Here is a my third draft of some key ideas that I consider important when teaching maths, as an addition/response to @pepsmccrea's post here.
In-the-moment pedagogy, awareness, attention, noticing; indeed almost anything by John Mason. Anyone who teaches maths should read any articles or books by him and also Dick Tahta, David Pimm, Anne Watson, David Wheeler, Dave Hewitt, Paul Dowling, Valerie Walkerdine - plus many others in back catalogues of Mathematics Teaching and For the Learning of Mathematics.
Conjecture/revision, negotiation of meaning, students as sense-makers not mistake-makers; encouraging students to present their work even/especially when it is only a partial solution, being happy to revise these partial understandings publicly; allowing students a voice with which they can validly speak and listening to them (see also this excellent book).
Starting points and points of departure: choosing problems/prompts that contain minimal information and so allow a range of responses, allow space for inquiry and mathematical play, sometimes also through the use of student generated examples. I am currently exploring what might be gleaned from teaching using structured improvisation and dramatic techniques.
Girls in maths (see also this article by Valerie Walkerdine). I have concerns about the universality/patriarchy of the rational a la The Mastery of Reason, also this by David Wheeler and this book by Foucault; see also Dowling's work on the myths of relevance and participation in school mathematics.
Language/semiotics (David Pimm, Brian Rotman, MAK Halliday,...) and connected thoughts around metaphor/metonymy - for example these articles by David Pimm, Dick Tahta, Alf Coles, Anna Sfard. The book Metaphors We Live By is very interesting.
General concepts applied to mathematical education: philosophy, sociology and psychology (see also this edition of For the Learning of Mathematics). I feel there is much to be explored/applied from psychoanalysis to education, for example transference and also this.
Complexity and the dilemmas of teaching: Deborah Ball, Magdalene Lampert (also try these papers), David Cohen. I like the idea of dilemmas not deficits in dialogic teaching - see this also. I find the instructional triangle model useful in considering multiple perspectives and interactions in the classroom.
Students working as a group, not in a group (i.e. true collaborative/co-operative learning) is a fundamental part of learning maths, but is often not implemented well in classrooms; the work of Ilana Horn, Malcolm Swan and Elizabeth Cohen explains how to do this effectively.
Understanding how to teach particular topics in mathematics by developing one's mathematics knowledge for teaching, for example the problem solving methods of Polya, Alan Schoenfeld, Paul Zeitz.
Trying to understand your students and consider their lives outside the classroom; I found this on urban classroom culture fascinating, and also this book on sociolinguistics. I am deeply interested in the construction of identity, for example this on gender, and also the links between socio-economic and academic status (Paul Willis, Annette Lareau, Ilana Horn, Elizabeth Cohen) and the negative effects of setting and differentiation. I see teaching mathematics as human development.
Building trust with children; treating them humanely, being aware of possibly inhumane educational approaches. This is linked to thoughts around authority - see the work of Jo Boaler, Beth Herbel-Eisenmann, Michel Foucault. There is a lot more to be said here - see upcoming posts for teachlikeahuman. Maths education is becoming ever more political.
I have been working on building a learning community in my classroom this year after reading about communities of practice, situated learning and apprenticeship - see also this book and this one. I view my role as a teacher as working together with students, of being alongside them.
Finally: I am not convinced by CLT. I prefer theories of embodied cognition, for example in this book; I find the mind/body dualism and the notion of education as filling the brain with knowledge hugely problematic.
Well, this is a start; I would be interested in your thoughts...