1. Mathematical thinking in adolescence: possible shifts of perspective Anne Watson University of Oxford Nottingham, November 2007

2. Mathematical thinking Thinking about mathematics The thinking that is required in order to understand ‘hard’ concepts The thinking that is required to work mathematically The thinking that ‘real’ mathematicians do

3. Mathematical thinking in adolescence The thinking that is required in order to understand the essential conceptual shifts in secondary school mathematics The thinking that is required to adapt and apply mathematical knowledge at school level

4. Shifts to empowerment in mathematics Discrete – continuous Additive - multiplicative Rules – tools Linear – non-linear Procedure – meaning Example – generality Percept – concept Operations – inverses Pattern – relationship Relationship – properties Conjecture – proof Result – objectify result Result –objectivify procedure/method Intuitive – deductive Inductive – deductive

5. Who were they? Year 9 class, above average prior attainment, mixed comprehensive Summer term after SATs

6. Task To find pairs of numbers of the form a + √b which, when multiplied together, give integer answers

7. What they knew ‘grid’ multiplication for numbers and algebra squares and square roots in simple cases, and use of √

8. Grid multiplication X z+3 2z 2z 2 6z -z-3

9. What did they do? Reach for the calculator! (7+ √19) (√17 + 3) (7 + √18) (√18 + 3) (7 + √18) (√17 + 3) (7 + √17) (√17 + 3) (4 + √4) (5 + √5) (√8 + √8) (12 + √69) (8 + √12) (10 + √6) (2 + √3) (√2 + √3) (2 + √3) (3 + √2) (2 + √2) (3 + √3) (a + √2) (b + √8) (2 + √2)(2 + √2)

10. Other classes Year 9 average and below average prior attainment Average were better at using negative signs, so several ‘found’ answers Below average ‘found’ that square numbers were more useful than ‘unsquare’ numbers

11. Adolescence identity belonging being heard being in charge being supported feeling powerful understanding the world negotiating authority arguing in ways which make adults listen sex

12. Adolescence identity belonging being heard being in charge being supported feeling powerful understanding the world negotiating authority arguing in ways which make adults listen »My examples: »shared with group »choice of recording method »generate their own characteristics »friends; calculator »calculator; my examples »can check answers; don’t need teacher »can justify answers

13. Further features The grid as domain, support, authority Grid has syntactic and semantic function –Tells you what to do symbolically –Also has mathematical meaning as physical model of distributivity in 2 dimensions Shift from empirical view of examples to structural view happened, for some, without teacher intervention Grid provides scaffold for example generation AND window on examples generated

14. Shifts to empowerment in mathematics Discrete – continuous √ Additive - multiplicative √ Rules – tools √ Linear – non-linear Procedure – meaning √ Example – generality √ Percept – concept √ Operations – inverses √ Pattern – relationship √ Relationship – properties √ Conjecture – proof Result – objectify result √ Result –objectivify procedure/method √ Intuitive – deductive √ Inductive – deductive √

15. Mathematical thinking There is a need to become more articulate about specific kinds of shifts in thinking which are required to learn secondary mathematics There is a need to identify methods-in- classrooms which seem to ensure these shifts are made by a large majority of students There is a need to understand such methods to identify common characteristics

16. Future plans Continue fine-grained classroom work Continue fine-grained analysis of mathematical activity Connecting very fine-grained differences with brain-and-eye function to understand more about expert/novice response to task layout and sequencing anne.watson@education.ox.ac.uk