Progress in Mathematical Thinking John Mason SMC Stirling

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

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 2

What is Progress? /Perceived change in – Behaviour (what people do) – Affect (what people feel about what they are doing; motivation; dispositions; initiative; confidence; selfefficacy 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 3

In Between /How many circles could there be between the two shown? Range of permissibl e change /How many numbers could there be between 1. 50 and 1. 59 1. 500 and 1. 5987 4 Discrete & Continuou s

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

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

7 Seeing through the particular to a generality

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

Doug French Fractional Parts 9

Making Mathematical Sense 10

Triangle Count 11

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

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

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

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 15

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

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 17

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

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 v. Discrete & Continuous v. It just is – I was told it – It must be because 19

My Website & Further Reading /Mcs. open. ac. uk/jhm 3 go to Presentations /New Edition of Thinking Mathematically due in April 60 new problems related to the curriculum 20