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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.

How confident are you that you are familiar with:

1

2

3

4

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, 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 pupils 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 pupils, and their levels of attainment.

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

2.

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

1

2

3

4

a.

demonstrate, model and explain mathematical ideas?

Demonstrating, explaining and modelling involves showing pupils how to do something or providing an image to help them to understand something

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

Some examples of practical models and images are:

place value cards, place value charts and multibase materials to show how whole numbers and decimals can be partitioned in different ways;

pegboards and other rectangular arrays to demonstrate the meaning of multiplication, including the commutative principle.

In the same way that working with physical quantities which represent mathematical ideas and relationships helps pupils to build correct abstract concepts of those ideas and relationships, so carrying out practical work in dealing with uncertainty helps them to develop correct conceptual understanding of probability. Electronic simulations will allow a large amount of experimental data to be collected very quickly, but working through a small number of trials directly themselves first is vital for pupils to appreciate what is actually happening in a simulation.

Multiple representations can be one of the most powerful mathematical tools available to mathematicians at any level. The different perspectives offered by the various available representations can offer illuminating insights and can consolidate the understanding of each. For example, the work on finding the nth term of a linear sequence to straight line graphs where the constant difference between terms corresponds to the constant rate of change (the gradient) of the graph, and the y intercept corresponds to the constant in the nth term expression (the equivalence of n = 0 and x = 0 in the two representations)

Demonstrating key ideas by the choice of vivid examples is an important skill for teachers to develop with experience. Pupils fail to appreciate the subtlety of the meaning of concepts at times. For example, they learn that pie charts are most appropriate for showing percentages. If they then have a set of data where the values are percentages, they may choose a pie chart when it is not appropriate because they fail to grasp the distinction between data that represent percentages of different quantities, as in the case illustrated below, and a graphical representation showing each data item as the proportion of the total of those data items. Moreover, in this situation, a package such as Excel will happily ignore a negative sign indicating a decrease and draw an impressive looking pie chart which is meaningless.

Percentage change in pass rates

English

10%

Maths

-5%

Science

12%

The appropriate graph for this data is shown below.

Percentage change in pass rates

English

10%

Maths

-5%

Science

12%

b.

use whole class discussion?

Whole-class discussion offers good opportunities for speaking and listening activities. The programmes of study for the National Curriculum describe the activities that should be available across the curriculum. Pupils are expected to meet different types of talk, such as telling, narrating, asking and answering questions, informing, explaining, instructing, describing, recounting, exploring, suggesting, hypothesising, speculating, collaborating, arguing, persuading, imagining, expressing feelings, reporting, summarising, evaluating, …

Some pupils find this difficult, and it may be useful to start the discussion in small groups within which each person has a number. Then in the whole-class discussion one representative from each group (the number being randomly chosen) can report on the group discussion and then the whole-class discussion can open out. This way, pupils see the process as transparent and they are not being ‘picked on’ at any stage, but everyone will contribute over time.

The choice of context is important. Many groups all reporting on the same task is likely to become repetitive very quickly as the first few reports will probably contain many of the remaining ideas. However, having groups investigate different aspects of a single problem and then report on it to their peers can provide a quite different dynamic. Since the tasks are all related, everyone knows something about the background to each report but have not looked at that aspect in depth themselves.

Another approach to achieve similar goals would be to ask different groups to do presentations from different perspectives. An example context might be a third world country announcing that it is banning major pharmaceutical companies from running drug trials in its country, and so it will lose the flow of subsidised drugs it currently has access to. The mother of an ill child in the country, the chairman of the company which distributes the subsidised drugs at the moment, the leader of the opposition, the wife of a man left paralysed after taking part in a drug trial last year because it was the only way he could earn money to feed his family, ... will have different views of the decision and can provide arguments from data to support their view. Another context might be a discussion about the different perspectives of how a water utility should be operated: should it maximise profit, minimise waste, maximise customer satisfaction?

c.

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

Closed questions like What are four threes? or What is the perimeter of this rectangle? have a single right answer. They have a place in mathematics as there are many instances in which there is a single right answer, although there may often be more than one way to get to that right answer. In some instances, discussion of different approaches may lead to greater insights – even in relatively simple situations such as calculating the perimeter of a rectangle, pupils may add the four sides, add length and width and multiply by 2, double each of the length and width and then add. The different strategies pupils choose to do a numerical example with can be used to reinforce simple algebraic rules involved in expansion and factorisation.

Open questions like: You know that 7 × 6 = 42. What else can you work out?
What is the same about 16 and 36?
have many possible responses. Pupils 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 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.

The first of these has a finite number of possible three-digit numbers which can be written down and tested for divisibility, giving a complete solution with a little patience and effort. However, there are strategies available which provide some real insights into the nature of divisibility by 4 which may be drawn out in class discussion by an expert teacher. Following up pupils’ answers with ‘Explain’, ‘Why?’, ‘What makes you think that?’ and ‘Tell me more’ provides greater challenge.

