The Number to Algebra Framework

Through KS2 and early KS3, pupils should study number in ways that expose mathematical structure. Through number lines, counters, area models, structural approaches to calculation, scaled multiplicative representations, and explicit attention to the laws of arithmetic, ideally pupils develop a connected understanding of how numbers behave and how operations work.

Unfortunately, this is often not the case. The approaches to number they are exposed to in too large a proportion of UK classrooms are much impoverished compared to this ideal. Without developing this rich understanding of number, many pupils struggle through to grasp procedures. Never gaining the fluency and ability to reason that underpins later maths. These pupils are those who leave school, after over a decade of formal maths education without reaching the basic standard to allow them to confidently function mathematically in society.

This understanding of number is not only an end in itself. It is the foundation for algebra, where the same structures are expressed in general form. The pupil who understands that 13 × 17 = (10 + 3)(10 + 7) is ready to understand that $(x + 3) (x + 7) = x^{2} + 10 x + 21$ . The pupil who understands zero pairs with counters is ready to simplify $3 x - 5 + 2 x + 5$ . The pupil who understands that 18 × 5 = (20 − 2) × 5 is ready to see the distributive law at work in $4 (x + 2) = 4 x + 8$ .

This framework covers 17 concept areas, organised by conceptual dependency and connected through four major chains: the arithmetic-to-algebra spine, the equality-to-algebra spine, the proportional reasoning pathway, and the generalisation pathway.

Many of us maths teachers were also taught in an impoverished way. We may have been successful in mathematics, confident with our own ability with these concepts, but teaching them successfully to pupils with the full range of aptitude for maths requires more. We need a flexible and deep grasp of the connected web of number concepts and how they underpin Algebra. This doesn’t come automatically, even through experience, it requires clarity that comes through thinking deeply.

The purpose of the framework is to support teachers to first ensure their own rich understanding, so that they can teach better. It aims to provide a shared language, a shared set of representations, and a shared understanding of why particular approaches are used. Hopefully, the result is that pupils experience mathematics as a connected discipline. One in which the arithmetic they learn in KS2 and early KS3, and the algebra they go on to meet are not seen as separate strands, but two views of the same structure.

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The Number to Algebra Framework

Concept Reference