Directing Mathematical Attention

Take a comparative bar model, selected specifically to expose the difference between two quantities. The architect and artist within the teacher have fulfilled their responsibilities. The representation has been carefully chosen, the lesson thoughtfully sequenced, and the mathematics made visible. Yet understanding is not guaranteed. The representation contains multiple features, relationships and pieces of information, any of which may capture a pupil’s attention. Unless attention is deliberately directed, what pupils take from the lesson may depend less on the teacher’s intentions and more on chance. This is where the conductor becomes essential.

Whilst the roles of the architect and artist may provide a degree of comfort and control within a lesson, I would argue that the conductor is ultimately responsible for whether the intended learning is actually realised. More concerningly, a perfectly chosen representation and expertly designed lesson can contribute to the development of misconceptions if attention is not directed towards the mathematics that matters.

Consider the teaching of multiplying by 10. Through careful representation and explanation, a diligent teacher may successfully reveal the underlying structure of place value. However, unless attention is deliberately directed, a pupil may instead focus on a superficial pattern: the product appears to contain the same digits with an additional zero. The lesson may appear successful, with correct answers produced, yet the pupil has attended to the wrong feature of the mathematics. Rather than understanding that each digit becomes ten times greater and shifts one place to the left, they have constructed a rule based upon the appearance of an extra zero.

The consequence of this misplaced attention may not become apparent until much later. When decimals are introduced, the rule begins to fail and the misconception is exposed. By this point, further learning may have been built upon an insecure foundation. The lesson itself was not poorly planned, the representation was not poorly chosen, and the mathematics was not hidden. The issue arose because the pupil attended to something different from that which the teacher intended. The mathematics may be visible, but it is the teacher’s responsibility to ensure that pupils are looking in the right place.

Visibility and Attention Are Not the Same Thing

The discussion in my previous post centred around making the mathematics visible through the careful choice of representation. This post is about what pupils attend to. Visibility and attention are not the same thing. To simply make something visible does not mean that the intended learning will occur.

Representation can reveal the structure; attention determines whether pupils see it.

Two pupils looking at the same representation in the same lesson may leave with two very different understandings. Research consistently shows that information, such as the mathematics made visible through our representations, must first be attended to before it can be meaningfully processed and encoded into working memory. To put it simply, learning does not occur because valuable information has been presented in a suitable way; it occurs because attention has been directed towards it.

In fact, cognitive science increasingly points towards attention as a critical gateway to learning, determining which aspects of an experience are processed, connected and ultimately remembered. This is particularly significant in mathematics classrooms. At any given moment, pupils are presented with numbers, representations, vocabulary, procedures and relationships. Whilst all of these may be visible, only some of them will receive attention. It is those features that are most likely to shape the understanding pupils leave the lesson with.

Moments of Noticing

Perhaps this is why moments of understanding often feel so sudden. A child who, moments earlier, appeared uncertain suddenly makes a connection. A relationship becomes apparent. A pattern emerges. An idea that previously seemed inaccessible now appears obvious. We often describe these as breakthroughs or light-bulb moments, but if we have carried out the three elements of our craft effectively, they are more accurately described as moments of noticing.

A child notices that two quarters occupy the same space as one half. A child notices that the gap within a comparative bar model represents the difference between two quantities. A child notices that multiplying by ten changes value through place value relationships rather than through the appearance of an additional zero. In each case, the mathematics was already present. The representation had exposed it. The learning occurred when attention settled upon it.

Mathematical attention is not simply concentration, compliance or looking in the right direction. A pupil can appear attentive whilst attending to entirely the wrong thing. Mathematical attention is concerned with what occupies a pupil’s thinking in a given moment. It is the relationship they are considering, the structure they are attempting to make sense of, or the mathematical feature they are noticing. Ultimately, it is this attention that determines what becomes connected, understood and remembered.

Viewed through the Primary Maths Craft framework, this is the responsibility of the conductor. The architect may design the learning journey and the artist may reveal the mathematics through carefully selected representations, but it is the conductor who determines what pupils actually notice along the way.

The Responsibility of the Conductor

The examples above perhaps make the process sound deceptively simple. A child notices something important, a connection is formed and understanding develops. Yet anyone who has stood in front of a primary classroom knows that directing attention is rarely so straightforward.

