mathematics

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