Structure of Attention

Inviting learners to say ‘what is the same and what is different’ about several entries in a Structural Variation Grid, or simply to ‘say what you see’ initiates a movement of their attention.  They may gaze at the whole (and be aware that there appear to be missing or hidden entries in a grid) or at the whole of a particular element such as a cell entry; they may discern details such as particular entries, or details within an entry (such as two rows to each cell, or the presence of various mathematical signs); they may recognise relationships within a cell (such as an equation) and between cells (such as all having two factors or the upper part being a calculation and the lower part an answer or vice versa); they may perhaps perceive some relationships as properties which apply across all visible cells and so might apply to all cells; they may even be able to reason on the basis of those properties in order to justify their prediction of what will appear in different cells, or where a particular entry is to be found.

As a teacher with a class, the problem is that different learners may be attending in different ways.  If as teacher you are attending in one way, say talking about properties of cells, when learners are busy discerning details or recognising relationships between particular entries, there may be a mismatch and consequent breakdown in communication.  By being aware of what you are attending to, and how, you can either direct learner attention appropriately, or put your own focus of attention to one side and try to enter the experience of some of the learners.

What is Attention?

For William James, philosopher and psychologist, attention is what makes it possible to perceive, conceive, distinguish and remember.  It is the basis of all our psychological functioning (James 1890 p 424). As might be expected, he deals with a number of important issues concerning attention in general. For example, he argues on the basis of experiments that attention is not simply what the eyes are looking at, or indeed any other particular source of sense im¬pressions (p 438). He links attention to anticipative imagination (p 439-411) as a prerequi¬site for discerning anything at all. James develops this theme of discernment, or discrimination, to make use of what he calls Helmholtz’s law, that

    we leave all impressions unnoticed which are valueless to us as signs by which to discriminate things (p 456).
In other words, we notice what we are attuned to discern (this theme has been articulated by psychologists, dramatists, novelists and artists since the end of the 19th century at the very leat). James goes on to discuss pedagogic implications such as that it is useful for teachers to work with learners to strengthen and attract their attention in order to improve motivation, since people engage with what catches their attention (James 1890, p 446). To do this re¬quires being aware of what in learners’ previous experience can be used as a basis of previ¬ous attention-experience, what John Dewey referred to as ‘psychologising the subject mat¬ter’ (Dewey 1902, p 12).

James sees attention as a form of ‘free energy’, since when you make an effort to attend to something you can sustain it for only very short periods before attention wanders (p 420) requiring a further expenditure of effort, but when attention is engaged it requires no energy expendi¬ture at all for it to remain focused for long periods of time.

I agree with James that ‘my experience is what I agree to attend to’ (his emphasis), although his wording might be taken to imply voluntary agreement, which is certainly not always the case. At each moment, as my attention shifts, I am the totality of that attention; the totality of my experience is my attention. Attention is not just as what puts me in touch with the world of my experience, but what creates and maintains that world. This is meant to include things of which I am subliminally or covertly aware, sometimes through body awareness, sometimes through social awareness, sometimes through emotional resonance, and sometimes through cognitive awareness.  None of these need be conscious. This makes attention difficult to study directly, because it is no good asking people ‘what are you attending to?’ since the very question alters the focus and locus of that attention.

Where I differ with James is in his metaphor of attention or consciousness as a flowing stream, for it seems to me that his own descriptions (e.g. James 1890 p456 quoting Müller), as well as my observations, lead to the conclusion that attention is briefly sharp and alert, and then slowly declines into absence of awareness until some fresh stimulus wakes it up again. The sense that we have of experience flowing by is actually much more episodic and fragmentary (Mason 1988), as attempts to reconstruct recent and distant experiences demonstrates all too clearly.

You can attend to things physically present and also to things not physically present (locus); you can ‘gaze’ while pondering, and you can concentrate very specifically on some small detail (focus). You can be aware of one single detail, and you can be multiply aware cognitively, multiply aware enactively, and multiply aware affectively (multiplicity). Once focused, attention can be diverted by rapid movement within your field of vision, especially if it is peripheral, and changes in other sense impressions can also attract your overt attention.

There are deep physiological questions about whether you actually attend to several things at once, or whether you rapidly cycle through a variety of foci, the way computers now do.  There is also an issue about whether consciousness directs behaviour or is subject to a ‘user illusion’ of being in charge, as Tor Norretranders (1998) proposes. Whatever may be the case, personal experience is sufficient to highlight important aspects of attention which can be used to improve both teaching and learning.

Whether attention is the subjective experience of physiological functioning, as Théodule Ribot (1890) would have it, or the engine for physiological response to environment, as William James (1890) proposes, there seem to be quite distinctive if subtly different forms of attention:

    Holding Wholes (gazing)
    Discerning Details (features & attributes)
    Recognising Relationships (part-part, part-whole)
    Perceiving Properties (leading to generalisation)
    Deducing from Definitions (reasoning on the basis of explicitly stated properties stated independently of particular objects)

Shifts between these are rapid, often subtle, but vital in order to engage in mathematical thinking. While gazing, some sudden movement, perhaps even apparent motion produced from circadian eye movement can suddenly switch attention to awareness of details amongst a mass of other, undiscerned detail.  As details are detected and discriminated, the mind automatically looks for relationships: differences and samenesses.  To do this requires something being relatively invariant as a background against which to detect change.  Recognising relationships tends to focus on particulars, whereas perceiving properties is a move to the more general, to the particular as exemplary or paradigmatic. Formalising in mathematics is the overt action which accompanies a shift from perceiving properties to taking certain properties as definitive and so as the basis for further reasoning. Discerning these subtle shifts in the structure of attention develops Marton’s notion of learning as discerning variation, because it provides a more detailed structure of what might attended to, and how.
Vor references see main SVGrids page.