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.
A medical professional who doctorifies about ‘making choices’ and ‘visualising’ is called a quack (or naturopath).
Well done Greg! I've tried for ages to articulate this point well, but you seem to have hit it on the head. The notion of an abstract "mathematising" as understanding is a vacuous one when it entirely discounts the role of procedure. In fact, the human brain seems to me to be WIRED for procedures and to unhitch understanding from procedure is like asking people to grasp walking without legs. Why would we play down procedure or expect our performance thereof to be replaced by something else without actually undermining understanding? Turns out that's largely what understanding *is* in mathematics -- it is an overall grasp of a logical sequence that comes through familiarity with that sequence through recitation and repetition. I also think of it like trying to appreciate poetry without reading it aloud. The best you can do is to use imagination and the "inner voice" to attempt to replicate the reading of poetry aloud ... if you kill that too, you've nothing with which to appreciate the poetry, because it is intrinsically bound up with how it sounds.
Many of my arguments about the development of mathematical schemata revolve around the mastery of procedure. What does it mean to a student to add 10 three-digit numbers? Or 3 ten-digit numbers? If a student has procedural mastery, they understand this *without* (I mean prior to) actually doing it, because the procedure plays out in their numerical schema: It is a cascade of single-digit additions linked by carries. The student who only has had a basketful of misc. "strategies" may well not even grasp the totality of the task before them. They may not approach it with confidence that they have the requisite "parts". They cannot visualize the sweep of the calculation in front of them, and they have no clear picture of how much work it will be. Those having mastered the calculation see all these things at a glance.
And I feel I must repeatedly underscore the point is that the student grasps this PRIOR to actually performing the skill. But the grasp is itself reliant on the familiarity that comes from having done it. Talking to the no-mechanics crowd it's hard to impress on them the critical role procedural familiarity plays in actual understanding.