Field Sciences
Review of Mathematics and Statistics and Probability Level 2 achievement and unit standards
Unit standards
Subfield 
Domain 
ID 
Mathematics 
Algebra 
5246, 5248, 5253 
Calculus 
5244, 5260, 5261  
Geometry 
5245, 5249  
Mathematical Processes 
5243  
Mathematical Studies 
7558, 7560, 7569, 12318, 12323, 12327, 12329  
Number 
12331  
Trigonometry 
5251, 5255, 90291, 90292  
Statistics and Probability 
Probability 
5250 
Statistics 
5247, 7564, 12332, 12333 
Achievement standards
Domain 
ID 
Subject reference 
Algebra 
90284 
Mathematics 2.1 
90285 
Mathematics 2.2  
90290 
Mathematics 2.7  
90806 
CAS Mathematics 2.1  
90809 
CAS Mathematics 2.4  
Calculus 
90286 
Mathematics 2.3 
90807 
CAS Mathematics 2.2  
Geometry 
90287 
Mathematics 2.4 
Trigonometry 
90291 
Mathematics 2.8 
90292 
Mathematics 2.9  
90808 
CAS Mathematics 2.3  
Probability 
90289 
Mathematics 2.6 
Statistics 
90288 
Mathematics 2.5 
The Ministry of Education and NZQA National Qualifications Services have completed the review of the achievement and unit standards listed above.
New registration date November 2011
Date new versions published November 2011
Planned review date December 2014
Summary of review and consultation process
In 2008 the Ministry of Education (MoE) and NZQA began to review achievement and unit standards in light of the revised New Zealand Curriculum (NZC). This Alignment of Standards (AoS) review also addressed duplication of outcomes, credit parity, fairness, consistency, and coherence. The AoS review was guided by the revised NZC itself and the Principles for Standards Review. A copy of the NZC is available at: http://nzcurriculum.tki.org.nz/Curriculumdocuments/TheNewZealandCurriculum.
Teacher subject associations were involved in the review, and draft achievement standards were the focus of wide consultation, especially with secondary schools and teachers. Extensive resources, including student exemplars, were also developed to support these standards, and are available on the MoE and/or the NZQA websites.
The review of unit standards included consultation with tertiary providers to assess continued relevance and likely future use of the standards. Unit standards that duplicate achievement standard outcomes and those without the likelihood of future tertiary use were recommended for expiry.
National consultation was undertaken in 2010, with the results analysed by Research New Zealand. The responses were generally positive.
The review of these Level 2 unit and achievement standards was completed in time for implementation in schools in 2012. The review of unit and achievement standards at Level 1 was completed in time for implementation in schools in 2011. Standards at Level 3 will be implemented in 2013.
Main changes resulting from the review
· All NZC Level 7 (NZQF Level 2) outcomes derived from the NZC are now assessed using achievement standards, and there are no longer any unit standards linked to the NZC.
· Existing achievement standards were reviewed and new achievement standards were developed to align with the NZC. See table below.
· Grading criteria for achievement standards were reviewed in accordance with the Standards Review Guidelines.
· Unit standards that recognised similar outcomes as achievement standards were recommended for expiry. See table below.
For a more detailed description of the review of, and the changes to, the Mathematics and Statistics and Probability standards see the appendix at the end of this report.
Impact on Consent and Moderation Requirements (CMR)
(Formerly known as AMAP)
All new achievement standards have been registered on CMR 0233.
Impact on registered qualifications
Key to type of impact  
Affected 
The qualification lists a reviewed classification (domain or subfield) in an elective set The qualification lists a standard that has changes to level or credits The qualification lists a C or D category standard 
Not materially affected 
The qualification lists a standard that has a new title The qualification lists a standard that has a new classification 
The following table identifies qualifications developed by other SSBs that are Affected by the outcome of this review. The SSBs have been advised that the qualifications require revision.
Ref 
Qualification Title 
ID 
SSB Name 
0129 
National Certificate in Metal Casting (Technology) 
5251 
Competenz 
0432 
National Certificate in Surveying (Assistant) (Level 3) 
5245, 5246, 5251, 12318 
InfraTrain New Zealand

