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Mathematics Teaching Self-evaluation Tools

You are viewing a limited version of the NCETMâ€™s self-evaluation tools. Any answers you save during this session will be removed after seven days. Log in or sign up to view, and use, the full version of the tools.

For the following questions, select the statement which most accurately matches your level of confidence (1 is not confident and 4 is very confident) or choose from the alternatives detailed in the question. You do not have to answer all questions. Your answers will be saved so you can exit and come back to your self-evaluation at any time. Click Save and Results to view the next steps for questions you have answered.

1.

1. How confident are you that you are familiar with

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a.

a. the teaching strategies that lead to successful mathematical learning?

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

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

A good teacher will select appropriate teaching and learning activities and approaches for each lesson. These will take into account the mathematical topic, ages and interests of the children and their existing levels of attainment.

The range of teaching strategies is discussed in more detail in the questions in the next section.

2.

2. How confident are you that you know how and when it is appropriate to:

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a.

a. demonstrate, model and explain mathematical ideas?

Demonstrating, explaining and modelling involves 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.

Some examples of practical models and images are:

beads, counters and pegs to help children learn to count, represent quantities, to develop their own images of numbers, subitise (recognise the amounts of objects, when small, merely by looking, rather than explicitly counting) and develop early calculation skills.

number tracks and lines, and grids such as 100-squares to help children develop their understanding of 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

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

What this looks like in the classroom

Subitising is a key step in helping children to move from seeing and manipulating practically to seeing and manipulating through imagining. It is the immediate recognition of a small number of objects without needing to count them. This is a key skill in the development of children’s ability to count. Children can be helped to improve their ability to subitise by being shown sets of counters for a short period and asked to say what they saw and how many they saw.

For example, children are shown the counters below for one second:

Those children who can subitise will be able to tell you that there are five counters without having to count them all individually. They might say that they saw a group of 4 counters and another counter. They might say that they saw a group of two, another group of two and a single counter.

This sort of activity will help children to develop an image of the numbers and count them ‘in their head’. If they are shown counters arranged in familiar ways they will recognise the number even more quickly. For example, children will recognise six counters swiftly if they are arranged the same way as on dice or dominoes, rather than in a random arrangement:

b.

b. use whole class discussion?

Whole-class discussion offers good opportunities for speaking and listening activities. 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, 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…

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 some children are having difficulties with an aspect of their work. This would be a good time discuss these with the children, 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 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 become dominated by the more confident children, with other children becoming passive as a result. Careful use of paired work and talk can help to ensure that all children remain involved and contribute. It is then possible to follow this up in a whole class discussion.

During the whole class discussion, children are exposed to different perspectives that broaden their thinking. The teacher’s role is to ask questions and scaffold the discussion so that everyone can understand. Eventually you want all children to develop more efficient strategies and a deeper understanding of mathematical ideas because they have been exposed to their classmates' ideas and strategies.

What this looks like in the classroom

A Year 1 class was asked to share with a partner their strategy for adding 8 + 8. Denzie took 8 cubes and placed them on the table. Then she took out another 8 cubes, and started counting by pointing to the cubes. She lost track of which cubes she had counted because she was not physically moving the cubes, and said that the answer was 13.

Her partner, Ellie also took out 8 cubes. Then she took another 8 cubes and added them to the first 8 cubes to make one large pile. She gently moved each block to the other side of the table and counted them one at a time. After counting all the cubes, she decided that the answer was 16. Noticing that the children had different answers, the teacher asked Denzie and Ellie to work out which answer was correct. They started by counting out two piles of 8 cubes. Ellie gathered the cubes together and moved them while Denzie counted. They both agreed that the answer should be 16. Even though Denzie changed her mind that the answer should be 16, she did not realise that she needed to find more efficient ways to keep track of her counting.

At another table, Joshua and Jermaine used completely different strategies. Joshua took out only 8 cubes. He explained to Jermaine: I started with the number 8 and counted up to get the answer (16). Jermaine told Joshua that he worked out the answer in a different way. He told Joshua that 8 + 2 equals 10. So he re-conceptualised the problem as 10 + 6 = 16.

The partner discussion allowed the children to explain their thinking to their partners, but it would have been useful for them to hear how other children solved the problem. Ellie and Denzie could have demonstrated how they counted 16 cubes; Joshua could have shown why he counted 8 cubes, which was more efficient than Ellie's method. Jermaine could explain how his approach was similar to Joshua’s. The rest of the class could listen, ask questions, explain their thinking, and make their own connections to develop their understanding of addition strategies and their number sense.

c.

c. use open questions with more than one possible answer to challenge children and encourage them to think?

Closed questions such as:

What are four 3s?

have a single correct answer.

Open questions such as:

The answer is 12. What was the question?

What is the same about 16 and 36?

You know that 9 + 7 = 16. What else can you work out?

have many possible responses. Children can be encouraged to give different answers and can suggest ideas according to their level of skills and understanding.

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

What is 15 ÷ 3?

Can become

Using the digits 0, 1, 2, 3, 4, 5,
make two-digit numbers that have no remainder when you divide them by 3.

How many sides has this shape?

can become:

Draw some different shapes with the same number of sides as this one.

Complete this addition table.

can become:

Investigate possible ways to complete this addition table.

What this looks like in the classroom

You can take forward children's learning by listening carefully to their answers and responding with both open and closed questions.

Open questions such as:

How did you…?

Why does…?

How can we be sure that…?

Explain why…

What makes you think that?

How would you…?

What if…?

will help them to reflect upon their own mathematical thinking, to develop their ability to convince and prove.

It is tempting to ask a series of closed questions to check that children have understood a particular topic. For example,

What is 12 + 5?

