The main teaching strategies used in effective mathematics lessons are: sharing learning goals, demonstrating and modelling, instructing, explaining how and why, illustrating, developing and consolidating, discussing, questioning, intervening, evaluating pupils’ responses, giving feedback and summarising.

These strategies are interwoven with providing varied opportunities for children to discuss, practise, investigate and solve problems, working independently as individuals, or collaboratively in pairs or small groups.

A good teacher will select teaching and learning activities and approaches as appropriate for different mathematical topics, the ages and interests of the children and their levels of attainment.

A range of teaching strategies is discussed in more detail in the questions that follow.

Related information and links

What makes some teachers of mathematics more effective than others? This is a short summary of research into effective teaching.

Demonstrating, explaining and modelling involve showing children how to do something or providing an image to help them to understand something.

Models and images help children to understand the structure of numbers, and how they can be partitioned and combined.

Careful choice of models and images can help children develop conceptual understanding, but it is important that the model or image you use helps children see the underlying structure that you want them to think about and doesn’t unintentionally limit the development of their understanding.

For example, if the only visual model of fractions that children see is the ‘chocolate bar’ image:

then they may fail to understand sufficiently the importance of fractions of a whole being equal sized divisions of the whole. They also need to see and discuss other shapes that have been divided up in different ways in order to explore this essential concept, for example images such as:

or

Images like these have the potential to help children develop a deeper conceptual understanding of what fractions are as they discuss why in each case the left hand image represents thirds but the right hand image does not.

Models and images for understanding and manipulating numbers in Years 1 to 3 is a very useful resource with a wide range of models and images to use in your teaching.

What this looks like in the classroom

Some examples of practical models and images are:

• number lines, tracks, grids such as a 100-square, to help children to see how numbers relate to one another;
• bead strings and counting frames, with each group of 10 distinguishable from the next, to give a linear image of tens and ones, and to provide model of finding complements of two-digit numbers;
• place value cards, place value charts and multibase materials to show how whole numbers and decimals can be partitioned in different ways;
• £1, 10p and 1p coins to show how each digit changes when multiples of 1p, 10p or £1 are added or subtracted;
• pegboards and other rectangular arrays to demonstrate the meaning of multiplication, including the commutative principle;
• empty number lines to support, record and explain calculations, e.g. 48 + 36 = 84

or:

Diagrams are another form of model. For example, these diagrams all model ideas associated with multiplication:

7 × 3 = (5 + 2) × 3
= (5 × 3) + (2 × 3)
= 15 + 6
= 21

12 × 6

Related information and links

How can practical resources support the development of mathematical understanding? is part of an NCETM CPD module.

Models and images - are you stuck in a rut? has ideas for some different models and images to support learning.

From Objects and Images to Mathematical Ideas and Models in Mind are NRICH discussion papers.

Whole-class discussion offers good opportunities for speaking and listening activities. There are many different ways that talk is used, for example:

telling, narrating, asking and answering questions, informing, explaining, instructing, describing, recounting, exploring, suggesting, hypothesising, speculating, collaborating, arguing, persuading, imagining, expressing feelings, reporting, summarising, evaluating…

and all of these have their place in the maths classroom. Skilled teachers think carefully about what types of talk they want the children to use in whole class discussions and plan their questions and prompts accordingly.

Used carefully, there is a place for whole class discussion at any stage of a lesson. An initial stage of teacher exposition can be enriched by focused whole class discussion, enabling you to get feedback on whether the children are following your main teaching points before you go further with the lesson content.

During a lesson you may notice that a lot of the children are having difficulties with one aspect of the work they are doing. This would be a good time to stop the lesson and discuss with the children what the difficulties are, asking them to try to explain what their problems are, sharing and discussing some of the errors before moving on.

Whole class discussion is also appropriate in a final plenary when you want to highlight the main learning points of the lesson, assess how successful the children’s learning has been and help the children reflect on what they have been learning.

However, whole class discussion can easily become dominated by a small number of more confident children, with other children then becoming passive. This is likely then to mean that most children will not benefit from the discussion, and you will get an inaccurate picture of learning across the whole class. Careful use of pair talk or the use of mini whiteboards are ways that all children can be encouraged to remain active in the discussion.

Effective discussion in a classroom requires a supportive learning environment. Children need to know that it is not a problem if they are wrong, and that it is okay if someone disagrees with them. Skilful teachers know that establishing this kind of ethos is essential.

What this looks like in the classroom

One way to prompt quality whole class discussion is to ask children to say whether they agree or disagree with something that another child has said – and what their reasons are.

For example, when working on the properties of 2D shapes you could prompt a discussion by asking the children:

“Is a square also a rectangle?”

Get them to discuss this in pairs and then take some feedback from some of the children. A child might say ‘No – rectangles are long and thin, they aren’t the same as squares’ and then you can ask others if they agree or disagree - and why. This might lead to a discussion of what a rectangle is, what its definition is. Write on the board what the children say and use these as further prompts to see if others agree or disagree.