Unlike an orchestra, our classrooms are not filled with carefully selected experts, each trained to perform a specific role. They are filled with children carrying different experiences, different misconceptions, different interests, different motivations and different levels of understanding. Some arrive ready to engage with the mathematics immediately. Others are distracted by events that happened five minutes earlier, five days earlier or even five years earlier. Some are searching for patterns. Some are searching for procedures. Some are simply searching for the answer.

This is why I believe the conductor represents the most complex element of the craft. The role extends far beyond presenting mathematics clearly or providing opportunities for pupils to notice. The conductor must first secure attention before directing it. Behaviour, engagement, interest, curiosity and participation are not separate from learning; they are often the very mechanisms through which attention becomes available in the first place.

Only once attention has been gathered can it be directed. Only once it has been directed can noticing occur, and only then can meaningful connections begin to form.

Perhaps this is what makes teaching such a remarkable craft. Thirty children may be sat in front of the same representation, hearing the same explanation and participating in the same lesson, yet each arrives with a different starting point. The work of the conductor is to bring these individuals together and guide them towards a shared mathematical destination. Not by chance, but through deliberate and responsive decision-making, moment by moment throughout the lesson.

Conducting Mathematical Attention

Returning to the comparative bar model from the opening example helps to illustrate what this looks like in practice. The representation itself has already fulfilled the role of the artist. The mathematics has been made visible. The relationship between the two quantities can be seen. The architect has also fulfilled their role, carefully selecting and sequencing a representation capable of exposing the concept of difference. Yet the responsibility of the conductor remains.

Without deliberate direction, pupils may attend to a variety of features within the representation. Some may simply notice that one quantity is larger than the other. Others may identify the numbers involved and immediately begin searching for an operation. Some may correctly infer that subtraction is required, but still fail to understand why. Whilst these observations may move pupils towards a correct answer, they do not necessarily move them towards understanding.

The conductor’s role is to direct attention towards the aspect of the representation that matters most. Attention must be drawn towards the gap, the missing section and the comparison between the two quantities. Questions such as, “Which part is different?”, “How is it different?” and “What does the missing section represent?” help to focus attention on the mathematical relationship rather than the numbers themselves.

Once attention settles upon these features, something interesting begins to happen. The operation is no longer chosen because it feels appropriate or because a remembered rule suggests it. Instead, the operation emerges naturally from the structure that has been exposed by the representation. Pupils do not simply arrive at subtraction; they begin to understand why subtraction is required.

The operation emerges from the structure.

The same principle applies beyond representations. If the conductor’s task is to direct attention, then one of the most powerful tools available is language. In many ways, language becomes the conductor’s baton. It signals where attention should be directed, what should receive emphasis and which mathematical ideas should be brought to the foreground.

Consider the difference between asking, “What is the answer?” and asking, “What do you notice?” The first directs attention towards an outcome. The second directs attention towards the mathematics itself. Similarly, questions such as, “What has changed?”, “What has stayed the same?” and “Why does that work?” encourage pupils to attend to relationships, patterns and structure rather than procedures alone.

Every question carries an instruction about where attention should be placed. The language we choose influences what pupils notice and, consequently, what they are likely to understand. If representations make mathematics visible, language often determines whether pupils look at the right part of it.

Every question tells pupils where to look.

Implications for Teaching

The further I develop the Primary Maths Craft framework, the more I find myself returning to the conductor. Whilst the architect provides structure and the artist reveals the mathematics, it is the conductor who ultimately determines whether that mathematics receives the attention it deserves.

The conductor decides emphasis. They bring important ideas to the foreground and prevent less important features from dominating pupils’ thinking. They ensure coherence between the representation, the language, the examples, the questioning and the behaviours that make learning possible. Viewed in this way, directing attention becomes far more than asking the right question at the right moment. It becomes the continual process of gathering, directing and sustaining attention towards the mathematics that matters.

This has significant implications for how we think about lesson design. It is easy to become consumed by the search for engaging activities, clever resources or memorable hooks. Yet perhaps a more useful question is not, “What activity shall I use?” but, “What mathematics must be noticed?” Closely followed by an equally important question: “What might pupils notice instead?”

The answers to those questions influence every instructional decision we make. They influence the representations we choose, the examples we construct, the language we use, the questions we ask and the classroom culture we establish.

The mathematics itself never changes. The representation may change. The language may change. The examples may change. The pupils certainly change. Yet the mathematics remains constant.

The experience of that mathematics, however, does not.

What pupils notice becomes what they understand. The craft of teaching lies in deciding what is noticed.