The Number Framework
At the core of the Numeracy Development Projects was The Number Framework. The framework was established to help teachers, parents, and students to understand the requirements of the Number knowledge and Number strategies sections of The New Zealand Curriculum.
In the two main sections to the framework, the distinction is made between strategy and knowledge. The Strategy section describes the mental processes students use to estimate answers and solve operational problems with numbers. The Knowledge section describes the key items of knowledge that students need to learn. It is important that students make progress in both sections of the framework.
The strategy section of the framework consists of a sequence of global stages. Progress through the stages indicates an expansion in knowledge and in the range of strategies that students have available.
The application of number knowledge and mental strategies is often described as 'number sense'. Strongly developed number sense leads to algebraic thinking.
The following table describes the key features of each strategy stage of the Number Framework.
Stage 0: Emergent
The student is unable to consistently count a given number of objects because they lack knowledge of counting sequences and/or one-to-one correspondence.
Stage 1: One-to-one counting
The student is able to count a set of objects or form sets of objects but cannot solve problems that involve joining and separating sets.
Stage 2: Counting from one on materials
The student is able to count a set of objects or form sets of objects to solve simple addition and subtraction problems.
Stage 3: Counting from one by imaging
The student is able to visualise sets of objects to solve simple addition and subtraction problems.
Stage 4: Advanced counting
The student uses counting on or counting back to solve simple addition or subtraction tasks.
Stage 5: Early additive part-whole
The student uses 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, making tens).
Stage 6: Advanced additive/early multiplicative part-whole
The student can estimate answers and solve addition and subtraction tasks involving whole numbers mentally by choosing appropriately from a broad range of advanced mental strategies (for example place value positioning, rounding and compensating or reversibility).
Stage 7: Advanced multiplicative part-whole
The student is 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, reversibility).
Stage 8: Advanced proportional part-whole
The student can estimate answers and solve problems involving the multiplication and division of fractions and decimals using mental strategies. These strategies involve recognising the effect of number size on the answer and converting decimals to fractions where appropriate. These students have strongly developed number sense and algebraic thinking.