Levels 1 to 5
This set of exemplars describes a possible progression in number: the development in students' understanding of number strategies.
Number strategies are the mental strategies that students use to estimate answers and solve problems within the operational domains of addition and subtraction, multiplication and division, and proportions and ratios.
Counting from one
 At this stage in the number strategy progression, students are able to count a set of objects or form sets of objects to solve simple addition and subtraction problems. They solve problems by counting all the objects.
Advanced counting
 At this stage in the number strategy progression, students are able to use counting on or counting back to solve simple addition or subtraction problems.
Early additive
 At this stage in the number strategy progression, students are able to use a limited range of mental strategies to estimate answers and solve addition or subtraction problems. These strategies involve deriving the answer from known basic facts (for example, doubles, fives, and making tens).
Advanced additive and early multiplicative
 At this stage in the number strategy progression, students are able to choose appropriately from a broad range of advanced mental strategies to estimate answers and solve addition and subtraction problems involving whole numbers (for example, place value positioning, rounding, compensating, and reversibility). They use a combination of known facts and a limited range of mental strategies to derive answers to multiplication and division problems (for example, doubling, rounding, and reversibility).
Advanced multiplicative
 At this stage in the number strategy progression, students are able to choose appropriately from a broad range of mental strategies to estimate answers and solve multiplication and division problems. These strategies involve partitioning one or more of the factors (for example, place value partitioning, rounding and compensating, and reversibility).
Advanced proportional
 At this stage in the number strategy progression, students are able to use mental strategies to estimate answers and solve problems involving the multiplication and division of fractions and decimals. These strategies involve recognising the effect of number size on the answer, and converting decimals to fractions where appropriate.
Background to the number strategy exemplars
The content for this set of exemplars is paraphrased from material used in the Ministry of Education's Numeracy Professional Development Projects.
The Number Framework
A key concept in the Numeracy Professional Development Projects is The Number Framework. This has been established to help teachers, parents, and students to understand the requirements of the Number strand from Mathematics in the New Zealand Curriculum. The framework relates to most of the achievement aims and objectives in levels 1 to 4. There are two main sections to the framework: strategy and knowledge.
Strategy
This describes the mental processes students use to estimate answers and solve operational problems with numbers.
Knowledge
The describes the key items of knowledge that students need to learn.
It's important that students make progress in both sections of the framework. Strong knowledge is essential for students to broaden their strategies across a full range of numbers, and knowledge is often an essential prerequisite for the development of more advanced strategies.
Reference: Ministry of Education (2003). Book 1: The Number Framework. Numeracy Professional Development Projects 2003. Wellington: Ministry of Education, page 1.
This set of exemplars is designed to give a snapshot of each stage in the strategy section of the Number Framework. These stages have been mapped to the achievement objectives of the curriculum using a "best fit" approach, as summarised in the following table:
| Levels in the number strategy progression |
Stages in the Number Framework |
| Level 1 i: Counting from one |
1 One-to-one counting
2 Counting from one on materials
3 Counting from one by imaging
|
| Level 1 ii: Advanced counting |
4 Advanced counting |
| Level 2: Early additive |
5 Early additive part-whole |
| Level 3: Advanced additive and early multiplicative |
6 Advanced additive part-whole |
| Level 3/4: Advanced multiplicative |
7 Advanced multiplicative part-whole |
| Level 4/5: Advanced proportional |
8 Advanced proportional part-whole |
Activities and further information to support this progression are available in the Numeracy Professional Development Projects material: www.nzmaths.co.nz/numeracy/index.htm
Background to the task
The task chosen as a context for this set of exemplars allows enough scope to observe the development of students' mental strategies for operating with numbers through curriculum levels 1 to 5. The teacher asks the students to:
- use mental strategies to solve a problem
- explain how they got their answers.
The teacher increases the difficulty of the task by altering the complexity of the problem according to their level of development.
References
Department of Education (1985). Beginning School Mathematics: Cycles 1–8. Wellington: School Publications.
Ministry of Education (1992). Beginning School Mathematics: Cycles 9–11. Wellington: Learning Media, Ministry of Education.
Ministry of Education (2003). Book 1: The Number Framework. Numeracy Professional Development Projects 2003. Wellington: Ministry of Education.
Ministry of Education (2003). Book 2: The Diagnostic Interview. Numeracy Professional Development Projects 2003 (Draft). Wellington: Ministry of Education.
Ministry of Education (2003). Book 3: Getting Started. Numeracy Professional Development Projects 2003 (Draft). Wellington: Ministry of Education.
Ministry of Education (2003). Book 4: Teaching Number Knowledge. Numeracy Professional Development Projects 2003 (Draft). Wellington: Ministry of Education.
Ministry of Education (2003). Book 5: Teaching Addition, Subtraction, and Place Value. Numeracy Professional Development Projects 2003 (Draft). Wellington: Ministry of Education.
Ministry of Education (2003). Book 6: Teaching Multiplication and Division. Numeracy Professional Development Projects 2003. Wellington: Ministry of Education.
Ministry of Education (2003). Book 7: Teaching Fractions, Decimals, and Percentages. Numeracy Professional Development Projects 2003. Wellington: Ministry of Education.
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