Generalisation: Fostering

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GeneralisationFostering Supporting Algebraic
Thinking

• John Mason
• Trondheim
• Oct 2007

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Assumptions

• Generalisation lies at the very core of
  mathematics and mathematical thinking
• A lesson without the opportunity for learners to
  generalise is not a mathematics lesson!

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Whats The Difference?
What then would be the difference?
What then would be the difference?
First, add one to each
First, add one to the larger and subtract one
from the smaller
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Think Of A Number (Thoan)

• intrigues adolescents
• Displays power over numbers
• Introduces a device for dealing with
  as-yet-unknown numbers

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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?

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Powers

• Imagining Expressing
• Specialising Generalising
• Conjecturing Convincing
• Classifying Characterising
• Fixing Changing
• Stressing Ignoring
• Attending Intending

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Pattern Continuation

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Experiencing Generalisation

• Going with the grain enactive generalisation

• Going across the grain cognitive generalisation

• Pleasure in use of powers disposition
  affective generalisation(Helen Drury)

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Raise Your Hand When You See
Something which is 2/5 of something 3/4 of
something 5/2 of something 4/3 of
something 3/4 of 2/5 of something 3/4 of 4/3
of something 1 2/5 of something 1 3/4 of
something
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CopperPlate Multiplication
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Paper Folding
Shape?
Shape?
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What Would Happen If ?

• The tap wasnt turned off
• It never rained
• The power went off
• A nearby stream flooded
• You kept on cutting a piece of paper in half

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One More

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

• 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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Perforations
If someone claimedthere were 228 perforationsin
a sheet, how could you check?
How many holes for a sheet of r rows and c
columns of stamps?
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Consecutive Sums
Say What You See
Say What You See
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Worlds of Experience
enactive
iconic
symbolic
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Remainders of the Day (1)

• Write down a number which when you subtract 1 is
  divisible by 5
• and another
• and another
• Write down one which you think no-one else here
  will write down.

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Remainders of the Day (2)

• Write down a number which when you subtract 1 is
  divisible by 2
• and when you subtract 1 from the quotient, the
  result is divisible by 3
• and when you subtract 1 from that quotient the
  result is divisible by 4
• Why must any such number be divisible by 3?

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Remainders of the Day (3)

• Write down a number which is 1 more than a
  multiple of 2
• and which is 2 more than a multiple of 3
• and which is 3 more than a multiple of 4

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Remainders of the Day (4)

• Write down a number which is 1 more than a
  multiple of 2
• and 1 more than a multiple of 3
• and 1 more than a multiple of 4

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Four Odd Sums
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Slope Reading
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Cutting Chocolate Bars

• In how few cuts can you separate the bar into its
  pieces?
• You can only cut one piece at a time!

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