Unifying Principles

The framework is organised around six recurring principles. These are not slogans; they are the intellectual commitments that shape every concept section in the detailed reference layer. They describe the kind of mathematical understanding the framework aims to develop.

a. Number is structured

Numbers are not random facts to memorise. They are related through pattern, composition, and operation. The number $24$ is simultaneously $20 + 4$ , and $6 \times 4$ , and $8 \times 3$ , and $2^{3} \times 3$ , and $24 \times 1$ . Each of these representations reveals something different about the number’s structure, and a pupil who can move between them has a fundamentally richer understanding than one who can only recall that $6 \times 4 = 24$ as an isolated fact.

This principle shapes the framework’s approach to multiplication tables (relational fluency, not bare recital), to place value (flexible partitioning, not rigid column work), and to fractions (connected representations, not separate topics). Throughout the framework, the emphasis is on helping pupils see the internal structure of numbers and the relationships between them.

b. Operations have meaning

Addition, subtraction, multiplication, and division are not just procedures to perform. They describe relationships and transformations. Addition combines. Subtraction finds a difference, removes, or compares. Multiplication scales, creates arrays, and combines dimensions. Division shares, groups, and inverts multiplication.

Each operation has multiple meanings, and pupils who understand only one meaning, for example, multiplication as ‘repeated addition’ or subtraction as ‘take away’, will be limited when they encounter situations that require a different interpretation. The framework insists that teachers develop the full range of meanings for each operation, because this range is exactly what pupils need when the same operations reappear in algebraic form.

c. The laws of arithmetic are the grammar of the framework

Three laws govern how the four operations behave: commutativity (order), associativity (grouping), and distributivity (multiplication over addition and subtraction). These laws explain why methods work and when calculations can be reorganised. Every concept section in this framework is, in one way or another, an expression of these laws at work.

When a pupil calculates $18 \times 5$ as $(20 - 2) \times 5 = 100 - 10 = 90$ , they are applying the distributive law. When they recognise that $7 \times 8 = 8 \times 7$ , they are using commutativity. When they simplify $3 x + 5 x$ to $8 x$ , they are relying on the distributive structure: $3 x + 5 x = (3 + 5) x$ . These are not tricks; they are applications of the laws that govern arithmetic and algebra alike.

The framework treats these laws as explicit knowledge. Something to be named, discussed, and used deliberately, rather than as background rules pupils apply without awareness.

A note on priority of operations

Prioritising multiplication over addition is a sensible convention rather than a mathematical law. It is arbitrary because we could reverse the order without breaking mathematics, but it is necessary for communication as a shared social norm that keeps our notation unambiguous and elegant. Division has equal priority to multiplication and subtraction to addition as they can be understood as inverses: $- 5 = + (- 5)$ and $\div 3 = \times \frac{1}{3}$ . Working from left to right when at the same level of priority is another convention.

The mnemonic BIDMAS (or BODMAS, PEMDAS, etc) is, at best, an incomplete memory aid. At worst, it is a source of misconceptions. It can imply that division always precedes multiplication and that addition always precedes subtraction, neither of which is correct. It also implies that brackets are an operation to be carried out, rather than one form of grouping, the purpose of which is to aid in communicating intention in calculation without ambiguity.

For more discussion on this, see Priority of operations: Necessary or arbitrary? Foster, Francome, Shore, Hewitt & Sangwin (2024). For the Learning of Mathematics, 44(2), 24–26.

d. Algebra grows out of arithmetic

This is the framework’s central thesis. Algebra is not a separate topic that begins when letters appear. It is a way of expressing general truths about how numbers behave. If pupils understand directed number, they can work with positive and negative terms in expressions. If they understand the area model, they can expand brackets. If they understand that subtraction can be rewritten as adding the opposite, they can simplify algebraic expressions. The algebra is already there in the arithmetic; the letters make it visible.

Every concept section in the detailed reference layer includes a component called ‘The bridge to algebra’, which shows explicitly how the arithmetical idea extends into algebraic form. This is the framework’s most distinctive feature: it does not merely assert that algebra connects to arithmetic; it shows the connection, concept by concept, with parallel worked examples.

e. Fluency comes from understanding as well as practice

Secure recall matters. A pupil who cannot retrieve $7 \times 8 = 56$ quickly will be slowed down in everything from simplifying fractions to solving equations. But durable fluency, the kind that transfers to new situations and survives the passage of time, comes from seeing connections and using structure intelligently, not just from repetition.

The framework promotes relational fluency: knowing that if $7 \times 8 = 56$ , then $7 \times 16 = 112$ (by doubling), and $56 \div 7 = 8$ (by the inverse), and $70 \times 8 = 560$ (by place value). A pupil with relational fluency has a network of facts, not a list. When one fact is forgotten, others can regenerate it. This is more robust and more mathematically productive than isolated memorisation.

f. Representations reveal structure, and pupils should move fluently between them

The point of a mathematical representation, a number line, an area model, a bar diagram, a set of counters, a table, a graph, is not decoration or necessarily to support calculation. It is to make a mathematical idea visible. Different representations illuminate different aspects of the same idea. A number line shows order and distance. Counters show cancellation and zero pairs. An area model shows the distributive law. A graph shows rate of change. No single representation tells the whole story.

The framework expects that pupils will be able to move between words, diagrams, manipulatives, number lines, tables, symbols, and graphs. This expectation is delivered through the ‘Key representations’ component of every concept section in the reference layer, which identifies the specific models that are most useful for each concept and explains what each one is good for and where it has limitations.

Algebraic competence, in particular, often depends on translating between representations, not just manipulating symbols. A pupil who can see that $y = 2 x + 3$ describes a straight line, a table of values, a function machine, and a real-world situation has a far richer understanding than one who can only substitute and simplify.

Next page: Concept map

Introduction
Overview
Unifying principles
Concept map