**mathematics** a s s e s s m e n t r e s u l t s 2001 nemp

Mathematics

Assessment Results

2001

EARU

Terry Crooks

Lester Flockton

national education monitoring report 23

Mathematics

Assessment Results

2001

Terry Crooks

Lester Flockton

with extensive assistance from other members of the EARU team:

Lee Baker

Linda Doubleday

Liz Eley

Kathy Hamilton

Craig Paterson

James Rae

Miriam Richardson

Pamala Walrond

EARU

NATIONAL EDUCATION MONITORING REPORT 23

©2002 Ministry of Education, New Zealand

This report was prepared and published by The Educational Assessment Research Unit,

University of Otago, New Zealand, under contract to the Ministry of Education, New Zealand.

NATIONAL EDUCATION MONITORING REPORT 23

ISSN 1174-0000

ISBN 1–877182–34-6

NEMP REPORTS

1995 1 Science

2 Art

3 Graphs, Tables and Maps

Cycle 1

1996 4 Music

5 Aspects of Technology

6 Reading and Speaking

1997 7 Information Skills

8 Social Studies

9 Mathematics

1998 10 Listening and Viewing

11 Health and Physical Education

12 Writing

1999 13 Science

14 Art

15 Graphs, Tables and Maps

16 Māori Students’ Results

Cycle 2

2000 17 Music

18 Aspects of Technology

19 Reading and Speaking

20 Māori Students’ Results

2001 21 Information Skills

22 Social Studies

23 Mathematics

24 Māori Students’ Results

Forthcoming

2002 Listening and Viewing

Health and Physical Education

Writing

Māori Students’ Results

EDUCATIONAL ASSESSMENT RESEARCH UNIT

PO Box 56, Dunedin, New Zealand Fax 64 3 479 7550 Email earu@otago.ac.nz Web nemp.otago.ac.nz

MATHEMATICS

CONTENTS

1

Ac k n o w l e d g e m e n t s 2

Su m m a r y 3

Ch a p t e r 1 Ke y Fe a t u r e s o f Th e Na t i o n a l Ed u c a t i o n Mo n i t o r i n g Pr o j e c t 5

Ch a p t e r 2 as s e s s i n g Ma t h e m a t i c s 9

Ch a p t e r 3 nu m b e r 12

TREND TASKS

Ch a p t e r 4 Me a s u r e m e n t 27

Video Recorder 28 Party time 34 Team photo 37

Apples 29 What’s the time? 34 Broken Ruler 38

Better Buy 29 Supermarket Pebbles packet 38

Measures 30 Shopping 35 Fishing competition Y4 39

Bean estimates 31 Two Boxes 35 Fishing competition 39

Lump balance 31 Stamps 36 Milkogram 40

Measurement B 32 Bank Account 36 Running records 40

Measurement C 33 Money A 37 Link Tasks 13–23 41

TREND TASKS

Ch a p t e r 5 Ge o m e t r y 42

TREND TASKS

Subtraction facts 13

Division facts 13

Addition examples 14

Multiplication

examples 14

Calculator ordering 15

Pizza pieces 15

Girls and boys 16

Equivalents 16

Hedgehog 43

Cut cube 44

Flat shapes 44

Shapes and nets 45

Kumara basket 17

Speedo 17

Jack’s Cows 18

Motorway 19

Wallies 19

36 & 29 20

Fractions 20

Number items B 21

One cut 45

Paper folds 46

Line of symmetry 46

Whetu’s frame 47

Number items C 22

Number line Y4 23

Number line 23

Population Y4 24

Population 24

Strategy 25

Missing the point 25

Link tasks 1–12 26

Boats 47

Grid Plans 48

Link Tasks 24–29 49

Ch a p t e r 6 al g e b r a a n d s t a t i s t i c s 50

TREND TASKS

Train trucks 51

Algebra/Logic A 52

Algebra/Logic B 53

Statistic items A 55

TREND TASKS

TREND TASKS

Statistic items B 55

Farmyard race 56

Photo line-up 57

Bag of beans 57

Games 58

How far? 58

Link Tasks 30–36 59

Ch a p t e r 7 su r v e y s 60

Ch a p t e r 8 pe r f o r m a n c e o f Su b g r o u p s 63

Ch a p t e r 9 pacific Su b g r o u p s 66

Ap p e n d i x Th e s a m p l e o f s c h o o l s a n d s t u d e n t s in 2001 70

2 ACKNOWLEDGEMENTS

The Project directors acknowledge the vital support and contributions

of many people to this report, including:

➢ the very dedicated staff of the Educational Assessment Research Unit

➢ Dr David Philips and other staff members of the Ministry of Education

➢ members of the Project’s National Advisory Committee

➢ members of the Project’s Mathematics Advisory Panel

➢ principals and children of the schools where tasks were trialled

➢ principals, staff, and Board of Trustee members of the 286 schools included in the 2001 sample

➢ the 3153 children who participated in the assessments and their parents

➢ the 108 teachers who administered the assessments to the children

➢ the 44 senior tertiary students who assisted with the marking process

➢ the 166 teachers who assisted with the marking of tasks early in 2002

SUMMARY 3

New Zealand’s National Education Monitoring Project commenced

in 1993, with the task of assessing and reporting on

the achievement of New Zealand primary school children

in all areas of the school curriculum. Children are assessed

at two class levels: Year 4 (halfway through primary education)

and Year 8 (at the end of primary education). Different

curriculum areas and skills are assessed each year, over a

four year cycle. The main goal of national monitoring is to

provide detailed information about what children can do so

that patterns of performance can be recognised, successes

celebrated, and desirable changes to educational practices

and resources identified and implemented.

Each year, small random samples

of children are selected

nationally, then assessed in

their own schools by teachers

specially seconded and trained

for this work. Task instructions

are given orally by teachers,

through video presentations,

on laptop computers, or in

writing. Many of the assessment

tasks involve the children

in the use of equipment and

supplies. Their responses are

presented orally, by demonstration,

in writing, in computer

files, or through submission of

chapter 2

other physical products. Many

of the responses are recorded

on videotape for subsequent

analysis.

The use of many tasks with

both year 4 and year 8 students

allows comparisons of

the performance of year 4 and

8 students in 2001. Because

some tasks have been used

twice, in 1997 and again in

2001, trends in performance

across the four year period can

also be analysed.

In 2001, the third year of the

second cycle of

national monitoring,

three areas

were assessed:

**mathematics**, social

studies, and

information skills.

T h i s r e p o r t

presents details

and results of the

**mathematics** assessments.

Chapter 2 explains the place of **mathematics** in the New Zealand

curriculum and presents the **mathematics** framework. It

identifies five areas of knowledge

or curriculum strands

(number, measurement,

geometry, algebra, and statistics),

linked to five major

processes and skills. The

importance of attitudes and

motivation is also highlighted.

The assessment results

are arranged in chapters according

to the strands in the

curriculum, but with algebra

and statistics in one chapter

because of small numbers of

tasks in these two areas and

also space constraints in this

report.

chapter 3

Chapter 3 presents the students’

results on 35 number

tasks. Averaged across 229

task components administered

to both year 4 and year

8 students, 25 percent more

year 8 than year 4 students

succeeded with these components.

Year 8 students

performed better on every

component. As expected,

the differences were generally

larger on more difficult tasks. These often were

tasks that many year 4 students would not yet have

had much opportunity to learn in school.

There was evidence of modest improvement

between 1997 and 2001, especially for year 4

students. Averaged across 59 trend task components

attempted by year 4 students in both years, 5

percent more students succeeded in 2001 than in

1997. Gains occurred on 51 of the 59 components.

At year 8 level, with 106 trend task components

included, 3 percent more students succeeded in

2001 than in 1997. Gains occurred on 85 of the 106

components.

Students at both levels scored poorly in tasks involving

estimation and tasks involving fractions (especially

fractions other than halves and quarters).

Asked to work on computations such as 36 + 29 or

9 x 98, few students at both levels chose the simplification

of adjusting one of the numbers to a more

easily handled adjacent number (making the 29 into

30, or the 98 into 100). Most relied instead on the

standard algorithms for these tasks, indicating a lack

of deep understanding of number operations.

chapter 4

Chapter 4 presents results for 33 measurement

tasks. Averaged across 101 task components administered

to both year 4 and year 8 students, 25

percent more year 8 than year 4 students succeeded

with these components. Year 8 students performed

better on 95 of the 101 components.

There was little evidence of change between 1997

and 2001. Averaged across 41 trend task components

attempted by year 4 students in both years, 2

percent more students succeeded in 2001 than in

1997. Gains occurred on 25 of the 41 components.

At year 8 level, with 45 trend task components

included, 2 percent fewer students succeeded in

2001 than in 1997. Gains occurred on 15 of the 45

components.

At both levels, students were much more successful

at making or reading measurements than at making

good estimates of measurements. Also, many

who could measure satisfactorily were not able to

explain clearly their processes and strategies for

making and checking their measurements.

4 NEMP Report 23: Mathematics 2001

chapter 5

Chapter 5 presents results for sixteen

geometry tasks. Averaged across

41 task components administered

to both year 4 and year 8 students,

23 percent more year 8 than year 4

students succeeded with these components.

Year 8 students performed

better on all components.

There was little evidence of change

between 1997 and 2001 for year 4

students, but a small decline for year

8 students. Averaged across 13 trend

task components attempted by year 4

students in both years, 2 percent more

students succeeded in 2001 than in

1997. Gains occurred on 10 of the 13

components. At year 8 level, with 22

trend task components included, 5

percent fewer students succeeded in

2001 than in 1997. Gains occurred on

3 of the 22 components.

Many students were able to identify

the nets of three-dimensional objects

and to mirror a shape in a line of symmetry.

Students had less success with

visualising the internal structure and

cross sections of three-dimensional

objects, and with other spatial relationships

tasks in three dimensions.

chapter 6

Chapter 6 presents results for ten algebra

tasks and seven statistics tasks.

Averaged across 36 task components

administered to both year 4 and year

8 students, 28 percent more year 8

than year 4 students succeeded with

these components. Year 8 students

performed better on 35 of the 36

components.

There was evidence of substantial improvement

between 1997 and 2001

for year 4 students, but little change

over the same period for year 8 students.

Averaged across 15 trend task

components attempted by year 4 students

in both years, 9 percent more

students succeeded in 2001 than in

1997. Gains occurred on 14 of the 15

components. At year 8 level, with 28

trend task components included, 1

percent more students succeeded in

2001 than in 1997. Gains occurred on

16 of the 28 components.

chapter 7

Chapter 7

focuses on

the results

of a survey

that sought

information

from

students about their strategies

for, involvement in, and enjoyment

of **mathematics**. Mathematics

was the third most popular

option for year 4 students

and the fourth most popular

option for year 8 students. At

year 4 level its popularity remained

constant between 1997

and 2001, but at year 8 level it

was chosen by 9 percent fewer

students in 2001, while technology

and art gained substantially

from the 1997 results.

The students’ responses to

eleven rating items showed

the pattern found to date in all

subjects except technology;

year 8 students are less likely

to use the most positive rating

than year 4 students. In other

words, students become more

cautious about expressing high

enthusiasm and self-confidence

over the four additional years of

schooling.

Between 1997 and 2001, fewer

students at both year levels said

that they didn’t know how good

their teacher thought they were

at maths. This is a worthwhile

improvement. A higher proportion

of students at both levels

believed that their teachers

and parents thought that they

were good at **mathematics**. The

results for several of the rating

items suggested that student

enthusiasm for **mathematics**

was static or declined slightly

over the four year period.

chapter 9

chapter 8

Chapter 8 details the results of analyses

comparing the performance of different

demographic subgroups. Statistically significant

differences of task performance among

the subgroups based on school size, school

type or community size occurred for very

few tasks. There were differences among

the three geographic zone subgroups on

15 percent of the tasks for year 4 students,

but only 2 percent of the tasks for year 8

students. Boys performed better than girls

on 12 percent of the year 4 tasks and 5 percent

of the year 8 tasks, but girls performed

better than boys on 2 percent of the year 8

tasks. Non-Māori students performed better

than Māori students on 75 percent of the

year 4 tasks and 66 percent of the year 8

tasks. The SES index based on school deciles

showed the strongest pattern of differences,

with differences on 87 percent of the year 4

tasks and 76 percent of the year 8 tasks.

The 2001 results for the Māori/Non-Māori

and SES (school decile) comparisons are

very similar to the corresponding 1997 results.

In 1997 there were Māori/Non-Māori

differences on 80 percent of year 4 tasks

and 77 percent of year 8 tasks, and school

decile differences on 85 percent of year 4

tasks and 77 percent of year 8 tasks. The

most noticeable, although still relatively

small, changes from the 1997 results involve

the gains of boys relative to girls. In 2001,

year 4 boys performed better than girls on

12 percent of tasks (2 percent in 1997) and

worse on none (4 percent in 1997). Year 8

boys performed better than girls on 5 percent

of tasks (2 percent in 1997) and worse

on 2 percent (14 percent in 1997).

Chapter 9 reports the results of analyses of the achievement of Pacific Island

students. Additional sampling of schools with high proportions of Pacific

Island students permitted comparison of the achievement of Pacific Island,

Māori and other children attending schools that have more than 15 percent

Pacific Island students enrolled. The results apply only to such schools, but it

should be noted that about 75 percent of all Pacific students attend schools

in this category.

Year 4 Pacific students performed similarly to their Māori peers, but less

well than “other” students on 45 percent of the tasks. Year 8 Pacific students

performed similarly to their Māori peers, but less well than “other” students

on 27 percent of the tasks.

CHAPTER 1 5

THE NATIONAL EDUCATION MONITORING PROJECT

This chapter presents a concise outline of the rationale

and operating procedures for national monitoring,

together with some information about the reactions of

participants in the 2001 assessments. Detailed information

about the sample of students and schools is available

in the Appendix.

Purpose of national monitoring

The New Zealand Curriculum Framework (1993, p26)

states that the purpose of national monitoring is to provide

information on how well overall national standards

are being maintained, and where improvements

might be needed.

The focus of the National Education Monitoring Project

(NEMP) is on the educational achievements and attitudes

of New Zealand primary and intermediate school children.

NEMP provides a national “snapshot” of children’s

knowledge, skills and motivation, and a way to identify

which aspects are improving, staying constant, or declining.

This information allows successes to be celebrated

and priorities for curriculum change and teacher development

to be debated more effectively, with the goal

of helping to improve the education which children

receive.

Assessment and reporting procedures are designed to

provide a rich picture of what children can do and optimise

value to the educational community. The result is

a detailed national picture of student achievement. It is

neither feasible nor appropriate, given the purpose and

the approach used, to release information about individual

students or schools.

Monitoring at two class levels

National monitoring assesses and reports what children

know and can do at two levels in primary and intermediate

schools: year 4 (ages 8–9) and year 8 (ages

12–13).

a special sample of 120 children learning in Māori immersion

schools or classes is selected. Their achievement

can then be compared with the achievement of Māori

students in the main year 8 sample, whose education

is predominantly in English (these comparisons are not

reported here, but in a separate report that shows the

tasks in both Māori and English).

Three sets of tasks at each level

So that a considerable amount of information can be

gathered without placing too many demands on individual

students, different students attempt different tasks.

The 1440 students selected in the main sample at each

year level are divided into three groups of 480 students,

comprising four students from each of 120 schools.

Timing of assessments

The assessments take place in the second half of the

school year, between August and November. The year

8 assessments occur first, over a five week period. The

year 4 assessments follow, over a similar period. Each

student participates in about four hours of assessment

activities spread over one week.

Specially trained teacher administrators

The assessments are conducted by experienced teachers,

usually working in their own region of New Zealand.

They are selected from a national pool of applicants,

attend a week of specialist training in Wellington led

by senior Project staff, and then work in pairs to conduct

assessments of 60 children over five weeks. Their

employing school is fully funded by the Project to employ

a relief teacher during their secondment.

National samples of students

National monitoring information is gathered

using carefully selected random samples of

students, rather than all year 4 and year 8

students. This enables a relatively extensive

exploration of students’ achievement,

far more detailed than would be possible if

all students were to be assessed. The main

national samples of 1440 year 4 children

and 1440 year 8 children represent about

2.5 percent of the children at those levels in

New Zealand schools, large enough samples

to give a trustworthy national picture. Additional

samples of 120 children at each level

allow the achievement of Pacific students to

be assessed and reported. At year 8 level only,

6 NEMP Report 23: Mathematics 2001

y e a r

New Zealand Curriculum

1 1999 Art

Science

(1995) Information Skills: graphs, tables, maps,

charts and diagrams

Language: reading and speaking

2 2000 Aspects of Technology

(1996) Music

Mathematics: numeracy skills

3 2001 Social Studies

4

(1997) Information Skills library, research

2002 Language: writing, listening, viewing

(1998) Health and Physical Education

Communication skills

Problem-solving skills

Self-management and competitive skills

Social and co-operative skills

Work and study skills

Attitudes

Four year assessment cycle

Each year, the assessments cover about one quarter of

the national curriculum for primary schools. The New

Zealand Curriculum Framework is the blueprint for

the school curriculum. It places emphasis on seven

essential learning areas, eight essential skills, and a variety

of attitudes and values. National monitoring aims to

address all of these areas, rather than restrict itself to

preselected priority areas.

The first four year cycle of assessments began in 1995

and was completed in 1998. The second cycle runs from

1999 to 2002. The areas covered each year and the

reports produced to date are listed inside the front cover

of this report. Similar cycles of assessment are expected

to be repeated in subsequent four year periods.

About one third of the tasks are kept constant from one

cycle to the next. This re-use of tasks allows trends in

achievement across a four year interval to be observed

and reported.

Important learning outcomes assessed

The assessment tasks emphasize aspects of the curriculum

which are particularly important to life in our

community, and which are likely to be of enduring

importance to students. Care is taken to achieve balanced

coverage of important skills, knowledge and understandings

within the various curriculum strands, but without

attempting to slavishly follow the finer details of current

curriculum statements. Such details change from time to

time, whereas national monitoring needs to take a longterm

perspective if it is to achieve its goals.

Wide range of task difficulty

National monitoring aims to show what students know

and can do. Because children at any particular class level

vary greatly in educational development, tasks spanning

multiple levels of the curriculum need to be included if

all children are to enjoy some success and all children

are to experience some challenge. Many tasks include

several aspects, progressing from aspects most children

can handle well to aspects that are less straightforward.

Engaging task approaches

Special care is taken to use tasks and approaches that

interest students and stimulate them to do their best.

Students’ individual efforts are not reported and have

no obvious consequences for them. This means that

worthwhile and engaging tasks are needed to ensure that

students’ results represent their capabilities rather than

their level of motivation. One helpful factor is that extensive

use is made of equipment and supplies which allow

students to be involved in “hands-on” activities. Presenting

some of the tasks on video or computer also allows

the use of richer stimulus material, and standardizes the

presentation of those tasks.

Positive students’ reactions to tasks

At the conclusion of each assessment session, students

completed evaluation forms in which they identified

tasks that they particularly enjoyed and tasks that did

not appeal. Averaged across all tasks in the 2001 assessments,

80 percent of year 4 students indicated that they

particularly enjoyed the tasks. The range across 106 tasks

was from 94 percent down to 55 percent. As usual,

year 8 students were more demanding. On average 70

percent of them indicated that they particularly enjoyed

the tasks, with a range across 124 tasks from 93 percent

down to 37 percent. The students’ parents and teachers

also reacted very positively to the tasks and assessment

approaches.

Chapter 1: The National Education Monitoring Project 7

Appropriate support for students

A key goal in project planning is to minimise the extent

to which student strengths or weaknesses in one area

of the curriculum might unduly influence their assessed

performance in other areas. For instance, skills in reading

and writing often play a key role in success or failure

in paper-and-pencil test areas such as science, social

studies, or even **mathematics**. In national monitoring,

a majority of tasks are presented orally by teachers, on

videotape, or on computer, and most answers are given

orally or by demonstration rather than in writing. Where

reading or writing skills are required to perform tasks in

areas other than reading and writing, teachers are happy

to help students to understand these tasks or to communicate

their responses. Teachers are working with no

more than four students at a time, so are readily available

to help individuals.

