Teaching mathematics in an interesting and challenging way has always been a concern for and of mathematics teachers. Making it even more difficult is that today's mathematics teachers are experiencing major changes not only in the mathematics content they teach, but also in the way they teach. Nearly all of these teachers came through school when mathematics consisted of a collection of facts and skills to be memorised or mastered by a relatively homogeneous group of students taught using a lecture approach. Now teachers are called on to teach new, more challenging mathematics to a very diverse audience using active learning approaches designed to develop understanding. Increasingly the use of computer assisted visualisation techniques is becoming popular amongst the younger generation of teachers who are more familiar and comfortable in the digital environment "With the web." As a young teacher was heard to remark, "we should be able to cater to every differentiation needed in the classroom."

In many ways visualisation in mathematics is reflective of mathematical understanding, insight and reasoning. Mathematics draws from reality and represents it as abstractions. Often the representations are presented in a visual form. Mathematics educators consider visualisation as an aide to understanding a concept or a problem and that in order visualise the problem this should be presented as an image or a diagram. A mathematician called Norma Presmeg of the University of Illinois and a specialist on visualisation in mathematics education, located visual imagery in the mind and theorises that a visual image is a mental image (close your eyes and consider a mental arithmetic problem, e.g. multiplying 2 x 2 x 2. Describe what you see as your mental picture). She goes on to list five kinds of imagery, namely: pictorial imagery (pictures in the mind), pattern imagery (charts and graphs), memory images (formulas), kinaesthetic imagery (fingers walking), and dynamic imagery (moving imagery, e.g. a fast moving car).

A large part of mathematics visualisation involves the use of Applets, a Java programme. What are applets?

Applying visualisation techniques to practice can be as simple or as complex as a teacher wants it to be. George Malaty of the University of Joensuu, Finland considers that visualisation can be made to help learners beyond just presenting a static demonstration of a principle. He considers visualisation is also helpful to motivating students to go beyond static appreciation to promoting causal thinking. In his article, "The Role of Visualisation in Mathematics Education: Can Visualisation Promote the Causal Thinking" (2008) he presents an example of how this can be achieved. You are encouraged to read this article and complete the following activity.

In an interesting article a teacher trainer George Whitley of Toronto Canada, reflectively queries the veracity of the many claims regarding the use of visualisation in teaching mathematics. For your next activity I would like you read the article "Visualisation in Mathematics: Claims and Questions towards a Research Programme" (2004). I am hoping that the questions being raised may stimulate you to consider a study on the use of visualisation in your work as a teacher.