Progress in Understanding Mathematics Assessment (PUMA) Interim Manual for Autumn tests – Years 3 to 6

Colin McCarty & Caroline Cooke

Copyright  2014 Hodder and Stoughton Ltd. Photocopying is prohibited. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, without permission in writing from the publisher. This publication is excluded from the reprographic licensing scheme administered by the Copyright Licensing Agency Ltd and may not be photocopied. Printed in England for Hodder Education, part of Hachette UK, 338 Euston Road, London NW1 3BH.

2

Contents 1 Introduction This interim manual What is PUMA? PUMA Curriculum Maps Why use PUMA? What does PUMA provide?

2 Administering the PUMA Tests When to test How to test Group size Timings Preparation Test conditions Administration

04 04 04 04 05 05

06 06 06 06 06 06 06 06

3 Answers and mark schemes

07

Marking and recording results PUMA 3 – Autumn PUMA 4 – Autumn PUMA 5 – Autumn PUMA 6 – Autumn

07 08 10 12 15

4 Obtaining and interpreting test scores Summative measures Reporting progress – the PUMA Scale Predicting future performance

5 Standardised score conversion tables PUMA 3 – Autumn PUMA 4 – Autumn PUMA 5 – Autumn PUMA 6 – Autumn

18 18 20 22

24 24 25 26 28

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1

Introduction

This interim manual for PUMA Autumn We have brought forward publication of the PUMA Autumn tests, so that you can start assessing Mathematics in the Autumn term, as you start using the new National Curriculum. To support you in using the PUMA Autumn tests, we have published this free teacher guidance which provides everything you need to administer and mark the Autumn tests. More extensive teacher guidance, including information relating to the PUMA Spring and Summer tests will be provided in the full PUMA Manuals – Stages 1 and 2, which will be published in March 2015, together with the PUMA tests for Spring and Summer. The PUMA Manuals will also include the following information, to assist you when using PUMA across the whole school year:     

Diagnostic and formative information Pupil profile sheets for each term, to enable you to review patterns of strengths and weaknesses across the year. Further information about interpreting and analysing results Technical information about the standardisation Teacher scripts and mark schemes for the PUMA Spring and Summer tests.

In the meantime, should you have any queries about using the PUMA Autumn tests, please email [email protected].

What is PUMA? Progress in Understanding Mathematics Assessment (PUMA) is a suite of tests written to the new National Curriculum. PUMA is designed to be used toward the end of each term, to measure and monitor pupils’ progress term by term, providing reliable, predictive and diagnostic information. The autumn test is a wider span test than the more focused Spring and Summer tests, with questions relating to mathematics covered in earlier years; it should be used to baseline the children (who will have only had at best one term of teaching on this year’s curriculum). PUMA is designed for whole-class use, with pupils of all abilities. The tests are easy and quick to administer – each taking between 30–60 minutes, depending on the year – and are straightforward to mark.

PUMA Curriculum Maps The PUMA tests provide thorough coverage of the new National Curriculum Programme of Study for the particular year. We have created Curriculum Maps, breaking down the Programme of Study for the year term by term. These Curriculum Maps help to define what PUMA assesses each term. The Curriculum Maps can be downloaded from www.hoddereducation.co.uk/puma. Schools taking part in the standardisation of PUMA followed these Curriculum Maps to guide them in delivering the new National Curriculum, before it was statutory. This made the standardisation of the PUMA tests a valid assessment of the new National Curriculum.

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Why use PUMA? PUMA provides reliable summative information. For example:  



PUMA uniquely provides three carefully designed tests for each year, enabling you to follow the progress of your pupils from term to term, as well as year to year throughout primary school. Marks have been calibrated onto the PUMA Scale to allow you to follow progress term by term and compare progress to national norms – see page 20 for further details. It allows you to predict what score they should obtain in subsequent terms and so set meaningful targets. However, if you need to establish a National Curriculum level for each pupil, PUMA tests are calibrated to indicate National Curriculum levels.