To further free teachers to concentrate on providing

appropriate guidance and help to students, so that the

students achieve their best efforts, teachers are not

asked to record judgements on the work the students are

doing. All marking and analysis is done later, when the

students’ work has reached the Project office in Dunedin.

Some of the work comes on paper, but much of it arrives

recorded on videotape. In 2001, about half of the students’

work came in that form, on a total of about 4000

videotapes. The video recordings give a detailed picture

of what students and teachers did and said, allowing rich

analysis of both process and task achievement.

Four task approaches used

In 2001, four task approaches were used. Each student

was expected to spend about an hour working in each

format. The four approaches were:

➢ One-to-one interview. Each student worked individually

with a teacher, with the whole session

recorded on videotape.

➢ Stations. Four students, working independently,

moved around a series of stations where tasks had

been set up. This session was not videotaped.

➢ Team. Four students worked collaboratively,

supervised by a teacher, on some tasks. This was

recorded on videotape.

➢ Independent. Four students worked individually

on some paper-and-pencil tasks.

the Project. Given that 108 teachers served as teacher

administrators in 2001, or about half a percent of all primary

teachers, the Project is making a major contribution

to the professional development of teachers in assessment

knowledge and skills. This contribution will steadily

grow, since preference for appointment each year

is given to teachers who have not previously served as

teacher administrators. The total after seven years is 690

different teachers.

Marking arrangements

The marking and analysis of the students’ work occurs

in Dunedin. The marking process includes extensive

discussion of initial examples and careful checks of the

consistency of marking by different markers.

Tasks which can be marked objectively or with modest

amounts of professional experience usually are marked

by senior tertiary students, most of whom have completed

two to four years of preservice preparation for

primary school teaching. Forty-four student markers

worked on the 2001 tasks, employed 5 hours per day

for 6 weeks.

The tasks that require higher levels of professional judgement

are marked by teachers, selected from throughout

New Zealand. In 2001, approximately 60 percent of the

teachers who applied were appointed: a total of 166.

Most teachers worked either mornings or afternoons for

one week. Teacher professional development through

participation in the marking process is another substantial

benefit from national monitoring. In evaluations of

their experiences on a four point scale (“dissatisfied” to

“highly satisfied”), 75 to 97 percent of the teachers who

marked student work in 2001 chose “highly satisfied” in

response to questions about:

➢ the extent to which marking was professionally

satisfying and interesting;

➢ its contribution to professional development in

the area of assessment;

➢ whether they would recommend NEMP marking

work to colleagues;

➢ whether they would be happy to do NEMP marking

again.

Professional development benefits for teacher

administrators

The teacher administrators reported that they found

their training and assessment work very stimulating and

professionally enriching. Working so closely with interesting

tasks administered to 60 children in at least five

schools offered valuable insights. Some teachers have

reported major changes in their teaching and assessment

practices as a result of their experiences working with

8 NEMP Report 23: Mathematics 2001

Analysis of results

The results are analysed and reported task by task.

Although the emphasis is on the overall national picture,

some attention is also given to possible differences in

performance patterns for different demographic groups

and categories of school. The variables considered are:

➢ Student gender: male, female

➢ Student ethnicity: Māori, non-Māori

➢ Geographical zone: Greater Auckland, other

North Island, South Island

➢ Size of community: urban area over 100,000,

community of 10,000 to 100,000, rural area or

town of less than 10,000

➢ Socio-economic index for the school: bottom three

deciles, middle four deciles, highest three deciles

➢ Size of school:

year 4 s c h o o l s less than 20 Y4 students 20–35 Y4 students more than 35 Y4 students

year 8 s c h o o l s less than 35 Y8 students 35–150 Y8 students more than 150 Y8 students

➢ Type of school (for year 8 sample only): Full primary

school, intermediate school (some students

were in other types of schools, but too few to

allow separate analysis)

Categories containing fewer children, such as Asian students

or female Māori students, were not used because

the resulting statistics would be based on the performance

of less than 70 children, and would therefore be

too unreliable.

A further subgroup analysis has also been included. This

compares the performance of Pacific,

Māori and other students attending

schools with 15 percent or more Pacific

students enrolled. Schools in this category

within the main samples are combined

with the supplementary samples

of 10 schools with 20 percent or more

Pacific students enrolled. The resulting

samples include about 105 students

attempting each task: typically about 50

Pacific students, 25 Māori students and

30 other students.

Funding arrangements

National monitoring is funded by the

Ministry of Education, and organised by

the Educational Assessment Research

Unit at the University of Otago, under

the direction of Associate Professor

Terry Crooks and Lester Flockton. The current contract

runs until 2003. The cost is about $2.5 million per year,

less than one tenth of a percent of the budget allocation

for primary and secondary education. Almost half of the

funding is used to pay for the time and expenses of the

teachers who assist with the assessments as task developers,

teacher administrators or markers.

Reviews by international scholars

In June 1996, three scholars from the United States and

England, with distinguished international reputations in

the field of educational assessment, accepted an invitation

from the Project directors to visit the Project. They

conducted a thorough review of the progress of the

Project, with particular attention to the procedures and

tasks used in 1995 and the results emerging. At the end

of their review, they prepared a report which concluded

as follows:

The National Education Monitoring Project is well conceived

and admirably implemented. Decisions about design, task

development, scoring, and reporting have been made thoughtfully.

The work is of exceptionally high quality and displays

considerable originality. We believe that the project has considerable

potential for advancing the understanding of and public

debate about the educational achievement of New Zealand

students. It may also serve as a model for national and/or state

monitoring in other countries.

Professors Paul Black, Michael Kane & Robert Linn, 1996

A further review was conducted late in 1998 by another

distinguished panel (Professors Elliot Eisner, Caroline

Gipps and Wynne Harlen). Amid very helpful suggestions

for further refinements and investigations, they

commented that:

We want to acknowledge publicly that the overall design of

NEMP is very well thought through. … The vast majority of

tasks are well designed, engaging to students and consistent

with good assessment principles in making clear to students

what is expected of them.

Further information

A more extended description of national monitoring,

including detailed information about task development

procedures, is available in:

Flockton, L. (1999). School-wide Assessment: National

Education Monitoring Project. Wellington: New Zealand

Council for Educational Research.

CHAPTER 2 9

Assessing Mathematics

The aims of **mathematics** education, like those of other

learning areas, are developed and shaped to reflect

understandings and processes that are meaningful,

important and useful to individuals and society. Just

as knowledge expands, circumstances alter, and needs

change with time, so too is the content and structure of

**mathematics** programmes adjusted and refined from

time to time to reflect current needs and future visions

for learners. Expecting students to get right answers

in the shortest possible time with the least amount of

thinking is no longer a prime goal of **mathematics** education.

For most students a major aim is to help them

develop attitudes and abilities to be flexible, creative

thinkers who can cope with open-ended real-world

problems. This requires them to become confident in

their understanding and application of mathematical

ideas, procedures and processes.

Because much conceptual knowledge and skill in

**mathematics** takes time to develop, fundamental ideas

introduced at the early years of schooling are repeatedly

elaborated on and extended as students progress

through their years at school. It is appropriate, therefore,

that assessment in **mathematics** include a substantial

proportion of tasks which allow us to observe the

extent of progress in conceptual knowledge and skill

over time.

Although conceptual understanding is clearly one of

the major goals of **mathematics** education, students’

capacity for exploring, applying and communicating

their mathematical understandings within real-world

contexts is also important. Mathematics education is

very much concerned with such matters as students’

confidence, interest and inventiveness in working with

a range of mathematical ideas. The NEMP assessment

framework recognises this by making provision for

students to demonstrate their mathematical skills

through a range of situations which involve them in

asking questions, making connections, and applying

understandings and processes to novel as well as familiar

situations. While the place for assessing confidence

and efficiency in basic knowledge of facts is recognised

in NEMP assessments, there is also a substantial focus

on thinking, reasoning and problem-solving skills

which require more open tasks that allow students to

demonstrate their number sense, reason, make decisions

and explain.

Framework for assessment of **mathematics**

National monitoring task frameworks are developed

with the project’s curriculum advisory panels. These

frameworks have two key purposes. They provide

a valuable guideline structure for the development

and selection of tasks, and

they bring into focus those

important dimensions of the

learning domain which are

arguably the basis for valid

analyses of students’ skills,

knowledge, understandings

and attitudes.

The assessment frameworks

are intended to be flexible and

broad enough to encourage

and enable the development of

tasks that lead to meaningful

descriptions of what students

know and can do. They are

also designed to help ensure

a balanced representation of

important learning outcomes.

The **mathematics** framework

has a central organising theme

and five areas of knowledge

linked to five major processes

and skills. Key aspects of

content are listed under each

major heading and attention

is drawn in the final section of

the framework to the importance

of students’ attitudes

and motivation.

10 NEMP Report 23: Mathematics 2001

NEMP MATHEMATICS FRAMEWORK

Confident understanding and application of mathematical ideas, procedures and processes

Areas of Knowledge

NUMBER

• numbers and the way they are represented

• operations on number

MEASUREMENT

• systems of measurement and their use

• using various measuring devices including instruments

and personal references

• approximate nature of measurement

GEOMETRY

• geometrical relations in 2 and 3 dimensions, and

their occurrence in the environment

• spatial awareness (e.g. position)

• recognition and use of geometrical properties of

everyday objects (e.g. symmetry and transformation)

• use of geometric models as aids to solving problems

ALGEBRA

• patterns and relationships in **mathematics** and the

real world

• symbols, notation, and graphs and diagrams representing

mathematical relationships, concepts and

generalisations

• properties/principles of number operations

STATISTICS

• collection, organisation, analysis and display of

statistical data

• estimation of probabilities and use of probabilities for

prediction

• critical interpretation of others’ data

seeing connections

between content areas,

processes and skills

LINKING

Processes and Skills

PROBLEM SOLVING

(puzzles to investigations)

understanding the problem, interpreting, experimenting,

hypothesising, evaluating, using strategies

and creativity

LOGICAL REASONING

classifying, interpreting, developing and refuting

arguments, using inductive and deductive procedures

INFORMATION

gathering, collating, processing, presenting

COMPUTATION

calculating with accuracy and efficiency, estimating

and approximating, using mental processes, pencil

and paper, or calculators as appropriate

COMMUNICATING

questioning, presenting, explaining and justifying,

discussing, collaborating

Attitudes and Motivation

valuing

perseverance

interest and enjoyment

confidence and willingness to take risks

voluntary engagement

The most important message emerging from the use of

the framework is the pervasive interrelatedness that exists

among **mathematics** understandings, skills and attitudes.

To regard each as a discrete entity of learning, whether for

teaching or assessment purposes, assumes clear cut boundaries

that frequently do not exist. In developing and administering

tasks, it was sometimes difficult to assign tasks

specifically to one aspect rather than another. However, for

purposes of reporting assessment information, tasks were

allocated to particular categories according to the balance

of emphasis. The results are arranged in chapters according

to the strands in the curriculum. Because a relatively

smaller number of tasks were concerned with algebra and

statistics (and space in this report was at a premium), the

algebra and statistics tasks have been grouped together in

one chapter.

The Choice of Tasks for National Monitoring

The choice of tasks for national monitoring is guided by a

number of educational and practical considerations. Uppermost

in any decisions relating to the choice or administration

of a task is the central consideration of validity

and the effect that a whole range of decisions can have

on this key attribute. Tasks are chosen because they

provide a good representation of important knowledge

and skills, but also because they meet a number

of requirements to do with their administration and

presentation. For example:

➢ Each task with its associated materials needs

to be structured to ensure a high level of consistency

in the way it is presented by specially

trained teacher administrators to students of

wide ranging backgrounds and abilities, and in

diverse settings throughout New Zealand.

➢ Tasks need to span the expected range of capabilities

of Year 4 and 8 students and to allow

the most able students to show the extent of

their abilities while also giving the least able the

opportunity to show what they can do.

Chapter 2: Assessing Information Skills 11

➢ Materials for tasks need to be sufficiently portable,

economical, safe and within the handling capabilities

of students. Task materials also need to have

meaning for students.

➢ The time needed for completing an individual task

has to be balanced against the total time available

for all of the assessment tasks, without denying

students sufficient opportunity to demonstrate

their capabilities.

➢ Each task needs to be capable of sustaining the

attention and effort of students if they are to

produce responses that truly indicate what they

know and can do. Since neither the student nor

the school receives immediate or specific feedback

on performance, the motivational potential

of the assessment is critical.

➢ Tasks need to avoid unnecessary bias on the

grounds of gender, culture or social background

while accepting that it is appropriate to have

tasks that reflect the interests of particular groups

within the community.

National monitoring **mathematics** assessment

tasks and survey

One hundred and one **mathematics** tasks were administered,

together with an interview questionnaire that

investigated students’ interests, attitudes and involvement

in **mathematics**.

Thirty-seven tasks were administered in one-to-one interview

settings, where students used materials and visual

information. Two tasks were presented in team or group

situations involving small groups of students working

together. Twenty-four tasks were attempted in a stations

arrangement, where students worked independently on

a series of tasks, some presented on lap-top computers.

The final thirty-eight tasks were administered in an independent

approach, where students sat at desks or tables

and worked through a series of paper-and-pencil tasks.

Forty-three of the tasks were identical for year 4 and

year 8 students. A further twenty-eight tasks included

common components for both years, together with more

challenging components for year 8 students and/or less

demanding components for year 4 students. Of the

remaining tasks, seven were specifically for year 4 students

and twenty-three for year 8 students. Some of these

single year tasks had parallel tasks at the other level, but

with different stimulus material or significantly different

instructions.

Trend Tasks

Thirty of the tasks were previously used, entirely or

in part, in the 1997 **mathematics** assessments. These

were called link tasks in the 1997 report, but were not

described in detail to avoid any distortions in the 2001

results that might have occurred if the tasks had been

widely available for use in schools since 1997. In the

current report, these tasks are called trend tasks and are

used to examine trends in student performance: whether

they have improved, stayed constant or declined over the

four year period since the 1997 assessments.

Link Tasks

To allow comparisons between the 2001 and 2005 assessments,

thirty-six of the tasks used for the first time in

2001 have been designated link tasks. Results of student

performance on these tasks are presented in this report,

but the tasks are described only in general terms because

they will be used again in 2005.

Marking methods

The students’ responses were assessed using specially

designed marking procedures. The criteria used had

been developed in advance by Project staff, but were

sometimes modified as a result of issues raised during the

marking. Tasks that required marker judgement and were

common to year 4 and year 8 were intermingled during

marking sessions, with the goal of ensuring that the same

scoring standards and procedures were used for both.

Task by task reporting

National monitoring assessment is reported task by task

so that results can be understood in relation to what the

students were asked to do.

Access Tasks

Teachers and principals have expressed considerable

interest in access to NEMP task materials and marking

instructions, so that they can use them within their own

schools. Some are interested in comparing the performance

of their own students to national results on some

aspects of the curriculum, while others want to use tasks

as models of good practice. Some would like to modify

tasks to suit their own purposes, while others want to

follow the original procedures as closely as possible.

There is obvious merit in making available carefully

developed tasks that are seen to be highly valid and

useful for assessing student learning.

Some of the tasks in this report cannot be made available

in this way. Link tasks must be saved for use in four

years time, and other tasks use copyright or expensive

resources that cannot be duplicated by NEMP and provided

economically to schools. There are also limitations

on how precisely a school’s administration and marking

of tasks can mirror the ways that they are administered

and marked by the Project. Nevertheless, a substantial

number of tasks are suitable to duplicate

for teachers and schools. In this report, these

access tasks are identified with the symbol above, and

can be purchased in a kit from the New Zealand Council

for Educational Research (P.O. Box 3237, Wellington

6000, New Zealand). Teachers are also encouraged to

use the NEMP web site (http://nemp.otago.ac.nz) to

view video clips and listen to audio material associated

with some of the tasks.

12 chapter 3

Number

The assessments included thirty-five tasks investigating students’ understandings,

processes and skills in the area of **mathematics** called number. Number includes

the ways numbers are represented, their value, operations on number, accuracy

and efficiency in calculating, estimating and making approximations.

Twelve tasks were identical for both year 4 and year 8. Thirteen tasks had overlapping

versions for year 4 and year 8 students, with some parts common to both

levels. Seven tasks were attempted by year 8 students only, and three by year 4

only. Eight are trend tasks (fully described with data for both 1997 and 2001),

fifteen are released tasks (fully described with data for 2001 only), and twelve are

link tasks (to be used again in 2005, so only partially described here).

The tasks are presented in the three sections: trend tasks, then released tasks and

finally link tasks. Within each section, tasks attempted (in whole or part) by both

year 4 and year 8 students are presented first, then tasks where year 4 and year 8

students did parallel tasks, followed by tasks attempted only by year 8 students.

Averaged across 229 task components administered to both year 4 and year 8

students, 25 percent more year 8 than year 4 students succeeded with these components.

Year 8 students performed better on every component. As expected, the

differences were generally larger on more difficult tasks — often tasks that many

year 4 students would not yet have had much opportunity to learn in school.

There was evidence of modest improvement between 1997 and 2001, especially

for year 4 students. Averaged across 59 trend task components attempted by year 4

students in both years, 5 percent more students succeeded in 2001 than in 1997.

Gains occurred on 51 of the 59 components. At year 8 level, with 106 trend task

components included, 3 percent more students succeeded in 2001 than in 1997.

Gains occurred on 85 of the 106 components.

Students at both levels scored poorly in tasks involving estimation and tasks involving

fractions (especially fractions other than halves and quarters). Asked to work

on computations such as 36 + 29 or 9 x 98, few students at both levels chose the

simplification of adjusting one of the numbers to a more easily handled adjacent

number (making the 29 into 30, or the 98 into 100). Most relied instead on the

standard algorithms for these tasks, indicating a lack of deep understanding of

number operations. The following percentages of year 8 students got ninety percent

or more of tested basic facts correct: 99 percent for addition, 95 percent for

subtraction, 86 percent for multiplication, and 65 percent for division. For year 4

students, the corresponding percentages were 84 percent, 53 percent, 26 percent,

and 11 percent.

Chapter 3: Number 13

trend

Subtraction Facts

trend

Division Facts

Approach: Station Level: Year 4 and year 8

Focus: Knowledge of subtraction facts.

Resources: Laptop computer with video recording showing

and saying each of the 30 items. Answers were

recorded on paper.

Questions/instructions

This activity uses the computer.

Click the Play button and watch the video that plays.

Approach: Station Level: Year 4 and year 8

Focus: Knowledge of division facts.

Resources: Laptop computer with video recording showing

and saying each of the 30 items. Answers were

recorded on paper.

Questions/instructions

This activity is done on the computer.

Click the Play button and watch the video that plays.

