Anne Watson Hong Kong 2011

 grasp formal structure  think logically in spatial, numerical and symbolic relationships  generalise rapidly and broadly  curtail mental processes  be flexible with mental processes  appreciate clarity and rationality  switch from direct to reverse trains of thought  memorise mathematical objects ◦ (Krutetski)

 ‘Higher achievement was associated with: ◦ asking ‘what if..?’ questions ◦ giving explanations ◦ testing conjectures ◦ checking answers for reasonableness ◦ splitting problems into subproblems  Not associated with: ◦ explicit teaching of problem-solving strategies ◦ making conjectures ◦ sharing strategies  Negatively associated with use of real life contexts for older students

 What activities can/cannot change students’ ways of thinking or objects of attention?  What activities require new ways of thinking?

– 35 a + b - a

 From number to structure  From calculation to relation

 grasp formal structure  think logically in spatial, numerical and symbolic relationships  generalise rapidly and broadly  curtail mental processes  be flexible with mental processes  appreciate clarity and rationality  switch from direct to reverse trains of thought  memorise mathematical objects

28 and and and and and and and and 46

 From physical to models  From symbols to images  From models to rules  From rules to tools  From answering questions to seeking similarities

 grasp formal structure  think logically in spatial, numerical and symbolic relationships  generalise rapidly and broadly  curtail mental processes  be flexible with mental processes  appreciate clarity and rationality  switch from direct to reverse trains of thought  memorise mathematical objects

 From visual response to thinking about properties  From ‘it looks like…’ to ‘it must be…’

 grasp formal structure  think logically in spatial, numerical and symbolic relationships  generalise rapidly and broadly  curtail mental processes  be flexible with mental processes  appreciate clarity and rationality  switch from direct to reverse trains of thought  memorise mathematical objects

 Describe  Draw on prior experience and repertoire  Informal induction  Visualise  Seek pattern  Compare, classify  Explore variation  Informal deduction  Create objects with one or more features  Exemplify  Express in ‘own words’

 Make or elicit statements  Ask learners to do things  Direct attention and suggest ways of seeing  Ask for learners to respond

 Discuss implications  Integrate and connect  Affirm This is where shifts can be made, talked about, embedded

 Vary the variables, adapt procedures, identify relationships, explain and justify, induction and prediction, deduction

 Associate ideas, generalise, abstract, objectify, formalise, define

 Adapt/ transform ideas, apply to more complex maths and to other contexts, prove, evaluate the process

 Remembering something familiar  Seeing something new  Public orientation towards concept, method and properties  Personal orientation towards concept, method or properties  Analysis, focus on outcomes and relationships, generalising  Indicate synthesis, connection, and associated language  Rigorous restatement (note reflection takes place over time, not in one lesson, several experiences over time)  Being familiar with a new object  Becoming fluent with procedures and repertoire (meanings, examples, objects..)

 Repertoire: terms; facts; definitions; techniques; procedures  Representations and how they relate  Examples to illustrate one or many features  Collections of examples  Comparison of objects  Characteristics & properties of classes of objects  Classification of objects  Variables; variation; covariation

 Between generalities and examples  From looking at change to looking at change mechanisms (functions)  Between various points of view  Between deduction and induction  Between domains of meaning and extreme values as sources of structural knowledge

 Visualise, seeing whole things  Analyse, describing, same/different  Abstraction, distinctions, relationships between parts  Informal deduction, generalising, identifying properties  Rigour, formal deduction, properties as new objects

 generalities - examples  making change - thinking about mechanisms  making change - undoing change  making change - reflecting on the results  following rules - using tools  different points of view - representations  representing - transforming  induction - deduction  using domains of meaning - using extreme values

 Methods: from proximal, ad hoc, and sensory and procedural methods of solution to abstract concepts  Reasoning: from inductive learning of structure to understanding and reasoning about abstract relations  Focus of responses: to focusing on properties instead of visible characteristics - verbal and kinaesthetic socialised responses to sensory stimuli are often inadequate for abstract tasks  Representations:from ideas that can be modelled iconically to those that can only be represented symbolically

Watson, A. (2010) Shifts of mathematical thinking in adolescence Watson, A. (2010) Shifts of mathematical thinking in adolescence Research in Mathematics Education 12 (2) Pages 133 – 148