0672 
National Certificate in Engineering and Technology (for the Design and Construction Sector) (Level 4) 
5244, 5253, 5260, 5261  
1235 
National Certificate in Engineering and Technology (Pharmaceutical and Allied Products) (Level 2) 
5251 
Plastics and Materials Processing Industry Training Organisation Incorporated 
1262 
National Certificate in Mechanical Engineering (Level 4) with strands in Fitting and Machining, General Engineering, Machining, Maintenance Engineering, Toolmaking, and Electricity Supply 
5251 
Competenz 
1683 
National Certificate in Rail Infrastructure (Level 2) 
5245 
Impact of changes on NCEA Exclusions List
For transition purposes, the following exclusions will apply for new achievement standards.
New achievement standard 
Excluded against each of these standards 
91256 (2.1) 
90287 (2.4), 90808 (CAS 2.3), 5245 
91257 (2.2) 
90285 (2.2), 90292 (2.9), 90806 (CAS 2.1), 90809 (CAS 2.4), 5253, 5255 
91258 (2.3) 
90290 (2.7), 90806 (CAS 2.1), 5248 
91259 (2.4) 
90291 (2.8), 90808 (CAS 2.3), 5251 
91260 (2.5) 
5249 
91261 (2.6) 
90284 (2.1), 90806 (CAS 2.1), 90809 (CAS 2.4), 5246 
91262 (2.7) 
90286 (2.3), 90807 (CAS 2.2), 5244, 5260, 5261 
91263 (2.8) 
12333 
91264 (2.9) 
90288 (2.5), 5247 
91265 (2.10) 
7564 
91267 (2.12) 
90289 (2.6), 5250 
91268 (2.13) 
90289 (2.6), 5250 
91269 (2.14) 
90284 (2.1), 90806 (CAS 2.1), 90809 (CAS 2.4), 5246 
Review Categories and changes to classification, title, level, and credits
The following summary shows the changes made to the standards as a result of the review. All changes are in bold. Where a new externally assessed achievement standard is registered, the following designation appears after the title [externally assessed]. Standards with an asterisk (*) appear in the table more than once.
Key to review category  
A 
Dates changed, but no other changes are made  the new version of the standard carries the same ID and a new version number 
B 
Changes made, but the overall outcome remains the same  the new version of the standard carries the same ID and a new version number 
C 
Major changes that necessitate the registration of a replacement achievement standard with a new ID 
D 
Achievement standard will expire and not be replaced 
Externally assessed achievement standards categorised as category C or D expire at the end of 
December 2011 
Internally assessed achievement standards and unit standards categorised as category C or D expire at the end of 
December 2012 
Sciences > Mathematics > Algebra
ID 
Title 
Level 
Credit 
Review Category 
5246 
Manipulate algebraic expressions and use algebraic methods to solve problems 
2 
4 
C 
90284

Manipulate algebraic expressions and solve equations 
2 
4 
C 
90806* 
Demonstrate an understanding of mathematical relationships 
2 
6 
C 
90809* 
Demonstrate understanding of straightforward mathematical processes 
2 
5 
C 
91261 
Apply algebraic methods in solving problems [externally assessed] 
2 
4 

91269 
Apply systems of equations in solving problems 
2 
2 

5248 
Use sequences and series to solve problems 
2 
2 
C 
90290

Solve straightforward problems involving arithmetic and geometric sequences 
2 
2 
C 
90806* 
Demonstrate an understanding of mathematical relationships 
2 
6 
C 
91258 
Apply sequences and series in solving problems 
2 
2 