16 – 8 =

12 – 6 =

9 – 8 =

49 ÷ 7 =

32 ÷ 8 =

9 ÷ 3 =

11 – 4 =

15 – 6 =

5 – 2 =

6 ÷ 1 =

14 ÷ 7 =

12 ÷ 4 =

10 – 8 =

7 – 3 =

15 – 8 =

0 ÷ 3 =

28 ÷ 4 =

27 ÷ 3 =

14 – 5 =

8 – 1 =

8 – 2 =

24 ÷ 8 =

5 ÷ 1 =

40 ÷ 8 =

7 – 1 =

4 – 4 =

3 – 2 =

48 ÷ 6 =

27 ÷ 9 =

16 ÷ 8 =

13 – 6 =

9 – 5 =

5 – 0 =

24 ÷ 3 =

14 ÷ 2 =

54 ÷ 9 =

2 – 1 =

6 – 5 =

10 – 6 =

8 ÷ 2 =

72 ÷ 9 =

40 ÷ 5 =

11 – 2 =

11 – 8 =

18 – 9 =

81 ÷ 9 =

15 ÷ 5 =

18 ÷ 2 =

1 – 1 =

9 – 4 =

11– 3 =

48 ÷ 8 =

25 ÷ 5 =

36 ÷ 6 =

10 – 5 =

9 – 2 =

10 – 1 =

4 ÷ 1 =

0 ÷ 7 =

28 ÷ 7 =

% responses

2001 (’97) 2001 (’97)

year 4 year 8

% responses

2001 2001 (’97)

year 4 year 8

total score: 30 27 (19) 65 (51)

27–29 26 (28) 30 (36)

24–26 13 (13) 2 (6)

21–23 9 (9) 1 (3)

18–20 10 (9) 1 (2)

15–17 5 (7) 0 (1)

12–14 3 (5) 1 (1)

9–11 3 (5) 0 (0)

6–8 1 (2) 0 (0)

3–5 1 (1) 0 (0)

0–2 2 (2) 0 (0)

Commentary

Year 8 students showed very good knowledge of subtraction

facts, with 95 percent of the students getting at least

90% of the questions right. Just over half of the year 4

students showed similar levels of knowledge. For both

year 4 and year 8 students there was a small improvement

between 1997 and 2001.

Total score: 30 5 33 (29)

27–29 6 32 (27)

24–26 7 11 (11)

21–23 7 9 (8)

18–20 8 2 (9)

15–17 15 4 (5)

12–14 11 3 (4)

9–11 12 2 (2)

6–8 10 2 (2)

3–5 8 1 (1)

0–2 11 1 (2)

Commentary

In 2001 this task was given to year 4 and year 8 students,

but in 1997 it was given only to year 8 students. Year 4

students varied enormously in performance, with 18 percent

getting at least 80% correct and 29 percent getting

less than 30% correct. Many year 4 classes do not study

division. Almost two thirds of the year 8 students got

at least 90% correct, with a small improvement in high

scores compared to the 1997 students.

14 NEMP Report 23: Mathematics 2001

Addition Examples

trend

Multiplication Examples

trend

Approach: Independent Level: Year 4 and year 8

Focus: Adding without a calculator.

Resources: None.

Questions/instructions

% responses

2001 (’97) 2001 (’97)

Write your answers in the white boxes.

year 4 year 8

You can use the shaded area to do your

working.

5

1. + 8

13 89 (91) 97 (95)

2. 6

3

8

7

+ 4

28 66 (72) 90 (91)

3. 42

+ 35

77 80 (79) 96 (96)

4. 87

+ 56

143 56 (51) 92 (90)

5. 327

+ 436

763 59 (59) 89 (90)

6. 5607

3294

6358

+ 7286

22545 • 69 (67)

Approach: Independent Level: Year 4 and year 8

Focus: Multiplying without a calculator.

Resources: None.

Questions/instructions

% responses

2001 (’97) 2001 (’97)

Write your answers in the white boxes.

year 4 year 8

You can use the shaded area to do your

working.

34

1. 2

68 65 (61) 94 (95)

2. 43

7

301 26 (18) 80 (76)

3. 412

3

1236 45 (34) 90 (88)

4. 726

5

3630 23 (19) 82 (76)

5. 3074

6

18444 11 (8) 65 (58)

6. 74

23

1702 • 61 (53)

7. 6243

34

212262 • 45 (37)

7. 36 +117 + 6 + 23 + 240= 422 • 62 (57)

• not asked for year 4

Total score: 7 • 40 (32)

6 • 33 (39)

5 36 (35) 16 (19)

4 21 (21) 7 (5)

3 18 (20) 2 (4)

2 12 (13) 1 (0)

1 9 (8) 1 (1)

0 4 (3) 0 (0

Commentary

At both year levels, the 2001 students performed very

similarly to the 1997 students. Comparative performance

on questions 3 and 4 shows that many year 4 students

struggled when required to rename (carry).

• not asked for year 4

Total score: 7 • 25 (20)

6 • 28 (23)

5 6 (5) 19 (20)

4 12 (8) 12 (15)

3 12 (7) 7 (9)

2 18 (18) 4 (7)

1 20 (25) 2 (4)

0 32 (37) 3 (2)

Commentary

Year 8 students performed much better than year 4 students

on multiplication tasks. There was a small improvement

between 1997 and 2001 for both years: 10 percent

more year 4 students got at least 3/5 correct, and 9 percent

more year 8 students got at least 5/7 correct.

Chapter 3: Number 15

trend

Calculator Ordering

trend

Pizza Pieces

Approach: One to one Level: Year 4 and year 8

Focus: Putting numbers into ascending order.

Resources: 6 pictures of red calculators; 6 pictures of blue

calculators; recording book.

Questions/instructions:

Arrange the red calculators in order A – F.

Here are 6 pictures of calculators with different answers.

Hand the student the red picture cards.

1. Put the calculators in order starting with the smallest

number and ending with the biggest number.

Allow time.

2. Now read the numbers out loud one

at a time starting with the smallest

number.

4010

41

% responses

2001 (’97) 2001 (’97)

year 4 year 8

0–1 1 (1) 1 (0)

A B C

Read correctly: calculator: F: 14 99 (95) 100 (100)

4100

number placed correctly: 6 70 (71) 95 (96)

D E F

J

.05

104

140

% responses

2001 (’97)

year 8

number placed correctly: 6 22 (21)

1.5

4–5 9 (10)

0–1 17 (19)

G H I

Read correctly: calculator G: 05 86 (81)

.501

K

1.26

.52

14

1.251

L

Commentary

Both for year 4 and year 8 students the results in 1997

and 2001 were very similar.

1.5

4–5 25 (23) 3 (4)

2–3 4 (5) 1 (0)

C: 41 99 (95) 100 (100)

E: 104 93 (88) 98 (97)

B: 140 92 (87) 97 (97)

A: 4010 56 (43) 94 (93)

D: 4100 58 (42) 89 (88)

year 8 only: Arrange the blue calculators in order G – L.

Now here are 6 pictures of calculators with different

answers.

Hand the student the blue picture cards.

3. Now put these calculators in order starting with the

smallest number and ending with the biggest number.

Allow time.

4. Now read these numbers out loud one at a

time starting with the smallest number,

and I’ll write them down.

2–3 52 (50)

J: .501 75 (73)

K .52 75 (70)

L: 1.251 82 (82)

H: 1.26 81 (80)

I: 1.5 97 (96)

Approach: One to one Level: Year 4 and year 8

Focus: Understanding fractions and calculating with them.

Resources: Two model pizzas in sections on plates.

Questions/instructions:

Here are 2 whole pizzas for a family dinner. This one

is a pepperoni pizza and this one is a ham and pineapple

pizza. After dinner, some of each pizza was left

over.

Remove 2 segments of the pepperoni pizza and one segment

of the ham and pineapple pizza.

Do not use fractional terms at this point.

1. How much of the pepperoni pizza

is left?

prompt: (if answer not given as fraction)

What fraction or part is left?

2. How much of the ham and pineapple

pizza is left?

prompt: (if answer not given as fraction)

What fraction or part is left?

% responses

2001 (’97) 2001 (’97)

year 4 year 8

3. Altogether, how much pizza is left?

1 or or of total 51 (46) 81 (76)

Now we are going to think about 2 different ways of

using up the pizza that is left over.

4. If 4 children had a quarter piece of pizza each,

then how much would be left? year 4 year 8

prompt: You can move the pieces of

pizza around to help you work it out. 52 (49) 73 (75)

Ensure the students are still looking at the 2 segments of

pepperoni pizza and the 3 segments of the ham and pineapple

pizza.

5. This time imagine that the two of us are going to have

an equal share of all of the pizza that is left. What fraction

or part of a whole pizza do we each get?

prompt: You can move the pieces of pizza year 4 year 8

around to help you work it out.

• 10 (8)

2 quarters and one eighth • 9 (13)

6. Can you explain to me how you

worked that out? not marked

80 (75) 97 (97)

54 (54) 89 (89)

• not asked for year 4

Commentary

On questions 1–4, on average about 25 percent more

year 8 than year 4 students succeeded. Year 4 students

performed a little better in 2001 than in 1997, but there

was no consistent pattern of change for year 8 students.

16 NEMP Report 23: Mathematics 2001

Girls and Boys

trend

Equivalents

trend

Approach: One to one Level: Year 4

Focus: Solving number problems using physical objects.

Resources: Two cue cards, 5 tens rods, 15 ones cubes,

recording book.

Questions/instructions:

Room 1

26 children

Show cue card 1: Room 1 – 26 children

Room 1 at Tupai School has 26 children.

Place 5 rods and 15 cubes in front of the student. % responses

1. Can you show me 26 using these

2001 (’97)

rods and cubes?

year 4

arranged 2 tens rods and 6 ones cubes 92 (89)

2. Is it possible that there could be the same

number of girls as boys in that class?

yes, initially 70 (63)

prompt: If the student says “no”, ask why? yes 8 (10)

3. Use the rods and cubes to show me how

many boys and how many girls there would

be in Room 1 if there was the same number

of girls as boys.

prompt: You could use two of the tens rods and six

of the ones cubes.

arranged 2 groups of

1 tens rod and 3 ones cubes 79 (76)

4. Now tell me how many girls there are,

and how many boys there are. 13 80 (73)

Put all of the rods and cubes back together.

Room 2 Show cue card 2:

32 children Room 2 – 32 children.

Room 2 at Tupai School

has 32 children.

5. Can you show me 32 using the rods and

cubes?

arranged 3 tens rods and

2 ones cubes 91 (88)

arranged 2 tens rods and 12 ones cubes 3 (4)

6. Could there be the same number of girls as

boys in that class? yes, initially 56 (49)

prompt: If the student says “no”, ask why? yes 6 (8)

7. Use the rods and cubes to show me how

many girls and how many boys there

would be in Room 2 if there was the same

number of girls as boys.

arranged 2 groups of 1 tens rod and

6 ones cubes 55 (43)

8. Now tell me how many girls there are,

and how many boys there are. 16 61(46)

Commentary

The year 4 students enjoyed high success with this task

until they were required to rename from tens to ones

in part 7. The 2001 students had a small but consistent

advantage over the 1997 students in parts 1 to 6, and a

larger advantage in parts 7 and 8.

Approach: Independent Level: Year 8

Focus: Conversion among fractions, decimals and percentages.

Resources: None.

Questions/instructions:

% responses

2001 (’97)

Fill in the empty boxes so that each row has an

year 8

equivalent fraction, percentage and decimal.

1.

Common Fraction Percentage

Decimal

50 79 (77)

0.5 67 (65)

2. 53 (50)

20%

0.2 69 (61)

3. 35 (31)

4.

40 64 (59)

0.4

25 59 (56)

0.25 52 (44)

5. 51 (47)

10 63 (61)

0.1

6. 57 (54)

75%

0.75 63 (58)

Total score: 12 24 (21)

9–11 22 (20)

6–8 17 (16)

3–5 16 (18)

0–2 21 (25)

Commentary

On average, 59 percent of year 8 students succeeded

with each conversion, 4 percent more than in 1997.

Although this improvement was slight, it occurred for

all 12 conversions.

Chapter 3: Number 17

Approach: Station Level: Year 4 and year 8

Focus: Computation.

Resources: 3 picture cards.

Questions/instructions:

% responses

y4 y8

1. Look at picture card 1.

There are 19 kumara in each kete.

Kumara Basket

How many kumara are there altogether?

2. Look at picture card 2.

There are 20 kumara altogether in these

kete. Each kete holds the same amount.

How many kumara in each kete?

57 40 80

5 33 61

Approach: Independent Level: Year 4 and year 8

Focus: Addition and subtraction, place value.

Resources: None.

Questions/instructions:

A trip meter on a speedo shows how many

kilometres a car travels.

1. Write what the trip meter will show if the

car travels one more kilometre than:

[The trip meter and response blank

was presented like this:]

Speedo

% responses

y4 y8

1 9 9 7 52 84

2. Write what the trip meter will show if the

car travels ten more kilometres than:

2 0 0 6 11 61

3. Write what the trip meter will show if the car

travels one hundred more kilometres

than: 2 0 9 6 8 45

4. Write what the trip will show if the car travels

one thousand more kilometres than:

5. Write what the trip meter showed

one kilometre before this:

6. Write what the trip meter showed

ten kilometres before this:

2 9 9 6 24 67

3 4 0 1 34 78

3 3 9 2 7 52

7. Write what the trip meter showed

one hundred kilometres before this:

3. Look at picture card 3.

There are 15 kumara in the first kete, 14 in

the second, and 21 in the last kete.

How many kumara are there altogether?

If you want to, you can work out

your answers here.

50 55 91

3 3 0 2 17 68

8. Write what the trip meter showed

one thousand kilometres before this:

2 4 0 2 21 68

Total score: 8 2 26

6–7 7 31

4–5 10 16

2–3 21 12

0–1 60 15

Commentary

On average about 35 percent more year 8 than year 4

students succeeded on each question.

Commentary

Most year 4 students showed a low level of understanding

of place value, and very few succeeded when renaming

was required. About 60 percent of year 8 students gave

correct answers to 6 or more of the 8 components.

18 NEMP Report 23: Mathematics 2001

Jack’s Cows

Approach: One to one Level: Year 4 and year 8

Focus: Analysing a task and adding.

Resources: Comic strip card, calculator, recording book.

Questions/instructions:

Place comic strip in front of student.

Here is a maths problem about a farmer and

his farm animals. It is written as a comic

strip. Follow the words as I read it to you.

Read comic strip to student.

1. What is the farmer asking you to do?

% responses

y4 y8

find total number of cows

(add up all the cows) 24 37

find how many cows will be in the yard 45 46

2. What information does he give you to

work this out?

the number of cows in each place 37 57

3. What information does he give you that you

don’t need for working out the answer?

The number of dogs, deer and sheep

in various places all three 23 41

two of three 20 32

one of three 14 9

“other animals” 3 3

The farmer wants to know how many cows

there will be in the yard.

4. How would you work out the answer?

% responses

y4 y8

add up the three numbers for cows 26 59

Give the student pencil, calculator, and recording

sheet.

5. Now try to work out the answer.

You can use any of these things.

Adding numbers: using the calculator 68 55

6. What is the answer?

using pencil/paper 23 38

mentally 7 6

correct 48 73

Chapter 3: Number 19

Wallies

Approach: One to one Level: Year 4 and year 8

Focus: Estimation.

Resources: Picture of “Wallies,” recording book.

Motorway

Approach: One to one Level: Year 4 and year 8

Focus: Approximate calculations.

Resources: Picture of busy motorway, recording book.

Question/instructions:

Put the Wallies picture in front of student.

About how many Wallies do you think are

shown in this picture?

Don’t count them. Just make your best

estimate.

% responses

y4 y8

Student estimate: 400–650 8 16

[Actual number of Wallies

approximately 525]

650–750 1 2

300–399 8 9

750–3000 29 35

100–299 38 25

other 16 13

Commentary

Only 27 percent of year 8 and 17 percent of year 4 students

made a reasonably accurate estimate (within ± 50

percent of the actual value).

Questions/instructions

Show student photo

This picture shows a busy motorway. During

the day time, about 98 cars go down this

road every minute.

% responses

y4 y8

1. About how many cars would go down the

road in 9 minutes?

882 2 18

900 4 17

any other answer between 850 and 899 5 13

2. Explain to me how you got your answer.

multiply 98 9 31 48

multiply 100 9, then subtract 9 2 1 9

multiply 100 9,

then say answer is a little less than that 2 3

multiply 100 9, then say answer is near that 4 16

Commentary

Only 48 percent of year 8 students and 11 percent of

year 4 students made an accurate calculation or good

approximation. In part, this was because many students

tried to use the standard multiplication approach with

the original numbers, rather than adjusting the 98 to

100. Only 28 percent of year 8 students and 7 percent of

year 4 students used that simplification.

20 NEMP Report 23: Mathematics 2001

36 and 29

Approach: One to one Level: Year 4 and year 8

Focus: Multiple strategies for adding two numbers.

Resources: Card with 2 numbers.

Show student the number card.

% responses

y4 y8

year 8 version

36 29

Here are two numbers, 36 and 29. If you had

to add the two numbers, and you didn’t have

a calculator, how would you work it out?

Try to think of three different ways you

could work it out, and explain each way to

me.

Encourage student to think of, and explain,

3 ways.

They are not asked to work out answers.

year 4 version

Here are two numbers, 36 and 29. If you had

to add the two numbers, and you didn’t have

a calculator, how would you work it out?

Try to think of one way you could work it

out, and explain it to me.

Encourage student to think of, and explain a way

of working it out.

They are not asked to work out answers.

If the student succeeds in explaining one way

ask:

Is there another way could work out

36 plus 29?

Explain to me how you would do it.

Explained a satisfactory… first way 61 64

second way 16 33

third way • 21

Fractions

Approach: Independent Level: Year 4 and year 8

Focus: Calculations with fractions.

Resources: None.

Questions/instructions:

Instructions:

Write your answers in the white boxes.

You can use the shaded area to do your working.

1. +

% responses

y4 y8

= 24 33

2. + = 1 7 36

3. + + = 1 19 49

4. + = 1 • 40

5. + = • 44

6. + 4 = 4 • 56

7. 3 + 2 = 5 11 45

8. – = 5 24

9. – = 13 44

10. 6 – = 5 17 45

11. – = • 43

12. 5 = 2 • 39

13. 3 = 2 • 19

14. = • 25

15. 2 3 = 6 • 25

16. ÷ 2 = • 23

17. 6 ÷ = 18 • 10

18. 1 ÷ = 3 • 24

• not asked for year 4

Commentary

Less than two thirds of students at both levels could give

an oral description of a strategy that would work if carried

out correctly. Considering only the first strategy, the

standard addition algorithm was the most popular strategy

for both year levels (28 percent of year 8 students

and 18 percent of year 4 students). The simplification

of adjusting the 29 to 30 was suggested by 17 percent of

students at both levels.

• not asked for year 4

Commentary

The percentages recorded here are for getting the correct

answer reduced to standard form (eg. 1/2 rather

than 2/4). A few of the success rates would have been

almost doubled if unsimplified correct answers had been

included, but no more than 62 percent would have succeeded

with any of the tasks.

Chapter 3: Number 21

Approach: Independent Level: Year 4 and year 8

Focus: Understanding number and calculating.

Resources: None.

Questions/instructions:

% responses

y4 y8

Subtract one hundred

1. 400 300 73 95

2. 643 543 58 92

3. 40 000 39,900 • 47

Divide by one hundred

4. 1200 12 • 65

5. 50 0.5 • 34

6. 3.6 0.036 • 20

7. When a 3 digit number is added to a 3 digit

number the answer is

A always a 3 digit number.

B always a 4 digit number.

C either a 3 or 4 digit number. C • 50

D either a 3, 4 or 5 digit number.

8. Which number best describes the amount

of the box shaded?

A 0.05

B 0.25 C • 42

C 0.45

D 0.6

E 0.65

9. A school has 410 children.

97 children are away at camp.

About how many are still at school?

A 200

B 300

C 400 B 66 90

D 500

10. On the number line, which letter

best represents A G? C • 4

11. On the number line, which letter best

represents B F? G • 18

12. Which fraction matches the letter X

on the number line?

A

B

C

D D 20 49

• not asked for year 4

Number Items B

% responses

y4 y8

is a fraction between and 1.

Write two other fractions that would be

between and 1.

13. correct • 48

14. other correct • 45

15. Without working out the exact answer,

choose the best estimate for 87 0.09.

A a lot less than 87 A • 27

B a little less than 87

C a little more than 87 ✗ C • 42

D a lot more than 87

16. Which part of the circle is missing?

A

B

C C 39 81

D

17. A class has 25 pupils. come by bus,

come by bike.

How many do not come by bus or bike?

A 5

B 10 B • 62

C 15

D 20

18. Write a multiplication sentence

to find the number of circles.

