The mathematics I do – Doug Williams
I am Project Manager, here at the Mathematics Centre in Australia.
This website and all associated work, is centred on learning to work like a mathematician in happy, healthy, cheerful, productive, inspiring classrooms. It is the place to come for a world of alternatives to text-based learning in mathematics.
The mathematics I use at work
...relates to all levels for school and beyond and focuses on the work of a mathematician or, ‘Working Mathematically’.
The work of a mathematician begins with an interesting problem, so it doesn't matter whether the work on a particular day is from the number or chance, or space or algebra area of the curriculum. As long as I am working on a problem, then I can try to work like a mathematician. So the main reason for the existence of all those sections of the curriculum is to activate the process of working mathematically. Along the way we:
- realise that we ARE working like mathematicians
- become better at referring to and applying the process
- build up a toolbox of mathematical skills
- build up a toolbox of problem solving strategies.
You can see exactly what ‘working mathematically’ means in the Working Mathematically section of the Mathematics Centre website.
The mathematics I’ve used in the last week...
Well that's not really a fair question because I have been on summer holiday. I live in Australia and that’s what we do at this time of the year.
However, some of the problems I have had to face have been:
- what’s the latest I can get out of bed, eat a hearty breakfast and still feel like lunch around 13:00?
- will I burn off more of the breakfast/lunch calories by walking along the beach for an hour, swimming in the surf (between the flags of course) for an hour or surfing for an hour?
- how many different routes can I take to the video shop to return last night's DVD and pick up a new one for tonight?
But one recent summer, when a family member who is a teacher visited from Holland, I was asked to explain some of our work. The photo below shows how we started.
First you make a cube with the 3D Geoshape squares. Then you unfold it so that each piece remains connected to at least one other piece and the whole thing can lie flat on the table.
The mathematician’s questions that guide this investigation are:
- how many flat shapes can you make this way?
- how do you know when you have made them all?
You can find out more about this problem (called Cube Nets) and the 3D Geoshapes on the Mathematics Centre website.
As you can see in this photo we sorted this problem. It took about 30 minutes and we celebrated as you would expect in Australia when doing maths on a balcony in the setting sun. (I turned the barbie on a little later).
Mathematics which has amazed and/or surprised me...
...most often comes from someone asking the mathematician's question: What happens if...? For example, in the problem above, What happens if we use triangles instead of squares and begin with a tetrahedron ... or an icosahedron?
But perhaps the greatest surprise I have had from this question was when two Year 5 boys were tackling the problem called ‘Sphinx’. This shape is a sphinx:
The challenge in the problem is to join four sphinxes to make a larger sphinx. No, that doesn’t mean stack them on top of each other! Tyler and Michael did that eventually, as have hundreds, perhaps thousands, of students before and since. So they realised, with the help of their teacher that if the single sphinx was Size 1, then the new one must be Size 2. From that they realised they had a template for making Sizes 4, 8, 16, ... Then they asked the question no one else had asked: “What about Size 3?”
You can read their story on the Mathematics Centre website, and find out heaps about how this inspired so many other kids.
The part of mathematics I like least...
The part of mathematics I like best...
...is when I have been working with a problem and the solution just comes to me. Usually it's been days or longer that I have been fiddling around with the ‘stupid thing’ and then I ask a new question, or an old question in a new way, and it all just comes together. You really feel like pumping the air and yelling “YES!”
is ... hmmm, that’s sort of the wrong question for me. There are heaps of topics in mathematics that I don't know much about, but what irks me most is the impression school mathematics often gives that maths is about remembering and practising an apparently infinite set of rules.
A maths teacher I remember...
...is Mr. van den Driesen. That’s because he gave the class a problem one day and promised two bob to anyone who could solve it. (That’s like 20¢ but you could get a lot for that when I was a kid.) The problem has become known as the Farmer’s Puzzle and it’s now recorded as a Professor Morris puzzle:
In Year 7 it took four of us to work it out and we stayed inside at lunchtime to do it. We enjoyed spending the two shillings.
In those days I didn't know about the mathematician’s questions:
Now I do, and the problem is even more fun.
- how many solutions are there?
- how do you know when you have found them all?
Find out more about Professor Morris Puzzles and how to use them in a Poster Problem Clinic.
And another teacher I remember...
...is Dr Laslo at university. He took my projective geometry course. We were only a small group, so he took us all to his office and had us running strings all over the place as guidelines for making projections of various sorts. It all made sense. The same could not be said of the courses provided by many of the other maths lecturers I had over three years.
Other people/events that have influenced my attitude to mathematics include...
- David Davidson, my mathematics education tutor in my diploma of education course, who helped me realise that the primary role of a mathematics teacher is to use mathematics to help students feel better about themselves
- Max Stephens, originally my Year 12 mathematics teacher and later a colleague on the council of the mathematical association, who encouraged me to become more deeply involved in pre-service and in-service teacher training
- Geoff Giles, a world-renowned Scottish mathematics educator, who challenged students in canny ways using visual and concrete starting points. You can learn something of his way of thinking here and here
- Charles Lovitt, a partner for many years in the development of many aspects of the Mathematics Centre, who helped me articulate that the only reason for having mathematics in a school curriculum which matters, is to encourage all students at all levels to learn to work like a mathematician.