5253 
Sketch and describe graphs 
2 
3 
C 
5255 
Use trigonometry and trigonometrical graphs to solve equations and problems 
2 
3 
C 
90285 
Draw straightforward nonlinear graphs 
2 
3 
C 
90292 
Solve straightforward trigonometric equations 
2 
2 
C 
90806* 
Demonstrate an understanding of mathematical relationships 
2 
6 
C 
90809* 
Demonstrate understanding of straightforward mathematical processes 
2 
5 
C 
91257 
Apply graphical methods in solving problems 
2 
4 

Sciences > Mathematics > Calculus
ID 
Title 
Level 
Credit 
Review Category 
5244 
Demonstrate calculus skills 
2 
2 
C 
5260 
Find and use derivatives to solve problems involving rate of change 
2 
2 
C 
5261 
Find and use integrals to solve problems 
2 
2 
C 
90286 
Find and use straightforward derivatives and integrals 
2 
4 
C 
90807 
Demonstrate an understanding of calculus methods 
2 
4 
C 
91262 
Apply calculus methods in solving problems [externally assessed] 
2 
5 

Sciences > Mathematics > Geometry
ID 
Title 
Level 
Credit 
Review Category 
5245 
Solve coordinate geometry problems 
2 
2 
C 
90287 
Use coordinate geometry methods 
2 
2 
C 
90808* 
Demonstrate an understanding of processes involving trigonometry and coordinates 
2 
4 
C 
91256 
Apply coordinate geometry methods in solving problems 
2 
2 

5249 
Use networks to find optimal solutions to problems in geometry 
2 
2 
C 
91260 
Apply network methods in solving problems 
2 
2 

Sciences > Mathematics > Mathematical Processes
ID 
Title 
Level 
Credit 
Review Category 
5243 
Apply mathematical processes and skills in problems 
2 
7 
D 
Sciences > Mathematics > Mathematical Studies
ID 
Title 
Level 
Credit 
Review Category 
7558 
Demonstrate understanding of the mathematics of prime numbers and use it to solve problems 
2 
3 
D 
7560 
Demonstrate understanding of the mathematics of navigation and use it to solve problems 
2 
3 
D 
7569 
Plan, carry out and report on a mathematical study into a given area in mathematics 
2 
3 
D 
12318 
Use surveying techniques and mathematics to solve problems relating to maps or plans 
2 
3 
D 
12323 
Apply mathematics in the design of an object or process for a given purpose 
2 
3 
D 
12327 
Demonstrate understanding of geometry of different surfaces 
2 
3 
D 
12329 
Investigate an aspect of the history of a mathematics topic or mathematician and present results 
2 
2 
D 
Sciences > Mathematics > Number
ID 
Title 
Level 
Credit 
Review Category 
12331 
Investigate and report on the mathematics of a given project 
2 
3 
D 
Sciences > Mathematics > Trigonometry
ID 
Title 
Level 
Credit 
Review Category 
5251 
Choose and apply trigonometric methods to solve problems involving lengths and angles 
2 
3 
C 
90291 
Solve trigonometry problems requiring modelling of practical situations 
2 
2 
C 
90808* 
Demonstrate an understanding of processes involving trigonometry and coordinates 
2 
4 
C 
91259 
Apply trigonometric relationships in solving problems 
2 
3 

Sciences > Statistics and Probability > Probability
ID 
Title 
Level 
Credit 
Review Category 
5250 
Use probability techniques to solve problems 
2 
2 
C 
90289

Simulate probability situations, and apply the normal distribution 
2 
2 
C 
91267 
Apply probability methods in solving problems 
2 
4 

91268 
Investigate a situation involving elements of chance using a simulation 
2 
2 

Sciences > Statistics and Probability > Statistics
ID 
Title 
Level 
Credit 
Review Category 
5247 
Make and evaluate statements about populations based on sample data 
2 
3 
C 
90288 
Select a sample and use this to make an inference about the population 
2 
3 
C 
91264 
Use statistical methods to make an inference 
2 
4 

7564 
Plan, carry out and report on a statistical investigation into a given area 
2 
3 
C 
91265 
Conduct an experiment to investigate a situation using statistical methods 
2 
3 

