Once a pattern starts to emerge, it can be developed and expressed in words, pictures, icons such as clouds to stand for ‘the number I’m thinking of but am not going to tell you’, and even using letters as symbols for as-yet-unknown or unspecified numbers, as in traditional algebra. This description builds on distinctions proposed by Jerome Bruner (1966) who identified three modes of representation: enactive, iconic and symbolic. In designing the Open University course mentioned earlier, we found that these resonated with our experience, but that we wanted to elaborate on some of the ramifications. The result was a metaphorical interpretation of his distinctions, and three closely related frameworks. The basic framework remains the same as Bruner’s but with elaboration:

Enactive mode: | manipulating familiar and confidence-inspiring entities, whether they are physical (blocks, sticks, counters, rods, …), as Bruner suggested, or meta-physical (numerals as numbers, letters as variables or as generalities, familiar diagrams, screen manipulable objects, etc.); |

Iconic mode: | images, pictures and drawings which depict what they are (as in a cloud for a number I am thinking of or don’t yet know) as Bruner suggested, but including also a pre-articulated as-yet-inchoate ‘sense of’; |

Symbolic mode: | symbols whose use is a convention and so by their nature have to be explained, as Bruner suggested, but they are abstractly symbolic only so long as they remain unfamiliar. |

For example, Helen Drury (personal communication) working with a year 10 top set used the Factor Grid for the first time as a computer display in whole-class mode and then invited learners to fill in a blank grid for themselves. They had a choice either to try to fill in the expanded expressions above the factored versions immediately, or else to begin by writing out the factored cells before completing the expanded expressions.

Filling in the entries themselves afforded an opportunity for enactive subconscious awareness of patterns to be manifested and experienced. This is why paying attention to how you fill out a table or draw a picture enactively can be so useful when trying to articulate a generality: so useful in fact that a slogan such as Watch What You Do, along with Say What You See can be useful for reminding learners to do more than simply try to get answers. Learners who tried to do the expanded expressions immediately mostly struggled to find the complex patterns, especially in the constant term. They tended to enter all the x2s first, which is efficient, and which may direct attention to significant patterns, but it may also divert attention inappropriately, as with copy–and–complete. One learner spotted that opposite corners had opposite signs, and another described similarities with the arithmetical multiplication grid. Pedagogic decisions had to be made about whether to invite them to report their observations or to leave others to make similar discoveries. The important work involved trying to make sense of the upper and lower entries in each cell, leading to a deeper enactive awareness of how factoring quadratics works.

The use of MGA as an acronym illustrates the framework perfectly. Unless you are already familiar with MGA, you are likely to need to expand it in your mind, to read out the full form of words and then think about the meaning. Over time and with use you may find MGA becoming a useful shorthand for triggering actions and awareness in yourself, for making sense of experiences, and for communicating with others. The acronym can actually help you to articulate some observation or to describe some phenomenon.

In relation to Structural Variation Grids, filling out a grid for themselves from the contents of a sequence of cells in one row and another in one column, or even from the content of a few sporadically placed cells, enables learners to work with familiar entities (expressions in cells whose content is known) in order to get a sense of the overall structure of a particular grid. At a more meta-level, familiarity with one grid enables them to recall what they did previously when they tackle a new one, so that over time they get a sense of structure indicated by a two-way grid, and two-way grids as a format for arithmetic structure.

Hand in hand with MGA is the triple Do–Talk–Record (in the sense of writing-up not exploratory writing-down). Writing our course in the early 80s we were well aware of the importance of learner-learner talk and collaboration. This framework allowed us to remind teachers that pushing learners to make written records too quickly can be at best unproductive and frustrating for all concerned, and at worst, actually harmful. It is valuable if not essential to allow learners time to talk about what they have been doing, and indeed to get them doing things (enactively manipulating the familiar) so that there is something mathematical to talk about. Talking to others, trying to justify your ideas and conjectures is an excellent way to externalise your thinking, get it outside of yourself so that you can look at it critically. That makes it easier not to be indentified with your idea but to treat it as a conjecture to be modified.

Alongside these three frameworks we also found it useful to include something to remind teachers that learners do not usually master ideas on first exposure. So we suggested a triple of See––Experience––Master as a reminder that first encounters are a bit like seeing a fast vehicle go by. It takes repeated encounters to begin to discern details and to recognise relationships amongst those details. Only then does it make sense to try to achieve mastery, to develop facility and fluency and to minimise the amount of attention needed to carry out techniques and procedures.

Finally, we embedded these frameworks in the notion of a classroom rubric or ways of working which correspond to what is now described as socio-cultural practices of a community of practice (Lave & Wenger 1991) and as sociomathematical norms (Yackel & Cobb 1996). Fundamental to the effective functioning of a classroom ethos is a mathematical or conjecturing atmosphere (Mason, Stacey & Burton 1982, see also Mason & Johnston-Wilder 2004a). This is a way of working in which everything said is treated as a conjecture, uttered in order to think about it more clearly and to modify it as appropriate. Those who are very confident take the opportunity to listen and to suggest illustrative examples and counter-examples, and those who are not so confident take opportunities to try to express their thinking in order to help them clarify that thinking, just as in the Eleusis game.

[See Bibliography on main Grid Page for references]