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
Spaces for Explorationis an article Alf wrote in 2006, in
which he describes the Function Game and how he uses it with
Read a description of a first
lesson on Functions and Graphs, together with suggestions for