12332 
Demonstrate knowledge of measures and displays used to compare data sets 
2 
2 
D 
12333 
Demonstrate understanding of, and use, questionnaire design 
2 
3 
C 
91263 
Design a questionnaire 
2 
3 

91266 
Evaluate a statistically based report 
2 
2 
New 
Appendix
Development of Level 2 standards
Process of aligning standards with the New Zealand Curriculum
All of the Level 7 Achievement Objectives (AOs) have been included for assessment in the new suite of Level 2 standards.
The relationship between each standard and the achievement objectives is set out in Explanatory Note 1.
Addressing duplication
There is no duplication of achievement objectives within the aligned standards. Any unit or achievement standards that duplicate the final aligned standards have been designated expiring.
Addressing credit parity
Consideration has been given to the widespread opinion that the current Level 2 achievement standards cover too much for a oneyear course, and that programmes leading to the assessment of the current Algebra and Calculus standards are rushed. The credit value of the new Calculus standard 2.7 (AS91262) is therefore set at 5 credits and the number of Statistics standards has increased. The total credits now available at Level 2 are more than could reasonably be covered in one course. As with the Level 1 standards, courses will need to be designed according to the needs and capabilities of students by making choices from the matrix.
Some 2 credit standards remain on the matrix. It was considered that combining these into standards with a higher credit value could not overcome the design problem of having to assess two different achievement objectives in one standard. The advantage of retaining these 2 credit standards is that it makes credits more accessible to students, and allows more flexibility in designing courses.
External and internal assessment
A learning programme based on Level 7 curriculum objectives needs to provide foundation skills for both Level 8 'subjects': eg Mathematics (currently Calculus) and Statistics. The needs of both subjects influenced the decision to make 2.12 (AS91267) an externally assessed standard. Apart from this probability standard, the Statistics strand is better covered by internal standards since assessment of the Statistics achievement objectives is generally not appropriate for a timelimited examination session. In the Mathematics strand, Algebra and Calculus standards were designated as externally assessed so that the graphing standard 2.2 (AS91257) could be assessed internally, allowing for a variety of approaches and a greater flexibility in using technology. It was considered that the standards with lower credit values were better assessed internally to give schools greater flexibility to include these in alternative courses.
The design of the standards
According to the Principles for Standards Review, the stepup from Achieved to Merit to Excellence is described in terms of the level of thinking required, not in terms of specific content.
The titles of the standards provide clear and recognisable summaries of the content. Explanatory Note 1 always contains a reference to the New Zealand Curriculum document.
The level of thinking expected for the Achieved, Merit, and Excellence grades is defined in Explanatory Note 2. The first part of this definition describes the evidence required for the Achieved grade. This is tailored to each standard depending on the topic, but generally involves demonstrating knowledge of the relevant concepts and procedures in response to a straightforward problem and communicating appropriately. Where Explanatory Note 2 requires amplification, additional detail is supplied in Explanatory Notes 3, 4 or 5.
The descriptions of the evidence expected for Merit and Excellence are generic to all Mathematics standards because they are about the level of thinking, and are independent of the context or content. Two terms from the Solo Taxonomy have been used as convenient generic labels in the Achievement Criteria. These are 'relational thinking' for Merit and 'extended abstract thinking' for Excellence. These terms have been redefined by the Standards Writing Group (SWG) in Explanatory Note 2. The definitions given here have been crafted to reflect existing practice in NCEA assessment. The SWG looked at Merit grade questions from current assessment tasks and found that these questions consistently required one or more of: modelling, linking different concepts, working through a sequence of more than two steps, or showing understanding rather than just procedural knowledge. In addition, solutions needed to be related to a context or mathematical statements written to convey the working or reasoning. Similarly, the SWG looked at what is currently expected in Excellence questions and wrote a description of these qualities to define the term 'extended abstract thinking'.
The definition of 'problem' is intended to clarify that the term covers problems set in a purely mathematical context as well as problems which require applications in 'reallife' contexts. In an assessment task, an extended question may involve solving several 'problems'.