3 5=15 or 5 3=15 35 48

19. Bob has 123 stamps to put in his album.

If 25 stamps fit on each page, how many

pages will he need?

A 4

B 5 B 28 65

C 6

D 7

To cook a meal for

10 people I need:

2 chickens

10 kumara

30 yams

1000g peas

I want to cook a meal

for 5 people. Fill in the

amounts of food I need.

20. 1 chickens 50 83

21. 5 kumara 51 84

22. 15 yams 33 78

23. 500(g) peas 28 75

22 NEMP Report 23: Mathematics 2001

Number Items C

Approach: Independent Level: Year 4 and year 8

Focus: Understanding number and number operations.

Resources: None.

Questions/instructions:

% responses

y4 y8

Multiply by ten:

1. 6 60 64 92

2. 78 780 26 83

3. 3.14 31.4 • 39

Write in figures:

4. Five hundred and eighty 580 74 91

5. Two thousand five hundred and eighteen

2 518 63 89

6. Two hundred thousand and forty-three

200 043 11 50

7. Without working out the exact answer,

what is the best estimate for 21 19 ?

A 200

B 300

C 400 C • 54

D 500

8. Five children each have 15 marbles.

Which of these tells us how many marbles

they have altogether?

A 5 15

B 15 15 15 15 15 B 60 79

C 15 5

D 5 5 5 5 5

Estimate the number shown by each arrow.

% responses

13. Only one of the answers is correct.

y4 y8

Without calculating, decide which one it is.

A 45 1.05 = 39.65

B 4.5 6.5 = 292.5

C 87 1.076 = 93.61 C • 32

D 585 0.95 = 595.45 ✗ D • 32

14. What is the difference between and ?

A

B B 33 50

C

D 2

15. What fraction is equivalent to ?

A

B

C C • 69

D

16. A fruit bar 11cm long is cut into 2 equal

pieces. How long will each piece be?

5.5 (with or without “cm”) 35 81

How much of this area is shaded?

17. As a fraction: or equivalent 47 83

9. A 75–175 37 51

10. B 450–550 26 60

11. C 650–750 30 53

12. Which fraction represents the largest

amount?

A A • 56

B

C

D

18. As a decimal: 0.5 • 51

19. As a percentage: 50% • 70

20. Which of the following is closest

to 10 seconds?

A 9.1 seconds

B 9.7 seconds

C 9.9 seconds C 63 92

D 10.2 seconds

• not asked for year 4

Chapter 3: Number 23

Number Line Y4

Approach: One to one Level: Year 4

Focus: Matching fractions with a number line.

Resources: 3 cards, each with a fraction; number line (0–1),

recording book.

Questions/instructions:

Place number cards in front of student.

Here are some number cards.

Read what is on them to me.

Place number line in front of student.

Approach: One to one Level: Year 8

Focus: Matching fractions with a number line.

Resources: 8 cards, each with a fraction, decimal or percentage;

number line (0–1), recording book.

Questions/instructions:

Place number cards in front of student.

0.1

100%

0.7

Here are some number cards.

Read what is on them to me.

50%

Place number line in front of student.

Number Line

0.5

0.25

Here is a number line.

It starts at zero and ends at one.

Put the cards where you think they should

be on the number line.

% responses

y4

Here is a number line. It starts at zero and

ends at one.

Put the cards where you think they should

be on the number line.

% responses

y8

Record the numbers after the student has put

them on the number line.

Record the numbers after the student has put

them on the number line.

correct 36

correct 38

correct 44

Total score: 3 29

2 1

1 30

0 40

0.1

0.25

0.5

50%

0.7

100%

correct 54

correct 38

correct 26

correct 36

correct 62

correct 37

correct 27

correct 66

Total score: 7–8 22

5–6 13

3–4 18

0–2 47

Commentary

About 30 percent of year 4 students placed these three

fractions correctly on the number line.

Commentary

About one third of the year 8 students matched more

than half of the numbers to their correct places on the

number line.

24 NEMP Report 23: Mathematics 2001

Population Y4

Population

Approach: One to one Level: Year 4

Focus: Interpreting place value.

Resources: Population card.

Questions/instructions:

Place population card in front of student

% responses

y4 y8

Approach: One to one Level: Year 8

Focus: Interpreting place value.

Resources: New Zealand population card.

Questions/instructions:

Place population card in front of student

% responses

y4

y8

Number of people living

in a small town

2495

New Zealand’s Population

3 825 614

This card shows the number of people in a

small town.

1. Read the number to me.

read correctly with all usual place values 73

Point to the 4.

2. What does this 4 mean?

prompt: What does the 4 stand for?

Point to the 5.

3. What does this 5 mean?

hundreds or four hundred 69

prompt: What does the 5 stand for?

Point to the 2.

4. What does this 2 mean?

prompt: What does the 2 stand for?

Point to the 9.

5. What does this 9 mean?

ones or five ones 71

thousands or two thousand 68

prompt: What does the 9 stand for?

tens or nine tens or ninety 66

This card shows the population of New

Zealand.

1. Read the number to me.

read correctly with all usual place values 67

Point to the 4.

2. What does this 4 mean?

prompt: What does the 4 stand for?

Point to the 5.

3. What does this 5 mean?

prompt: What does the 5 stand for?

Point to the 6.

4. What does this 6 mean?

ones or 4 ones 84

thousands or five thousand 86

prompt: What does the 6 stand for?

Point to the 8.

5. What does this 8 mean?

prompt: What does the 8 stand for?

Point to the 3.

6. What does this 3 mean?

hundred or six hundred 91

hundreds of thousands

or eight hundred thousand 69

prompt: What does the 3 stand for?

millions or three million 87

Commentary

About two thirds of the year 4 students correctly interpreted

the place values of a 4 digit number. The parallel

task for year 8 students (Population) used a seven digit

number.

Commentary

The parallel task (Population Y4) used a 4 digit

number.

Chapter 3: Number 25

Strategy

Missing the Point

Approach: One to one Level: Year 8

Focus: Computation strategies.

Resources: Prompt card.

Questions/instructions:

Show card.

% responses

y4

y8

Approach: Independent Level: Year 8

Focus: Estimation.

Resources: None.

Questions/instructions:

% responses

y4

y8

17 x 6 = 102

19 x 6 =

This card tells you that 17 times 6 is 102.

If you already know that 17 times 6 equals

102, how would you work out 19 times 6.

Tell me how you would work it out.

prompt: Can you explain that a bit more

to me?

Strategy:

You have 17 groups of 6;

you need 2 more groups of 6;

that is 2 6 = 12 more;

102+12=114 45

You have 6 groups of 17;

you need 2 more per group;

that is 6 2= 12 more;

102 +12=114 7

You need to add 12 (6 2),

but not clearly explained why 9

Any method where 19 6

calculated directly 17

No workable strategy explained 22

Fred used the calculator to work out

786 divided by 13.

There is something wrong with the calculator.

It doesn’t show the decimal point.

Put the decimal point where you think it

should go. Make it nice and black.

after first 2 digits

(60.461538) 44

Commentary

About 60 percent of the students realised that the task

could be achieved without direct calculation of 19 6.

However more than 20 percent did not describe a workable

strategy.

Commentary

Less than half of the year 8 students correctly identified

the correct order of the magnitude for the result of this

division calculation.

26 NEMP Report 23: Mathematics 2001

Link tasks 1–12

% responses

y4 y8

% responses

y4 y8

% responses

y4 y8

LINK TASK 1

Approach:Station Level: Year 4 & year 8

Focus: Addition facts

Total score: 30 43 68

27–29 41 31

18–23 7 1

12–17 4 0

0–11 5 0

LINK TASK 5

Approach:One to one Level: Year 4 & year 8

Focus: Understanding place values.

Item: 1 79 97

2 67 91

3 79 95

4 69 92

LINK TASK 9

Approach:One to one Level: Year 4 & year 8

Focus: Fractions

Total score: 4–5 15 58

3 29 22

2 19 10

0–1 37 10

LINK TASK 2

Approach:Station Level: Year 4 & year 8

Focus: Multiplication facts

Total score: 30 7 47

LINK TASK 3

27–29 19 39

18–23 21 9

12–17 27 3

0–11 26 2

Approach:Independent Level: Year 4 & year 8

Focus: Subtraction facts

Item: 1 71 91

2 73 93

3 64 90

4 33 77

5 35 77

LINK TASK 6

Approach:One to one Level: Year 4 & year 8

Focus: Understanding number operations.

Item: 1 98 100

LINK TASK 7

2 80 93

3 65 85

4 47 75

5 84 98

6 57 84

Approach:One to one Level: Year 4 & year 8

Focus: Computation strategies

Total score: 5–6 3 9

3–4 12 34

1-2 56 49

0 29 8

LINK TASK 10

Approach:One to one Level: Year 4 & year 8

Focus: Various number items.

Score:

(9 common items) 6 40 5

LINK TASK 11

5 17 10

4 18 15

2–3 20 45

0–1 5 25

Approach:One to one Level: Year 8

Focus: Computation strategies.

Total score: 3 39

2 32

1 12

0 17

LINK TASK 4

Approach:Independent Level: Year 4 & year 8

Focus: Division facts

Item: 1 62 92

2 50 92

3 53 90

4 36 89

5 6 55

6 10 70

LINK TASK 8

Approach:One to one Level: Year 4 & year 8

Focus: Estimation and computation

Total score: 5 21 49

4 20 29

2–3 29 14

0–1 30 8

LINK TASK 12

Approach:One to one Level: Year 8

Focus: Number lines.

Total score: 6 19

5 20

4 17

3 16

2 14

0–1 14

chapter 4 27

measurement

The assessments included sixteen tasks investigating students’ understandings,

processes and skills in the area of **mathematics** called measurement. Measurement

includes knowledge, understanding and use of systems of measurement, the use of

measurement apparatus, and processes of predicting, calculating and recording.

This chapter includes tasks relating to money.

Sixteen tasks were identical for both year 4 and year 8. Seven tasks had overlapping

versions for year 4 and year 8 students, with some parts common to both levels.

Three tasks were attempted by year 4 students only, and seven by year 8 only.

Twelve are trend tasks (fully described with data for both 1997 and 2001), ten are

released tasks (fully described with data for 2001 only), and eleven are link tasks

(to be used again in 2005, so only partially described here).

The tasks are presented in the three sections: trend tasks, then released tasks and

finally link tasks. Within each section, tasks attempted (in whole or part) by both

year 4 and year 8 students are presented first, followed by parallel tasks, then tasks

attempted only by year 8 students.

Averaged across 101 task components administered to both year 4 and year 8

students, 25 percent more year 8 than year 4 students succeeded with these

components. Year 8 students performed better on 95 of the 101 components. As

expected, the differences were generally larger on more difficult tasks. These often

were tasks that many year 4 students would not yet have had much opportunity

to learn in school.

There was little evidence of change between 1997 and 2001. Averaged across

41 trend task components attempted by year 4 students in both years, 2 percent

more students succeeded in 2001 than in 1997. Gains occurred on 25 of the 41

components. At year 8 level, with 45 trend task components included, 2 percent

fewer students succeeded in 2001 than in 1997. Gains occurred on 15 of the 45

components.

A representative range of measurement systems, processes and applications was

covered in the set of tasks attempted by students. At both levels students’ skills

of reading measurements were substantially stronger than those of making good

estimations. Moderate to low

percentages of year 8 and year 4

students demonstrated abilities

to effectively explain processes

and strategies for making and

checking measurements.

Bean Estimates

28 NEMP Report 23: Mathematics 2001

Apples

trend

Better Buy

trend

Approach: One to one Level: Year 4 and year 8

Focus: Money calculations and change.

Resources: Advertisement card, calculator, $5, $10 and $20 notes.

Questions/instructions:

The calculator is not given to

the student until question 4.

Questions leading up to Q4 are

solved mentally.

In this activity we are using

some artificial money.

Apples at a shop cost $1.95

a kilogram.

Apples

$1.95 a kilo-

Show the advertisement card and hand out money.

I want to buy 5 kilograms of apples.

I have a $5 note, a $10 note and a $20

note.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

1. What is the smallest value note that

I could use to pay for the apples?

$10 47 (39) 86 (92)

2. Why did you choose that note?

calculation 2 (1) 11 (3)

estimation and elimination 27 (14) 46 (49)

estimation,

but not eliminating $20 option 18 (17) 28 (31)

3. How much change would you expect?

Hand student a calculator.

Use the calculator to work out the

next problem. Tell me what you are

doing with the calculator as you do it.

year 8

4. If I bought a 5 kilogram bag of

apples on special for $7.50, how

much would the apples cost per

kilogram?

25¢ 7 (8) 33 (35)

Ask the student to read out the answer

shown on the calculator. $1.50 • 57 (70)

year 4

4. If apples cost $1.50 a kilogram, how

much would five kilograms cost?

Ask the student to read out the answer

shown on the calculator. 26 (24) •

Commentary

Although most year 8 students realised that a $10 note

would be sufficient in question 1, few realised that the

total amount required would be 5x5 cents less than $10,

making the answer to question 3 easy.

Approach: One to one Level: Year 4 and year 8

Focus: Value for money (cost per unit).

Resources: 100g Pebbles labelled with price $1.30; 50g

Pebbles labelled with price 60¢; 20g Pebbles labelled with

price 30¢.

Questions/instructions:

Place the 100g and 50g boxes of Pebbles in front of the

student.

In this activity you

will be using some

boxes of Pebbles.

The big box holds

100 grams of

Pebbles and costs

$1.30. The smaller

box holds 50 grams

of Pebbles and costs 60 cents.

1. Which one is the better value for

money?

prompt Which box would give you more

Pebbles for the money?

2. Why is that box better value for

money?

3. How do you know that?

% responses

2001 (’97) 2001 (’97)

year 4 year 8

50g box 64 (68) 83 (82)

correct, clear explanation 13 (10) 62 (65)

on right track but vague 9 (14) 10 (13)

Place the 20g box of Pebbles in front of

the student.

4. This box costs 30 cents. Which is

the better buy —this 20g box or this

100g?

Point to the 20g box.

6. If I wanted 100g of Pebbles, how

many of these boxes would I need?

7. How did you work that out?

100g box • 76 (78)

5 • 86 (90)

correct and clear • 80 (83)

Commentary

The results show substantial progress from year 4 to

year 8, especially in ability to justify the choice made,

with little change from 1997 to 2001 at either level.

Chapter 4: Measurement 29

trend

Approach: One to one Level: Year 4 and year 8

Focus: Estimating measurements.

Resources: Ruler; measuring tape;objects: bolt, pencil, stick, ribbon.

Measures

Questions/instructions:

Do not let the student see a ruler or the measuring tape

until it is time to make the actual measurements.

In this activity I am going to ask you to estimate or

guess the length of some things. After that, you can

measure them to check your estimates. I’ll write

down the measurements you tell me. Each time, tell

me if the measurement is in millimetres, centimetres

or metres.

One at a time, introduce each of the objects in the order

given. Ask student to estimate all of the objects before

they are measured. Write student’s answers for each

% responses

measure.

2001 (’97) 2001 (’97)

Hand student the bolt. [70mm] year 4 year 8

1. How long do you think this is?

prompt: What are the units of measurement?

Record student answer.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

56–84mm 26 (27) 42 (38)

42–55mm or 85–98mm 20 (25) 29 (32)

Hand student the pencil. [144mm]

2. How long do you think this is?

prompt: What are the units of measurement?

115–173mm 20 (26) 36 (38)

89–114mm or 174–203mm 35 (35) 41 (40)

Hand student the stick. [302mm]

3. How long do you think this is?

prompt: What are the units of measurement?

240–360mm 33 (34) 63 (63)

180–239 or 361–420mm 19 (23) 20 (23)

Hand student the ribbon. [400mm]

4. How long do you think this is?

prompt: What are the units of measurement?

320–480mm 23 (24) 50 (44)

260–319 or 481–560mm 20 (24) 28 (35)

5. Now how tall do you think you are?

prompt: What are the units of measurement?

% responses

2001 (’97) 2001 (’97)

year 4 year 8

Closeness to teacher’s measurement

in question 6:

within 20% 45 (39) 74 (78)

Give the student the ruler

(not the measuring tape).

Now I want you to use this ruler to

measure the length of each of the four

things. As you measure them, tell me

what the length is, and I’ll write it

down. Remember to tell me the unit

of measurement.

within 20–40% 14 (10) 7 (9)

• bolt 65–75mm 79 (79) 95 (97)

• pencil 130–150mm 87 (88) 98 (98)

• stick 290-310mm 91 (93) 98 (99)

• ribbon 390–410mm 51 (56) 89 (93)

Now let’s see how tall you are.

Have student stand against a wall. Place

a piece of tape on the wall to show the

height. Give student the measuring tape.

6. Now measure your height and tell

me how tall you are so I can write

it down. Remember to tell me the

unit of measurement.

Record student answer. Check student’s

measurement and record actual height.

within 3cm 69 (76) 73 (77)

Commentary

Both year 4 and year 8 students were quite inaccurate

in estimating the lengths of four objects. Most year 8

students measured accurately, except where the length

to be measured was greater than the length of the measuring

device. There was little change in performance

between 1997 and 2001.

30 NEMP Report 23: Mathematics 2001

Bean Estimates

trend

Lump Balance

trend

Approach: One to one Level: Year 4 and year 8

Focus: Estimating (area related).

Resources: 3 trays: red, blue and green; 20 beans; recording book.

Questions/instructions:

In this activity I want you to estimate (or guess) how

many of these beans will fit into each of these trays.

Give the student the red tray and 1 bean.

1. What is the largest number of beans this

frame will hold if the beans are

lying flat with none on top of each

other? I’ll write your estimate on

the activity sheet.

Give the student the beans.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

9–11 34 (32) 33 (43)

Now use these beans to check your estimate by

arranging them on the tray, like this.

Show the student how to fit 10 beans in

the tray.

So, how many beans actually fit?

Now give the student the blue tray.

Ensure BOTH of the trays

are in front of student.

2. How many beans do you think this blue tray will

hold? This time we won’t use the beans, but you

can look at them. I will write your estimate on the

activity sheet.

year 4 year 8

18–22 59 (52) 71 (78)

Give student green tray — leave the red and blue trays in

front of the student.

3. How many beans do you think this green tray will

hold? We still won’t use the beans, but you can

look at them. I will write your estimate on the

activity sheet.

year 4 year 8

Record student answer.

36–44 36 (32) 49 (62)

Commentary

Only about one third of the students managed an accurate

estimate in question 1. The year 8 students showed

greater ability to use the actual value (demonstrated) to

improve their estimates in questions 2 and 3. The 2001

year 4 students did a little better than their 1997 counterparts,

but 2001 year 8 students did markedly worse.

Approach: One to one Level: Year 4 and year 8

Focus: Use of a balance.

Resources: Balance, box of raisins (42.5g), lump of plasticine.

Questions/instructions:

Check the adjustment of the balance before starting this task.

Place the balance, lump of plasticine & box of raisins

in front of the student.

1. I want you to use the balance to make a lump of

plasticine which is as heavy as this box of raisins.

If student just changes the shape of the plasticine:

prompt: You don’t have to use all of the plasticine.

If student doesn’t use the balance:

% responses

2001 (’97) 2001 (’97)

prompt: Remember you can use the balance.

year 4 year 8

successful 78 (75) 92 (93)

almost successful 13 (14) 6 (2)

2. How do you know that the lump of

plasticine and the box of raisins are

the same weight?

clear and correct explanation 29 (18) 48 (46)

fuzzy explanation, but on right track 45 (46) 45 (40)

Remove excess plasticine and give the student the lump

which they have made.

3. Now use this lump to make two pieces that each

weigh the same amount.

If halved visually without being checked on the balance, ask:

prompt: How do you know that each lump weighs the same amount?

prompt: Can you think of a way of checking

that out to see if they are the same? year 4 year 8

successful 78 (81) 93 (97)

almost successful 17 (13) 7 (2)

Place the two plasticine lumps in front of the student.

