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# Assessment in mathematics

## Making a judgement

At the end of a key stage, teachers should judge which level description best fits the pupil's performance. Each description should be considered alongside descriptions for adjacent levels. When making a judgement at the end of a key stage, you may wish to note the following points.

• You will arrive at judgements by taking into account strengths and weaknesses in performance across a range of contexts and over a period of time, rather than focusing on a single piece of work.

• A single piece of work will not cover all the expectations set out in a level description. It will probably provide partial evidence of attainment in one or two aspects of a level description. If you look at it alongside other pieces of work covering a range of contexts you will be able to make a judgement about which level best fits a pupil's overall performance.

### Giving pupils opportunities to demonstrate attainment

• Your pupils will need to use a range of forms of communication to show what they can do.

• In planning units of work and classroom approaches, you will need to provide opportunities for pupils to display their achievements in different ways, and to work in a range of situations.

### Recording

Although you will want to be able to explain why you have awarded particular levels to pupils at the end of the key stage, there is no requirement for judgements to be explained in a particular way or to be supported by detailed collections of evidence for each pupil. Decisions about collecting information, about its purpose and how it should be used are matters for teachers working within an agreed school policy.

## Progression in mathematics

The level descriptions indicate the progression in the knowledge, skills and understanding set out in the different sections of the programmes of study. These are:

#### At key stage 1

• number

• shape, space and measures.

At key stage 2

• number

• shape, space and measures

• handling data.

The mathematics programmes of study for key stages 1 and 2 and the primary framework for mathematics are fully aligned, with the framework providing a detailed basis for implementing the statutory requirements of the programmes of study.

Performance has been outlined here in terms of:

• progression by key stage

• progression by level.

### Progression by key stage

During key stage 1
Pupils develop their knowledge and understanding of mathematics through practical activity, exploration and discussion. They learn to count, read, write and order numbers to 100 and beyond. They develop a range of mental calculation skills and use these confidently in different settings. They learn about shape and space through practical activities, which build on their understanding of their immediate environment. They begin to grasp mathematical language, using it to talk about their methods and explain their reasoning when solving problems.

During key stage 2
Pupils use the number system more confidently. They move from counting reliably to calculating fluently with all four number operations. They always try to tackle a problem with mental methods before using any other approach. Pupils explore features of shape and space and develop their measuring skills in a range of contexts. They discuss and present their methods and reasoning using a wider range of mathematical language, diagrams and charts.

### Progression by level

#### Level 1

Typically, pupils:

• represent their work with objects or pictures and discuss it

• recognise and use a simple pattern or relationship

• count, order, add and subtract numbers when solving problems involving up to 10 objects and can read and write the numbers involved

• use everyday language to describe properties and positions

• measure and order objects using direct comparison, and order events

• sort objects and classify them, demonstrating the criterion they have used.

#### Level 2

Typically, pupils:

• select the mathematics they use in some classroom activities and discuss their work using mathematical language

• represent work using symbols and simple diagrams

• count sets of objects reliably, and use mental recall of addition and subtraction facts to 10

• use their understanding of place value to order numbers up to 100

• solve addition and subtraction problems

• use mental calculation strategies to solve number problems involving money and measures

• recognise sequences of numbers, including odd and even numbers

• use mathematical names for common 3-D and 2-D shapes and describe their properties

• distinguish between straight and turning movements, understand angle as a measurement of turn, and recognise right angles in turns

• use non-standard and standard units to measure length and mass

• sort objects and classify them using more than one criterion

• record results in simple lists, tables and block graphs, in order to communicate their findings.

