1. 1 Progress in Mathematical Thinking John Mason SMC Stirling Mar 6 2010

2. 2 Outline  What is progress in mathematical thinking?  Progress in what? –Performance (behaviour) –Conceptual appreciation and understanding; connectedness; articulacy (cognition) –Independence & Initiative (affect) –Ways of working individually and collectively(milieu)  Need for a sufficiently precise vocabulary –to make thinking, discussion and negotiation possible Tasks that reveal progress

3. 3 What is Progress?  Perceived change in –Behaviour (what people do) –Affect (what people feel about what they are doing; motivation; dispositions; initiative; confidence; self- efficacy etc.) –Cognition (Awareness, Key developmental Understandings, Critical Features) –Meta: Learning how to learn mathematics  These come about as learners –Discern what can vary over what range, and what must remain invariant –Discern details, recognise relationships, perceive properties and reason on the basis of agreed properties –Make fundamental shifts in both what they attend to and how they attend mathematically

4. 4 In Between  How many circles could there be between the two shown?  How many numbers could there be between 1.50 and 1.59 1.500 and 1.5987 Range of permissibl e change Discrete & Continuou s

5. 5 Difference of 2 write down 2 numbers with a difference of 2 write down the equation of two lines with slopes differing by 2 write down an integral over two different intervals whose values differ by 2 And another PrimarySecondaryUpper Secondary

6. 6 Seeing As ✎ Raise your hand when you can see something that is 1/3 of something; again differently again differently A ratio of 1 : 2 Range of permissibl e change Dimension s of possible variation Threshold Concept: Clarifying the unit ✎ What else can you ‘see as’? ✎ What assumptions are you making? 4/3 of something

7. 7 Seeing through the particular to a generality

8. 8 Dimensions-of-Possible- Variation Regional  Arrange the three coloured regions in order of area Generalise!

9. 9 Doug French Fractional Parts

10. 10 Making Mathematical Sense

11. 11 Triangle Count

12. 12 Reading a Diagram: Seeing As … x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

13. 13 Length-Angle Shifts  What 2D shapes have the property that there is a straight line that cuts them into two pieces each mathematically similar to the original?

14. 14 Tangential  At what point of y=e x does the tangent go through the origin?  What about y = e 2x ?  What about y = e 3x ?  What about y = e λx ?  What about y = μf(λx)?

15. 15 Progress in What?  Use of their own powers –To imagine & to express –To specialise & to generalise –To conjecture & to convince –To stress & to ignore –To persist and to let go  Enrichment of their accessible example spaces  Awareness of the pervasiveness of mathematical themes: –Doing & Undoing (inverses) –Invariance in the midst of change –Freedom & Constraint and of the opportunities to think mathematically outside of classrooms and of the opportunities to think mathematically outside of classrooms

16. 16 Natural Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Organising & Characterising  Stressing & Ignoring  Distinguishing & Connecting  Assenting & Asserting

17. 17 Conjectures  Progression can be seen in terms of  Dimensions-of-Possible-Variation & Range-of-Permissible-Change  Use of powers on own initiative –E.g. Specialising in order to re-Generalise  Construction tasks to reveal richness of accessible example spaces  Self-Constructed Tasks  Using Natural Powers to –Make sense of mathematics –Make mathematical sense

18. 18 Mathematical Themes  Invariance in the midst of change  Doing & Undoing  Freedom & Constraint  Extending & Restricting Meaning

19. 19 Reprise  What is progress and how is it revealed? –Use of powers –Initiative taken (assent-assert) –Disposition to enquire, to think mathematically outside of the classroom –Manifesting results of shifts in perspective  Discrete & Continuous  It just is – I was told it – It must be because

20. 20 My Website & Further Reading  Mcs.open.ac.uk/jhm3 go to Presentations  New Edition of Thinking Mathematically due in April 60 new problems related to the curriculum