### by Livia Mitson (May 2020)

One of the best websites for problem solving resources and rich tasks is median, which has resources written and collated by Don Steward, who died recently.

As my year 7 class have been experiencing a lot of mymaths and mathswatch recently, I thought we could try out having a go at a Don Steward problem, and we chose this one:

https://donsteward.blogspot.com/2009/04/chchchchanges.html

Consider this set of numbers

3, 23, 11, 7 and 31

What happens to the mean of the 5 numbers when (any) one of the five numbers is removed?

We decided to use it as a maths THUNK! – a problem where we didn’t focus on getting the right answer but on what we noticed.

We used live chat to discuss the maths that we had done – and there was some lovely noticing going on…..

I noticed that all of the five numbers were prime numbers

If you remove 3 or 31 the range changes

There is also no mode

And then some students worked out the median and got different answers, so they tried to discover where the mistake was….

(names changed)

Adam: the median is 14.5

Adam: to get the middle number I found the range and split it.

Bert: the median is the middle number

Curtis: all I know is it think the median of all 5 numbers is 11

Teacher: that’s interesting, I wonder what middle number means

Bert: so when you cross off numbers from both sides

Adam: but it doesn’t have to be in the sequence

Curtis: true

Bert: for example in 1234567 cross off 1 and 7 and then 2 and 6 and so on

Curtis: yeh

Teacher: that’s a good method

Adam: also 2, 2, 3, 4,4,5,6,6,6,7,7,9,11

The students had all completed a lesson on the median on an on-line website, and in theory had all understood the concept – but this discussion really brought home to me how much students need to discuss and “make sense” of what they have done.

What I found interesting about that was that Adam had interpreted the middle number as being the number halfway between the biggest and smallest number. And of course, there is a sense in which that _is_ the middle number. He also picked up that the median didn’t have to be part of the set of numbers (presumably from examples with four or six numbers in the set of data).

From this I got a sense of how complicated some maths concepts can be – the idea of the middle number is quite a fuzzy one, and in some ways I rather like Adam’s idea of finding the midpoint of the range.

From this discussion above it’s hard to tell whether Adam now has a more nuanced understanding of “the middle number” – but I will certainly be more wary of just saying “the middle number” in future.

The students who took part in the online chat enjoyed it, and we’ll be continuing with the maths THUNKS! on a weekly basis now.

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