1 Progress in Mathematical Thinking John Mason BMCE Manchester April 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 what aspect? –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 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

4 Difference of 2 write down 2 numbers with a difference of 2 Sketch the two lines with slopes differing by 2 write down an integral over two different intervals whose values differ by 2 And another PrimarySecondaryUpper Secondary Progression is visible in the range of choices exhibited; in the richness of the example space being sampled

5 Shifts Conjecture  Every technical term indicates a shift in perspective, in ways of perceiving;  The name or label serves as a reminder of trigger for that shift;  in order to use the term effectively, learners need to experience a similar shift

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 Seeing through the particular to a generality Hands up when you can 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 !

8 Dimensions-of-Possible- Variation Regional  Which is the smallest and which the largest shaded area? Generalise!

9 Doug French Fractional Parts

10 Making Mathematical Sense

11 Which way did the bicycle go?

12 Triangle Count

13 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)

14 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?

15 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)?

16 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 –Extending & Restricting Meaning and of opportunities to think mathematically outside of classrooms and of opportunities to think mathematically outside of classrooms

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

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

19 Conjecture: 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  Manifesting results of shifts in perspective –Discrete & Continuous –It just is – I was told it – It must be because –Seeing-As; Behaviour Disposition (affect) Cognition Assenting & Asserting Reacting & Responding Shifts Conjecture

20 Progress & The Psyche  Only behaviour is trainable  Only Awareness is educable  Only emotion is harnessable  So progress in mathematical thinking includes coordination of progress in all three aspects;  All classroom actions involve an element of each aspect

21 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

22 My Website & Further Reading  open.ac.uk  mcs.open.ac.uk/jhm3 go to Presentations  New Edition of Thinking Mathematically due end of April 77 new problems related to the curriculum Special conference price of £20 regularly £25 Special conference price of £20 regularly £25  Designing Mathematical Tasks (Tarquin)  Questions & Prompts (ATM)  Thinkers (ATM)  Fundamental Constructs in Maths Edn (Sage)  Researching Your Own Practice (Routledge)