'An essential teacher skill'
So how many mathematics teachers have had the experiencing of sharing a really cool bit of mathematics with their students only to be disappointed that their students didn't get caught up in the magic like they did? I know it has happened to me and these are some of the more salient moments of teaching. Mathematical ideas will grab different peoples attention in different ways because we are all different and this is just another factor that makes mathematics teaching such a challenge. In this blog I am trying to describe the process of taking a fascinating mathematical idea and turning it in to an activity so that it a) gives students an opportunity to experience the same fascination that we did, b) engages students in mathematical thinking and c) leads to some useful conclusions. As is so often true, I am going to find it easier to explain with an example. I am going to start by looking at the prime number visualisations shown here by the Ptolemy Primitaves application. This is just one of many awesome tools and ideas out there related to prime number visualisation. The video below gives a quick overview of the application.
So, if you haven't seen the application yet then you are probably thinking just how cool it is. (I am making an assumption about people who might be reading this blog post!). In the comments below, you can see people who are saying things like 'I am using this with my students' and this really sums up the issue I am trying to present in this blog post. Just how exactly are teachers 'Using' this with their students? I am sure there are loads of good answers to that question, I am just making the point that once we have found things like this then, our next and often most challenging job is to create activities that allow students to experience the mathematics. I have shown this to students before and they do seem interested and will play with it for a while, but without any associated activity there have been no real moments of mathematical thinking and discovery. It is almost like telling a joke in wrong order with the punchline first. This year I decided to do something I have wanted to for a long time with this wonderful app. I started by asking myself what things I would want my students to get out of playing with this idea Here is a list. I wanted students to.....
- Above all, be engaged in mathematical activity.
- Discover the 'coding' of the diagrams for themselves and not tell them it was anything to do with prime numbers.
- To see for themselves how numbers can be expressed as products of their prime numbers.
- To reason with each other about why the above is true.
- To understand how and why there is more than one way to show the same numbers in this way.
- To understnd how they could create a visualisation for any number this way.
For any of the above things to happen there needed to be an activity with an objective as an entry point for students and, not being able to settle on one, I created three that I thought could be used together or could be used to differentiate.
- I captured an screen shot of each of the first 30 intergers, jumbled them up and asked students to work out which image should go with which number.
- I made a similar sheet of 30 images, but made it clear that there were not 30 different numbers. Numbers would appear either 2, 3 or 4 times in different ways. The objective here was to work out which pictures represented the numbers and then what the numbers were. The order in which those things happened was for students to decide.
- A third sheet was made of 30 different numbers (but not the first 30 integers) and the objective was to work out what the numbers were.
Each task was prepared with some prompts to help students get started. In the end , I decided to go with the 3rd sheet for everybody and gibve out the 1st and 2nd as and when I felt it might help a group or individual. I was completely delighted with the result which achieved, in different measures for different students, all of the objectives set out above. What I find most important is the reasoning that students use with each other when arguing a particular conjecture. The conversations were rich, the students were engaged and my role was to step in as and when required to make sure these things were true for as long as possible.
There is a rich irony in that I had a great fear that students would crack the code by working out that they could simple count the black dots. It is ironic because, for the activity to go well, I needed students to search for something more complex than a relatively simple solution that is just in front of them. Although it alone would not completely crack the code, it would have taken the sting out of the activity somewhat.
Thinking about and creating what seems like a simple activity took a long time, but I find this to be my favourite kind of work. The resourecs I made can be found here 'Prime Pictures'. Returning to the title of the blog, this, for me, is one of the very key elements of mathematics teaching - turning great ideas and tools into real activities that help us to engage students in mathematics. There are so many other examples of this and the one below is what is on my mind at the moment.
I have always loved this problem I recently come to know as Curry's paradox. This is nicely explained through the link. This is such a lovely bit of maths and I had never considered the range of cases to which it could apply. I read the link with relish and enjoyment and am left thinking 'mmmmm how can make this in to a great activity for students?'. It must be possible! (image below taken from 'Cut-the-knot')