Also, because it has a diagnostic capability, PUMA enables you to investigate some of the strengths and weaknesses of your pupils’ mathematics skills. To enable you to use the information in a diagnostic/formative way, total scores can be broken down into distinct aspects of mathematics, giving a useful profile which reflects the categories of the new National Curriculum. These are:  Number, place value and rounding  Addition, subtraction, multiplication and division; algebra  Fractions, decimals and percentages, ratio and proportion  Measures  Geometry: shapes position, direction, motion  Statistics, data handling.

What does PUMA provide? PUMA provides a standardised assessment of a pupil’s mathematics attainment, plus a profile that helps you to identify pupils who may need further teaching and practice, as well as helping you to identify where they are doing well. It provides four ‘global’ measures of mathematics attainment for each pupil:    

an age-standardised score (from which a percentile can be derived) a Mathematics age a score on the PUMA Scale National Curriculum sublevels and APP level.

Each test also gives a points score (widely used by local authorities). The PUMA test results have been statistically linked from term to term and year to year to enable you to track or predict progress through the whole primary phase. This also enables detailed comparison of individual patterns of performance against the norms and patterns for the term. Underpinning all this is the PUMA Scale: this gives the equivalent of a decimalised level which enables you to monitor small increments of progress from term to term. Although National Curriculum levels are no longer in use, these tests carry forward the standard from 2014, so that you have a measure that may be compared back to previous years, at this time of transition. The PUMA Scale acts as a common ‘spine’ on which all the PUMA tests across the primary phase are plotted (Table 4.3 on page 20 draws this all together). It provides the statistical basis for predicting pupil progress and future attainment, based on the termly performance data of over 10,000 pupils nationally.

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2

Administering the PUMA tests

When to test The PUMA tests have been designed to assess the National Curriculum objectives presented in the PUMA Curriculum Map for that term. They should, ideally, be used just before the end of term.

How to test Give each pupil a test booklet and ask them to write their names on the front cover. Before the test, tell pupils these key points:    

That pupils need to read the questions themselves, but weak readers may be given help with reading the question (see ‘Administration’ below). There will be some sections they can do easily, particularly the earlier questions. They shouldn’t worry if they find some questions difficult. They should just try their best and move on to see if they can answer some of the following questions. Ask them to write answers clearly. If they change their mind, they should cross or rub out the wrong answer and write in the new answer.

Group size You can administer the tests to a whole class or large group, if you feel comfortable doing so. With weaker Year 3 children, however, it may be better to work with small groups, with the TA also delivering the test – for example, five or six children of similar ability – so that pauses can be taken, if required.

Timings A maximum time limit of 50 minutes is advised for the Year 3 and 4 tests, and 60 minutes for the Year 5 and 6 tests. In the PUMA standardisation we found that it took well under one minute for a mark for most pupils, unless they were particularly hesitant or slow workers, where extra time may be allowed.

Preparation Each pupil will need the appropriate test booklet and a pencil or pen.

Test conditions For results to be reliable, it is important that the pupils work alone, without copying or discussing their answers.

Administration If any pupils are uncertain about what they need to do, you may give additional explanation to help them understand the requirements of the test. Do not, however, help with the mathematical content of the question or read out any of the actual numbers that form part of the question.

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3 Answers and mark schemes Marking and recording results Use the score box in the right-hand margin alongside each question in the test booklets to record marks. Some questions have more than one part, or attract more than one mark, so please follow the mark scheme carefully. No half-marks should be awarded. Beneath each score box there are code letters indicating the category of maths the question focuses on (using abbreviations for numeracy, operations, fractions, geometry, measures, statistics and problem solving). If you would like to profile the pupil’s performance, add up the number of marks they have obtained in each coded category and record them on the front cover of the test booklet. You can record total marks for the page at the bottom of each page in the test booklets. Then add together the page scores to find each pupil’s total raw score and record this at the bottom of the front cover. Please use your professional judgement when marking. For example, accept mirror reversals of single digits but not reversals of double-digit numbers. Equally, any clear indication of the answer is acceptable, irrespective of what was asked for (e.g. an object ticked instead of circled). When you have calculated the total raw score for each pupil, refer to the conversion tables to obtain:  the age-standardised score  Mathematics age  PUMA Scale score  plus National Curriculum sublevel and APP level, as required. See ‘Section 4: ‘Obtaining and interpreting test scores’ on page 18 for further details.

