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a rich task What teachers CAn do
Aspects of a rich taskIdeas for teacher supportStep into a problem even when the route to a solution is unclear (see definition of a problem below), getting started and exploring is made accessible to pupils of wide ranging abilities. Selecting appropriate tasks and problems for example those with a low threshold and a high ceiling.
Asking pupils to spend a little time on their own then working in pairs and then sharing ideas about what the problem is about and how to get started
Think ( Pair ( Share
Encouraging some general exploration of the situation before pinning things down
Considering and sharing different ways of representing the information
Thinking about things like this you have seen before.Pose as well as solve problems, make conjecturesIt is a challenge for pupils to pose their own problems so a first step is to model asking what if questions yourself.
Encourage learners to think about the things they can vary in a problem and to conjecture about the effect of any variation.
At the end of a problem ask what next? or If we had time what might we do next?
Highlight occasions where pupils do pose their own problems and share them with the group. Put unanswered questions and conjectures on a board.
Use a conjecture board. When pupils come up with a conjecture they write it up and get others to consider it and either prove it or find a counter exampleWork at a range of levelsEncouraging and discussing different ways of tackling a problem.
Interpreting and evaluating findings can offer opportunities to work at a range of levels.
Think about problems with open starting points, open middles and open ends these all contribute to allowing pupils to work at different levels.
Generalisation enables extension and the use of algebra can extend problems. Reflect on the algebra, when it is used, and how it represents underpinning structure of a problem. For example:
Why does .. generate a Fibonacci sequence?Extend knowledge or apply knowledge in new contextsSet problems that offer scope to extend knowledge or which are set in new contexts.
Ask questions of learners that encourage them to make connections:
Have you done something before that was similar?
What mathematics is in this problem?Allow for different methodsEncourage a range of representations at the start of the work.
Discuss ideas for different approaches.
Discuss different approaches, their effectiveness and efficiency at the end of the work.
Value different approaches as representing learners different understandings and levels of confidence.
Realise that methods used often reflect learners progress, areas of strength and weaknesses.
Aspects of a rich taskIdeas for teacher supportOffer opportunities to broaden students problemsolving skills Talk about what a pupil is doing. For example:
How will you collect the data?
Was that a good method, are there other ways that might have been more efficient?.
Can you be more systematic?
Can you generalise?Deepen and broaden mathematical content knowledge Use problems that offer challenging contexts in which can help develop content knowledge
Highlight the mixture of skills pupils are bringing to bear of problems:
In this problem you needed to be able to in order to .
Ask pupils what mathematics they used to tackle the problem, new things they have learnt and what they feel more confident about.Have potential to reveal underlying principles or make connections between areas of mathematics Problems like this might not be as engaging at first sight their fascination comes from the patterns or ideas they reveal as you work on them. For example:
The relationship between square and triangular numbers might come out of work on triangular numbers.
Games or problems that have the same underpinning mathematics (e.g. nim or variations on noughts and crosses)Include intriguing contextsUse games or challenges.
Use problems that reveal interesting patterns.
Identify mathematics in unfamiliar settings. When you notice some mathematics why not draw attention to it and use it. For example the sun shining through the window, arrangements of the desks, work on sports day such as laying out the track and recording results.
When you see something intriguing in some mathematics draw pupils attention to it. For example, an unexpected pattern in geometry or arithmetic that needs to be explained. That two shapes with the same volume look completely different. Make a note on the board and ask pupils to think about it and return to it at odd moments over a period of time.Offer opportunities to observe other people being mathematical or the role of mathematics within cultural settingsModel being stuck sometimes.
Allow pupils to ask and work on problems you do not know the answer to and say so. We will find out about this together
Use video and films related to mathematics being used or which put mathematics in historical and cultural contests. For example, when tackling a problem involving a Fibonacci sequence show some examples of its occurrence in the world around us. When talking about being stuck discuss what mathematicians do. When doing work on time look at how this has been measured in the past.
University of Cambridge
www.nrich.maths.org
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