Thinking Mathematically and Learning Mathematics Mathematically

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Thinking Mathematicallyand Learning
MathematicsMathematically
John Mason Greenwich Oct 2008
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Conjecturing Atmosphere

• Everything said is said in order to consider
  modifications that may be needed
• Those who know support those who are unsure by
  holding back or by asking revealing questions

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Up Down Sums
1 3 5 3 1
3 x 4 1
22 32


1 3 (2n1) 3 1

n (2n2) 1
(n1)2 n2

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One More

• What numbers are one more than the product of
  four consecutive integers?

Let a and b be any two numbers, one of them even.
Then ab/2 more than the product of any number, a
more than it, b more than it and ab more than
it, is a perfect square, of the number squared
plus ab times the number plus ab/2 squared.
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Remainders of the Day

• Write down a number that leaves a reminder of 1
  when divided by 3
• and another
• and another
• Choose two simple numbers of this type and
  multiply them together what remainder does it
  leave when divided by 3?
• Why?
• What is special about the 3?

What is special about the 1?
What is special about the 1?
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Primality

• What is the second positive non-prime after 1 in
  the system of numbers of the form 13n?
• 100 10 x 10 4 x 25
• What does this say about primes in the
  multiplicative system of numbers of the form 1
  3n?
• What is special about the 3?

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Inter-Rootal Distances

• Sketch a quadratic for which the inter-rootal
  distance is 2.
• and another
• and another
• How much freedom do you have?
• What are the dimensions of possible variation and
  the ranges of permissible change?
• If it is claimed that 1, 2, 3, 3, 4, 6 are the
  inter-rootal distances of a quartic, how would
  you check?

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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)

• Here there are 3 bags and two objects.
• There are 0,1,22 objects in the bags with 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?

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Statisticality

• write down five numbers whose mean is 5
• and whose mode is 6
• and whose median is 4

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ZigZags

• Sketch the graph of y x 1
• Sketch the graph of y x - 1 - 2
• Sketch the graph of y x 1 2 3
• What sorts of zigzags can you make, and not make?
• Characterise all the zigzags you can make using
  sequences of absolute values like this.

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Towards the Blanc Mange function
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Reading Graphs
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Examples

• Of what is x an example?
• Of what is y x2 and example?
• y b (x a)2 ?

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Functional Imagining

• Imagine a parabola
• Now imagine another one the other way up.
• Now put them in two planes at right angles to
  each other.
• Make the maximum of the downward parabola be on
  the upward parabola

• Now sweep your downward parabola along the
  upward parabola so that you get a surface

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MGA
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Powers

• Specialising Generalising
• Conjecturing Convincing
• Imagining Expressing
• Ordering Classifying
• Distinguishing Connecting
• Assenting Asserting

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Themes

• Doing Undoing
• Invariance Amidst Change
• Freedom Constraint
• Extending Restricting Meaning

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Teaching Trap Learning Trap

• Expecting the teacher to do for you what you can
  already do for yourself
• Learner Lust
• desire that the teacher teach
• desire that learning will be easy
• expectation that dong the tasks will produce
  learning
• allowing personal reluctance/uncertainty to
  drive behaviour

• Doing for the learners what they can already do
  for themselves
• Teacher Lust
• desire that the learner learn
• desire that the learner appreciate and understand
  
• Expectation that learner will go beyond the tasks
  as set
• allowing personal excitement to drive behaviour

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Human Psyche

• Training Behaviour
• Educating Awareness
• Harnessing Emotion
• Who does these?
• Teacher?
• Teacher with learners?
• Learners!

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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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Didactic Tension
The more clearly I indicate the behaviour sought
from learners, the less likely they are
togenerate that behaviour for themselves (Guy
Brousseau)
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Didactic Transposition
Expert awareness is transposed/transformed
into instruction in behaviour(Yves Chevellard)
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More Ideas
For Students
(1998) Learning Doing Mathematics (Second
revised edition), QED Books, York. (1982).
Thinking Mathematically, Addison Wesley, London
For Lecturers
(2002) Mathematics Teaching Practice a guide for
university and college lecturers, Horwood
Publishing, Chichester. (2008). Counter Examples
in Calculus. College Press, London.
http//mcs.open.ac.uk/jhm3 j.h.mason_at_open.ac.uk
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Modes of interaction
Expounding
Explaining
Exploring
Examining
Exercising
Expressing
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Expounding
Examining
Expressing
Content
Exploring
Exercising
Explaining