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PUMA 3 – Autumn answers and mark scheme Answer

Category

NC level

ops, PS geom

1b 2b

ops

1a

frac num num num

2b 2b 2a 2c

ops ops stat stat stat, PS ops, PS meas, PS meas, PS

2c 2b 2c 1a 2c 2a 2b 2c

11

8 Join boy to the cylinder and girl to the triangular prism. Both required. 8-5=3 3+6=9 Both required. No mark if more than these two ticked. Any four small triangles shaded (a) 40 (b) 30 27, 31 and 55 circled No mark if more than these three circled. (a) 60 (b) 11 (a) France (b) Greece and Turkey (c) 25 14 children (a) Pencil and Pad (b) A 20p, 5p and a 2p circled No mark for 10p, 10p, 5p, 2p. 14cm

meas

3b

12

A cross on the pentagon in the hexagons section

geom

2a

13

203, 223, 230, 302, 320

num

2b

14

Any two identical numbers Accept two zeros. 83 and 103

ops

2c

num

2c

445 and 145 Crosses drawn on both parallelogram and right-angled triangle only. 7 groups 573 Both 740 and 46 only

num geom

2c 3a

ops, PS ops num

3b 3c 3b

Square correctly completed within 2mm of correct vertex, even if ruler not used

geom

3b

1 2 3

4 5 6 7 8

9 10

15

16 17 18 19 20

8

21

6

ops

3c

ops

3b

frac

3a

geom

3b

4×8 36 20 × 3 4 12 ÷ 3 32 24 ÷ 4 60

22

8 All three correct for 2 marks, two correct for 1 mark. 15

23

24

30

num, PS

3c

25 26

845 + 1 (a) 40 minutes (b) any one section shaded on cake on left any two sections shaded on cake on right Both required Vertical faces need not be shaded. 60 120

ops meas

3a 3a

frac

3a

frac frac frac frac

3a 4c 3a 3a

ops meas, PS

3a 4b

27 28

¾ 1½ Both joined near enough to be unambiguous.

0

1

½ 29 30

¾

2

1½

118 30cm or 0.3m if unit altered

PUMA 3 – Autumn: Analysis of performance by category Category Number Operations Fractions Measures Geometry Statistics Total Problem solving

Number of marks 8 12 7 5 5 3 40 8

National average mark 4.4 6.6 1.6 1.5 1.8 2.4 17.9 4.2

National % 55 52 23 30 36 80 45 60 9

PUMA 4 – Autumn answers and mark scheme Answer 1 2 3 4 5 6 7 8 9 10 11

12

13

14 15 16 17

18 19 20

21 22

Category

NC level

20 47p (a) D (b) C (a) 25 children (b) 9 girls 16mph Any two rectangles shaded 250 24 and 48 Both required. C and D only

ops meas geom geom ops ops, PS meas frac num num

2c 2c 2b 2a 2c 2b 2a 2a 2b 2b

geom

3c

60 cherries

ops, PS num

2a 3c

num

3c

geom geom

3c 4b

meas num num

3c 2a 2a

stat stat, PS stat, PS frac

2c 3a 4c 3b

frac

3c

ops ops ops meas

3b 3b 3a 3a

ops frac

4c 3b

rounds to 150 rounds to 160 154 161 156

rounds to 170 169 172

All four numbers correct. 50 30 16 All three required. (a) Isosceles triangle and trapezium ticked, only. (b) Trapezium (correct spelling not required) Do not accept quadrilateral. 20 minutes 4 12 and 7 × 3 or 3 × 7 Both required. (a) Ben and Sarah (b) £15 (accept £15.00, but not £15.0) (c) £80 (accept £80.00, but not £80.0) 7.3 Accept 7 3/10 31/3 and 42/3 Both required. < < > 30 weeks 522 10

