1. Developing Mathematical Thinking John Mason Flötur, Selfoss Sept 2008

2. Some Throat Clearing • What you get from this session will be what you notice happening inside you • Everything said is to be treated as a conjecture, and tested in your experience • If you don’t engage in my tasks, you will get nothing!

3. Getting Going • If the difference of two numbers is even, then their product is the difference of two squares Specialisingin order to(re)generalise How often do you arrange for your students to use this power for themselves?

4. Bag Constructions (1) • Here there are three bags. If you compare any two of them, there is exactly one colour for which the difference in the numbers of that colour in the two bags is exactly 1. • For four bags, what is the least number of objects to meet the same constraint? • For four bags, what is the least number of colours to meet the same constraint? 17 objects 3 colours

5. Bag Constructions (2) • For b bags, how few objects can you use so that each pair of bags has the property that there are exactly two colours for which the difference in the numbers of that colour in the two bags is exactly 1. • Construct four bags such that for each pair, there is just one colour for which the total number of that colour in the two bags is 3.

6. Bag Constructions (3) • Here there are 3 bags and two objects. • There are [0,1,2;2] objects in the bags and 2 altogether • Given a sequence like [2,4,5,5;6] or [1,1,3,3;6] how can you tell if there is a corresponding set of bags? • In how many different ways can you put k objects in b bags?

8. Triangle Count

9. Attention • Holding Wholes (gazing) • Discerning Details • Recognising Relationships • Perceiving Properties • Reasoning on the basis of agreed properties

10. Doing & Undoing • What operation undoes ‘adding 3’? • What operation undoes ‘subtracting 4’? • What operation undoes ‘subtracting from 7’? • What are the analogues for multiplication? • What undoes multiplying by 3? • What undoes dividing by 2? • What undoes multiplying by 3/2? • What undoes dividing by 3/2?

11. With the Grain Across the Grain Tunja Sequences -1 x -1 – 1 = -2 x 0 0 x 0 – 1 = -1 x 1 1 x 1 – 1 = 0 x 2 2 x 2 – 1 = 1 x 3 3 x 3 – 1 = 2 x 4 3 x 5 4 x 4 – 1 =

12. 2 6 7 2 1 5 9 8 3 4 Sum( ) – Sum( ) = 0 Magic Square Reasoning What other configurationslike thisgive one sumequal to another? Try to describethem in words

13. Sum( ) – Sum( ) = 0 More Magic Square Reasoning

14. Map Drawing Problem • Two people both have a copy of the same map of Iceland. • One uses Reykjavik as the centre for a scaling by a factor of 1/3 • One uses Akureyri as the centre for a scaling by a factor of 1/3 • What is the same, and what is different about the maps they draw?

15. Some Mathematical Powers • Imagining & Expressing • Specialising & Generalising • Conjecturing & Convincing • Stressing & Ignoring • Ordering & Characterising • Seeing Sameness & Seeing Difference • Assenting & Asserting

16. Some Mathematical Themes • Doing and Undoing • Invariance in the midst of Change • Freedom & Constraint

17. Imagery Awareness (cognition) Will Emotions (affect) Body (enaction) HabitsPractices Structure of the Psyche

18. Language Patterns& prior Skills Imagery/Sense-of/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise;dispositions Techniques & Incantations Standard Confusions & Obstacles Structure of a Topic Emotion Behaviour Awareness Only Emotion is Harnessable Only Awareness is Educable Only Behaviour is Trainable

19. For More Details Thinkers (ATM, Derby) Questions & Prompts for Mathematical ThinkingSecondary & Primary versions (ATM, Derby) Mathematics as a Constructive Activity (Erlbaum) Structured Variation GridsStudies in Algebraic ThinkingOther PublicationsThis and other presentations http://mcs.open.ac.uk/jhm3j.h.mason@open.ac.uk