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

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 objects 3 colours  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?

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?

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8 Square Count

9 Triangle Count

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

11 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?

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

13 Magic Square Reasoning –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2

14 More Magic Square Reasoning –= 0Sum( )Sum( )

15 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?

16 Some Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Ordering & Characterising  Seeing Sameness & Seeing Difference  Assenting & Asserting

17 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint

18 Structure of the Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

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

20 For More Details Thinkers (ATM, Derby) Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby) Mathematics as a Constructive Activity (Erlbaum) Structured Variation Grids Studies in Algebraic Thinking Other Publications This and other presentations