1. Anne Watson Hong Kong 2011

2.  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)

3.  ‘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

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

5. 35 + 49 – 35 a + b - a

6.  From number to structure  From calculation to relation

7.  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

8. 28 and 34 280 and 340 2.8 and 3.4.00028 and.00034 1028 and 1034 38 and 44 -38 and -44 40 and 46

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

10.  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

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

14.  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

15.  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’

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

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

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

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

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

21.  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..)

22.  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

23.  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

24.  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

25.  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

26.  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

27. anne.watson@education.ox.ac.uk 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