Dimensions of Possible Variation

One of the essential features of Structural Variation Grids is that they permit rapid exposure to several examples which display variation in one or two different aspects.  These examples can be chosen according to pedagogic and didactical purposes according to the perceived needs of the learners. Human beings have brains which are well suited to detecting variation, especially systematic or structured variation, and the grids enable a quick succession of examples to be presented.

Ference Marton has for some years been developing the observation that what people discern is variation (Marton & Booth 1997, Marton & Trigwell 2000, Marton & Tsui 2004; see also Runesson 2005).  This has led him to propose that learning consists of discerning freshly, that is, of becoming aware of new dimensions of variation.  A dimension of variation is an aspect which can vary in an example and still it remains an example.  In other words, a concept or technique is understood to the extent that the person is aware of what can be varied and what must nevertheless remain invariant. Often it is relationships which are invariant rather than aspects of objects themselves. Anne Watson and I (Watson & Mason 2002, 2005) extended this idea to dimensions of possible variation to indicate that at any time teacher and learner may be aware of different aspects or dimensions which could be varied, even if they are not varied in the current situation. Furthermore, we noted that often in mathematics learners have a restricted notion of the range of permissible change in any specific dimension.  For example, when generalising sequences and grids, learners typically think of whole numbers while the teacher may be aware of rationals and reals as possibilities.

The vital aspect of variation as a description of learning is experiencing sufficient variation in sufficiently quick succession to be aware of it as variation of some feature, and hence as a dimension of possible variation.

We have found Marton’s idea of great interest and use for two basic reasons: first, it fits with our own view that invariance in the midst of change is a central theme of mathematics, and second, it proves to be fruitful for analysing learning and the potential for learning afforded by tasks, including sets of exercises (Watson & Mason in press).