Representation, Structure and the Emergence of Procedure

In my previous post, I described the teaching of maths as a craft through the analogy of the architect, the artist and the conductor. These are not abstract roles. They are realised through the decisions made by the teacher within, and prior to, every lesson.

It is now my aim to clarify the most important elements of that craft: to bring the decision-making and actions of a high-level maths teacher to the fore; to lay bare the craft, making it visible and, ultimately, replicable.

One of the most significant components of this craft is how mathematics is presented to pupils. The structure we design as the architect, and the way that structure is revealed as the artist, determines whether mathematics is understood, or whether a conceptually absent procedure is simply followed.

This is not to deny the importance of procedure. It is fundamental. It provides the reliability and precision that makes mathematics so powerful. However, it has historically been over-valued, particularly prior to the 2014 curriculum shift towards a mastery approach, where ‘how’ often took precedence over ‘why’. While practice has moved forward, there remains a lingering tendency to prioritise procedure over understanding.

The difference now is not in the importance of procedure, but in its origin. Where it was once something to remember, communicated through instruction, it should now materialise as a result of careful structural and representational decisions.

The architect and the artist

As the architect, the teacher ensures that the structure of mathematics is made visible. This means making explicit the relationships within mathematics — how quantities are composed, how they relate, and how they can be manipulated.

For example, by explicitly teaching 24 as 2 tens and 4 ones, we expose the structure of the number system: the digits represent quantities within a unit system. 2 tens = 20 and 4 ones = 4. This allows pupils to generalise. A pupil who understands why 2 ones + 2 ones = 4 ones can apply this elsewhere. For example, 2 fifths + 2 fifths = 4 fifths, because the unit remains constant.

Without this understanding, pupils may rely on surface features and produce misconceptions such as 2 fifths + 2 fifths = 4 tenths, incorrectly combining both numerators and denominators.

In this case, the error is not procedural — it is structural.

Another example highlighting the importance of structure is the use of part–part–whole models from an early stage to represent number bonds to ten. By partitioning a whole into parts, pupils begin to understand that numbers can be decomposed and recombined, while also securing number bonds and key relationships within 10.

A pupil who understands this structure will see that 7 + 5 can be considered as 7 + 3 + 2. By partitioning 5 into 3 and 2, the calculation can be restructured to make 10 before adding the remaining part. In this case, bridging through ten is not introduced as a strategy to remember — it is a logical consequence of the structure that has been made visible.

If understanding and planning for structure is the role of the architect, then representing that structure is the role of the artist. Structure is mathematics; representation is how that mathematics is made visible. As the artist, the teacher determines how the structure is seen. Carefully selected representations illuminate relationships within the concepts being taught.

What is presented is highly specific — it determines what pupils are able to see.

Classroom examples

Rounding

To illustrate the impact of structural and representational choices, consider rounding. An instruction often heard in classrooms is: ‘look at the digit next to …’. When presented in this way, pupils are directed towards a procedure with little understanding of the quantity being represented, or why the number rounds up or down. It narrows attention to a single digit, removing the need to consider the number as a whole.

Yes, this procedure requires some level of understanding, but when taught in isolation, it bypasses the underlying structure. An architect focused on structure, combined with an artist’s intent of making that structure visible, will instead use a number line. By positioning a number between two endpoints, rounding becomes a question of proximity. The structure is exposed.

The question shifts from: ‘What do I do?’ to ‘Which value is it closer to?’

The decision to round is justified, not remembered.

The procedure is not taught — it emerges. It begins with a physical representation, develops into a visualisation, and ultimately becomes internalised. Pupils can see the mathematics and understand what is happening when a number is rounded.

Without this structure, pupils may produce correct answers while holding fragile understanding. The rule is followed, but the reasoning is absent.

Equivalent fractions

Another example highlighting the importance of representational choice is the teaching of equivalent fractions. A well-intentioned teacher, aware of the importance of the concrete–pictorial–abstract approach, may still make a poor representational decision. Consider a lesson designed to help pupils understand that one half is equivalent to two quarters. A representation is needed that makes this relationship undeniable.

Using string, for example, may seem appropriate depending on the context. It can be manipulated, cut, and compared. However, it lacks precision. Equal parts are difficult to maintain and comparisons are unclear. The structure remains hidden.

What is required is a representation that makes the equivalence explicit: a simple strip of paper. The paper can be folded to show halves. Further folds reveal fourths. Crucially, both representations exist simultaneously on the same whole. Pupils can see that two quarters and one half occupy exactly the same space.

The equivalence is not told — it is revealed.

From this point, a procedure can emerge as a logical consequence. Any rule that follows is grounded in understanding. Pupils know what equivalent means because they have seen it.

There are, of course, other representations that could be used. A fraction wall, for example. However, this can easily become a matching exercise — finding pieces that fit — rather than understanding the relationship. The underlying mathematics is not revealed; it becomes a game of alignment.

What appears intuitive is not always what reveals the mathematics.