How many sides has a square?

These questions are useful as an initial quick check, however it is important to follow them up. Such questions can be opened up, e.g. for What is 12 + 5?

What other pairs of numbers will give the same answer?

Similarly, for: How many sides has a square?

What else can you tell me about a square?

It is important to give children sufficient time to think through their answers before expecting a response. There are two crucial periods for thinking time. The first is after you ask a question. The second is after a child responds to the question.

The first waiting period is important because it allows children to consider the question and formulate a response. The second waiting period is crucial to encouraging that child to continue their response or to given another child the opportunity to extend the idea.

d.

d. use higher order or more demanding questions to encourage children to explain, analyse and synthesise?

Different types of questions in mathematics demand different skills, from simple recall of facts to problem solving. However, questions that involve recalling and applying facts tend 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.

Recalling facts, e.g.

What is 3 add 4?

How many centimetres are there in a metre?

How many sides has a hexagon?

Applying facts

The answer is 15. What is the question?

You know that 5 + 9 = 14. What else can you work out?

How do you know that this is a triangle?

What odd numbers lie between 10 and 20?

Designing procedures

How could you count this pile of counters?

Are there other ways of doing this mental calculation?

How could you find the 10th odd number?

How shall we investigate the different rectangles you can make on a 4 by 4 pin-board?

Hypothesising or predicting

What are the next three shapes in this sequence?

Roughly, what is 48 + 62?

Look at this shape that is partly hidden. What do you think the shape is? Why?

Interpreting results

What does the graph tell you about favourite colours?

What does that tell you about numbers that end in 5 or 0?

Applying reasoning

Two numbers have a sum of 12 and a difference of 6. What are the numbers?

What is the biggest even number you can make with the digits 0, 2 and 5?

Seven coins in my purse total 23p. What could they be?

How many different numbers can you make with the digits 2, 3 and 4?

What this looks like in the classroom

The aim in the classroom should be to strike a balance in the type of questions used. Relying upon a limited range of closed questions will lead to children becoming bored. More demanding questions that encourage children to explain, analyse and synthesise will help to maintain attention. However it is important to avoid the promotion of a confusing variety of approaches. Higher order questions can be repeated with sufficient similarity for children to become familiar with their style so that they begin to use this approach in their own thinking and discussion.

e.

e. intervene in the independent work of an individual or group?

If possible, interventions should help children makes sense of a problem rather than tell them what to do. Discussion makes the link between doing something and knowing something.

So when children 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 picture?

Once they have clarified what the problem is about, help them to 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?

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?

Could you draw a diagram?

How many possibilities have you found? Are there any more?

Can you spot a pattern? How might this help?

When they quickly reach a conclusion, you might encourage them to extend the scope of their activity by suggesting:

How could you check your solution?

How can you be sure that you have found all the ways?

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… ?

What this looks like in the classroom

It is important to judge carefully the point at which to intervene, and what form the intervention should take. Children learn by doing things with each other and by talking to each other and adults about what is taking place at each stage of an activity. Learning to exchange ideas through discussion, rather than short, quick responses to teacher questions, is a critical part of learning mathematics. This means that it is important to observe children first before being tempted to intervene. Assess the phase of a problem and tailor questions appropriately.

For example, a group of 7-year olds were working on a problem in which they used interlocking cubes, to which the teacher had assigned different 'prices', to make model dogs worth £2.50.

Cube

Price

Blue

5p

White

10p

Yellow

20p

Pink

50p

Orange

£1

Mei-Ling had gathered together a pile of 25 cubes of assorted colours and was sitting looking at them without seemingly doing anything.

The teacher came to the table. His initial reaction was to assume that Mei-Ling was stuck in the early stages of working on the problem and could not get going. He was about to ask questions like: Would it help to draw something? Why did you choose those particular cubes? Before he could do this, Mei-Ling looked at her friend across the table and said: My dog's going to be too expensive. The teacher asked: How do you know? This led to Mei-Ling explaining that she had added up the value of the pink cubes by counting in 50s and had got to 300.

The teacher asked Mei-Ling what she was going to do next, and she explained that she was going to start again, but this time keep track of the total value of her cubes as she took them.

f.

f. summarise and review the learning points in a lesson or sequence of lessons?

Children generally do not remember what they have learned and apply it to new situations. Discussing learning points with the class at the end of a lesson helps children to crystallise what they have learned.

At the end of an activity, teachers often work with the whole class, summarising the learning points, questioning children and rectifying any remaining misunderstandings. This time is also an opportunity to talk about other examples that match a general statement using what they have learned in the lesson.

The final few minutes of a lesson, or of a sequence of lessons, are also 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 introduced or developed in the lesson;

stress what needs to be remembered;

make links to other work;

highlight the progress children have made and what they will go on to do next.

This final few minutes may vary in length, from two minutes on one day to ten minutes on another, depending on the lesson. The vital part is helping children to think about what they may have learned.

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

share with a partner three things they have learned in the lesson;

think of an important question to ask the rest of the class;

write key vocabulary in their ‘maths dictionary’;

describe ‘What I found difficult or easy was …’, ‘The most important/the most interesting part was …’, or ‘I need to get better at …’.

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:

enable children to present and discuss what they have done during the main teaching activity

help children generalise a rule from the work that has been done by different groups, pairs or individuals during the main teaching activity

summarise and reflect on the lesson

assess and evaluate the progress that children have made during the lesson, including whether or not they have achieved the learning objectives

expose, discuss and address common misconceptions

extend the learning that has taken place in the lesson by using and applying new facts or skills to solve a problem

make links to other areas of mathematics and other subjects

set any work to be done at home if appropriate, and discuss next steps.