Functions and Graphs is an example of what we call at Kingsfield a "common task" that all classes in a year group will work on at the same time, for anything between 2 and 6 weeks. These tasks are designed so that they cover a wide range of skills and content objectives - hence justifying the length of time spent on the task. We aim for the tasks to be motivated by the students' own questions as much as possible.
The first lesson of a common task usually starts off with a closed activity (in the case of Functions and Graphs, this is playing the Function Game) and often students get taught a new mathematical skill (e.g., how to create a graph from a function).
Student questions are motivated by looking at two or more contrasting examples. For example, at the point of having drawn two graphs from functions (e.g., one linear and one quadratic) students are able to ask meaningful mathematical questions (such as: "Which rules give straight lines, and which rules give curves?"). If questions are not forthcoming then the teacher has a challenge prepared
(e.g., "Given any rule, can you predict what the graph will look like without having to plot any points?").
At the heart of our work on common tasks is the process of encouraging students to think mathematically by forming "conjectures", which they work on by testing, finding counter-examples, and then modifying as needed. We are always pushing students to express conjectures algebraically where possible, e.g., offering notation; and getting students to think about why conjectures work (leading to proof).
We encourage students to put up their work on "common boards" for others to see. In the case of functions and graphs, we would get students to draw their graphs on paper and pin them up as they do them. There might also be a board or piece of flip chart paper to write down conjectures, and questions. These boards often provide a common focus for the whole class, e.g., at the start of a lesson the teacher might focus everyone on two contrasting examples from the students work.
As teachers we look out for occasions to motivate, from what the students are doing, the teaching of specific new skills - e.g., how to find the gradient and y-intercept of a graph. These skills can then be practised in the context of the on-going work of the common task. It is also important to practice the same skills in a different context, and a common task might be broken up with intervals of work on exam questions for example.
We have come to trust that, given appropriate starting points, students will come up with all the mathematical questions we would want from a context. We have also realised, as we get more experienced in working with these tasks, that we are able to notice and hear more and so students seem to get further and into more complex issues each year. We find working in this way students become enthused and motivated by studying mathematics, and learn to view it as a subject which makes sense and over which they have control.
Hear Tracy describe her classroom practice when working on the Function Game (MP3 File).
Spaces for Exploration is an article Alf wrote in 2006, in which he describes the Function Game and how he uses it with classes.
Read a description of a first lesson on Functions and Graphs, together with suggestions for follow-up lessons: