Research supporting the Standards Unit resources: Improving Learning in Mathematics (Malcolm Swan, 2005) shows that teaching mathematics is effective when it:

o     builds on the knowledge learners already have;

o     exposes and discusses common misconceptions;

o     uses higher-order questions;

o     uses cooperative small group work;

o     encourages reasoning rather than 'answer getting';

o     uses rich, collaborative tasks;

o     creates connections between topics;

o     uses technology in appropriate ways.

These principles should underlie all teaching, including planning, selecting teaching strategies, selecting suitable learning activities, selecting and using resources for learning, and monitoring learning.

Building on the knowledge learners already have
Good teachers are continually monitoring learning, gathering evidence about learners' knowledge and planning how to build on it.  They select and present tasks, and use strategies and resources that provide opportunities for learners to overcome obstacles, use familiar ideas in new contexts, develop ideas or gain sudden deeper understandings.

Exposing and discussing common misconceptions
Monitoring learning includes diagnosing misconceptions and obstacles to learning, and planning includes anticipating when and where they are possible or likely.  Tasks and resources can be selected to expose misconceptions, and there are strategies that encourage learners to discuss them and correct them naturally.

Using higher-order questions
When planning tasks, effective teachers think of higher-order questions that encourage learners to reason, explain their reasoning, and reflect on their own, and other people¹s, thinking.  They try to create a learning atmosphere that facilitates this, and in which activities will provide opportunities to ask these kinds of questions.  Learners' responses to such questions contribute to teachers' evidence about learning, and therefore observing responses to higher-order questions is part of monitoring learning.

Using cooperative small group work
Incorporating cooperative small group work into learners' experiences is part of planning.  There are many strategies that enhance small group work, such as starting with individual work followed by discussion in pairs that eventually join to form groups of four learners.  And there are many tasks that are best done, or that can only be done, by learners discussing, cooperating and reaching decisions in small groups.  Whether learners are working independently in small groups or individually, effective teaching includes observing and listening to learners as they discuss, write, and do things, intervening to prompt thinking.  Such observations can contribute significantly to the evidence of learning that is used to enable further learning.

Encourages reasoning rather than 'answer getting'
A conjecturing atmosphere encourages reasoning rather than 'answer getting'.  When learners' contributions to discussions, and responses to questions, are regarded as conjectures that may be modified, learners will contribute and respond confidently, without fear of being 'knocked down'.  Teachers' strategies may create or destroy a conjecturing atmosphere, and whether or not tasks and uses of resources are effective will depend on this.  When teaching encourages reasoning and explaining, learning is more easily monitored because there is more evidence.

Using rich, collaborative tasks
Good teachers plan to use rich, collaborative tasks. But the teacher's role is vital.  The 'richness' of a task depends on what a teacher does, and on the learning environment.  Therefore the richness of a task depends upon teaching strategies.  Because some tasks provide more opportunities than others for learners to make decisions, speculate, make and test hypotheses, prove or explain, reflect, discuss, interpret, search in new directions, invent, ask 'what if?' and 'what if not?' questions, be surprised, and enjoy themselves, they are potentially richer.  Good teachers recognise potentially rich tasks.  The richer the task is the more opportunities there will be to monitor learning.

Creating connections between topics
Teachers can plan to create connections between topics so that learners explore and understand ways in which mathematical ideas are linked.  Appropriate tasks, resources and activities can be selected so that seeing connections within mathematics enhances learning.  For example using algebra to help solve geometrical problems, or generating equivalent algebraic expressions by 'seeing' different structures in diagrams, is likely to deepen understanding of all the ideas involved.

Using technology in appropriate ways
Using technology effectively depends on understanding how it can provide opportunities for aspects of learning and doing mathematics, such as visualising, conjecturing, and making and testing hypotheses.  The learning needs to be planned for, and the learning atmosphere is crucial, as are the kinds of tasks and activities that learners engage in.  Learners may be contributing to class discussions generated by situations depicted, visualised, experimented with, reflected on and reasoned about using an interactive whiteboard.  Or they may be collaborating and discussing ideas in pairs or small groups working with spreadsheets or dynamic geometry software on computers.  Or learners may be using graphic calculators to explore number properties, patterns or relationships. Activities such as these, in which the resources are technological, enable the teacher to monitor learning by observing what learners do and say spontaneously, and how they respond to prompts and probing questions.

Planning, selecting teaching strategies, selecting suitable learning activities, selecting and using resources for learning, and monitoring learning are essential aspects of teaching that determine whether or not learners are given the best possible opportunities to learn and do mathematics.
 

Mathemapedia links

Taking action:

Research Sources

See also