A good mathematics teacher engages learners in mathematical activity and interacts with them to enhance the learning that they derive from the activity.  This is especially inportant if you want to develop learners mathematical thinking and independence.

The many aspects of doing mathematics include making decisions, visualising, reasoning, developing and using convincing arguments, discussing ideas, asking questions, investigating, remembering things, using techniques, generalising, communicating methods, findings and conclusions to different audiences, and applying mathematics in different contexts.

A good teacher selects tasks which, over time, engage learners in all these aspects of doing and learning mathematics, and enable the teacher to interact with the learner in different ways.

One task may provide more opportunities than another for acting in a particular way, such as speculating or inventing, and the likely diversity of activity will vary from one task to another. Teachers attempting to describe what makes a task 'rich' have suggested that it depends upon its accessibility, and to what extent it has the potential to involve making decisions, speculating, making and testing hypotheses, proving or explaining, reflecting, discussing, interpreting, searching in new directions, inventing, asking 'what if?' and 'what if not?' questions, being surprised, and enjoying oneself. (See Rich Tasks)

But a task is 'rich' only potentially.  The teacher's role is vital. The 'richness' of the task depends on what a teacher does, and on the learning environment. (See Task and Activity and Learning Environments) Whether they are working independently as individuals or collaboratively with others, a good teacher interacts with the learners in ways that support the various aspects of mathematical activity and the interactions between learners. (See Selecting Teaching Strategies (in Key Stage 3) and Talking in pairs)
  
A good teacher varies the kind of task that learners undertake, and there are many types of task to use with learners at Key Stage 3. (See Task Types)

For example, learners might be challenged to create examples of something, such as sets of numbers with a particular mean.  Or they might engage in a classification task such as explaining why each one of a set of objects might be considered the 'odd one out', or sorting representations of objects into cells of two-way grids and creating examples that will fit empty cells.  Or a task might involve matching different representations of the same object, explaining the connection; for example they might try to match algebraic expressions, word expressions and diagrams.

Tasks are often starting points for more substantial activities. For example, deciding whether statements are 'always', 'sometimes' or 'never' true, and giving explanations with examples and counterexamples, can lead to adapting statements so that they become 'always true' or 'never true'.

There are various kinds of task in which learners analyse chains of reasoning. One such task, that usually leads to earnest discussion, is set up by the teacher or a learner writing each step of a chain of reasoning on a separate card, and challenging learners, working collaboratively, to put the steps in order. Another rich task is challenging learners to find as many different solution strategies as they can for solving a simple problem such as a ratio problem. (See Mathematical Reasoning and What is a proof?)

Tasks in which learners create problems can generate further activity.  For example, learners having paired up, one learner devises a problem to which the other learner tries to find a solution.  If the solution is not the one expected, the learners naturally discuss why this is.  A different task is to create new questions by making small changes to a given question, and asking questions about the 'family' of possible questions, such as "For which numbers is the solution 'impossible'?" or "For which numbers is a solution 'impossible'?". (See Own problems)

Mathematical resources or teaching aids can generate rich tasks. (See Selecting and Using Resources for Learning (at Key Stage 3))

For example, a teacher might challenge learners to join Mathematical Activity Tiles edge to edge to form 3-D shapes with particular properties, such as having a particular number of edges. Or they might tackle problems stimulated by still images, such as images from Richard Phillips' 'Problem pictures' CD-ROM.

Stretching rubber bands around pins on a geoboard is an immediate, non-committing, way for learners to explore shapes.  For example learners might investigate the perimeters of different shapes with the same area and with their corners on pins.  Dime symmetry shapes generate tasks involving visualising that are very simple to explain, but not simple to complete.  For example the teacher places two pieces together on an OHP surface and challenges learners to describe ways of placing a third piece to form a symmetrical shape.  Other tasks that help learners develop 'mind pictures' include inviting learners to visualise short image sequences using simple prompts such as: 'Think of six points. Can you align some or all of them? How many ways can you find?' Learners are encouraged to talk about their images. (See Imaging & Expressing)

Learners can use graphic calculators to generate and explore relations between numbers.  For example, what happens if the 'start numbers' and the 'steps' of a sequence are altered?  Tasks in which learners model situations using still images and dynamic geometry software are examples of using mathematics  as a tool in a 'real' context. For example learners investigate possible shapes and equations of curves to fit the cables of a suspension bridge in a photograph. (See Dynamic Geometry).

Tasks of most types, such as the types described above, can be made accessible to pupils with learning difficulties. For example, if the task is to create new questions by changing a given question, the teacher chooses an accessible question. Or, in an 'odd one out' task each object is such that it might be the 'odd one out' for obvious, or for less obvious, reasons. However, good teachers take care not to put a ceiling on anyone's potential learning.  Therefore they try to give all learners tasks that allow the learner to surprise the teacher.  Good teachers also try to build on life skills that learners bring to the classroom. (See Turning Can't Into Didn't, Intelligence and Low Attaining Students, and Mathematical Powers).

Learners with difficulties are often able to access tasks with the support of physical materials that other learners may not need. For example, they might use coloured counters to explore simple ratios, or mirrors and coloured pegs on pegboards to explore reflections.

When selecting a task a good teacher is aware of the types of errors or inappropriate generalisations that learners might make.  If they arise, misconceptions can be probed and discussed.  (See Types Of Errors and Misconceptions)

Many tasks can be made more challenging or extended by asking 'what if?' and 'what if not?' questions.  For example, 'what if the centre of rotation is not at the origin?', 'what if the line need not be straight?'.  Good teachers expect high attaining learners to sustain extended investigations, and to make conjectures which they try to prove, or to justify using steps of reasoning.  Good teachers are aware that even high attaining learners may be hampered by anxiety, and try to remove unnecessary sources of stress, such as the feeling that getting a task completed is a 'race'. (See Learners' anxieties about mathematics).

A good homework task is not usually just continuing a task started in the lesson, although it may be a separate related task.  It might be a task that prepares learners for discussion in the next lesson.

Probes and prompts:

Taking action:

Case Studies

Research Sources