3. Principles of progression
In the Mathematics and Numeracy Area of Learning and Experience (Area), the model of progression is based on the development of five interdependent proficiencies, outlined below. This model of progression can be considered as both longitudinal and cross-sectional. To ensure progress in any mathematics learning, proficiencies should be developed and connected in time and should also develop over time.
Each proficiency may relate to multiple principles, and these are set out below.
The following interdependent proficiencies have been used in developing the descriptions of learning and are central to progression at each stage of mathematics learning. Numeracy involves applying and connecting these proficiencies in a range of real-life contexts, across the curriculum.
Mathematical concepts and ideas should be built on, deepened and connected as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding through being able to explain and express concepts, find examples (or non-examples) and by being able to represent a concept in different ways, flowing between different representations including verbal, concrete, visual, digital and abstract.
An increasing breadth of knowledge is achieved through the learners being introduced to new mathematical concepts, and depth of knowledge is achieved through learners being able to represent, connect and apply a concept in different ways and in different situations. The concepts that learners are introduced to will become increasingly complex, and understanding the way in which concepts connect will contribute to a growing understanding of the ideas within this Area. An understanding of how mathematical concepts underpin learning help learners make connections and transfer learning into new contexts.
Communication using symbols
Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility with the application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.
The introduction and application of a new concept will involve developing an understanding of how symbols or expressions are abstract representations that succinctly describe a range of situations, thus contributing to a growing understanding of the nature of mathematics. The introduction of new symbols will add to the breadth of knowledge and the communication with symbols will contribute to refinement and growing sophistication in the use and application of skills.
As learners experience, understand and effectively apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow, meaning that facts, relationships and techniques learned previously should become firmly established, memorable and usable.
Development of fluency and accuracy reflects the refinement and a growing sophistication in the use and application of skills.
As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal explanations, visual or concrete representations to abstract representations involving symbols and conventions.
Refinement and growing sophistication in the use and application of skills will be demonstrated through the application of increasingly sophisticated logical reasoning. The development of an understanding of relationships between mathematical concepts and the development of justifications and proofs, leads to a growing understanding of the nature of mathematics and helps learners make connections and transfer learning into new contexts. The development of justifications and proof help support the increasing effectiveness of learners.
Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to develop strategies to be able to solve them.
Recognising mathematical structure within a problem and formulating problems mathematically in order to be able to solve them relies on an understanding of the ideas and disciplines within areas of learning and experience alongside a depth of knowledge. It also supports making connections and transferring learning into new contexts and developing increasing effectiveness as a learner. The recognition of the power of mathematics in enabling the representation of situations should lead to a growing appreciation of the usefulness of mathematics.