The inclusion of the term 'in solving problems' in the titles of some of these standards is intended to convey that acceptable evidence could come from a partially successful solution to a problem. Communication of the process of solving a problem may yield the required evidence of knowledge and thinking, even though a correct final solution to the problem is not obtained.
What has changed?
· The matrix is numbered in the order of the curriculum objectives.
· More choice of Statistics is now offered.
· Only three standards are assessed externally.
Comments on individual draft standards
2.1 (AS91256), Apply coordinate geometry methods in solving problems
This is internally assessed and has been assigned 2 credits.
2.2 (AS91257), Apply graphical methods in solving problems
The trigonometric models have been included with the other graphs and the standard has been assigned 4 credits. The emphasis has moved to the interpretation and formation of graphical models rather than simply drawing graphs. Solving problems could involve the solution of trigonometric equations. This is internally assessed and has been assigned 4 credits and technology is expected to be used in assessment.
2.3 (AS91258), Apply sequences and series in solving problems
This is internally assessed and has been assigned 2 credits.
2.4 (AS91259), Apply trigonometric relationships in solving problems
The standard includes circular measure and triangle rules and does not require practical measuring. It is internally assessed and has been assigned 3 credits.
2.5 (AS91260), Apply network methods in solving problems
This is internally assessed and has been assigned 2 credits.
2.6 (AS91261), Apply algebraic methods in solving problems
This standard is externally assessed and therefore the technology involved in assessment is limited to a graphic calculator. It has been assigned 4 credits to reflect the teaching time required.
2.7 (AS91262), Apply calculus methods in solving problems
This standard is externally assessed and therefore the technology involved in assessment is limited to a graphic calculator. It has been assigned 5 credits to reflect the teaching time required. This credit value was considered fair even though area by integration is no longer included in this standard. The intention is to give increased teaching and learning time to the fundamental concepts of this topic which are new at this level.
2.8 (AS91263), Design a questionnaire
This standard involves a detailed coverage of the questionnaire design process.
2.9 (AS91264), Use statistical methods to make an inference
In addition to replacing unit standard 5247 and AS90288, this standard covers aspects of 7564. The context is a comparative situation rather than a summary situation as this allows richer analysis at this level. Students will be given population data to sample from; they will not be expected to conduct a survey, however the population size must be large enough as to show a need to sample.
2.10 (AS91265), Conduct an experiment to investigate a situation using statistical methods
This standard involves planning and conducting an experiment. While it covers aspects of 7564, it focuses more on the experimental process of statistics.
2.11 (AS91266), Evaluate a statistically based report
This is a new standard reflecting the learning objectives of the statistical literacy thread of the New Zealand Curriculum at Level 7.
2.12 (AS91267), Apply probability methods in solving problems
This standard focuses on risk and relative risk. The theoretical aspects of 5250 and AS90289 are covered here. It is externally assessed.
While the AO for this standard refers to investigating situations involving elements of chance, the idea of an investigation has been seen as a learning progression to develop the concepts of solving probability problems, and the intent in this standard is to then assess whether students are able to solve them. This also applies to the idea of comparing distributions, but this aspect of the AO has been retained in the standard to help teachers see that the development of the understanding of normal distributions is needed so that students can solve some probability problems using normal distribution models.
As this is not an investigation standard this standard was seen as appropriate for external assessment.
The terminology for describing Achieved, Merit and Excellence has been aligned to the standards in the Mathematical strand, the terminology used in the Statistics strand standards reflects the use of an investigative cycle.
2.13 (AS91268), Investigate a situation involving elements of chance using a simulation
This standard covers the practical aspects of 5250 and AS90289. The title emphasises the need for students to use the results of their simulation and promotes the notion of an investigation.
2.14 (AS91269), Apply systems of equations in solving problems
This standard is internally assessed and has been assigned 2 credits.