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77 useless websites4/27/2024 There are times when context really helps students to get a sense of the underlying mathematics, but there are also times when context can get in the way. Certainly, they have important applications to statistics, and elsewhere, but if you are dealing with the AM-GM inequality, for instance, there is no reason to think that the quantities being averaged must constitute some kind of 'realistic data set'. The arithmetic and geometric means (as well as the harmonic mean and other kinds of mean) are all essentially (and certainly were originally) pure mathematics concepts. In such a task, there is a purpose - something you want to find out from the data - and the focus becomes less on the nitty gritty of adding up and dividing and more on asking meaningful questions and using the mathematics to figure out meaningful answers (i.e., mathematical modelling).īut I don’t see that kind of work as an alternative to tasks like the ones above. This all seems very valuable to me, and I am all for students grappling with realistic, messy datasets, with all the opportunities they present for data cleaning, examining outliers and using descriptive statistics to get a handle on what’s going on. And would it really make sense to calculate summary statistics from data sets containing so few numbers? How meaningful is a median when there are just 5 data values altogether? People who object in this kind of way would be much happier if all of the data points had a couple of decimal places, and ideally would like us to have 500 data points, rather than 5, and handle them in a modern fashion using technology. 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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