Some teachers use the ‘no hands up’ approach as a way of encouraging all to take an active part in whole class discussion. Children are not allowed to put their hands up. Instead names are drawn at random when there is a question to answer – teachers often use lolly sticks for this, with each lolly stick having a child’s name on it. Again, a positive learning environment is needed – children need to know that it is acceptable if they don’t know the answer.

Related information and links

Volleyball NOT Ping Pong is a Mathemapedia discussion about how to generate purposeful discussion in the classroom.

Dialogues is an account of the way some KS2 teachers developed their understanding of discussion in maths lessons.

NCETM Evidence Bulletin - Cognitive acceleration includes consideration of the place of whole class discussion.

A closed question such as

• What are four threes?

has a single right answer.

Open questions such as:

• The answer is 12. What is the question?
• You know that 7 × 6 = 42. What else can you work out?
• What is the same about 16 and 36?

have many possible responses. Children can be encouraged to give different answers and can suggest ideas according to their level of skills and understanding. Such questions encourage the use of higher order thinking skills.

The distinction between open and closed questions is important, but it is helpful also to see how open and closed question can work together to enrich learning. For example, this question:

Is 12 + 13 bigger or smaller than 1?

is a closed question since there is a single right answer. But it can quickly lead to discussion of how we can be sure of the answer.

A sequence of questions such as:

• What 2D shape has four sides…
• And two lines of symmetry…
• And no right angles?

leads to a unique answer (a rhombus) but at each stage there are various possibilities that have to be considered and accepted or rejected as possible solutions.

What this looks like in the classroom

Closed questions can often be turned into open questions to make them more worthwhile. For example:

What is 324 ÷ 4?
can become:
Use the digits 0, 1, 2, 3, 4, 5.
Make three-digit numbers that have no remainder when divided by 4.

Find the perimeter of this rectangle

can become:

Draw some rectangles with the same perimeter as this one.

Complete this multiplication table

can become:

Investigate the possible ways of completing this multiplication table.

Following up children’s answers with ‘Explain’, ‘Why?’, ‘What makes you think that?’ and ‘Tell me more’ provides greater challenge.

Related information and links

Effective questioning is a CPD resource to use with colleagues.

Open and Closed Questions explores the relationship between open and closed questions in greater depth.

NCETM Evidence Bulletin - Questions, questions, questions

Using Questioning to Stimulate Mathematical Thinking is an NRICH discussion paper

In the Teachers TV video clip Questions the late Ted Wragg discusses his many years of research into the questions teachers use in classrooms.

Different types of questions in mathematics demand different skills, from simple recall of facts to problem solving. There is a tendency for questions that involve recalling and applying facts to be used more often than questions that require a higher level of thinking. Aim to use all types of questions over a period and encourage children to give more than short answers by explaining or justifying their answers.

The concept of higher order questioning relates to Bloom’s taxonomy. Bloom’s revised taxonomy outlines six types of thinking, all of which can be related to the types of questions teachers ask:

An NCETM funded research project looking at the use of Bloom’s revised taxonomy to develop the way secondary teachers asked questions in maths lessons devised a set of examples to help colleagues. For example, when developing understanding of prime factors they devised these examples:

Remembering:

• What is a factor?
• What is a prime number?

Understanding:

• Why is 7 prime?

Applying:

• What are the prime factors of 125? 81? 343?
• What do you notice?
• Can a prime number be a multiple of 4?

Analysing:

• How do you go about finding the prime factors of a given number?

Evaluating:

• Can you think of a number that has one repeated prime factor?

Or all different prime factors?

• Which numbers less than 100 have exactly three factors?
• What number up to 100 has the most factors?

Creating:

• The sum of four even numbers is a multiple of four. When is this statement true? When is it false?

What this looks like in the classroom

Recalling facts

• What are three fives?
• How many centimetres are there in a metre?
• How many faces has a cube?

Applying facts

• The answer is 15. What is the question?
• You know that 5 + 9 = 14. What else can you work out?
• Is this triangle equilateral? How do you know?
• What are the factors of 42?

Designing procedures

• How might we count this pile of counters?
• Are there other ways of doing this mental calculation?
• How shall we investigate the different triangles we can make on a 3 by 3 pinboard?
• How could we find the 20th triangular number?

Hypothesising or predicting

• What are the next three numbers in this sequence?
• Roughly, what is 48 × 62?
• Do you think that there will be a connection between eye colour and hair colour?
• How many tiles will be in the next diagram? And the next?

Interpreting results

• What does that tell us about numbers that end in 5 or 0?
• What does the graph tell us about the most common shoe size?
• What can we conclude about the perimeters of rectangles with the same areas?
• So what can we say about the sum of the angles in a triangle?