Pupils are adept at looking for hidden signals in the way teachers ask questions. ‘Are you sure?’ will often be interpreted as code for ‘you are wrong’, but if it is used both when the pupil is right and when they are wrong and they have to learn to think about it and decide whether they are sure or not, it can be a powerful learning experience for them.

The second rectangle problem has an infinite number of possible solutions, and it is immensely valuable for pupils to experience situations such as this – and this example shows that you do not need to be using sophisticated mathematics as the context for one of the most fundamental concepts in mathematics, infinity. It also allows the development of other key ideas, such as the capacity for generalisation. You cannot list all possible solutions to this problem in the way that you can the divisibility problem, but you can summarise all of them by a single rule identifying the width that the rectangle has to be given the length, and the range of possible widths that the rule is valid for.

With any class preparing for GCSE, this could lead on to an interesting investigation about the possible areas of a rectangle with fixed perimeter – with opportunities for using ICT to calculate multiple values, and use of graphical representations to make sense of the information gathered.

Central to the motivation for using open questions is to help pupils to learn to think for themselves, and a key aspect of this is to see opportunities for development of related activities for those who have made substantial progress on the initial task. For example, the investigation of the perimeter of rectangles with a fixed area is a substantially harder one analytically, but not arithmetically, and it provides an accessible way in for pupils to see why straight lines and quadratics are not sufficient to describe even relatively simple situations.

d.

use higher order or more demanding questions to encourage pupils 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 pupils to give more than short answers by explaining or justifying their answers.

Recalling facts

What is 60% as a decimal?

How many centimetres are there in a metre?

How many vertices has a cube?

What is a parallelogram?

Applying facts

How many centimetres are there is 8.3 metres?

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

How do you know that this triangle is equilateral?

What are the factors of 42?

Is every rhombus a parallelogram?

Designing procedures

Are there other ways of doing this mental calculation?

How shall you investigate the different triangles you can make on a 3 by 3 pinboard?

How could you find the 20th number in the arithmetic sequence 5, 8, 11, 14, 17, ...?

What properties do I need to define in constructing a quadrilateral in dynamic geometry software to ensure that it is a parallelogram?

Hypothesising or predicting

What are the next three numbers in this sequence? 1, 4, 8, ... (Different rules are possible.)

Roughly, what is 48 × 62?

Do you think that there will be a connection between age and the proportion of pupils who have part-time jobs?

How many tiles will be in the next diagram? And the next?

Interpreting results

What does that tell youabout numbers that end in 5 or 0?

What does the graph tell youabout the proportion of pupils who were drinking alcohol in 1994? (data in Drug use, smoking and drinking among young people in England in 2005, http://www.ic.nhs.uk/datasets)

Proportions of pupils who had an alcoholic drink
during the week before the survey (done in 1994)

What can you conclude about the perimeters of rectangles that have the same areas?

So what can you say about the sum of the angles in a n-sided polygon?

Applying reasoning

What is the largest amount of money I can have in coins without being able to make £1?

In how many different ways can four pupils sit at a round table?

Why is the sum of two consecutive triangular numbers always a square number?

If the radius of the medium-sized circle is 1 cm, what are the radii of the largest and smallest circles?

e.

intervene in the independent work of an individual or group?

When pupils 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.

For example, when pupils 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?

In the example of making a number of rectangles with the same perimeter, choosing a reasonably large value for the perimeter, and one which is even, will make it easier for them to produce a number of examples since integer pairs for length and breadth are the most obvious ones. Weak pupils may produce random rectangles and work out the perimeter of each, then listing the ones with the right perimeter but discarding it completely if it is not right . Getting them to draw one of their rectangles which did not work and asking them how it would need to change to be right can give them a concrete way to approach the problem: seeing that they can work out a length to make for a particular width may help them to start to generate more solutions in a systematic way. For stronger classes, encouraging an algebraic formulation of the problem may make it more accessible to pupils.

Once they have clarified what the problem is about, you might help pupils 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?

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 them systematically?

Is there a pattern? How could you use it?

For example, if pupils have found all the integer solutions to the rectangle problem they might plot them on a graph using width and length as the two axes and see that they lie on a straight line (x + y = ½ P). This could provide an opening for the teacher to ask about whether there are any other rectangles which lie in the gaps between the plotted cases (which have non-integer dimensions). It might also help the pupil’s recognition that once intermediate cases are allowed (and the obvious ones are when the dimensions end in 0.5) there is no limit to how many cases can be found, and that a generalised rule can be found which is actually much more powerful than any list could be.

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

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 (used a different perimeter – what would stay the same, what would change?), changed the rules (started with a rectangle rather than a square)?

What if the area of the rectangle had to stay the same – what could you say about the perimeter?

f.

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

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

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

The final few minutes of a lesson, or of a sequence of lessons, again with the whole class, 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;

stress what needs to be remembered;

make links to other work;

highlight the progress pupils 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 final review may vary in length, from two minutes on one day to ten minutes on another, depending on the lesson. 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 the pupils. 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.