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Trying out a Simple Goal Free Problem in HE Maths

What are Goal Free Problems?

I have recently been reading "How I wish I'd Taught Maths" by Craig Barton. I'm still not even half way through the book but it is full of great ideas. Very importantly these ideas are also backed up by research and Mr. Barton makes it very easy for readers to find the research in question. So today I tried something nearly new to me. I say nearly new because I have done something similar before but never had a name for it. In the past when students have been struggling with trigonometry problems I have always advised them to find out whatever they can and then see if that helps with the question. This is probably a simplified version of the goal free problem.The idea of the goal free problem is that there is no fixed 'answer' that students need to reach. Instead they explore the question and see what 'answers' they can come up with.

Today's Goal Free Problem

I decided to try it out today with a first year Business Maths module. We have been working on an introduction to statistics i.e. measures of central tendency, measures of variability and a little bit of work on histograms. I asked the students to think of a number between ten and forty and to write it down. Then we went around the class and collected the numbers. This left us with the following dataset.
20, 35, 20, 30, 20, 30, 17, 21, 15, 27, 12, 28
(There are more than twelve students in the class but hey it's nine o'clock on Monday morning so there were stragglers who didn't get to enter their number)
Under this list on the board I added the following

What Can You Tell me about this Data?


I thought there would be more backlash against this idea. I expected students to complain that they didn't know what to do. In fact they just put their heads down and started working.... and kept working..... and kept working. I was very surprised at how long the students stayed on task (or no task as the case may be). There was at least fifteen minutes of almost silent work. It took all my self restraint to stop myself from interrupting and asking how it was going or interjecting with ideas (I'm also trying to get into the silent teacher idea, but that is HARD).
When I eventually got the idea that some students were running out of steam (or I just couldn't handle the silence any more) we gathered the ideas together on the white board. Students had done the expected mean, mode, median, range, interquartile range. Some had also created grouped frequency tables and histograms. Some had even used the formula method for grouped frequency tables to estimate the mode and median. This led to a nice conversation about whether this was an appropriate choice or not.
I liked the fact that the students were all working at their own level and pace. I hope that everyone got something from it and possibly stayed in their own ideal learning range (zone of proximal development).
I'll definitely try to work some more goal free problems into my teaching.
The silent teacher thing is going to take more work. I realised that I talk A LOT when students are trying to process other information.  But that might be a blog post for another day!
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