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

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Presentation transcript:

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

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

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

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 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 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 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  What was your progress in that task?

10 Fractions

11 What was your progress in that task?

12 Triangle Count

13 What was your progress in that task?

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 Tangential  At what point of y = x 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  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 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 My Website & Further Reading  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)