4. This time I want you to try to make a lump that is

one and a half times as heavy as one of these lumps.

prompt: How do you know that one lump is one and a half

times as heavy as the other?

prompt: Can you think of a way of checking

that? successful 5 (4) 12

year 4 year 8

(19)

almost successful 5 (2) 9 (7)

Total score: 5–6 8 (5) 20 (25)

4 62 (65) 68 (67)

3 15 (11) 9 (2)

0–2 15 (19) 3 (6)

Commentary

Year 4 students were much less successful than year 8

students in the explanation (question 2). The 1997 and

2001 results were very similar overall.

Chapter 4: Measurement 31

trend

Video Recorder

Approach: Independent Level: Year 4 and year 8

Focus: Relating am and pm times to 24 hour clock times.

Resources: None.

Questions/instructions:

Write the numbers on the video recorder

clocks that are the starting times for the

programmes.

T.V. TIMES

8:15 am What Now

10:00 am Kids Time

12:00 pm Movie Time

3:45 pm Cartoons

5:30 pm Spot On

% responses

2001 (’97) 2001 (’97)

year 4 year 8

10:00 67 (79) 86 (95)

T.V. TIMES

9:15 am What Now

10:30 am Kids Time

2:00 pm Movie Time

1:00 pm Cartoons

8:30 pm Spot On

14:00 7 (6) 68 (65)

✗ 2:00 54 (67) 20 (29)

T.V. TIMES

9:30 am What Now

10:40 am Kids Time

11:00 am Movie Time

5:45 pm Cartoons

7:30 pm Spot On

19:30 5 (4) 64 (67)

✗ 7:30 57 (69) 22 (27)

Total score: 5 4 (4) 60 (63)

4 1 (1) 5 (2)

3 53 (63) 21 (29)

2 5 (7) 3 (2)

1 9 (9) 3 (1)

0 28 (16) 8 (3)

Commentary

Almost all year 4 students failed to convert pm times

to correct 24 hour time. Overall, performance of both

year 4 and year 8 students was slightly lower in 2001

than in 1997.

32 NEMP Report 23: Mathematics 2001

Measurement Items B

trend

Approach: Independent Level: Year 4 and year 8

Focus: Various measurement items.

Resources: None

Questions/instructions:

% responses

2001 (’97) 2001 (’97)

year 4 year 8

% responses

2001 (’97) 2001 (’97)

year 4 year 8

1. Draw a square that is four times the

area of the square shown.

Tennis

Balls

Golf

Balls

Rubber

Balls

2. Explain how you know that it is four

times as big. not marked

4x4 39 (•) 62 (63)

✗ 8x8 3 (•) 19 (13)

5. Linda has three large boxes all the

same size and three different kinds

of balls as shown below. If she fills

each box with the kind of balls

shown, which box will have the

fewest balls in it?

A The box with the tennis balls. A 48 (48) 75 (80)

B The box with the golf balls.

C The box with the rubber balls.

D You can’t tell.

3. An airport clock shows that the

time in Sydney is 3.00 p.m. New

Zealand time is 2 hours ahead of

this time.What is the time in New

Zealand?

5.00 pm 51

(57)

88

(86)

5.00 21 5

6. The cups shown contain different

amounts of water.

Which cup has about 200 millilitres

of water in it?

A A

B B

C C

A 63 (51) 84 (80)

4. What number would the minute

hand point to at 10 past 8?

2 41 (39) 86 (85)

Chapter 4: Measurement 33

Approach: One to one Level: Year 4 and year 8

Focus: Understanding a variety of measurements.

Resources: None.

Questions/instructions:

trend

Measurement Items C

% responses

2001 (’97) 2001 (’97)

year 4 year 8

1. The map shows the travel times

between places on a bush walk. If

John and Kate leave the camp at

9.00 a.m. and stop for 20 minutes at

the waterfall, about what time will

they arrive at the cave?

A 2:00 p.m.

B 3:00 p.m.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

C 4:00 p.m. C • 51 (55)

2. I get out of bed at 7.45 a.m. I take

3 minutes to get dressed, 15 minutes

for breakfast, and 8 minutes to get

to school. What time do I get to

school? 8.11 with or without “am” • 63 (62)

3. How many hours are equal to

150 minutes?

A 1

B 2

C 2 C 30 (24) 56 (56)

D 3

4. The area of the

triangle is

A 6 square units

B 9 square units

C 12 square units B 23 (20) 69 (66)

D 42 square units

5. If the area of the shaded

triangle shown here is

4 square centimetres,

what is the area of the

entire square?

A 4 square centimetres

B 8 square centimetres B 26 (•) 50 (52)

C 12 square centimetres

D 16 square centimetres

E Not sure

6. What is the area of the square?

9cm 2 0 (•) 27 (26)

9

with incorrect or no unit 3 (•) 15 (16)

7. What is the reading on this scale?

3.2 with or without “cm” • 32 (28)

✗ 3.1 • 51 (•)

8. The radius of the circle is 5.

What is the perimeter of the square?

A 10

B 20

C 40 C • 23 (•)

D 25

9. The time is 7.25a.m. I take 15

minutes for breakfast, 3 minutes to

get dressed and 8 minutes to get to

school.What time do I get to school?

7.51 with or without “am” 29 (21) •

10. Which unit would you use to

measure the length of a pencil?

A centimetres A 88 (84) •

B metres

C kilometres

11. About how far could you walk in

one day?

A centimetres

B metres

C kilometres C 57 (69) •

12. The height of a person is about:

A 2 metres A 56 (52) •

B 20 metres

C 200 metres

D 2000 metres

13. About how long is a bed?

A 10 cm

B 100 cm

C 200 cm C 32 (29) •

D 500cm

14. What is the most likely weight of a

pencil?

A 5 grams A 59 (57) •

B 5 kilograms

C 5 litres

D 5 centimetres

34 NEMP Report 23: Mathematics 2001

Party Time

trend

What’s the time?

trend

Approach: Station Level: Year 4

Focus: Adding costs and calculating change, with calculator

available.

Resources: Priced packets of candles ($1.00),

balloons ($1.25), sweet bags ($1.10)

and invitations ($ .95); calculator.

Approach: Independent Level: Year 4

Focus: Reading analogue clocks.

Resources: None.

Questions/instructions:

Draw a ring around the two clocks that are showing the

same time.

Questions/instructions:

You have some shopping for a birthday

party.

balloons

invitation cards

candles

sweet bags

1. How much do the 4 packets cost

altogether?

Answer: $4.30 42 (46)

2. If you had $5 to pay for the

shopping,how much change

should you get?

Answer:

correct, given answer in 1 38 (31)

3. If you had $10 to pay for the

shopping,how much change

should you get?

Answer:

correct, given answer in 1 32 (32)

Total score: 3 21 (19)

% responses

2001 (’97) 2001 (’97)

year 4

2 16 (13)

1 16 (26)

0 47 (42)

Commentary

Even with a calculator available, less than half the students

correctly totalled the four prices. Fewer calculated

the change appropriately.

% responses

2001 (’97)

year 4

top left & bottom right 38 (41)

Commentary

In 2001, 30 percent of students misread the top left clock

as 3:00 and matched it with the top centre clock. The

percentage of students choosing the correct pair was

very similar in 1997 and 2001.

Chapter 4: Measurement 35

trend

Supermarket Shopping

trend

Two Boxes

Approach: Station Level: Year 8

Focus: Adding costs and calculating change with calculator available.

Resources: Picture of supermarket brochure; calculator.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

Approach: Station Level: Year 8

Focus: Measuring boxes and calculating volume.

Resources: Gold box, silver box, ruler, calculator.

Questions/instructions:

In this activity you will be finding out

which box has the greater volume.

Measure the outside of the boxes.

1. Work out the volume (capacity) of

the gold box.

% responses

2001 (’97)

year 8

Questions/instructions:

You have a brochure from a shop that

tells you the prices of some food.

Here is a shopping list. Work out how

much this food will cost.

Shopping List

1 packet of bacon

1 packet of ice cream

1 packet of peas

2 packets of butter

% responses

2001 (’97)

year 8

Answer: 160 cm 3

within 10% and correct units 18 (11)

within 10% but units not correct 12 (18)

2. Work out the volume (capacity) of

the silver box.

Answer: 216cm 3

within 10% and correct units 16 (12)

within 10% but units not correct 14 (17)

3. Which box has the greater volume—

the gold box or the silver box?

silver box 64 (63)

1. This food will cost: $13.30 58 (66)

2. You have $20 to buy this food.

How much change would you get?

correct, given answer in 1 83 (87)

Commentary

The 2001 students were a little less successful than the

1997 students.

Commentary

Only 30 percent of students in 2001 calculated the

volume of each box correctly. This compared with

29 percent in 1997, but more students used the correct

units in 2001 than in 1997.

36 NEMP Report 23: Mathematics 2001

Stamps

Bank Account

B

Approach: Station Level: Year 4 and year 8

Focus: Money calculations.

Resources: 3 cards of stamps, 5¢, 10¢, 20¢.

% responses

y4 y8

A

Amount of money

$25

$20

$15

$10

$5

Approach: One to one Level: Year 4 and year 8

Focus: Relating money transactions to a graph of them.

Resources: Graph, ruler (to align graph bars with axis markings).

Questions/instructions:

Put graph and ruler in front of student.

This graph shows someone’s bank account.

Point to the word dollars.

Up this side is the amount of money the person has.

Point to the word time.

Along the bottom are the days of a week.

Have a careful look at the graph then tell me a story to

explain what is happening with the money.

% responses

Point to the beginning of the graph.

y4 y8

Story gives daily balance:

Bank Account

Monday Tuesday Wednesday Thursday Friday Saturday

Days

Monday:

$10 64 71

Tuesday:

$20 60 59

Wednesday:

$20 57 49

Thursday:

$16 5 12

15 6 9

Friday:

$16 4 11

15 5 9

Saturday:

$10 51 44

C

Story explains deposits and withdrawals:

Monday —> Tuesday $10 deposit 14 31

other or unspecified 15 26

Wednesday —> Thursday $4 withdrawal 0 11

other or unspecified withdrawal 28 45

Friday —> Saturday $6 withdrawal 0 8

other, or unspecified withdrawal 25 45

Questions/instructions:

Look at the 3 cards of stamps.

Which set of stamps would cost the most?

Tick the best answer.

% responses

y4 y8

A A 35 71

B

C

✓

Total score: 10–14 2 9

8–9 8 13

6–7 14 23

4–5 33 21

2–3 14 15

0–1 29 19

Commentary

Less than half of the year 8 students and quarter of the

year 4 students gave a fairly full, accurate account of the

graph.

Chapter 4: Measurement 37

Approach: Independent Level: Year 4 and year 8

Focus: Money calculations.

Resources: None.

Questions/instructions:

1. y e a r 4

In a sale is taken off the price of everything.

How much will you save on something

which used to cost $2.00?

Money A

% responses

y4 y8

50 (with or without “c”) 15 •

y e a r 8

In a sale is taken off the price of everything.

How much will you save on something

which used to cost $2.40?

60 (with or without “c”) • 40

2. $2.50 is divided equally between 2 children.

How much will each child get?

1.25 (with ot without “$”) 48 91

3. Sonny bought 3 Play Station games at $98

each. How could he work out how much

he spent?

A 3 $100, minus $2

B 3 $100, minus $3

C 3 $100, minus $6 C 21 61

D 3 $100, minus $12

Approach: Station Level: Year 4 and year 8

Focus: Placing measurements in order.

Resources: Team card, 5 stickers.

Simon

1.33m

Jamie

1.26m

Questions/instructions:

Swimming Team

Te Aroha

1.28m

Team Photo

Te Aroha

1.28m

Simon

1.33m

Cleo

1.40m

Jamie

1.26m

Rangi

1.37m

Cleo

1.40m

Rangi

1.37m

These children are in a swimming team.

They are getting ready for a team photo.

Use the stickers to show the heights in order

from tallest to shortest.

Tallest

Shortest

% responses

y4 y8

Cleo is above Rangi 93 99

Rangi is above Simon 90 98

Simon is above Te Aroha 92 98

Te Aroha is above Jamie 92 98

Total score: 4 81 96

2–3 16 3

0–1 3 1

Commentary

In question 1, many students in both years gave the discounted

price, not the amount of discount.

Commentary

Most year 4 and year 8 students succeeded fully with

this task.

Chapter 4: Measurement 39

Fishing Competition Y4

Approach: Station Level: Year 4

Focus: Comparing weights.

Resources: 6 pictures of fish on weighing scales.

Questions/instructions:

Here are the weights of fish.

Find the 3 heaviest fish and write down their

colours.

pink 93

blue 89

red 81

% responses

y4 y8

Fishing Competition

Approach: Station Level: Year 8

Focus: Comparing weights.

Resources: 6 pictures of fish on weighing scales.

Questions/instructions:

Here are the weights of fish.

Find the 3 heaviest fish and write down their

weights.

heaviest:

2nd heaviest:

3rd heaviest:

% responses

y4

y8

4kg 92

4 4

3.3kg 15

3.3 1

3.2kg 30

3.2 2

2.8kg 38

2.8 2

Commentary

Most students recorded the units (kg) as well as the

number.

40 NEMP Report 23: Mathematics 2001

Milkogram

Approach: Station Level: Year 8

Focus: Money computations.

Resources: Milkogram sheet.

Running Records

Approach: Independent Level: Year 8

Focus: Comparing times (minutes, seconds and decimals).

Resources: None.

Questions/instructions:

% responses

This table shows times for the men’s

y4

y8

Olympic 1500-metre race.

They were recorded between 1960 and

2000. Use the table to answer the questions.

Olympic men’s 1500-metre times

1960 Herb Elliott 3:35.6

1964 Peter Snell 3:38.1

1968 Kip Keino 3:34.9

1972 Pekka Vasala 3:36.3

1976 John Walker 3:39.17

1980 Sebastian Coe 3:38.4

1984 Sebastian Coe 3:32.53

1988 Peter Rono 3:35.96

1992 Fermin Cacho 3:40.12

1996 Noureddine Morceli 3:35.78

2000 Noah Ngeny 3:32.07

Questions/instructions:

Circle the best answer.

1. Look at the milk prices for 600ml glass

bottle of Standard Milk, and 300ml carton

of Standard Milk.

Which is the best value for money?

% responses

y8

A 600ml glass bottle of Standard Milk. A 70

B 300ml carton of Standard Milk.

C Both are the same value for money.

1. What was the fastest time?

3:32.07/Noah Ngeny 83

2. What was the second fastest time?

3:32.53/Seb Coe 79

3. What was the slowest time?

3:40.12/Fermin Cacho 71

2. Look at the milk prices for 2 litre of

Standard Milk, and 1 litre of Standard Milk.

Which is the best value for money?

A 2 litre of Standard Milk.

B 1 litre Standard Milk.

C They are the same value for money. C 48

3. Look at the cream prices.

Which is the best value for money?

A 2 litre plastic. A 51

B 1 litre plastic.

C 500ml plastic.

Chapter 4: Measurement 41

Link task 13

Approach: One to one

Level: Year 4 and year 8

Focus: Calculating change.

Link task 14

Approach: One to one

Level: Year 4 and year 8

Focus: Money computations.

% responses

y4 y8

Item: 1 52 88

2 40 84

3 37 83

4 18 73

5 39 81

6 16 67

Item: 1 54 78

2 30 76

Link task 17

Approach: Independent

Level: Year 4 and year 8

Focus: Temperatures.

Link task 18

Approach: One to one

Level: Year 4 and year 8

Focus: Estimating weights.

% responses

y4 y8

Item: 1 36 80

2 46 88

3 29 72

Item: 1 8 20

2 6 22

3 3 13

4 12 31

Link tasks 13 – 23

Link task 21

Approach: Independent

Level: Year 4 and year 8

Focus: Assorted measurement items.

% responses

y4 y8

Item: 1 95 94

2 60 78

3 30 68

4 64 85

5 9 60

6 26 43

7 43 71

8 19 65

Link task 15

Approach: One to one

Level: Year 4 and year 8

Focus: Calendar and date calculations.

Item: 1 37 71

2 23 46

3 17 59

4 21 64

5 20 56

6 17 54

Link task 19

Approach: Independent

Level: Year 4 and year 8

Focus: Length.

Item: 1 47 82

2 47 82

3 47 82

4 47 82

Link task 22

Approach: Station

Level: Year 8

Focus: Discount computations.

Total score: 12–15 23

9–11 18

6–8 16

3–5 20

0–2 23

Link task 16

Approach: One to one

Level: Year 4 and year 8

Focus: Clock times.

Total score: 12 6 43

10–11 18 35

8–9 20 12

6–7 29 9

4–5 21 1

0–3 6 0

Link task 20

Approach: Station

Level: Year 4 and year 8

Focus: Money computations.

Item: 1 • 53

2 26 75

3 60 86

4 37 •

Link task 23

Approach: Independent

Level: Year 8

Focus: Adding weights.

Total score: 4 15

3 18

2 20

1 18

0 29

42 chapter 5

geometry

The assessments included sixteen tasks investigating students’ understandings,

processes and skills in the area of **mathematics** called geometry. Geometry is

concerned with geometrical relations in two and three dimensions, and their

occurrence in the environment. It also involves recognition of the geometrical

properties of everyday objects and the use of geometric models as aids to solving

problems.

Eleven tasks were identical for both year 4 and year 8. One task had overlapping

versions for year 4 and year 8 students, with some parts common to both levels.

Four tasks were attempted only by year 8 students. Three are trend tasks (fully

described with data for both 1997 and 2001), seven are released tasks (fully

described with data for 2001 only), and six are link tasks (to be used again in 2005,

so only partially described here).

The tasks are presented in the three sections: trend tasks, then released tasks and

finally link tasks. Within each section, tasks attempted (in whole or part) by both

year 4 and year 8 students are presented first, followed by tasks attempted only by

year 4 students and then tasks attempted only by year 8 students.

Averaged across 41 task components administered to both year 4 and year 8 students,

23 percent more year 8 than year 4 students succeeded with these components.

Year 8 students performed better on all components.

There was little evidence of change between 1997 and 2001 for year 4 students,

but a small decline for year 8 students. Averaged across 13 trend task components

attempted by year 4 students in both years, 2 percent more students succeeded in

2001 than in 1997. Gains occurred on 10 of the 13 components. At year 8 level,

with 22 trend task components included, 5 percent fewer students succeeded in

2001 than in 1997. Gains occurred on 3 of the 22 components.

Many students were able to identify the nets of three-dimensional objects and to

mirror a shape in a line of symmetry. Students had less success with visualising the

internal structure and cross sections of three-dimensional objects, and with other

spatial relationships tasks in three dimensions. Many year 8 students had limited

capability to use and interpret angle measurements expressed in degrees.

Chapter 5: Geometry 43

trend

Hedgehog

Approach: One to one Level: Year 4 and year 8

Focus: Understanding rotation and angles.

Resources: Garden picture with metal washer in the middle;

model hedgehog to fit washer; recording book.

Questions/instructions:

Here is a hedgehog and its garden.

Hand student the hedgehog.

Put the hedgehog in the centre of the garden so it is

facing the tree. I want you to move the hedgehog the

way I tell you.

Write each student response on the recording sheet. After

every turn, ask student to turn the hedgehog to face the

tree again.

1. First turn the hedgehog clockwise a

quarter-turn. What is it facing?

