Mathematics is certainly a subject that can develop good thinkers. This development needs to be planned for as it does not always happen by chance. Thinking skills are essential for using and applying mathematics and efficient and creative problem solving, communication and reasoning.

These enable pupils to locate and collect relevant information, to sort, classify, sequence, compare and contrast and to analyse part/whole relationships.

These enable pupils to give reasons for opinions and actions, to draw inferences and make deductions, to use precise language to explain what they think and to make judgements and decisions informed by reason or evidence.

These enable pupils to ask relevant questions, to pose and define problems, to plan what to do and how to research, to predict outcomes and anticipate consequences and to test conclusions and improve ideas.

These enable pupils to generate and extend ideas, to suggest hypotheses, to apply imagination and to look for alternative innovative outcomes.

These enable pupils to evaluate information, to judge the value of what they read, hear and do, to develop criteria for judging the value of their own and othersâ€™ work or ideas and to have confidence in their judgements.

In the document Leading in Learning the UK Department for Education and Skills suggested ten teaching strategies to address these thinking skills, each strategy being suitable for developing one or more of the thinking skills.

The following links provide resources on the Transum website to support these strategies. [Click the buttons to see the links]

Advance organisers are devices used to enable pupils to orient themselves to a topic through what they already know. They are organisational frameworks that teachers present to pupils before teaching a topic to prepare them for what they are about to learn. It could be: a handout outlining what will be covered in the topic; concept map; spider diagram; flow chart; story or anecdote; or study guide. The chosen advance organiser should help pupils access what they already know about a topic and focus them on the new information.

Examples

- Code Cracking View this presentation on code cracking before learning the techniques and practising the skills.
- Maths Videos Introduce a new topic to the class with a video showing the 'big picture'.
- Probability Line Before learning how to do probability calculations make sure the pupils understand the 'big picture' ideas by aranging these statements in order of likelyhood.
- Reverse Bar Chart Learn how to update this unusual graph in preparation for lerning the times tables.

An analogy, in this context, is being used to describe a teaching device that helps pupils understand an unfamiliar concept or process by comparing it with familiar objects or processes.

Examples

- Maths Analogies Match the mathematical phrases to the pictures of everyday ideas and concepts.
- Stable Scales A lesson Starter and a self-marking exercise based on the notion of old-fashioned weighing scales representing balanced equations.
- Washing Line A washing line can be used to represent the continuum of probability between 0 and 1.
- Fraction Wall The traditional fraction wall diagram showing the relationship between simple fractions.
- Composite Shapes Using letters to show how the areas of composite shapes are calculated.

In life, we spend a lot of time either making things or constructing messages (communicating with people) - both can be regarded as products. These products are usually designed for a particular audience with a particular purpose, although these are not always clearly defined. This strategy enables pupils to give consideration to audience and purpose. The audience could be people of a particular age, from a particular region or with a common interest. The purpose could be to entertain, inform, explain, persuade, serve a practical need or a decorative function.

Examples

- Hurdles Race A distance/time graph is presented in an animated way while pupils interpret the meaning of the graph for an (imagined) TV audience.
- Maths Words Members of the class give clues so that the person with their back to the screen can guess the word.
- Wordles Construct cartoon-like images of mathematical words which communicates their meanings.
- Sum Story Make up a real life story for the given calculations.
- The Story of ... Be creative and come up with as many facts about a number as you can think of to tell its story.
- Maths Display Inspirational ideas for your classroom walls communicating a range of mathematical ideas.

Classifying is a thinking skill we use naturally to organise information and ideas. It is a vital skill for processing information and for the ability to use and apply information in new ways. A common way of setting up a classification task is by means of a card sort, although it can also be carried out using objects rather than cards. Pupils work together to sort these into groups that have shared characteristics, which establish criteria for a classification group. Having to consider and justify their criteria helps them to develop their skills and understanding.

Examples

- True or False Arrange the given statements in groups to show whether they are always true, sometimes true or false.
- Satisfaction Is it possible to rearrange the numbers, row and column headings so that the table is mathematically correct?
- Satisfy An easier version of Satisfaction (above).
- Venn Diagram Arrange the numbers 1 to 16 in the three intersecting circles above to show the sets of even, prime and square numbers.
- Happy Numbers Classify numbers as happy or unhappy. A drag and drop exercise.
- Pesto Students classify numbers randomly appearing on the screen by holding up cards.

In this strategy pupils work in small teams to recreate a map, picture, diagram, photograph, advertisement, poem, sheet of music or other item that has some obvious physical structure. Each team sends one member at a time to look at the image for 10 seconds. They return to their group and start to reproduce the original. After a short period of time, the next representative from the group looks at the map for 10 seconds. After each turn, groups reflect and plan the next visit. After a few turns each, pupils are asked to compare their versions with the original.