Applying reasoning

• Seven coins in my purse total 23p. What could they be?
• What is the biggest even number you can make with the digits 0, 3, 5 and 8?
• In how many different ways can four children sit at a round table?
• Why is the sum of two consecutive triangular numbers always a square number?

Related information and links

Using Questioning to Stimulate Mathematical Thinking

NCETM Evidence Bulletin - Questions, questions, questions

Questioning the use of Bloom's Taxonomy is a report of a research project by a group of secondary teachers which explored the use of Bloom’s taxonomy as a way to help them develop their questioning in maths lessons. You can also read their full final report and their examples of different kinds of questions suggested by Bloom’s taxonomy.

When children are working independently, there will be times when their reasoning and flow of ideas falter. If possible, interventions should prompt their thinking, rather than tell them what to do. It is also important to give them time to think before intervening.

It is helpful to think about the concept of ‘metacognitive questioning’. Metacognition means ‘thinking about thinking’. If a child is having difficulty with a maths problem a question such as ‘Which bit are you OK about, which bit are you finding hard?’ encourages the child to reflect on their own learning, to think about their own thinking. One of the aims of metacognitive questioning is to encourage children to take greater ownership of their own learning, to develop as independent learners.

What this looks like in the classroom

For example, when they are in the early stages of working on a problem and cannot get going, you might ask:

• What do you need to find out?
• What information do you have?
• Can you get any more information?
• How could you organise the information?
• Would it help to draw a diagram?

Once they have clarified what the problem is about, you might help them attack the problem by asking:

• What did you do last time?
• So what is different this time?
• Could you try a particular case?
• What would happen with a simpler problem?
• Why not make a guess and see what happens?
• Would it help to work backwards?
• Is there anything that you can eliminate? What won’t work, for example?

When a problem has been partly solved, you might ask:

• What do you still need to do?
• Is there any information you haven’t used yet?
• Would it help to arrange things in order?
• Would a graph help? Or a diagram?
• How many possibilities have you found? Are there any more? Could you count the possibilities systematically?
• Is there a pattern? How could you use it?

• How could you check your solution?
• How can you be sure that you have found all the possibilities?
• Can you explain your results?
• What if you changed the numbers, changed the rules, started with a rectangle rather than a square, …?
• What if you could only use …?

Related information and links

Metacognitive Questioning is a Mathemapedia entry which discusses the use of metacognitive questioning in maths lessons.

If you would like to explore the concept of metacognition further, the Research Gateway has links to research papers. Put ‘metacognition’ into the Research Gateway’s search facility.

How can we help pupils discuss mathematical problems together effectively? explores ways to encourage purposeful collaborative working and group discussion in maths lessons.

Children generally do not remember every detail of a lesson. A summary of the learning points can help them to crystallise what they have learned.

At the end of an activity, teachers often work with the whole class, summarising the learning points, going through some of the problems tackled, questioning children and rectifying any remaining misunderstandings. This time is also an opportunity to generalise from examples generated during the activity.

However, plenary sessions like this can take place at any stage of a lesson. Sometimes it is clear during a lesson that a number of children are having difficulty with some aspect of the work, or appear to have a shared misunderstanding. You might decide to summarise and review what they have been doing either as a group or with the whole class.

Equally you can start a lesson by getting the children to summarise the previous lesson or lessons.

However, it is easy in such plenaries for the teacher to dominate, and it is easy for it to become the teacher who is reviewing and summarising, not the children. This is likely to mean that the plenary has little if any impact on learning. It is important to use whole class discussion strategies that encourage every child to participate, for example using pair talk or mini-whiteboards.

What this looks like in the classroom

The final few minutes of a lesson, or of a sequence of lessons, again with the whole class, can provide a chance to:

• draw together what has been learned, highlighting the most important rather than the most recent learning points;
• summarise key facts, ideas and vocabulary;
• stress what needs to be remembered;
• make links to other work;
• highlight the progress children have made, suggest how they might improve their work in the future and remind them about their personal targets;
• explain what the class will go on to do next;
• when appropriate, set homework to extend or consolidate class work or to prepare for future lessons.

This review may vary in length depending on the lesson and the age of the children. The vital part is helping pupils to think about what they may have learned.

There are different ways to do this but a key feature is the involvement of children. For example, they can be asked to:

• write down three facts or ideas they have learned in the lesson, then share these with a partner, adding to the list;
• think of an important question to ask the rest of the class;
• write key words and definitions in their ‘maths dictionary’;
• describe ‘What I found difficult or easy was …’, ‘The most important/the most interesting part was …’, ‘I need to get better at …’;
• suggest a way in which ideas from the lesson might be used in other subjects or outside school.

Related information and links

ICT in the classroom - ICT to enhance the plenary

Ideas for plenaries in mathematics (.doc) and 28 Plenary ideas for mathematics (.ppt) are rich sources of ideas for plenaries from the Suffolk maths team.