The Numeracy Development Projects & Number Framework
Teachers are key figures in changing the way in which mathematics is taught and learned in schools. Their subject matter and pedagogical knowledge are critical factors in the teaching of mathematics for understanding. The effective teacher of mathematics has a thorough and deep understanding of the subject matter to be taught, how students are likely to learn it, and the difficulties and misunderstandings they are likely to encounter.
The focus of the Numeracy Development Projects is improving student performance in mathematics through improving the professional capability of teachers. A key feature of the project is its dynamic and evolutionary approach to implementation. This ensures that the project can be informed by developing understandings about mathematics learning and effective professional development and that flexibility in approach and sector involvement is maximised.
The diagram below illustrates the relationship within the projects between in-service teacher educators, teachers and principals, and children.
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The Number Framework
At the core of the Numeracy Development Projects is The Number Framework. The framework has been 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.
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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. |
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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. |
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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. |
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Stage 3: Counting from one by imaging |
The student is able to visualise sets of objects to solve simple addition and subtraction problems. |
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Stage 4: Advanced counting |
The student uses counting on or counting back to solve simple addition or subtraction tasks. |
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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). |
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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). |
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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). |
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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. |
