## Wednesday, 6 January 2016

### Mode - Most unappreciated?

For me, the mode is one of the most unappreciated averages that we teach in maths education. Perhaps because of its simplicity, perhaps because sometimes we feel it can contradict the idea of 'centralcy' that we look for in a good average, or perhaps because it isn't often talked about formally in many everyday situations, but rarely does mode get significant lesson time beyond its introduction in primary school. Often it is paired with median in a unit which then devotes a whole lesson or more to mean calculations; occasionally it is lumped into a general 'averages' lesson and becomes almost a footnote when looking at lists of numbers to get pupils to look at the list ready to find median and mean. In my opinion this is a real shame as the mode can be one of the most versatile and available averages, so I thought I would share with you some things that I like to do with the mode...

1) Mode from different representations

I love introducing the idea of mode as the most frequent item of data, and then challenging pupils to identify it in lots of different representations, such as bar charts, pie charts, tally charts, bar line graphs etc - recently I gave pupils a sheet with these images on and challenged them to find the mode from each situation:

We got a load of misconceptions out of the way here; a mode of 4 or a mode of 7 from the frequency table, a mode of 6 from the bar line chart, what happens when 241 and 242 appear the same number of times; stimulated a lot of discussion and conflict and led to some real understanding.

2) Make up a list of data.

A nice pre-cursor to more complicated problem solving is to just give pupils a mode and to ask pupils to come up with different lists of numbers that satisfy the conditions. As a simple example, the question might well be something like "4 numbers have a mode of 3, Give a possible list of the 4 numbers." This can then be complicated in the following ways:
• 4 numbers have a mode of 3. What is the maximum number of 3s in the list? What is the minimum number of 3s?
• 4 positive whole numbers have a mode of 3. All of the numbers are 3 or less. Write all of the possible lists of numbers.
• 4 positive whole numbers have a mode of 3. The numbers add up to 10. Write down the four numbers.
• 4 positive whole numbers have a mode of 3. What is the minimum total that the four numbers can have? What about the maximum total if all of the numbers are less than or equal to 3? Less than or equal to 5?
amongst other similar examples.

These sorts of questions are nice to get pupils thinking and reasoning with mode; it is lovely to see them reason that the third list cannot have two 3s or realising that the fourth list cannot sum to 8.

3) Modal mystery

Similar to above, designed to promote reasoning around the mode, these sorts of questions are lots of fun to throw at kids:

2           .............         1         ..................          3

The above list of 5 numbers has 2 values missing. What could the mode be? What could it be if 3 is the highest number? If 1 is the lowest number? What about if we change the 3 to a 1?

4) Real life modes

Although rarely referred to formally as mode, a lot of statistics encountered in real life boil down to a mode. Whether it is votes on a popular TV reality show (X Factor, Strictly come Dancing, I am a Celebrity et al...) or likes on Instagram, giving a couple of examples like this and asking for more from pupils personal experience is a lovely thing to do with mode, because once you start to think about it, you can come up with loads!

So please, when you are teaching averages, don't just skip over the mode; there is so much more to come from this most common of measures.