Examples

- Mathterpiece Groups of pupils are asked to memorise a picture made up of geometrical shapes.
- Pairs A memory game which can be played with two teams.
- Kim's Game This is the Maths version of the traditional memory game.
- Ancient Mysteries This activity requires students to memorise fifteen numbers in a three by five grid.
- Memor Sea Mental activity in which the students have to memorise numbers.
- Number Recall Can you improve your ability to remember telephone numbers?
- Shopping List A quick quiz about five items on a shopping list written 40 years ago.
- Memory This a main Transum topic page with links to starters, activities, videos and investigations.

Living graphs and fortune lines are strategies that relate to graphical representation. Both strategies require pupils to consider how one variable relates to another, such as the heart rate of a football player over the period of a match or the mood of Hamlet during different episodes of the play. In Living graphs a line graph is presented, together with a set of related statements. Pupils have to position the statements on the graph and give reasons to justify their decisions. In Fortune lines pupils are asked to suggest a scale and then to plot the fortunes or emotions of one or more individuals over a sequence of episodes in time, and then to justify their decisions.

Examples

- Helicopter View A good introduction to distance-time graphs.
- Hurdles Race A distance/time graph is presented in an animated way while pupils interpret the meaning of the graph for an (imagined) TV audience.
- Cartoon Scatter Graph Place the cartoon characters on the scatter graph according to their height and age.
- Human Scatter Graphs Pupils move to positions in the room according to their data relative to the walls as axes.
- Graphs A list of lesson starters and pupil online activities.

In a mystery pupils are presented with between 15 and 20 items of data on slips of paper about a situation where there is a single open question or problem for them to resolve. The statements can be general or background information, specific details and sometimes 'red herrings' or irrelevant information, but always there is an element of uncertainty or ambiguity. Pupils work in groups to read and sort the statements, link information on different cards and come up with a solution to the mystery question. Later they are asked to explain their answer.

Examples

- Pattern Clues Use the clues given to create a colourful pattern in the grid provided. Not all of the clues are needed to complete the pattern.
- Scheduling Puzzle Arrange the schedule taking note of all the restrictions.
- Mystery Numbers Strange clues for everyday number expressions.
- Scaramouche Can you work out from the five clues given what the mystery number is?
- Tran Towers An online activity designed to stimulate a range of thinking skills by challenging learners to find their way through the rooms of this vast building and solve mathematical problems on the way.
- Tran Tunnels Can you find your way through this puzzle ridden maze to find Goldberg's magic harpsichord?

This very basic but powerful technique involves providing pupils with a photograph or other visual image as a source of information and asking them to annotate or label it. They are asked to make links to what they already know, whether from previous work or general knowledge, and should suggest a title or overall heading for the image. There are variations around this basic approach. As with other thinking strategies, it is important for pupils to be able to explain their thinking to others.

Examples

- Great Dodecahedron Pupils look at the picture an describe what the can see. They are not allowed to use their hands to point but must describe any shapes they can see in the picture so that the rest of the class can understand what they are describing.
- Start Wars Day Your classroom has been visited by a giant Jedi Warrior. He has left only his hand print on the white board.
- Missing Square Puzzle Why is there one square missing in the second triangle?
- Optical Illusions This flexible visual aid can be used as a stimulus for mathematical discussion.
- Hot Estimates Estimate the number of chillies in this photograph. What strategies can be used to make a good estimate?

Relational diagrams provide a clear and accurate medium through which pupils can communicate their thinking. They illustrate the meaning that pupils give to terms that stand for classes of objects or concepts. Pupils are able to use overlapping, separate or subsumed shapes to show whether all, some or none of the terms of a particular class belong to another class. The visual simplicity of relational diagrams makes the explanation of the relationships easy to understand and more likely to be remembered.

Examples

- Satisfaction Is it possible to rearrange the numbers, row and column headings so that the table is mathematically correct?
- Venn Diagram Arrange the numbers 1 to 16 in the three intersecting circles above to show the sets of even, prime and square numbers.
- Polygons Pupils construct relational diagrams for quadrilaterals, polygons and equal-sided shapes. They then place particular shapes into the diagram.

We use summarising naturally, for example, when recounting an event. But effective summarising, selecting salient points and presenting them in a concise and ordered manner, is a skill that needs to be developed. Pupils who tend to give narrative accounts when they summarise need to make the step to sifting out themes and main messages. The basic idea is for pupils to find the main threads in the information and make connections between these threads.

Examples

- Sequences Finding the rule for a given sequence is the ultimate 'summary'.
- Mental Methods Pupils devise, explain and practise techniques for performing mathematical calculations in their heads.
- Strategy Games Finding the winning strategy is a way of sumarising how a game is played.