Developing Mathematical Thinking

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Developing Mathematical Thinking
John Mason Flötur, Selfoss Sept 2008
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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 dont engage in my tasks, you will get
  nothing!

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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?
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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
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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.

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Bag Constructions (3)

• Here there are 3 bags and two objects.
• There are 0,1,22 objects in the bags and 2
  altogether
• Given a sequence like 2,4,5,56 or 1,1,3,36
  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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(No Transcript)
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Square Count
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Triangle Count
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Attention

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

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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?

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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
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Magic Square Reasoning
What other configurationslike thisgive one
sumequal to another?
Try to describethem in words
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More Magic Square Reasoning
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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?

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Some Mathematical Powers

• Imagining Expressing
• Specialising Generalising
• Conjecturing Convincing
• Stressing Ignoring
• Ordering Characterising
• Seeing Sameness Seeing Difference
• Assenting Asserting

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Some Mathematical Themes

• Doing and Undoing
• Invariance in the midst of Change
• Freedom Constraint

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Structure of the Psyche
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Structure of a Topic
Emotion
Behaviour
Awareness
Only Emotion is Harnessable
Only Behaviour is Trainable
Only Awareness is Educable
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For More Details
Thinkers (ATM, Derby) Questions Prompts for
Mathematical Thinking Secondary 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_at_open.ac.uk