% responses

2001 (’97) 2001 (’97)

year 4 year 8

flowers 63 (56) 92 (89)

2. Now turn the hedgehog clockwise a onethird

turn. What is it facing?

cat 18 (13) 40 (40)

3. Now turn the hedgehog anti-clockwise

a half-turn. What is it facing?

ladybird 65 (58) 85 (91)

turned anti-clockwise 77 (74) 94 (95)

4. Now turn the hedgehog anti-clockwise

a three-quarters turn.What is it

facing?

flowers 39 (38) 63 (77)

year 8

5. Now turn the hedgehog 90° to the

left. What is it facing?

spider • 60 (67)

6. Now turn the hedgehog 360° to

your left. What is it facing?

tree • 74 (84)

turned left • 91 (93)

7. Turn the hedgehog 30° to the right.

What is it facing?

butterfly • 53 (58)

8. Turn the hedgehog 270° to the

right. What is it facing?

spider • 38 (44)

9. What directions could you give me

if I wanted to turn the hedgehog to

face the frog?

prompt: Could you be more specific?

appropriate directions • 35 (39)

year 4

10. Now turn the hedgehog 1 /12 to the

right. What is it facing?

11. Now turn the hedgehog three-quarters

around to the left. What is it

facing?

12. What directions could you give me

if I wanted to turn the hedgehog to

face the frog?

% responses

2001 (’97) 2001 (’97)

year 4 year 8

butterfly 17 (16) •

flowers 37 (41) •

appropriate directions 48 (45)

•

Commentary

Students were much more successful with 1/4 and 1/2

turns than others such as 1/3, 3/4, and 1/12. Year 4 students

did equally well or slightly better in 2001 than in

1997, but year 8 students did a little worse in 2001 than

in 1997.

Chapter 5: Geometry 45

Shapes and Nets

Approach: One to one Level: Year 4 and year 8

Focus: Matching 3D objects and their nets.

Resources: None.

Questions/instructions:

Draw a line from each 3D shape to the net

that would make the shape.

% responses

y4 y8

68 85

Approach: One to one Level: Year 4 and year 8

Focus: Identifying shapes of cross-sections.

Resources: None.

Questions/instructions:

The block of cheese has been

cut through with one straight

cut. The cut made a square

cross section.

Draw the cross section for each

of these items of food:

One Cut

% responses

y4 y8

58 83

Potato

oval 38 62

56 83

Chocolate bar

triangle 32 60

73 88

Banana

circle 18 48

hexagon

or similar 3 4

39 67

Pear

pear

shaped

2D image 24 54

rectangle 16 41

42 65

Cheese

rectangle 14 41

60 86

Total score: 7 22 54

5–6 29 27

3–4 18 8

1–2 17 2

0 14 9

Commentary

On average, about 25 percent more year 8 than year 4

students correctly identified the net for each 3D object.

Cake

Total score: 12 2 14

9–11 9 25

6–8 10 13

3–5 11 9

1–2 16 12

0 52 27

Commentary

Students found it hardest to identify the rectangular

cross-sections for the cheese and cake. One quarter of

the year 8 students and half of the year 4 students got

none correct.

46 NEMP Report 23: Mathematics 2001

Paper Folds

Approach: Station Level: Year 4 and year 8

Focus: Spatial relationships.

Resources: Folded, stapled paper with hole punched.

Questions/instructions:

Look at the folded piece of paper.

DON’T take the staples out.

It was folded into 4 like this:

Line of Symmetry

Approach: Independent Level: Year 4 and year 8

Focus: Reflection.

Resources: None.

Questions/instructions:

Draw the other half of the shape by using its

line of symmetry as a starting point.

A hole was punched in it like this:

Where would the holes be when the paper

was unfolded?

Draw your answer here:

% responses

y4 y8

Drew 4 (and only 4) holes in correct places

(near centre fold) (as illustrated) 25 74

% responses

y4 y8

Completely correct 66 94

Chapter 5: Geometry 47

Whetu’s Frame

Approach: Independent Level: Year 4 and year 8

Focus: Edges and corners of 3D object.

Resources: None.

Questions/instructions:

Whetu wants to make a cube shaped frame.

She will make it with plastic piping. Each

edge will be 1 metre long.

Approach: Station Level: Year 8

Focus: Measuring angles.

Resources: Protractor.

Questions/instructions:

How many degrees did the boats turn as they

changed course?

White boat

Boats

% responses

y8

30±2 50

✗ 150±2 13

1. How much pipe does she need?

2. What is the smallest number of corners

she needs to use to make the frame?

% responses

y4 y8

12 metres 14 43

12 31 30

8 44 73

Black Boat

110±2 33

✗ 70±2 32

48 NEMP Report 23: Mathematics 2001

Grid Plans

Approach: One to one Level: Year 8

Focus: Different views (plans) of a 3D object.

Resources: Model made with multilink cubes.

Questions/instructions:

% responses

y4 y8

Front

% responses

y8

74

In front of you is a model made with blocks.

On the grids below, draw the shape of each view.

The first one has been done for you.

Right

74

Back

Left

46

Top

73

Chapter 5: Geometry 49

% responses

y4 y8

% responses

y4 y8

Link Tasks 24 – 29

% responses

y4 y8

Link task 24

Approach: One to one

Level: Year 4 and year 8

Focus: Describing 3D objects.

Total score: 12-27 9 13

Link task 25

Approach: Independent

Level: Year 4 and year 8

Focus: Nets of 3D objects.

9-11 14 26

6-8 31 35

3-5 38 22

0-2 8 4

Item: 1 32 61

2 74 88

3 25 40

4 80 91

5 44 77

6 79 94

Link task 26

Approach: Station

Level: Year 4 and year 8

Focus: Spatial relationships.

Total score: 5 29 38

Link task 27

Approach: Station

Level: Year 4 and year 8

Focus: Lines of symmetry.

4 22 30

3 13 10

2 8 5

1 9 4

0 19 13

Item: 1 66 75

2 47 69

Link task 28

Approach: Independent

Level: Year 4 and year 8

Focus: Drawing 2D objects.

Link task 29

Approach: Independent

Level: Year 8

Focus: Angles.

Item: 1 22 57

2 77 88

3 84 94

4 25 41

Item: 1 51

2 50

3 71

50 chapter 6

algebra and statistics

The assessments included seventeen tasks investigating students’ understandings,

processes and skills in the areas of **mathematics** called algebra and statistics.

Because of limited numbers of tasks in these two areas, and space constraints in

this report, these two strands of the curriculum are presented in one chapter.

Algebra involves patterns and relationships in **mathematics** in the real world, the

use of symbols, notation and graphs and diagrams to represent mathematical relationships

and ideas, and the use of algebraic expressions for solving problems.

Statistics is concerned with the collection, organisation and analysis of data, and

the estimation of probabilities and use of probabilities for prediction.

Four tasks were identical for both year 4 and year 8. Seven tasks had overlapping

versions for year 4 and year 8 students, with some parts common to both levels.

One task was attempted by year 4 students only, and five by year 8 only. Seven are

trend tasks (fully described with data for both 1997 and 2001), three are released

tasks (fully described with data for 2001 only), and seven are link tasks (to be used

again in 2005, so only partially described here).

The tasks are presented in the three sections: trend tasks, then released tasks and

finally link tasks. Within each section, tasks attempted (in whole or part) by both

year 4 and year 8 students are presented first, followed by tasks attempted only by

year 4 students and then tasks attempted only by year 8 students.

Averaged across 36 task components administered to both year 4 and year 8 students,

28 percent more year 8 than year 4 students succeeded with these components.

Year 8 students performed better on 35 of the 36 components.

There was evidence of substantial improvement between 1997 and 2001 for year

4 students, but little change over the same period for year 8 students. Averaged

across 15 trend task components attempted by year 4 students in both years, 9

percent more students succeeded in 2001 than in 1997. Gains occurred on 14 of

the 15 components. At year 8 level, with 28 trend task components included, 1

percent more students succeeded in 2001 than in 1997. Gains occurred on 16 of

the 28 components.

Chapter 6: Algebra and Statistics 51

trend

Train Trucks

Approach: One to one Level: Year 4 and year 8

Focus: Patterns/sequences.

Resources: Yellow/blue train; red/green train, garage.

Questions/instructions

Show the yellow/blue train.

The trucks on the train are being

painted yellow and blue. You will see

that there is a pattern in the way they

are being painted, and that not all of

the trucks have been painted yet.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

1. If the painters keep on painting the

trucks, what colour will truck 18 be?

(Point to number 18) blue 97 (95) 97 (97)

Now I’ll put a garage over some of the

trucks.

Cover numbers 12 to 25. Leave garage on

through to question 6.

2. What colour will they paint truck 21?

yellow 81 (76) 87 (83)

3. What colour will they paint truck 26?

blue 69 (69) 85 (78)

4. If the train went on to have more

trucks, what colour would they

paint truck 39? yellow 64 (59) 83 (75)

5. How did you work that out?

not marked

6. If the train had 60 trucks, how many

would be painted yellow?

30 57 (53) 84 (86)

7. If Batman came swooping down

and landed on a truck, do you think

he would definitely land on a blue

truck, he might land on a blue truck,

or he wouldn’t land on a blue truck?

might 74 (62) 90 (88)

% responses

2001 (’97) 2001 (’97)

8. What is the chance of him landing

year 4 year 8

on a blue truck?

50%, 50/50, equal • 72 (62)

Show second train with [green] and

[red] trucks.

Here is another train, and it has its

trucks painted in a different pattern.

9. What colour will truck 18 be?

red 92 (90) 93 (91)

10. If the train went on to have more

trucks, what colour would they

paint truck 42?

red • 50 (53)

11. How did you work that out?

not marked

12. How many green trucks would

there be all together up to 42? 28 • 20 (17)

13. If Batman came swooping down

and landed on a truck of this train,

do you think he would definitely

land on a green truck, he might land

on a green truck, or he wouldn’t

land on a green truck?

might 59 (51) 74 (73)

14. What is the chance of him landing

on a green truck?

2/3, 2 out of 3, 67% or similar • 23 (26)

Commentary

The 2001 year 4 students were a little more successful

than their 1997 counterparts. The results for year 8 students

were more mixed.

52 NEMP Report 23: Mathematics 2001

Algebra/Logic Items A

trend

Approach: Independent Level: Year 4 and year 8

Focus: Varied algebra items.

Resources: None.

% responses

Questions/instructions

2001 (’97) 2001 (’97)

year 4 year 8

1.How many tiles will you need to

make the pattern for …

10 people? [year 8] 52 • 31(•)

5 people? [year 4] 27 17 (•) •

2. Which statement is true?

A 442 > 436 A 56 (41) 70 (71)

B 352 > 759

C 518 > 819

D 883 < 794

3. What rule is used to get the numbers

in Column B from the numbers

in Column A?

Column A Column B

12 3

16 4

24 6

40 10

5. If = , then n =

A 20

B 30

% responses

2001 (’97) 2001 (’97)

year 4 year 8

C 40 C • 34 (35)

D 50

6. What is the least whole number x

for which 2x > 11?

A 5

B 6 B • 30 (29)

C 22

D 23

7. x < 6

x is a whole number.

Write down the solution set of this

sentence.

0,1,2,3,4,5 • 3 (3)

1,2,3,4,5 • 5 (5)

A Divide the number

in Column A by 4. A 27 (•) 61 (60)

B Multiply the number

in Column A by 4.

C Subtract 9 from the number

in Column A.

D Add 9 to the number

in Column A.

4. 3 ( + 5) = 30

The number in this box should be

A 2

B 5 B • 69 (74)

C 10

D 22

Commentary

Only one trend item was available for year 4 students,

with 15 percent more students succeeding in 2001 than

in 1997. Very little change was evident overall on the six

trend items for year 8 students.

Chapter 6: Algebra and Statistics 53

trend

Algebra / Logic Items B

Approach: Independent Level: Year 4 and year 8

Focus: Varied algebra items.

Resources: None.

Questions/instructions

% responses

2001 (’97) 2001 (’97)

1. Write a rule for each number pattern

using and , then complete

year 4 year 8

them. The first two are done for

you.

1 2 3 4 5 6 7

9 18 27 36 45 54 63

% responses

2001 (’97) 2001 (’97)

year 4 year 8

10. Squares 1 2 3 4 5 6

Matches 4 7 10 13 16 19 23 (•) 60 (•)

11. Predict the number of matches

needed to make 20 squares. 61 • 22 (•)

1 2 3 4 5 6 7

4 7 10 13

1. 7 • 55 (•)

1 2 3 4 5 6 7

7 14 21 28

1 2 3 4 5 6 7

1 5 9 13 17 21 25

1 2 3 4 5 6 7

4 14 24 34 44 54 64 • 39 (•)

1 2 3 4 5 6 7

7.5 8 8.5 9

35 42 49 • 49 (•)

2. ( 4) –3 (or equivalent) • 7 (•)

3. ( 10) –6 (or equivalent) • 8 (•)

4. ( .5) +7 (or equivalent) • 2 (•)

Use the machine to finish putting

numbers in the spaces on the card.

9.5 10 10.5 • 36 (•)

Number Number

in out

5. 2 9 39 (•) 67 (•)

6. 7 34 30 (•) 64 (•)

7. 12 59 28 (•) 62 (•)

8. 3 14 23 (•) 58 (•)

9. 10 49 • 52 (•)

year 4 year 8

12. How many squares are in

the 5th building? 25 15 (•) 42 (•)

13. How many squares are in the

10th building in this pattern? 100 (•) 11 (•)

14. Lee delivers newspapers. Let x

represent the number of newspapers

that Lee delivers each day.

Which statement represents the

total number of newspapers that

Lee delivers in 5 days?

A 5 x A 35 (21) 69 (66)

B 5 x

C x 5

D ( x x ) 5

15.The cost to rent a motorbike is

given by the following formula:

Cost = ($3 number of hours) $2

Fill in this table:

Time in Hours Cost in $

1 5

4 14

5 17

both correct • 48 (47)

one correct • 16 (11)

54 NEMP Report 23: Mathematics 2001

% responses

16. Write > or or < to make this 2001 (’97) 2001 (’97)

statement true:

year 4 year 8

Approach:

Level:

Focus: 456 8 = 456 = • 18 (•)

Resources:

17. Here is a number sentence:

% responses

2001 (’97) 2001 (’97)

3 < 14

year 4 year 8

Which number could go in

the to make the sentence true?

A 4 A 52 (37) •

B 5

C 12

D 13

18. Which number sentence is true?

A 321 < 123

B 321 < 321

C 321 > 123 C 47 (33) •

D 321 = 123

19 Fill in the empty spaces on the grid

to finish the pattern.

Approach:

Focus:

Resources:

20. Which is true?

A 16 < 15

B 16 > 17

C 17 < 16

Level:

% responses

2001 (’97) 2001 (’97)

year 4 year 8

% responses

2001 (’97) 2001 (’97)

year 4 year 8

D 17 > 16 D 56 (43) •

21. It costs $3 for every hour to rent a

bike.

Fill in this table:

Time in Hours Cost in $

1 3

4 12

5 15

both correct 40 (•)

one correct 19 (•)

year 4 year 8

Number correct: 6 28 (20) •

5 9 (7) •

4 18 (13) •

3 10 (9) •

2 15 (14) •

1 10 (12) •

0 10 (25) •

Commentary

This collection of items included some used previously in

1997 and others new in 2001. On five trend items, year 4

students in 2001 performed substantially better than the

1997 students. Year 8 students performed similarly in

1997 and 2001 on the two year 8 trend items.

Chapter 6: Algebra and Statistics 55

trend

Statistics Items A

trend

Statistics Items B

Approach: Independent Level: Year 4 and year 8

Focus: Combinations and probability.

Resources: None.

Questions/instructions

1. Each cup can be paired with each

saucer. How may different pairings

of a cup and saucer can be made?

A 2

B 3

C 5

% responses

2001 (’97) 2001 (’97)

year 4 year 8

D 6 D 15 (•) 56 (56)

Approach: Independent Level: Year 4 and year 8

Focus: Probability and statistics.

Resources: None.

Questions/instructions

% responses

2001 (’97) 2001 (’97)

year 4 year 8

1. In a bag of marbles, are red, are

blue, are green, and are yellow.

If a marble is taken from the bag

without looking, it is most likely to

be:

A red A 33 (•) 71 (76)

B blue

C green

D yellow

2. Here are the ages of five children:

13, 8, 6, 4, 4.

What is the average (mean) age

of these children? 7 • 34 (33)

2. There is only one red marble in

each of the bags shown below.

Without looking, you are to pick

a marble out of one of the bags.

Which bag would give you the

greatest chance of picking the red

marble?

10 marbles 100 marbles 1000 marbles

A Bag with 10 marbles. A 49 (•) 76 (74)

B Bag with 100 marbles.

C Bag with 1000 marbles.

D It makes no difference.

3. Maria made a survey of the students

in her class. She found that 60%

of the students know how to use

one brand of computer, and 40%

knew how to use a different brand

of computer. She said that because

it added up to 100%, it meant that

everybody in the class knew how to

use a computer.

Explain to Maria why she is right or

wrong.

If possible, use a diagram.

Clear explanation that

Maria is wrong: with diagram • 1 (2)

without diagram • 1 (2)

Commentary

The results for year 8 students were almost identical in

1997 and 2001. These tasks were not administered to

year 4 students in 1997.

Commentary

Item 3 was very difficult for the year 8 students. The 2001

results for year 8 students were very similar to the1997

results.

56 NEMP Report 23: Mathematics 2001

Farmyard Race

trend

Approach: Team Level: Year 4

Focus: Logic.

Resources: Placement mat, 8 plastic animals, 4 clue cards.

% responses

2001 (’97) 2001 (’97)

year 4 year 8

% responses

2001 (’97) 2001 (’97)

year 4 year 8

Sheep and piglet ran together

until the end when piglet

tripped.

Dog was third to cross the finish

line.

Questions/instructions

Put out placement mat.

These farm animals had a race.

Show the animals and name each one.

Stand them in the middle of the table.

The white horse saw four

legs beat him home.

Goat wanted to bite Mother

pig’s tail as he followed her

across the finish line.

Your team is going to put the animals in the order that

they came in the race. Put the animals that your team

thinks came first in this box marked 1st, the animal

your team thinks came second in this box marked

2nd, and so on. Use these clues to find out the order

they finished.

Allocate one clue card to each team member.

You can start now.

When they say they have finished, record the order.

Be sure to distinguish

between mother pig and

piglet, white horse and

brown horse.

Now go back and

check one more time

to make sure that what

you’ve done fits with

all of the clues.

The white horse finished

before the goat.

Piglet finished second to last.

2 animals came between

mother pig and her piglet.

The brown horse led the

pack until he stopped to eat.

% responses

2001 (’97)

year 4

Final order fitted clues: all 8 33 (20)

[8 clues] 7 22 (33)

6 24 (23)

5 14 (17)

< 5 7 (7)

Strategy rating: very good 17 (23)

Cooperation level:

good 48 (33)

moderate 29 (34)

poor 6 (10)

good all involved most of time 50 (42)

moderate all involved some time 38 (50)

poor 12 (8)

Commentary

Overall, the 2001 students performed a little better than the 1997 students on this logic task.

Chapter 6: Algebra and Statistics 57

trend

Photo Line-Up

Approach: Team Level: Year 8

Focus: Logic.

Resources: Placement mat, 12 picture cards (numbered on back), 4 clue cards.

Questions/instructions

Sue is taller than Hone.

Ben is standing somewhere

between Tasi and Sim.

There is only one person

Arrange the cards numerically

from 1-12 before giving them

to the students.

Here are the members of a sports team.

There are 12 people in the team.

Spread the picture cards out for the team to view.

Bag of Beans

between Sim and Tuku.

The team is going to have its photograph taken, but

they need to be arranged in a line from the tallest to

the shortest. Here is a mat for placing them on.

Allocate one clue card to each team member.

You will need to use the clues on these cards to place

the people in a row, from tallest to shortest. See how

well you can work together to find out the order

of the team. Discuss the best way to work together

before you start.

When the group agrees it has finished its arrangement

write down the order of the names, then say:

Now check your cards through one more time to

make sure that the order you have them in matches

the clues.

Approach: One to one Level: Year 4 and year 8

Focus: Probability.

Resources:

Questions/instructions

Zip is taller than Mary, but

shorter then Hone.

