1. 1 Progress in Mathematical Thinking Portugal MSc June 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 Outline  What is progress in mathematical thinking?  Progress in … –Performance (behaviour) –Understanding; connection; being able to talk about (cognition) –Independence and initiative (affect + will) –Ways of working (milieu)  Language –for thinking and discussing Tasks that reveal progress and provide new language

3. 3 Structure of human psyche: chariot metaphor Behaviour (enaction) Emotion (affect) Awareness (cognition) Will Mental Imagery Habits

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 Using coordinate notation, write down two points whose distance apart is two units And another And two numbers whose ‘distance apart’ is between 1.5 and 2 PrimarySecondary Progression is visible in the range of choices exhibited; in the richness of the example space being sampled And another And two points whose distance apart is between 1.5 and 2 And another

6. 6 Progress through shifts  Every technical term indicates a shift in ‘ways of seeing’  The name is a reminder of that shift  To use the term effectively, learners need to experience the shift

7. 7 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 ✎ What else can you see? ✎ What assumptions are you making? 4/3 of something

8. 8 Seeing the general through an example Can you see something that is: One fifth of something One fourth of something One fourth of something take away one fifth of the same thing Now Generalise !

9. 9  What was your progress in that task?

10. 10 Fractions

11. 11 What was your progress in that task?

12. 12 Triangle Count

13. 13 What was your progress in that task?

14. 14 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)

15. 15 Tangential  At what point of y = x 2 + 1 does the tangent go through the origin?  What about y = 4x 2 + 1?  What about y = 9x 2 + 1?  When y = (λx) 2 + 1, what is the locus of that point (as λvaries) ?  What about y = f(λx)?

16. 16  Progress in mathematics means: –Getting better at … –Knowing more about... –Being able to … –Taking initiative to … –Contributing to an atmosphere (milieu) conducive to mathematical thinking …

17. 17 Progress in What?  Use of ability –To imagine & to express –To specialise & to generalise –To conjecture & to convince –To stress & to ignore –To persist and to let go  Use of mathematical themes: –Doing & Undoing (inverses) –Invariance and Variation –Freedom & Constraint –Extending & Restricting Meaning

18. 18 My Website & Further Reading  j.h.mason @ open.ac.uk  mcs.open.ac.uk/jhm3 go to Presentations  Designing Mathematical Tasks (Tarquin)  Questions & Prompts (ATM)  Thinkers (ATM)  Fundamental Constructs in Maths Edn (Sage)  Researching Your Own Practice (Routledge)