10

23

1

/4

frac

4c

frac

4c

meas frac geom num

3a 4c 4c 3c

num, PS

3a

ops, PS ops, PS

4b

0.5

0.25 1

/2

0.04 0.4

4

/10

0.14

24 25 26 27 28

8 jugs 15 eggs (8, 3) 10 735 Accept with or without space or comma. 958 or 938

29

(a) 9 flags

30

(b) 8 flags 8575

ops

4b 4b

PUMA 4 – Autumn: Analysis of performance by category Category Number Operations Fractions Measures Geometry Statistics Total Problem solving

Number of marks 8 11 7 5 6 3 40 7

National average mark 5.0 4.8 2.2 2.4 2.5 1.5 18.3 2.3

National % 50 43 32 48 42 49 46 33

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PUMA 5 – Autumn answers and mark scheme Answer

Category

NC level

1

35 pebbles

ops

2a

2

Any four rectangles shaded.

frac

3b

3

(a) 7 children (b) 22 children

stat, PS stat, PS

2a 2a

4

2¾ and 1¼

frac

4b

5

Bottom right

geom

3c

num num ops, PS meas frac

2a 3c 3c 4c 3c

num

3b

frac

3b

meas

3a

mirror line

6 7 8 9 10

(a) 70 (b) Any one of 32, 33 or 34 52 sweets 4.5cm2 or 4½ cm2 1.1 and 2.1 Both required. All four correct.

456

0 to 200

192

201 to 400

758

902

89

401 to 600

601 to 800

over 800

11

3

12

(a) £3.54 (b) £6.46 Accept follow through, i.e. if the answer to (a) and (b) add up to £10 (c) 8

/8

meas, PS 4c meas, PS 3b

12

13

717

ops

3a

14

(a) 764 (b) 647 or 627

num, PS num, PS

2a 3c

15

1, 2, 3, 5, 6, 10, 15, 30 All required for the mark. Do not penalise incorrect ordering, the instruction is there to assist marking.

ops

4b

16

(a) 3 (b) 50

ops ops

3a 4c

17

(a) 6 edges (b) 8 faces (c) 12 vertices

geom, PS 3a geom, PS 4a geom 4a

18

(a) 20 (b) 75 (c) can’t tell

stat stat stat

3c 3a 3b

19

4336

ops, PS

4a

20

(a) 60 (b) 90 (c) 40

frac frac frac

3b 4a 5c

21

592

num

3a

22

5 cups

meas

3b

23

1 hr 40 min

meas

4a

ops

3b 3a

24

× 9 3 6 25

26

8 72 24 48

3 27 9 18

4 36 12 24

ops

8 or 9 for 2 marks; 6 or 7 correct for 1 mark. 1 mark for both nearest 100 i.e. 36 300 and 79 800 1 mark for both nearest 1000 i.e. 36 000 and 80 000

num num

4b 4a

(a) 100 (b) 7500

ops ops

4b 4b

13

27

2

/100

1

/2

2

0.5 0.25 0.2 0.4 0.02

/5

28

(a) A and C only (b) C and E only

29 30

23 and 29 only 2 (a) /3 < (b)

3

/4

>

5 8

/6

/12

frac

3b

frac

5c

geom geom

4b 5c

ops frac frac

4a 4b 4b

31

a = 3cm b = 7cm Both required.

meas

5c

32

7656

ops

5c

PUMA 5 – Autumn: Analysis of performance by category Category Number Operations Fractions Measures Geometry Statistics Total Problem solving

Number of marks 8 13 11 7 6 5 50 10

National average mark 4.3 5.0 4.0 2.3 2.0 3.1 20.7 5.5

National % 54 39 36 33 33 63 41 55

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PUMA 6 – Autumn answers and mark scheme Answer

Category

NC level

1

1052 and 1152 Both required.

num

3c

2

4¾ and 5¼ respectively Both required.

frac

3c

3

(a) 22 (b) 90

ops, SP frac

3c 3b

4

Rectangle and both triangles ticked only

geom

3b

5

£3.75 Accept £3.75p, £3-75, £3-75p. Do not accept £375 or £375p.

meas

3b

6

95km/h Accept 94-96

meas

3c

7

819

ops

3b

8

9 and 32 15 and 45

ops ops, PS

3c 3b

num

3b

frac

3a

geom geom

4c 4a

num

4c

Both required. Both required.

