Saturday, 28 March 2015

Learn the formula or learn the idea (part 2): Finding the mean

I wrote in my previous post about whether students should learn formulas or in some cases (such as finding sector areas) simply learn the underlying ideas. Just learning a formula can quickly lead to students' losing sight of the interconnected underlying ideas and not being able to adapt their thinking to different situations. In this post, I consider the mean of a data set.

Look up a definition of (arithmetic) mean and you can usually expect to see something like: 'the result when the sum of some quantities is divided by their number'. However, this is a definition of a particular rule for finding the mean, rather than the underlying idea. To me, the key underlying idea of the mean could be described as 'levelling out'.

Given the set of numbers 4, 7, 6 and 3, the mean could be considered the values of each of these numbers after a process of 'levelling out'. This can be seen visually by the following representation:


The 7 'gives' two to the 3 and the 6 'gives' one to the 4. This 'levels out' the numbers and the level amount, 5, is the mean. In this instance the result is an integer, but it is soon clear that this is not always the case. Once the idea of 'levelling out' is established, it can be introduced that one useful way to work out the mean is to collect all the numbers together and distribute them evenly - ie. to add the numbers and divide by how many there are. Again, diagrams are useful to see how this is equivalent to 'levelling out':


Understanding the mean as 'levelling out' rather than as 'the result when the sum of the numbers is divided by how many there are' allows students to be more adaptable in approaching problems. For example, how would you go about finding the mean of these sets of numbers?

(a) 173, 174, 175, 176, 177

(b) 346, 348, 348, 348, 348, 348, 349, 349

In both cases, 'levelling out' by some numbers 'giving' to others is clearly more efficient than adding and dividing. Students who have latched on too readily to the universal rule 'add up the numbers and divide by how many there are' at the expense of developing a deep and connected understanding  of the underlying idea are disadvantaged by their inflexibility. To put the idea centre stage in the lesson, rather than the arbitrary word 'mean', I would suggest the title of the lesson should be 'Levelling Out', rather than eg. 'Finding the Mean' (although, of course this important key word would be prominent later on in the lesson.)

This idea of 'levelling out' can then also be adapted to other number work. It can be used to quickly add a series of consecutive numbers, for example:

49 + 50 + 51

Or indeed, (with a little more mental effort!) numbers in any linear sequence:

71 + 74 + 77

It seems natural to want to summarise ideas with a generalised method or to encapsulate them with an easy to remember soundbite. When this happens, we need to be careful that we do not actually undermine our students' powers of mathematical thinking.

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