Jan is next to Sim.

Jo is taller than the person

wearing glasses.

Bag of beans: 8 blue, 8 green, 8 orange, recording book.

% responses

y4 y8

Take beans out of bag, and place on the table.

1. I want you to choose 6 beans to put back

into the bag. Choose them so that

I would always pull out a green bean if I

put my hand in to take one out.

6 green beans in bag 68 89

Cooperation level:

Jo is shorter than Mary.

Tuku is standing beside

both Terri and Jan.

There are seven people

standing between Tuku

and Jo.

Hone is taller than Mary.

Sim is shorter than Terri,

but taller than Ben.

The tallest person’s name

starts with the letter “T.”

% responses

2001 (’97)

year 8

Final order fitted clues: all 15 81 (73)

[15 clues] 14 6 (8)

13 6 (7)

10–12 4 (9)

58 NEMP Report 23: Mathematics 2001

Games

Approach: Independent Level: Year 8

Focus: Combinations.

Resources: none.

Questions/instructions

6 teams play in a table

tennis competition.

Each team plays each of

the other teams once.

1. What is the total number of games played

in the competition?

% responses

y4 y8

15 15

✗ 30 19

How Far?

Approach: Independent Level: Year 8

Focus: Series.

Resources: None.

Questions/instructions

Show how you worked that out:

Complete the table to find out

the kilometres the car will travel

with different amounts of fuel.

Litres 8 12 16 20 24 28 32 36 40

Kilometres

computational method ( not listing games) 34

systematic listing of games 29

Seemingly random listing of games 7

% responses

y8

all correct 49

1 or 2 errors 5

Commentary

Only 15 percent of year 8 students determined the

correct answer, with many of those who used a computational

method counting each combination (game)

twice.

Chapter 6: Algebra and Statistics 59

% responses

y4 y8

% responses

y4 y8

Link Tasks 30 – 36

% responses

y4 y8

Link task 30

Approach: Independent

Level: Year 4 and year 8

Focus: Series.

Link task 31

Approach: Independent

Level: Year 4 and year 8

Focus: Various algebra items.

Item: 1 62 91

2 22 67

3 21 61

4 44 78

5 3 32

Item: 1 20 75

2 18 74

3 19 38

Link task 32

Approach: Station

Level: Year 4 and year 8

Focus: Series.

Link task 33

Approach: One to one

Level: Year 4 and year 8

Focus: Probability.

Item: 1 64 84

2 • 12

3 • 30

4 • 17

Item: 1 79 94

2 70 93

3 2 32

4 2 43

Link task 34

Approach: Independent

Level: Year 4 and year 8

Focus: Combinations.

Link task 35

Approach: Independent

Level: Year 8

Focus: Statistics.

Link task 36

Approach: Independent

Level: Year 8

Focus: Series.

Item: 1 13 56

2 19 53

Item: 1 57

Total score: 7-8 2

5-6 8

3-4 20

1-2 36

0 34

60 Chapter 7

**mathematics** surveys

Students’ attitudes, interests and liking for

a subject have a strong bearing on their

achievement. The Mathematics Survey

sought information from students about their

curriculum preferences and perceptions of

their own achievement. The questions were

the same for year 4 and year 8 students. The

survey was administered to the students in an

independent session (four students working

individually on tasks, supported by a teacher).

The questions were read to year 4 students,

and also to individual year 8 students who

requested this help. Writing help was available

if requested.

The survey included eleven items which

asked students to record a rating response by

circling their choice, two items which asked

them to select three preferences from a list,

one item which asked them to nominate

up to six activities, and three items which

invited them to write comments.

The students were first asked to select their

three favourite school subjects from a list of

twelve subjects. The results are shown below, together

with the corresponding 1997 results.

Three Favourites:

Percentages of students rating subjects

among their 3 favourites

% responses

2001 (’97) 2001 (’97)

year 4 year 8

Subject: Art 64 (68) 52 (43)

Physical Education 49 (47) 62 (57)

Mathematics 42 (42) 26 (35)

Reading 33 (30) 18 (16)

Writing 31 (19) 13 (13)

Music 27 (27) 22 (25)

Science 20 (22) 25 (23)

Technology 9 (10) 46 (30)

Māori 8 (9) 6 (11)

Social Studies 4 (5) 13 (16)

Speaking 3 (4) 8 (9)

Health 1 (3) 4 (3)

Mathematics was the third most popular option for year

4 students and the fourth most popular option for year 8

students. At year 4 level its popularity remained constant

between 1997 and 2001, but at year 8 level it was chosen

by 9 percent fewer students while technology and art

gained substantially over the four year period.

Students were presented with a list of nine **mathematics**

activities and asked to nominate up to three that they

liked doing at school. The responses are shown below,

in percentage order for year 4 students. Comparative figures

are given for 1997, but it should be noted that four

additional choices were available in 1997 so the percentages

are not strictly comparable.

The most notable changes from year 4 to year 8 are that

“maths problems and puzzles” are substantially more

popular at year 8 level, while “work in my maths book”

is substantially less popular at year 8 level. Comparing

the 1997 and 2001 results, “maths problems and puzzles”

and “using equipment” became more popular at

both levels.

Maths activities students like

doing at school:

% responses

2001 (’97) 2001 (’97)

year 4 year 8

Doing maths work sheets 41 (41) 33 (30)

Work in my maths book 40 (34) 22 (21)

Maths problems and puzzles 39 (30) 60 (43)

Using equipment 35 (21) 43 (27)

Maths tests 30 (23) 16 (16)

Using a calculator 29 (31) 27 (26)

Maths competitions 22 (18) 25 (17)

Using maths textbooks 14 (11) 17 (14)

Something else 5 (3) 10 (7)

Chapter 7: Surveys 61

An opene

n d e d

q u e s t i o n

asked stud

e n t s t o

nominate

what they

considered

to be

some very

important

t h i n g s a

p e r s o n

n e e d s t o

learn or do

to be good at maths. They were asked to try to think

of three things. Their responses were coded into nine

categories and the results shown in the table below are

percentage totals from the sets of three ideas. Because

some students nominated two or three things that were

coded into the same category (e.g. practising addition,

subtraction and multiplication) the percentage could

have exceeded 100. Basic facts and tables were seen by

students in both years to be most important, but this in

part will have arisen because some students referred

separately to two or more of addition, subtraction,

multiplication and division facts.

Important for learning

and being good at maths

Activities nominated by students as being very

important for learning maths or for being very

good at maths.

% responses

y4 y8

Basic facts and tables 79 85

Classroom behaviours

seeking help, discussing with others, paying

attention 33 28

Work skills

practise, study, revision, homework 32 25

Personal attributes

good attitudes, concentration, focus, enjoyment 33 29

Maths knowledge

algebra, money, percentages,

use of calculators, etc. 12 26

Intelligence

thinking, being brainy, being smart,

being able to understand 18 25

Skills and abilities in related subjects

reading, writing 11 9

Problem solving skills 1 8

Other factors 6 6

A second open-ended question asked students “What

are some interesting maths things you do in your own

time?” Their responses were coded into seven categories,

and the results shown in the table are percentage

totals, out of those students who responded. Year 4 students

placed more emphasis on basic facts and tables,

while year 8 students made more diverse choices.

Maths activities students do in their

own time.

% responses

y4 y8

Basic facts and tables 56 21

Puzzles, quizzes and games 23 24

Maths homework 7 10

Math skills (excluding basic facts) 9 25

Life skills maths

Counting money, banking, calculating

animal feed, fencing for paddocks, etc. 3 15

None 8 16

Other 8 12

The third open-ended question asked, “If you have

something really hard to do in maths, what do you

do?” Students’ responses were coded into seven categories,

and the results shown in the table are percentage

totals, out of those students who responded. Year

8 students were more inclined to ask for help, while

year 4 students were a little more likely to keep trying

by themselves.

Strategies students use when they

have something in maths that is very

hard to do.

% responses

y4 y8

Ask a teacher 31 42

Try harder; persevere 33 24

Ask for help

No specific people indicated 16 25

Ask family/friends for help 6 22

Quit/nothing 8 4

Guess 3 1

Other 10 9

Rating Items

Responses to the eight rating items are presented in separate

tables for year 4 and year 8 students.

The student responses to the rating items showed the

pattern found to date in all subjects except technology:

year 8 students are less likely to use the most positive

rating than year 4 students. In other words, students

62 NEMP Report 23: Mathematics 2001

year 4 **mathematics** survey 2001 (’97) year 8 **mathematics** survey 2001 (’97)

1. Would you like to do more, the same or less

maths at school?

more about the same less

38 (36) 39 (46) 23 (18)

don’t know

2. How much do you like doing maths at school?

51 (52) 30 (31) 10 (10) 9 (7)

3. How good do you think you are at maths?

41 (40) 45 (46) 10 (11) 4 (3)

4. How good does your teacher

think you are at maths?

46 (37) 25 (29) 5 (5) 1 (1) 23 (28)

5. How good does your Mum or Dad

think you are at maths?

65 (60) 15 (19) 4 (3) 1 (1) 15 (16)

6. How much do you like doing maths on your

own?

53 (•) 23 (•) 14 (•) 10 (•)

7. How much do you like doing maths with others?

55 (•) 27 (•) 9 (•) 9 (•)

8. How much do you like helping others

with their maths?

56 (•) 25 (•) 9 (•) 10 (•)

9. How do you feel about doing things in maths

you haven’t tried before?

47 (39) 28 (35) 15 (20) 10 (6)

10. How much do you like doing maths

in your own time (not at school)?

37 (41) 23 (26) 16 (14) 24 (19)

11. Do you want to keep learning maths

when you grow up?

yes maybe / not sure no

51 (54) 41 (41) 8 (5)

1. Would you like to do more, the same or less

maths at school?

more about the same less

13 (14) 59 (63) 28 (23)

don’t know

2. How much do you like doing maths at school?

26 (25) 40 (49) 23 (18) 11 (8)

3. How good do you think you are at maths?

22 (14) 58 (60) 16 (22) 4 (4)

4. How good does your teacher

think you are at maths?

20 (15) 34 (36) 10 (6) 3 (2) 33 (41)

5. How good does your Mum or Dad

think you are at maths?

35 (26) 32 (39) 7 (9) 1 (2) 25 (24)

6. How much do you like doing maths on your

own?

23 (•) 42 (•) 21 (•) 14 (•)

7. How much do you like doing maths with others?

49 (•) 34 (•) 11 (•) 6 (•)

8. How much do you like helping others

with their maths?

30 (•) 40 (•) 20 (•) 10 (•)

9. How do you feel about doing things in maths

you haven’t tried before?

33 (26) 38 (46) 21 (22) 8 (6)

10. How much do you like doing maths

in your own time (not at school)?

9 (13) 22 (28) 33 (33) 36 (26)

11. Do you want to keep learning maths

when you grow up?

yes maybe / not sure no

39 (43) 54 (53) 7 (4)

become more cautious about expressing high enthusiasm and self-confidence over the four additional years of

schooling. Between 1997 and 2001, fewer students at both year levels said that they didn’t know how good their

teacher thought they were at maths. This is a worthwhile improvement. A higher proportion of students at both

levels believed that their teachers and parents thought that they were good at **mathematics**. Student enthusiasm for

**mathematics** was static or declined slightly.

c Chapter 8 63

P

performance of Subgroups

Although national monitoring has been designed primarily

to present an overall national picture of student

achievement, there is some provision for reporting

on performance differences among subgroups of the

sample. Seven demographic variables are available for

creating subgroups, with students divided into two

or three subgroups on each variable, as detailed in

Chapter 1 (p5).

The analyses of the relative performance of subgroups

used an overall score for each task, created by adding

scores for the most important components of the task.

Where only two subgroups were compared, differences

in task performance between the two subgroups were

checked for statistical significance using t-tests. Where

three subgroups were compared, one way analysis of

variance was used to check for statistically significant

differences among the three subgroups.

Because the number of students included in each analysis

was quite large (approximately 450), the statistical tests

were quite sensitive to small differences. To reduce the

likelihood of attention being drawn to unimportant differences,

the critical level for statistical significance was set

at p = .01 (so that differences this large or larger among

the subgroups would not be expected by chance in more

than one percent of cases). For team tasks, the critical

level was raised to p = .05, because of the smaller sample

size (120 teams, rather than about 450 students).

For the first four of the seven demographic variables,

statistically significant differences among the subgroups

were found for no more than 12 percent of the tasks at

both year 4 and year 8. For the remaining three variables,

statistically significant differences were found on more

than 12 percent of the tasks at one or both levels. In the

report below, all “differences” mentioned are statistically

significant (to save space, the words “statistically significant”

are omitted).

Community Size

Results were compared for students living in communities

containing over 100,000 people (main centres),

communities containing 10,000 to 100,000 people (provincial

cities), and communities containing less than

10,000 people (rural areas).

For year 4 students, there was a difference among the

three subgroups on 1 of the 78 tasks. Students from

rural areas scored lowest on Multiplication Examples

(p14). There was also a difference on one question of

the Mathematics Survey (p62): students from provincial

cities were least positive and students from rural areas

most positive on question 8 (how good their Mum or Dad

thought they were at maths).

For year 8 students, there was a difference among the

three subgroups on 1 of the 94 tasks. Students from provincial

cities scored lowest on Link Task 36 (p59). There

were no differences on questions of the Mathematics

Survey.

School Size

Results were compared from students in larger, medium

size, and small schools (exact definitions were given in

Chapter 1).

For year 4 students, there were differences among the

three subgroups on 2 of the 78 tasks. Students attending

small schools scored highest on One Cut (p46) and Link

Task 26 (p49). There was also a difference on one question

of the Mathematics Survey (p62), with students

from large schools most positive and students from small

schools least positive on question 10 (how much they

liked doing **mathematics** in their own time).

For year 8 students there were differences among the

three subgroups on 2 of the 94 tasks. Students from

small schools scored highest on How Far? (p58), and

students from large schools scored highest on Link Task

18 (p41). There were no differences on questions of the

Mathematics Survey.

School Type

Results were compared for year 8 students attending full

primary schools and year 8 students attending intermediate

schools. Differences between the two subgroups

were found on 2 of the 94 tasks. Students from full

primary schools scored higher than did students from

intermediate schools on Lump Balance (p31) and Bank

Account (p36). There were no differences on questions

of the Mathematics Survey.

64 NEMP Report 23: Mathematics 2001

Gender

Results achieved by male and female students were

compared.

For year 4 students, there were differences between boys

and girls on 9 of the 77 tasks.

Boys scored higher than girls on all 9 tasks: Division

Facts (p13), Speedo (p17), Number Items C (p22),

Link Task 9 (p26), Link Task 10 (p26), Apples (p29),

Measurement Items C (p33), Link Task 16 (p41), and

Link Task 17 (p41). There were no differences on questions

of the Mathematics Survey.

For year 8 students, there were differences between

boys and girls on 7 of the 93 tasks. Girls scored higher

than boys on two tasks: Addition Examples (p14), and

Link Task 15 (p41). However, boys scored higher than

girls on Broken Ruler (p38), Link Task 14 (p41), Link

Task 18 (p41), How Far? (p58), and Link Task 34 (p59).

Boys were more positive than girls on one question of

the Mathematics Survey (p62): how good their teacher

thought they were at maths (question 4).

Zone

Results achieved by students from Auckland, the rest of

the North Island, and the South Island were compared.

For year 4 students, there were differences among the

three subgroups on 12 of the 78 tasks. Students from the

South Island scored highest and students from Auckland

scored lowest on 6 tasks: Jack’s Cows (p18), Link Task

9 (p26), Lump Balance (p31), Link Task 17 (p41),

Farmyard Race (p56), and Link Task 31 (p59). Students

from the South Island scored higher than the other two

groups on 4 tasks: Population [Y4] (p24), and Link Tasks

30, 33 and 34 (p59). Students from Auckland scored

lowest and students from elsewhere in the North Island

highest on two tasks: One Cut (p46) and Link Task 26

(p49). There were also four differences on questions of

the Mathematics Survey (p62). Students from Auckland

were most positive and students from the South Island

least positive on question 2 (how much they liked doing

maths at school), question 1 (would they like to do more

maths at school), question 9 (how they felt about doing

maths they haven’t tried before), and question 10 (how

much they liked doing maths in their own time).

For year 8 students, there were differences among the

three subgroups on 2 of the 94 tasks. Students from

the North Island other than Auckland scored lowest

on Link Task 24 (p49) and Link Task 33 (p59), with

students from the South Island scoring highest on the

latter task. There were no differences on questions of the

Mathematics Survey.

Chapter 8: Performance of subgroups 65

Student Ethnicity

Results achieved by Māori and non-Māori students were

compared.

For year 4 students, there were differences in performance

on 58 of the 77 tasks. In each case, non-Māori

students scored higher than Māori students. Because of

the large number of tasks involved, they are not listed

here. There was also a difference on one question of the

Mathematics Survey (p62): Māori students were less

positive than non-Māori students on question 3 (how

good they thought they were at maths).

For year 8 students, there were differences in performance

on 61 of the 93 tasks. In each case, non-Māori

students scored higher than Māori students. Because of

the large number of tasks involved, they are not listed

here. There was also a difference on one question of the

Mathematics Survey (p62): Māori students were more

positive than non-Māori students on question 2 (how

much they like doing maths at school).

Socio-Economic Index

Schools are categorised by the Ministry of Education

based on census data for the census mesh blocks where

children attending the schools live. The SES index takes

into account household income levels, categories of

employment, and the ethnic mix in the census mesh

blocks. The SES index uses ten subdivisions, each containing

ten percent of schools (deciles 1 to 10). For our

purposes, the bottom three deciles (1-3) formed the

low SES group, the middle four deciles (4-7) formed

the medium SES group, and the top three deciles (8-10)

formed the high SES group. Results were compared for

students attending schools in each of these three SES

groups.

For year 4 students, there were differences among the

three subgroups on 68 of the 78 tasks. Because of the

number of tasks involved, the specific tasks are not listed

here. In each case, performance was

lowest for students in the low SES

group. Students in the high SES group

generally performed better than

students in the medium SES group,

but these differences often were

smaller. There was also a difference

on one question of the Mathematics

Survey (p62), with students from

low SES schools reporting greater

enjoyment of doing maths at school

(question 2).

For year 8 students, there were differences

among the three subgroups

on 71 of the 94 tasks. Because of the

number of tasks involved, the specific

tasks are not listed here. In each case,

performance was lowest for students

in the low SES group. In most cases,

students in the high SES group also

performed better than students in the

medium SES group. On the Mathematics Survey (p62),

there was a difference on one question. Students from

low SES schools were least positive on question 5 (how

good their mum or dad thought they were at maths).

Summary

Statistically significant differences of task performance

among the subgroups based on school size, school type

or community size occurred for very few tasks (at about

the 1 percent level likely to occur by chance). There

were differences among the three geographic zone subgroups

on 15 percent of the tasks for year 4 students, but

only 2 percent of the tasks for year 8 students. Boys performed

better than girls on 12 percent of the year 4 tasks

and 5 percent of the year 8 tasks, but girls performed

better than boys on 2 percent of the year 8 tasks. Non-

Māori students performed better than Māori students on

75 percent of the year 4 tasks and 66 percent for the year

8 tasks. The SES index based on school deciles showed

the strongest pattern of differences, with differences on

87 percent of the year 4 tasks and 76 percent of the year

8 tasks.

The 2001 results for the Māori /Non-Māori and SES

(school decile) comparisons are very similar to the

corresponding 1997 results. In 1997 there were Māori

/Non-Māori differences on 80 percent of year 4 tasks and

77 percent of year 8 tasks, and school decile differences

on 85 percent of year 4 tasks and 77 percent of year 8

tasks. The most noticeable, although still relatively small,

changes from the 1997 results involve the comparative

performance of boys and girls. In 2001, year 4 boys performed

better than girls on 12 percent of tasks (2 percent

in 1997) and worse on none (4 percent in 1997). Year

8 boys performed better than girls on 5 percent of tasks

(2 percent in 1997) and worse on 2 percent (14 percent

in 1997).

