It's all in the game
A) There are many different activities that can be used to encourage mathematical thinking skills and building pupils' confidence in maths.
"Play" doesn't always mean a game. Aren't the happiest of mathematicians "playing" with ideas, chatting about these ideas to launch further exploration?
As a member of the Association of Teachers of Mathematics (ATM), I receive a regular bulletin via email. Looking at the courses, I thought you might find these two interesting: "Developing mathematical thinking" and "Interactive Mathematics Teaching"
(www.atm.org.ukcoursesindex.htmlNoNoimt). These are delivered by Barbara Ball, the ATM's professional officer in maths. I asked Barbara what kind of problems and games encourage mathematical thinking. She replied that learners are more likely to think mathematically if they are working on challenging games, in which they are:
* thinking about and working on significant mathematical ideas;
* asking their own questions;
* talking about their ideas with other people (not just teachers!) and learning from the discussion;
* understanding what they are doing, with no jargon to bar their way;
* not limited in what they can (or are allowed to) do with the ideas they are exploring;
* putting several steps or ideas together in order to solve problems;
* choosing their own techniques;
* identifying what they need to know and what they need to be able to do;
* making their own mathematical connections;
* making conjectures;
* seeing the need for explanation and proof (for their conjectures).
At college, I discovered that if you knew how the game of Nim worked you could create a winning strategy. Isn't that what we try to do when we play games: work out the best strategy, making and proving conjectures as a part of that process - as in cribbage, if you know the winning combinations and have an understanding of probability then your chances of winning are greatly improved.
Barbara also sent me a copy of an article she had written for MT181 (ATM's quarterly publication), a special issue on mathematical thinking. She describes an activity in which the audience each have four cubes. They are told: "Join two of your cubes together, face to face. What is the surface area of the shape you have made?"
They then join four cubes to make as many different surface areas as possible. This generalises to include any number of cubes joined face to face.
* What is the maximum surface area possible?
* What is the minimum surface area possible?
* What surface areas between the maximum and minimum are possible?
"What is likable about this task is that the first question is easy - most students can offer an explanation of the result - while the second question is much harder, and I do not know the answer to the third question. Yet all three questions are easy to understand and do not require a high level of mathematical knowledge and skills; what they do demand is a high level of mathematical thinking."
Barbara continues: "When I first worked on this problem with a class I had assumed that you must be able to get all the even numbers between the minimum and maximum values for surface area, and I thought one particular girl was just being lazy when she submitted her report about models made from eight cubes, with illustrations of all the different surface areas except 26. I know better now!"
There is an excellent series of articles by Jenni Way - "Learning Mathematics Through Games"
(nrich.maths.orgpublicviewer.php?obj_id=2489part=index). She presents some maths games that Year 8 pupils might enjoy.
Andrew Griffin of Tarquin suggests games such as MathMagic, Triolet and Katamino. For strategy games and a variety of card and dice-based games go to www.tarquinbooks.com and search under "games". Games and puzzle links are also available at www.virtualteacher.com.auclassroomgossip.html (you may need a Java browser plug-in). A book called Questions and prompts for mathematical thinking by A Watson and J Mason (published by ATM) may be of interest too.