Teaching mathematics as a contextual application of mathematical

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

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

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.

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

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!)

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