Why calculators are in the picture
The contradiction between calculator worship and mathematical understanding
One of the worst moves made in recent times in Australian mathematics education was to change the format of the national numeracy assessments in Years 7 and 9. Previously, students were required to sit two papers - one without a calculator and one with. In 2017, that changed so that students only had to answer eight questions without a calculator.
Regardless of what is stated in the Australian Curriculum, schools know they will be held primarily accountable for the results of these national assessments and so a signal was sent that solving problems without a calculator was of reducing importance. This is consistent with a global maths education orthodoxy that favours the use of calculators. Sparked perhaps inevitably by an American trend towards calculator use from the late 1980s onward, it is now considered a truism that calculators should be an integral part of maths teaching across the anglosphere, although England seems to have bucked this trend in recent years.
The situation in East Asia is different, with calculator use seemingly less of a priority in states such as Singapore. Interestingly, the Programme for International Student Assessment (PISA) is neutral about calculator use and simply advises participating states to allow or ban the use of calculators in their mathematics assessments based upon the usual practice in that state. They also survey students on calculator use although I am finding this data hard to track down. A 2005 analysis based on data from PISA and another international survey, TIMSS, found that U.S. students were more reliant on calculators than a comparison group of countries and that they were also worse at maths.
Oddly, another part of the global maths education orthodoxy asserts the primacy of conceptual understanding. Interestingly, while teachers in the U.S., Australia, China and Hong Kong all agree that understanding is important, U.S. and Australian teachers tend to suggest it must come before procedural knowledge whereas East Asian teachers are more relaxed about that. What role do calculators play in this debate? Well, it is hard to see how punching the buttons on a calculator so that it can perform an operation you cannot see is a route to conceptually understanding that operation. How can it be? And the research actually supports the East Asian view - conceptual understanding and procedural knowledge have a two-way relationship. If you want to understand the mathematics then it helps if you can do the mathematics. It therefore follows that getting a calculator to do the mathematics for you is not going to help.
So why the orthodoxy in favour of calculators given their seemingly paradoxical relationship to the need to develop conceptual understanding? A clue can be found in the backlash to England’s 2014 decision to ban calculator use in an assessment for 11-year-olds, with one expert claiming that, “Removing national tests where pupils can use calculators will place greater emphasis on the testing of calculation skills and less on the assessment of mathematical reasoning.”
Can you see what is happening here? There is some other plane of mathematics beyond mere calculation - mathematical reasoning - and it is this that should be the focus. It is this that we want students to understand. When you realise that people genuinely do believe in a higher plane of mathematics, the motive to quickly dispose of lower order mathematics by employing technology makes more sense. It also explains why the concept of ‘mathematising’ which involves ‘making choices’ and ‘visualising’ has crept in to the draft of the new Australian Curriculum. It’s these sorts of a thing that we want students to understand, not mere calculation!
It is hard to think of a good analogy because in any other field, such an approach would seem bizarre. It would be like downplaying the mere technical aspects of playing tennis and asserting the existence of something called ‘tennisifying’ that is about, well, ‘making choices’ and ‘visualising’. Or perhaps we give budding chefs a Thermomix to make their risotto for them and suggest they instead develop the skill of ‘cookification’ that involves, well, ‘making choices’ and ‘visualising’.
You get the picture. Or maybe you don’t. More visualising practice for you.