#### Level 3

Typically, pupils:

• try different approaches to problems to overcome difficulties

• organise their work and check results

• discuss their mathematical work and explain their thinking

• use and interpret mathematical symbols and diagrams

• show understanding of place value in numbers up to 1000, use decimal notation and recognise negative numbers, in contexts such as money and temperature

• use mental recall of addition and subtraction facts to 20 in solving problems involving larger numbers

• use mental recall of the 2, 3, 4, 5 and 10 multiplication tables and derive the associated division facts

• solve whole-number problems involving multiplication or division, including those that give rise to remainders

• use simple fractions that are several parts of a whole and recognise when two simple fractions are equivalent

• classify 3-D and 2-D shapes in various ways

• use non-standard units, standard metric units of length, capacity and mass, and standard units of time, in a range of contexts

• extract and interpret information presented in simple tables and lists

• construct and interpret bar charts and pictograms.

#### Level 4

Typically, pupils:

• are developing strategies for solving problems and present information and results in a clear and organised way

• use their understanding of place value to multiply and divide whole numbers by 10 or 100

• use a range of mental methods of computation with the four operations, including mental recall of multiplication facts up to 10 10 and quick derivation of corresponding division facts

• use efficient written methods of addition and subtraction and of short multiplication and division

• check the reasonableness of their results by reference to their knowledge of the context or to the size of the numbers

• recognise approximate proportions of a whole and use simple fractions and percentages to describe these

• recognise and describe number patterns, and relationships including multiple, factor and square and begin to use simple formulae expressed in words

• use and interpret coordinates in the first quadrant

• make 3-D mathematical models and draw common 2-D shapes in different orientations on grids

• reflect simple shapes in a mirror line

• choose and use appropriate units and instruments, interpreting, with appropriate accuracy, numbers on a range of measuring instruments

• find perimeters of simple shapes and find areas by counting squares

• collect discrete data, group data where appropriate, draw and interpret frequency diagrams and construct and interpret simple line graphs.

Level 5

Typically, pupils:

• identify and obtain necessary information to solve problems and check their results

• show understanding of situations by describing them mathematically using symbols, words and diagrams and draw conclusions of their own explaining their reasoning

• use their understanding of place value to multiply and divide whole numbers and decimals

• order, add and subtract negative numbers in contextÂ· use all four operations with decimals to two places

• can reduce a fraction to its simplest form and solve simple problems involving ratio and direct proportion. They calculate fractional or percentage parts of quantities and measurements, using a calculator where appropriate

• understand and use appropriate non-calculator methods to solve problems that involve multiplying and dividing any three-digit number by any two-digit number. They check their solutions by applying inverse operations or estimating using approximations

• construct, express in symbolic form, and use simple formulae involving one or two operations and use brackets appropriately

• use and interpret coordinates in all four quadrants

• measure and draw angles to the nearest degree, and use language associated with angle. They know the angle sum of a triangle and that of angles at a point

• identify all the symmetries of 2-D shapes

• know the rough metric equivalents of imperial units still in daily use and convert one metric unit to another

• make sensible estimates of a range of measures

• understand and use the formula for the area of a rectangle

• understand and use the mean of discrete data and compare two simple distributions, using the range and one of the mode, median or mean

• interpret graphs and diagrams, including pie charts, and draw conclusions

• understand and use the probability scale from 0 to 1.

The attainment targets in mathematics set out the knowledge, skills and understanding that pupils of different abilities and maturities are expected to have by the end of each key stage. Attainment targets consist of eight level descriptions of increasing difficulty, plus a description of exceptional performance above level 8. Each level description describes the type and range of performance that pupils working at that level should characteristically demonstrate.

The level descriptions provide the basis for making judgements about pupils' performance at the end of a key stage.

The majority of pupils are expected to work at:

• levels 1-3 in key stage 1 and attain level 2 at the end of the key stage

• levels 2-5 in key stage 2 and attain level 4 at the end of the key stage.

By indicating expectations at particular levels and by charting broad progression in the subject, the level descriptions can also inform planning, teaching and assessment. Please note, the level descriptions are not designed to be used to 'level' individual pieces of work.

This content relates to the 1999 programmes of study and attainment targets.