66 chapter 9

pacific subgroups

A new feature in National Monitoring since 1999 has

been the commitment to look directly at the achievement

of Pacific students in New Zealand primary and

intermediate schools. These students were among the

samples in NEMP assessments between 1995 and 1998,

but not in sufficient numbers to allow their results to

be reported separately. At the request of the Ministry of

Education, NEMP now selects special additional samples

of 120 year 4 students and 120 year 8 students to allow

the achievement of Pacific students to be assessed and

reported. The augmented samples are too small, however,

to allow separate reporting on students from different

Pacific nations (such as Samoa, Tonga, and Fiji).

The augmented samples are drawn from schools with at

least 15 percent Pacific students. Schools in this category

comprise about 10 percent of New Zealand schools and

include about 15 percent of all students. About 75 percent

of Pacific students attend such schools.

All schools in the main NEMP year 8 sample that had 15

percent or more Pacific students (as classified in school

records) were selected. All other schools nationally with

at least 12 year 8 students and at least 15 percent Pacific

students in their total roll were identified, and an additional

random sample of 10 schools drawn from this list.

A similar procedure was followed at year 4 level, except

that schools already chosen at year 8 level were excluded

from the sampling list. From each specially sampled

school, 12 students (in 3 groups of 4) were sampled, confirmed

and assessed using exactly the same procedures

as in the main sample. The students’ performances were

also scored in the same manner as the performances of

students in the main sample.

The results for Pacific, Māori, and other students in the

schools with more than 15 percent Pacific students were

then compared. Because all of the schools chosen for

these analyses have at least 15 percent Pacific students,

the results only apply to students at schools like these.

Differences among the three ethnic groups of students

were checked for statistical significance using one way

analysis of variance on the overall scores for each task

attempted by individual students (team tasks were

excluded). Each analysis compared the performance of

about 50 Pacific students, 30 Māori students and 30 other

students. The critical level for statistical significance was

set at p = .05 (so that differences this large or larger

among the subgroups would not be expected by chance

in more than five percent of cases). Where statistical

significance occurred, Tukey tests were used to identify

which groups differed significantly.

The mean scores for each group on each task are presented

in the tables below, together with the standard

deviations for all students in this sample. Statistically

significant differences are clearly indicated.

YEAR 4

Average (mean) marks for year 4 students, attending

schools enrolling at least fifteen percent Pacific students,

who are classified as Pacific students, Māori students or

other students

Statistically significant (p

Chapter 9: Pacific subgroups 67

Link Task 9 1.4 1.7 2.0 1.0

Link Task 10 1.9 2.5 3.8 2.0

Apples 0.6 0.6 0.8 0.7

Better Buy 0.7 0.7 1.2 1.3

Measures 7.2 9.4 11.2 4.3

Bean Estimates 0.8 1.0 1.1 0.8

Lump Balance 2.7 2.8 3.5 1.3

Video Recorder 1.3 1.6 2.1 1.4

Measurement Items B 2.1 2.1 3.4 1.8

Measurement Items C 2.8 2.3 3.6 1.8

Party Time 0.5 0.5 1.0 1.0

What’s the Time 0.7 0.9 1.2 0.8

Stamps 0.3 0.2 0.4 0.4

Bank Account 1.9 0.8 3.4 2.3

Money A 0.3 0.4 1.0 0.7

Team Photo 3.3 3.4 3.6 1.0

Broken Ruler 0.4 0.6 1.2 0.9

Pebbles Packet 1.4 1.9 2.4 1.9

Fishing Competition (Y4) 2.2 2.3 2.6 1.0

Link Task 13 1.2 2.4 3.0 3.3

Link Task 14 0.9 1.1 1.4 1.3

Link Task 15 0.5 0.6 1.0 1.2

Link Task 16 5.4 5.6 6.5 2.3

Link Task 17 0.5 0.7 1.3 1.2

Link Task 18 3.1 3.1 3.8 1.9

Link Task 19 1.1 2.2 4.1 3.0

Link Task 20 1.0 1.0 1.3 0.9

Link Task 21 5.1 5.9 6.6 2.3

Cut Cube 0.2 0.5 0.7 0.7

Hedgehog 1.8 2.2 3.3 1.8

Shapes & Nets 2.3 1.8 3.9 2.2

One Cut 1.0 1.0 1.7 2.3

Paper Folds 0.0 0.0 0.2 0.3

Line of Symmetry 0.4 0.6 0.6 0.5

Whetu’s Frame 0.3 0.5 0.6 0.7

Link Task 24 4.4 4.4 6.2 3.1

Link Task 25 2.9 2.8 3.5 0.9

Link Task 26 2.2 2.0 1.8 1.8

Link Task 27 1.6 1.4 1.8 1.0

Link Task 28 2.4 2.6 2.8 1.3

Train Trucks 5.0 5.5 5.7 1.8

Algebra Items A 0.5 0.7 0.9 0.8

Algebra Items B 5.2 5.1 9.1 3.4

Statistics Items A 0.4 0.4 0.5 0.6

Statistics Items B 0.2 0.2 0.3 0.4

Bag of Beans 0.6 0.7 1.0 0.7

Link Task 30 2.0 2.3 2.6 2.1

Link Task 31 1.0 1.0 2.6 1.3

Link Task 32 0.4 0.4 0.7 0.5

Link Task 33 1.2 1.6 2.1 1.4

Link Task 34 0.3 0.2 0.5 0.7

For year 4 students, there were statistically significant

differences in performance among the three groups on

39 of the 77 tasks:

➢ on 14 tasks both Pacific and Māori students scored

lower than “other” students;

➢ on 21 tasks only Pacific students scored lower than

“other” students;

➢ on 4 tasks only Māori students scored lower than

“other” students.

Thus Pacific students scored lower than “other” students

on 45 percent of the tasks and Māori students scored

lower than “other” students on 23 percent of the tasks..

On the Year 4 Mathematics Survey (p62), there was a

statistically significant difference on 1 of the 11 rating

items. The Māori students were more positive than the

“other” students on question 2 (how much they liked

doing maths at school).

68 NEMP Report 23: Mathematics 2001

YEAR 8

Average (mean) marks for year 8 students, attending

schools enrolling at least fifteen percent Pacific students,

who are classified as Pacific students, Māori students or

other students

Statistically significant (p

Chapter 9: Pacific subgroups 69

Cut Cube 1.4 1.6 1.8 1.5

Hedgehog 5.3 6.1 5.2 2.4

Flat Shapes 4.1 4.3 6.5 2.2

Shapes & Nets 4.5 3.7 5.3 2.7

One Cut 3.7 3.7 3.9 4.4

Paper Folds 0.7 0.6 0.7 0.5

Line of Symmetry 0.9 0.8 0.9 0.3

Whetu’s Frame 1.3 1.1 1.7 1.0

Grid Plans 2.0 1.9 2.5 1.7

Boats 1.5 1.5 2.1 1.7

Link Task 24 6.4 6.4 6.8 3.3

Link Task 25 4.2 3.8 4.5 1.2

Link Task 26 3.2 3.3 3.9 1.9

Link Task 27 2.0 2.0 2.1 0.9

Link Task 28 3.3 3.1 3.6 1.3

Link Task 29 1.1 0.7 1.3 1.1

Train Trucks 7.4 11.0 8.6 2.6

Algebra Items A 2.5 1.9 2.7 1.5

Algebra Items B 5.6 3.7 7.8 4.2

Statistics Items A 0.7 0.9 1.3 0.7

Statistics Items B 0.6 0.6 1.1 0.8

Bag of Beans 1.3 1.6 1.6 0.6

Games 0.8 1.2 1.9 1.5

How Far? 0.4 0.3 1.0 0.8

Link Task 30 6.0 5.6 6.4 2.9

Link Task 31 1.2 1.8 2.0 1.4

Link Task 32 1.4 0.9 1.7 1.3

Link Task 33 2.8 2.9 3.7 1.6

Link Task 34 0.6 1.4 1.7 1.4

Link Task 35 0.5 0.4 0.6 0.5

Link Task 36 1.1 1.3 1.2 1.4

For year 8 students, there were statistically significant

differences in performance among the three groups on

35 of the 93 tasks:

➢ on 11 tasks both Pacific and Māori students scored

lower than “other” students;

➢ on 13 tasks only Pacific students scored lower than

“other” students;

➢ on 6 tasks only Māori students scored lower than

“other” students;

➢ on 2 tasks Māori students scored lower than both

Pacific and “other” students;

➢ on 1 task Pacific and “other” students scored lower

than Māori students;

➢ on 1 task Pacific students scored lower than just Māori

students;

➢ on 1 task Pacific students scored lower than both

Māori and “other” students.

Thus Pacific students scored lower than “other” students

on 27 percent of tasks (and never higher), and Māori

students scored lower than “other” students on 20 percent

of tasks (and higher on 1 percent of tasks). Pacific

students scored lower than Māori students on 3 percent

of tasks and higher on 2 percent of tasks.

On the Year 8 Mathematics Survey (p62), there was a

statistically significant difference on 1 of the 11 rating

items. The Pacific students were more positive than both

Māori and “other” students on Question 8 (How much do

you like helping others with maths?).

Summary

In schools with more than 15 percent Pacific students,

Year 4 Pacific students performed similarly to their Māori

peers, but less well than “other” students on 45 percent

of the tasks. Year 8 Pacific students performed similarly

to their Māori peers, but less well than “other” students

on 27 percent of the tasks.

70 Appendix

Main samples

In 2001, 2869 children from 254 schools were in the

main samples to participate in national monitoring.

About half were in year 4, the other half in year 8. At each

level, 120 schools were selected randomly from national

lists of state, integrated and private schools teaching at

that level, with their probability of selection proportional

to the number of students enrolled in the level. The process

used ensured that each region was fairly represented.

Schools with fewer than four students enrolled at the

given level were excluded from these main samples, as

were special schools and Māori immersion schools (such

as Kura Kaupapa Māori ).

Early in May 2001, the Ministry of Education provided

computer files containing lists of eligible schools with

year 4 and year 8 students, organised by region and

district, including year 4 and year 8 roll numbers drawn

from school statistical returns based on enrolments at 1

March 2001.

From these lists, we randomly selected 120 schools with

year 4 students and 120 schools with year 8 students.

Schools with four students in year 4 or 8 had about a

one percent chance of being selected, while some of the

largest intermediate (year 7 and 8) schools had a more

than 90 percent chance of inclusion. In the six cases

where the same school was chosen at both year 4 and

year 8 level, a replacement year 4 school of similar size

was chosen from the same region and district, type and

size of school.

Additional samples

From 1999 onwards, national monitoring has included

additional samples of students to allow the performance

of special categories of students to be reported.

To allow results for Pacific students to be compared with

those of Māori students and other students, 10 additional

schools were selected at year 4 level and 10 at year 8

level. These were selected randomly from schools that

had not been selected in the main sample, had at least 15

percent Pacific students attending the school, and had at

least 12 students at the relevant year level.

To allow results for Māori students learning in Māori

immersion programmes to be compared with results

for Māori children learning in English, 10 additional

schools were selected at year 8 level only. They were

selected from Māori immersion schools (such as Kura

Kaupapa Māori ) that had at least 4 year 8 students, and

from other schools that had at least 4 year 8 students in

classes classified as Level 1 immersion (80 to 100 percent

of instruction taking place in Māori ). Only students that

the schools reported to be in at least their fifth year of

immersion education were included in the sampling

process.

Pairing small schools

At the year 8 level, 9 of the 120 chosen schools in the

main sample had less than 12 year 8 students. For each

of these schools, we identified the nearest small school

meeting our criteria to be paired with the first school.

Wherever possible, schools with 8 to 11 students were

paired with schools with 4 to 7 students, and vice versa.

However, the travelling distances between the schools

were also taken into account. Four of the 10 schools in

the year 8 Māori immersion sample also needed to be

paired with other schools of the same type.

Similar pairing procedures were followed at the year 4

level. Five pairs were required in the main sample of 120

schools.

Contacting schools

Late in May, we attempted to telephone the principals

or acting principals of all schools in the year 8 samples

(excluding the 15 schools in the Māori immersion

sample). We made contact with all schools within a

week.

In our telephone calls with the principals, we briefly

explained the purpose of national monitoring, the

safeguards for schools and students, and the practical

demands that participation would make on schools and

students. We informed the principals about the materials

which would be arriving in the school (a copy of a 20

minute NEMP videotape plus copies for all staff and trustees

of the general NEMP brochure and the information

booklet for sample schools). We asked the principals to

consult with their staff and Board of Trustees and confirm

their participation by the end of June.

A similar procedure was followed at the end of July with

the principals of the schools selected in the year 4 samples,

and they were asked to respond to the invitation by

the end of August. The principals of the 14 schools in the

Māori immersion sample at year 8 level were contacted in

the middle of August and asked to respond by the middle

of September. They were sent brochures in both Māori

and English.

Response from schools

Of the 288 schools originally invited to participate, 282

agreed. Three schools in the year 8 sample declined to

participate: an intermediate school because of major

building work, an independent school because of a clash

with a drama production involving all year 8 students,

and a very small paired school because of the high level

of teaching in Māori in that school. The first two were

replaced within their districts by schools of similar size.

The paired school was not replaced: instead, additional

pupils were selected from the other school in the pair.

An independent school in the Year 4 sample declined to

participate, and was replaced by a school of similar size

in the same district. In the Māori Immersion sample, a

school chose not to participate and was replaced by a

nearby school, while a very small paired school lost stu-

Appendix 71

dents and was replaced by selecting additional students

from its paired school.

Sampling of students

With their confirmation of participation, each school sent

a list of the names of all year 4 or year 8 students on their

roll. Using computer generated random numbers, we

randomly selected the required number of students (12,

or 4 plus 8 in a pair of small schools), at the same time

clustering them into random groups of four students. The

schools were then sent a list of their selected students

and invited to inform us if special care would be needed

in assessing any of those children (e.g. children with disabilities

or limited skills in English).

At the year 8 level, we received 110 comments from

schools about particular students. In 58 cases, we randomly

selected replacement students because the children

initially selected had left the school between the

time the roll was provided and the start of the assessment

programme in the school, or were expected to be away

throughout the assessment week, or had been included

in the roll by mistake. The remaining 52 comments concerned

children with special needs. Each such child was

discussed with the school and a decision agreed. Nine

students were replaced because they were very recent

immigrants or overseas students who had extremely limited

English language skills. Eight students were replaced

because they had disabilities or other problems of such

seriousness that it was agreed that the students would

be placed at risk if they participated. Participation was

agreed upon for the remaining 35 students, but a special

note was prepared to give additional guidance to the

teachers who would assess them.

In the corresponding operation at year 4 level, we

received 123 comments from schools about particular

students. Thirty-six students originally selected were

replaced because they had left the school, were not

actually year 4 students, or were expected to be away

throughout the assessment week. Two students were

replaced because they attended a satellite school more

than 60 minutes travel from the main school. Ten students

were replaced because of their NESB status and

very limited English. Sixteen students were replaced

because they had disabilities or other problems of such

seriousness the students appeared to be at risk if they participated.

Special notes for the assessing teachers were

made about 59 children retained in the sample.

Communication with parents

Following these discussions with the school, Project staff

prepared letters to all of the parents, including a copy of

the NEMP brochure, and asked the schools to address

the letters and mail them. Parents were told they could

obtain further information from Project staff (using an

0800 number) or their school principal, and advised that

they had the right to ask that their child be excluded

from the assessment.

At the year 8 level, we received a number of phone calls

including several from students wanting more information

about what would be involved. Three children were

replaced as a result of these contacts, one at the child’s

request and two at the parents’ request (one family

would not allow their child to view videos or use computers

on religious grounds, the other simply requested

that their child not participate).

At the year 4 level we also received several phone

calls from parents. Some wanted details confirmed or

explained (notably about reasons for selection). Three

children were replaced at parents’ request.

Practical arrangement with schools

On the basis of preferences expressed by the schools,

we then allocated each school to one of the five assessment

weeks available and gave them contact information

for the two teachers who would come to the school for

a week to conduct the assessments. We also provided

information about the assessment schedule and the space

and furniture requirements, offering to pay for hire of a

nearby facility if the school was too crowded to accommodate

the assessment programme. This proved necessary

in several cases.

Results of the sampling process

As a result of the considerable care taken, and the attractiveness

of the assessment arrangements to schools and

children, the attrition from the initial sample was quite

low. Only about two percent of selected schools did not

participate, and less than two percent of the originally

sampled children had to be replaced for reasons other

than their transfer to another school or planned absence

for the assessment week. The sample can be regarded

as very representative of the population from which it

was chosen (all children in New Zealand schools at the

two class levels except the one to two percent in special

schools or schools with less than four year 4 or year 8

children).

Of course, not all the children in the samples actually

could be assessed. Five year 8 students and 13 year 4

students left school at short notice and could not be

replaced. A parent withdrew one year 8 student too late

to be replaced. One NESB year 8 student was judged by

the teacher administrators to be too limited in English

language skills, and another was a year 7 student. A further

16 year 8 students, 11 year 4 students, and 2 Māori

immersion students were absent from school throughout

the assessment week. Some others were absent

from school for some of their assessment sessions, and

a small percentage of performances were lost because

of malfunctions in the video recording process. Some of

the students ran out of time to complete the schedules

of tasks. Nevertheless, for many tasks over 95 percent of

the student sample were assessed. No task had less than

90 percent of the student sample assessed. Given the

complexity of the Project, this is a very acceptable level

of participation.

72 NEMP Report 23: Mathematics 2001

Composition of the sample

Because of the sampling approach used, regions were

fairly represented in the sample, in approximate proportion

to the number of school children in the regions.

Region

Percentages of students from each region

region % of year 4 sample % of year 8 sample

Northland 4.2 4.2

Auckland 32.6 30.8

Waikato 10.0 10.0

Bay of Plenty/Poverty Bay 8.3 8.3

Hawkes Bay 3.3 4.2

Taranaki 3.3 3.3

Wanganui/Manawatu 5.8 5.8

Wellington/Wairarapa 10.8 10.8

Nelson/Marlborough/West Coast 4.2 4.2

Canterbury 10.8 11.7

Otago 4.2 4.2

Southland 2.5 2.5

Demography

demographic variables:

p e r c e n t a g e s o f s t u d e n t s in e a c h c a t e g o r y

variable category % year 4 sample % year 8 sample

Gender Male 49 50

Female 51 50

Ethnicity Non-Māori 82 80

Māori 18 20

Geographic Zone Greater Auckland 31 30

Other North Island 47 47

South Island 22 23

Community Size > 100,000 57 57

10,000 – 100,000 26 29

< 10,000 17 14

School SES Index Bottom 30 percent 27 20

Middle 40 percent 32 46

Top 30 percent 41 36

Size of School < 20 y4 students 13

20 – 35 y4 students 20

> 35 y4 students 67

150 y8 students 49

Type of School Full Primary 32

Intermediate 54

Other (not analysed) 14

Mathematics is pervasive. We encounter and

use mathematical ideas and processes in our

ordinary everyday lives, and in varying

degrees of sophistication it is used in all

fields of industry, commerce, the sciences and

technology.

In order to fully understand the world around us and

exercise effective control over our own affairs, we all

need to develop mathematical understandings, skills

and attitudes.

ISSN 1174-0000

ISBN 1-877182-34-6

National monitoring provides a “snapshot” of

what New Zealand children can do at two levels

in primary and intermediate schools: ages 8–9 and

ages 12–13.

The main purposes for national monitoring are:

• to meet public accountability and information

requirements by identifying and reporting patterns

and trends in educational performance

• to provide high quality, detailed information which

policy makers, curriculum planners and educators

can use to debate and review educational practices

and resourcing.

**mathematics** a s s e s s m e n t r e s u l t s 2001 nemp