Reasoning Reasonably in Mathematics

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Reasoning Reasonably in Mathematics
Promoting Mathematical Thinking
John Mason Schools Network Warwick June 2012
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Outline

• Some Tasks to engage in
• On which to reflect
• In a conjecturing atmosphere
• Everything said must be tested in your experience
• Make Contact with
• Natural powers used to reason mathematically
• Mathematical themes
• Pedagogic choices
• Straw Poll please raise your hand if you
• Teach proof or reasoning
• to previously high attaining students
• to previously low attaining students

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Secret Places
Homage to Tom OBrien (1938 2010)

• One of the places around the table is a secret
  place.
• If you click near a place, the colour will tell
  you whether you are hot or cold
• Hot means that the secret place is there or else
  one place either side
• Cold means that it is at least two places away
  either side

What is your best strategy to locate the secret
place?
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Reflexive Stance

• What did you notice yourself doing?
• What actions were effective?
• What actions were not effective?

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Magic Square Reasoning
What other configurationslike thisgive one
sumequal to another?
Try to describethem in words
Any colour-symmetric arrangement?
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More Magic Square Reasoning
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Reflexive Stance

• What did you notice yourself doing?
• What actions were effective?
• What actions were ineffective?

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Convincing Justifying

• Justifying actions
• Social aspect
• convincing yourself
• convincing a friend
• convincing a sceptic
• Learning to be sceptical of others reasoning
• Learning to be sceptical of your own reasoning

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Four Consecutives

• Write down four consecutive numbers and add them
  up
• and another
• and another
• Now be more extreme!
• What is the same, and what is different about
  your answers?
• What numbers can be expressed as the sum of four
  consecutive numbers?

Express 424 as a sum of four consecutive numbers
Doing Undoing
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Carpet Theorem
How are the red and blue areas related?
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Doug Frenchs Fractional Parts
Whats the same and what different? What can be
varied? What is the whole?
Construct your own
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Square Sums
Imagine a triangle Imagine the
circumcentre Drop perpendiculars to the
edges On each half of each edge, put a square
outwards, one yellow, one cyan.
More generally
Does
The sum of the Yellow areas the sum of the cyan
areas
???!!??
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Polygon Perimeter Projections

• Imagine a square
• Imagine a point traversing the perimeter of the
  square
• Imagine projections of that point onto the x and
  y axes

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Withdrawing from the Action

• What actions did you carry out?
• What mde you think of those?
• Which ones were effective (for possible use in
  the future)?
• Did I
• Gaze (holding wholes)?
• Discern Details
• Recognise Relationships in the particular?
• Perceive Properties as instantiated
• Reason on the basis of properties?

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Eyeball Reasoning
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Square Deductions
3b-3a
3(3b-3a) 3ab 12a 8b So 3a 2b
a3b
3ab
b
a
For an overall square 4a 4b 2a 5b So 2a b
a2b
2ab
ab
For n squares upper left n(3b - 3a) 3a b So
3a(n 1) b(3n - 1) But not also 2a b
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Square Deductions
To be consistent, 3x y x x 3y y So 3x
3y
x3y
3xy
y
x
2xy
xy
x2y
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Rectangle Deduction
(x7y)/4
Suppose the rectangles aretwo by one
(x7y)/8
(7xy)/4
y
x
(x3y)/2
(3xy)/2
(xy)/2
xy
(x3y)/4
(3xy)/4
(x7y)/8 y (7x y)/4 x 21x 13y
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Wasons cards

• Each card has a letter on one side and a numeral
  on the other.
• Which 2 cards must be turned over in order to
  verify that
• on the back of a vowel there is always an even
  number?

A
2
B
3
Which cards must be turned over to verify
thaton the back of a red vowel there is a blue
even number
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Diamond Multiplication
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Seven Circles
How many different angles can you discern, using
only the red points?
How do you know you have them all?
How many different quadrilaterals?
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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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Reflective Stance

• What did you notice yourself doing?
• Checking?
• Interpreting?
• Specialising?
• Being systematic?
• What forms of attention did you notice?
• Gazing (holding wholes)?
• Discerning Details?
• Recognising Relationships?
• Perceiving Properties?
• Reasoning on the basis of Properties?

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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 2 objects.
• 0,1,22means that there are 0, 1 and 2 objects
  in three bags and 3 objects 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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Reasoning Types

• Logical Deduction
• Arithmetic reasoning into the unknown
• Algebra reasoning from the unknown
• Empirical (needs justification)
• Exhaustion of cases
• Contradiction
• (Induction)

Issue is often what can I assume? what can I
use? what do I know?
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Attention
Holding Wholes (gazing) Discerning Details
(discriminating) Recognising Relationships (in a
situation) Perceiving properties (as being
instantiated) Reasoning On The Basis of Agreed
Properties
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Powers

• Imagining Expressing
• Specialisng Generalising
• Stressing Ignoring
• Conjecturing Convincing (what is the status
  of what you say?)
• Organising Characterising

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Themes

• Doing and Undoing
• Invariance in the Midst of Change
• Freedom Constraint
• Extending Restricting

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Follow Up

• mcs.open.ac.uk / jhm3 presentations applets
• j.h.mason _at_ open.ac.uk