1. Teaching mathematics as a contextual application of mathematical modes of enquiry Anne Watson (& Bill Barton) BSRLM,Cambridge

2. Strong claims Fluent personal mathematical knowledge, modes of enquiry and methods, brought to mind habitually in mathematical contexts, form a basis for mathematics teaching Understanding mathematical enquiry by engaging with it yourself enables you to imagine a range of possible student responses

3. Things you learn by engaging in mathematical practices How ‘errors’ are made - reduces the need to learn about individual errors. Understanding that epistemological obstacles are inherent in mathematics - need to learn notation, interpretation, abstraction. To learn scientific, counter-intuitive, concepts we have experience of adopting new ways of thinking supported by language, diagram and other tools.

4. Teaching as contextual application of modes of mathematical enquiry: Reading mathematical text Identifying possible variables and relationships Using examples and hypothesising structural and inductive generalisations Interrelating formal, scientific, knowledge with intuitive and everyday knowledge Shifting between representations which present different affordances Imagining, through understanding what can be generalised, misrecognised, varied etc., what students might construe

5. What we are not saying: Personal fluent knowledge is enough for good teaching All teachers should have fluent personal knowledge (though that would be good to have!)

6. What we are saying: Fluent personal knowledge and ongoing mathematical enquiry, for example into curriculum maths, contribute coherently to teaching through use of modes of mathematical enquiry Typographies of mathematical knowledge for teaching are comparatively fragmented, minimal ‘coping’ tools and should also include the possibility of personal ongoing mathematical enquiry to increase knowledge and experience