Sometimes, teachers object that when you are calculating a mean of a real-life dataset the data points are very unlikely to be nice, neat positive integers. Write down 5 positive integers with a mean of 7 and a median of 4. Write down 5 positive integers with a mean of 7. If you are happy that you can have a number like 8, all on its own, which isn't a measure of anything in any particular unit, then it ought to be OK to have a length of 8 or an area of 8 too.Ī similar issue arises when people object to tasks like: The abstract concept of area can be used to solve real-world problems, like painting walls and laying carpets, but there is also just the abstract notion of area, which is measured in dimensionless numbers. Contexts are very important, as are the applications of mathematics, but I am not convinced that everything is always made clearer by setting it in context. ![]() The debate around units seems to be one where both sides think that the other side is demonstrating some kind of dangerous misconception. ![]() When we calculate the area enclosed between the curve $y=x^2$ and the curve $y=x(2-x)$, the answer is $\frac dx=45°,$$ which makes no sense at all! (How would you respond to the question: "When you do a definite trigonometric integral, should you give the answer in radians or degrees?") In pure mathematics, these things are dimensionless numbers. ![]() I am quite happy to have a line segment of length 4 or a rectangle of area of 8. “You can’t have an area of 8,” someone said – “it has to be 8 somethings, like 8 centimetres squared.” The whole question was completely unspecified – what on earth is a “4 by 2” rectangle – imagine going into a shop to buy a carpet that is “4 by 2” – without some units it is completely useless! Some of the teachers were complaining that the question was ‘wrong’, because the question writer had apparently ‘forgotten the units’! This was seen as ironic, because we are always telling students, “Don’t forget to include the units”, and yet here was a situation where this error had apparently been made in the writing of the question.
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