9

positioned closer to middle i.e. 2500, than 2000 or 3000 Accept arrows in the range 2250 and 2750 10

3/4, 0.52 and 55% only

11

number of edges

number of faces

number of vertices

cube

12

6

8

triangular prism

9

5

6

square based pyramid

8

5

5

All correct for 2 marks; any 2 rows correct for 1 mark. 12

Only 164g and 159g ticked

15

13

ops ops

3c 3a

7 or 8 correct for 2 marks; 5 or 6 correct for 1 mark. 22cm

ops

3b

15

(a) Range 85-86cm (b) Range 45-46 months (c) Range 2-3cm (d) Range 16.5-17.5cm

stat stat, PS stat, PS stat, PS

3a 4b 4b 6c

16

12020

ops

3b

17 18

8 weeks 3 (a) /10

ops frac

4b 4c

1

frac

5b

meas

3b

meas

4c

ops, PS ops num frac, PS ops frac

4c 4a 4c 4b 4b 4a

num num

3a 4c

ops

5b

meas meas

5a 4a

14

×

12

9

7

8

96

72

56

11

132

99

77

6

72

54

42

(b) /3 19

20 21 22 23 24 25

26 27

kilograms 5kg 2.4kg

grams 5000g 2400g

½kg

500g

12.5kg or 12 500g 12½ kg All correct for 2 marks; 2 correct for 1 mark. (a) £128 (b) 30 people 400 000 and 600 000 Tom 253 61/8, 6¼, 6½, 65/8, 6¾ All required. (a) 3000 (b) 60 1, 2, 3 and 6 only. Ignore the omission of 1 (a) 54cm2 (b) 48cm

16

28

Always

Sometimes

Multiples of 7 are even… Prime numbers have exactly two factors… Square numbers have an odd number of factors…

Never

ops, PS

3a

ops, PS

6c

frac

3a

  

30

All correct for 2 marks; 2 correct for 1 mark. 590 1000 0.59 All correct for 2 marks; 2 correct for 1 mark. Top vertex

31

16

frac

32

28cm

meas, PS 5b

33

(8, 10)

geom, PS 5a

34

(a) 76300 (b) 88 508 Accept answer for part (b) if it is 12 208 greater than the answer for part (a). 999 940

ops ops

5c 5a

num, PS

6c

29

35

4a frac geom, PS 5b

5c

PUMA 6 – Autumn: Analysis of performance by category Category Number Operations Fractions Measures Geometry Statistics Total Problem solving

Number of marks 7 17 10 7 5 4 50 13

National average mark 3.8 8.7 4.3 3.0 1.7 1.5 23.0 5.8

National % 54 51 43 43 34 38 46 32

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4

Obtaining and interpreting test scores

Summative measures The results obtained from PUMA will enable you to report pupil performance in terms of:     

Age-standardised scores (see Section 5) Mathematics ages (Table 4.2) National Curriculum sublevels (Table 4.3) APP level, subdivided as high, secure and low (Table 4.3) The PUMA Scale (Table 4.3).

For a swift overview, you could compare how well a pupil has done by comparison to Table 4.1, which shows average scores for each year group, by gender, for each PUMA test. You can also compare your own class average raw scores against these averages. Table 4.1: Average test scores by term and gender Year 3 4 5 6

Autumn test Boys Girls Total 17.9 18.4 17.3 18.3 18.6 18.0 20.6 21.8 19.3 23.0 23.8 22.3

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Mathematics ages Many teachers use ‘Mathematics ages’ as a quick reference: they shows the average chronological age of the pupils who obtained each particular raw score (i.e. the chronological age at which this level of performance is typical). For more detailed comparative information, however, and especially for tracking progress over time, please refer to standardised scores. Table 4.2: Mathematics ages for the Autumn term PUMA 3 – Autumn raw score 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Mathematics age 11:3

PUMA 6 – Autumn raw score 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Mathematics age