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Becoming aware of something unexpected can be a form of awakening: sharpening attention, enriching noticing, opening up fresh possibilities of action, and educating awareness so as to enable a sensitive response to similar situations in the future. However it can also trigger tunnel vision (few if any actions available) or freezing (no actions available), and so reaction based on automated habitual actions. A variety of types of unexpected classroom incidents are used to support the case that surprise is a release of energy, that disturbance is what initiates learning, and that what matters even more than noticing an opportunity is how that opportunity is taken up, how the released energy is used. Drawing on the perspective of multiple selves, of human beings as transformers of energy, on a six-fold structure of the human psyche, and on the Discipline of Noticing, proposals are made as to how it is not only possible but desirable to prepare for the unexpected.

This article discusses the concepts of "key ideas" and "memorability" and how they relate to the metric "width of a proof" put forward by the Fields medalist Timothy Gowers (2007) in a recent essay entitled "Mathematics, memory and mental arithmetic". The paper looks at the meaning of these concepts and attempts to show whether and how they are congruent with other aspects of proof discussed in literature on the teaching of proof and proving. It also explores how the insights behind "width of a proof" and "memorability" might in themselves mesh with various teaching perspectives.

Given a family of "p" greater than or equal to 3 points in the plane, some three of them have the property that the smallest circle encompassing them encompasses all "p" points. Similarly, we show that for "p" greater than or equal to 3 circles, there are three of them such that the smallest circle encompassing them encompasses all the circles, and that there are three circles for which the smallest circle encompassing and tangent to them encompasses all the circles.

We propose that what teachers can say or write about their views of mathematics and about a range of pedagogic strategies and didactic tactics that they use is of minor importance compared to what comes to mind moment by moment when they are planning or leading a lesson. Beginning with a quote from a teacher that provoked a focus on the role and nature of in-the-moment choices made in the midst of teaching, we offer some comments provoked in us. Following a concise history of attempts to articulate "mathematics needed for teaching", three further brief accounts of incidents led us to propose that what really matters, in the words of Heidegger and others, is one's "mathematical being", because it orients awareness and is the source and basis for choices, whether consciously or unconsciously made. The development of mathematical being is described using the discourse of awareness in the sense of Gattegno, making use of the discipline of noticing to educate awareness. These are then used to extend the analysis of prepositional suffixes for the gerund "knowing" begun by Ryle in order to focus on the development of "connective tissue" between mathematical awareness and in-the-moment pedagogical choices of actions. (Contains 2 notes.)