I am struck by the number of people who want an answer to the question “Why should we bother at all learning any sort of arithmetic”. The reasoning that goes before this argues that “I’ll always have a calculator to hand”.
Quite possibly, this is so. If you always do indeed have a calculator to hand – and on the occasions you do not then you hope never to be asked for any arithmetic skills, then you would think there is no problem and that you are correct. But I observe that you DO need to be able to identify when you have an answer from the calculator that is not right. I observe people working a till (POS machine) who, confronted with something like a bill at McDonalds for £80, can see that this might be wrong, possibly by a factor of ten, but have no idea how to identify what has gone wrong. It is far more likely to be operator error than machine error. So while one might indeed be excused a future requirement to be provably competent at written arithmetic, the related skills of estimation and being able to deduce what error has occurred actually become more important. of these, it is teh ability to recognise a wrong answer that is the essential to be learned at school. Quite what form that skill takes is unclear, but what teachers are asked to deliver (or were while I was still teaching) is some ability to predict a result, preferably before reaching for the calculator.
Within the maths syllabus, the process called estimation generally reduces each number in a calculation to one or two significant figures, resulting in having one or two sig fig of answer - and the decimal point in the right place. I would like there to be non-calculator questions where correct answers are chosen, where the decimal point is to be inserted among the digits and where the fault in the calculation is identified; I am prepared to agree that the demand for the skill of arithmetic is reduced, but I think it is still needed for the estimation process. I see this as understanding Number more than being able to process it with a high degree of precision.
Coupled with these skills is the requirement to check an answer for sense. I think this on its own will serve all teachers of mathematics very well in attempting to connect the ‘Maths’ to the ‘real world’. We were given a problem, we found a way to express this as mathematics (to model the problem), we have a result, so does this meet our needs? If it does not, we need to adjust our model. This is fundamental to modelling and, while the modelling process is usually taught in Y12, it seems to me that this could be one of those diagrams that belongs on a classroom wall year round, being a fundamental concept of the subject.
DJS 20180824
