Wonder, the Rainbow and the Aesthetics of Rare Experiences NATHALIE SINCLAIR, ANNE WATSON Aesthetics, the pleasure of the sensory or intellectual cognisance of fit, or grasp of beauty It is a rare and wonderful experience to pick up a book found accidentally to find it speaks directly to one's senses. It is rarer still to find a colleague who has also found the same book, again fortuitously, and responded similruly. This piece describes a book which sounds as if it has little to do with our field, but which turns out to have much to do with math- ematics, with education, and mathematics education in pruticulru. Both of us have used it in our work First, we explain aspects of our work to which this book speaks, before detailing how Toward wonder I (Anne) work with new teachers in a context defined by targets, statements, learning objectives, structmes fot accountability, the frequent ascription of level numbers to students, testing and league tables. Oddly, within this appru- ent gridlock of heavily-framed practice, there is also the requirement that teachers should: plan opportunities to contribute to pupils' spiritual development (DfEE, 1998, Section B4d, p 12) There is bewilderment among mathematics educators at this requirement. First, no one is sure what it means; second, being sme would mean definition, which should, in the cmrent climate, lead to measurability which seems contra- dictory to the notion of spiritual development; third, the tequirement seems to contradict the assumptions of mea- surability by suggesting there is something in education which cannot be measured In mathematics, educators have tended to respond to the demand for attention to a spiritual dimension by appealing to the sense of awe and wonder which can be generated by experience of the inirulte. At a high school level of mathe- matics, the equation ein + 1 = 0 is also offered as an opportunity for awe. Other candidates for the role of awe- inspirer are the identity cos 2 x = 1, the PythagOrean proof of the irrationality of -/2 and geometrical generalities such as the co-incidence of the medians of a ttiangle An alternative track to awe and wonder might be to suggest that mathematics can describe patterns apperuing in nature. But how does one generate awe in students if they are unable to appreciate the beauty which others see in these structtues, or who can see beauty, but do not fmd it enhanced by mathematical expression? Not only do UK teachers search for ways to induct their pupils into these aesthetic perceptions, but educators also search for ways to induct new teachers similruly, so that they may be able to incorpo- rate such opportunities into their· teaching I had a growing disaffection with this pedestrian approach to awe and wonder in mathematics, as if there were common sites for expressing awe, like scenic viewpoints seen from a tourist bus, whose position can be recorded on the cuniculum as one passes by, en route for something else Spontaneous appreciation of beauty and elegance in mathe- matics was not, for me, engendered by occasional gasps at nice results, nor by passing appeals to natmal or constructed phenomena such as the patterns in sunflowers or the mathe- matics of tiling ***** I (Nathalie) was looking for a more adequate account of motivation in mathematics education and, in particular, one that acknowledged its roots in wonder and the aesthetic I saw the growing emphases on relevance (usually taken to mean connection with 'real-life'), problem solving and interdisciplinruy activities (mathematics tluough music, rut, sports, etc.) as attempts both to justify mathematics teaching and to motivate student interest and achievement, by dwelling within their existing horizons. These attempts seem largely to ignore what might be intrinsically satisfying for students in mathematical acti- vity: how can their tendencies for wonder and exploration, their desires to express themselves, their need to understand themselves, be fulfilled by mathematics? Or, as Dewey (1913) might insist, what is it about mathematical activity that lies in the direction of the student's growth? Indirectly, many mathematics educators appeal to the notion of aesthetics in their discussions of the importance and value of mathematics. (Marion Walter (2001), in her article in the previous issue of this journal, provides a more direct appeal tluough her integrated, aesthetic-mathematical response to geometry and geometrical rut) Yet few of them explicitly address the role of aesthetics in mathematics leruning. Those who do, often focus on the aesthetic judge- ments of beauty and elegance made about mathematical objects (theorems, proofs, equations, definitions, etc.), and are concerned with the extent to which students can appre- ciate such judgements. Very few of them mention the possibility of students' aesthetic experience in mathematics learning, one that is synoptic and integrative, that has aesthetic quality and is suffused by a distinctive emotion. Such experiences are not only familiar to mathematicians, whether amateur or professional, but they also represent one of the animating purposes of mathematical knowledge - why we do and value mathematics Are these experiences available to students, and what makes them possible? Perhaps if they were to begin in wonder . . ? For the Learning of Mathematics 21, 3 (November, 2001) FLM Publishing Association, Kingston, Ontario, Canada 39