framed this debate as one between educational research and “political intervention” (Boorman, 2015).
This is particularly interesting given that the research findings were relatively consistent some time ago. From the Calculator-Aware Number Curriculum report, considering the effects of providing unrestricted access to calculators for primary pupils: ‘As children play with their calculators, they find out a great deal about how numbers behave…Most children in the project have also decided for themselves that they do no need, or want, to be dependent on calculators. The project has seen a great flowering of mental calculation’ (Shuard et al, 1991). Teachers participating in the project found that pupils, given the freedom to explore using a calculator, used it not only to check mental calculations or perform complex calculations, but as ‘a resource for generating mathematics’ (Shuard et al, 1991); it opened up the range of processes, concepts and ideas that could be explored in primary mathematics lessons. Several other projects concurred with its conclusion: ‘used properly from an early age, calculators can greatly enrich the number experience of learners’ (Duffin, 1997).
Much of the controversy surrounding the use of calculators appears therefore to be surrounding this idea of proper use, and in particular assessment. Much of the literature suggests proper useis pupils being able to compute on their own first; then calculators can “relieve students of cumbersome computation, allowing them to concentrate on more meaningful mathematical activities” (Ballheim, 1999, in Ruthven 2009)
Criticism of calculator use for mathematics learning seems to focus on three areas: reliance (sometimes called ‘black box thinking’), focusing on answers, and lack of writing down workings.
The calculator in UK policy
As for assessment, UK policy seems bafflingly different for different age groups: in 2014 for the new-design SATS exams, Year 6 pupils were not allowed to use calculators on any of the papers – contrast this with current UK GCSE assessment practice, which has prescribed calculator use in between 50% and 100% of its specifications and “It is left to the individual awarding bodies to determine how the content is allocated to calculator and non-calculator assessment” (Grayson, 2013). The differences seem very stark by the time pupils reach A-Level, and have been for some time – this is from a Department for Education - a ministerial department responsi... More report in 1982, quoted in Pimm & Johnston-Wilder (2004):
‘In A-level work there was a very widespread use of the pocket calculator as a substitute for mathematical tables. It was disappointing to find much less use being made of them in other examination courses in mathematics. The fact that some examination rubrics would not allow their use during the actual examination was often interpreted to mean that they cannot be used at any time during the course. Much valuable mathematical activity can be derived from the use of these devices beyond the more obvious purposes for which they are largely used at the present time.’
This highlights beautifully the difficulties – past and present – of assessment being the primary driver of classroom practice, and the inconsistencies which might occur as a result of failing to take research into account. Pimm and Johnston-Wilderconclude “the comment in the HMI quotation about resources is still as significant today as it was then.”
Anne Watson from Oxford University responded to the SATs changes thus:
‘In fact, students who use calculators regularly in lessons score as high or higher in tests, taken without calculators, compared to those who do not. On the whole, the use of calculators as an integral mathematical tool has been shown to be beneficial, particularly in the development of mathematical problem solving. It is a pity that current policy is retrogressive in this respect.’
Criticism of calculator use for mathematics learning seems to focus on three areas: reliance (sometimes called ‘black box thinking’), focusing on answers, and lack of writing down workings – yet ‘Using a calculator is far from being the unthinking process of popular repute’ (Ruthven, 2009) and the findings of the Calculator-Aware Curriculum project starkly undermine these assumptions. Other researchers emphatically agree: ‘It might be thought that possession of a calculator brings with it the ability to perform any ordinary calculation. Far from it. Effective use of a calculator requires an understanding of number. (McIntosh, 1990).
In the seminal Cockroft report, now some 35 years old, it is equally clear: ‘the availability of a calculator in no way reduces the need for mathematical understanding on the part of the person who is using it.’ (Cockroft, 1982). ‘Much of the criticism which has been levelled at calculator use has claimed that children’s numeracy skills will decline. No evidence in the research has been found to support these assertions’ (Sparrow et al, 1994). Boorman calls these issues ‘political fears about calculators being harmful,’ and refers to recent steps away from using calculators in the primary classroom as a ‘prohibition’ (Boorman, 2015). There are also socio-economic factors at play here which deserve more exploration – for example some students (white, female) are more likely to use calculators than others (Crowe & Ma, 2010).
Calculators, in order to be used effectively to stimulate mathematical understanding, cannot simply be ‘improvised around a conventional curriculum’ but must be an integral part of the design of a curriculum.
Using calculators effectively
Calculators, in order to be used effectively to stimulate mathematical understanding, cannot simply be ‘improvised around a conventional curriculum’ (Ruthven, 2009) but must be an integral part of the design of a curriculum, requiring ‘careful planning, particularly of curriculum sequences to underpin continuity and progression in children’s learning’ (ibid). Of course, electronic calculators are not the only tool for calculation we have in the classroom – and, seen within a broader context of technological advance, it is interesting to note that some ‘devices are purely about enhanced or substitute performance, while others have greater or lesser pedagogic intent present in their design .
Some, such as pocket calculators, are to be found all over the adult world, while others, such as geoboards or Dienes multibase blocks, are only to be found in school classrooms. Some computer software, such as the spreadsheet, was primarily designed for business applications, while others, such as recent interactive geometry environments, were designed with mathematics classrooms specifically in mind.’ (Pimm & Johnston–Wilder, 2004). It may be worthwhile asking what calculators are actually for outside the mathematics classroom, as well as inside it.
As I suggested at the beginning of this article, part of the issue with consistent adoption and integration of calculators into mathematics teaching seems to be that ‘mathematics curricula should be able to respond flexibly to technological change’ (Mann & Tall, in Pimms & Johnston-Wilder, 2004) – and, due to the current political and bureaucratic processes surrounding curriculum writing in the UK; ‘The reality over the subsequent decade has once more differed’ (ibid). Unless something changes soon, we seem to be stuck in ‘an apparently increasing dichotomy between empirical research and the status of calculators in the National Curriculum’ (Boorman, 2015).