Title: Generalisation: Fostering
1GeneralisationFostering Supporting Algebraic
Thinking
- John Mason
- Trondheim
- Oct 2007
2Assumptions
- 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!
3Whats 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
4Think Of A Number (Thoan)
- intrigues adolescents
- Displays power over numbers
- Introduces a device for dealing with
as-yet-unknown numbers
5Four 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?
6Powers
- Imagining Expressing
- Specialising Generalising
- Conjecturing Convincing
- Classifying Characterising
- Fixing Changing
- Stressing Ignoring
- Attending Intending
7Pattern Continuation
8Experiencing Generalisation
- Going with the grain enactive generalisation
- Going across the grain cognitive generalisation
- Pleasure in use of powers disposition
affective generalisation(Helen Drury)
9Raise 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
10CopperPlate Multiplication
11Paper Folding
Shape?
Shape?
12What 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
13One 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.
14Perforations
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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16Consecutive Sums
Say What You See
Say What You See
17Worlds of Experience
enactive
iconic
symbolic
18Remainders 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.
19Remainders 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?
20Remainders 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
-
21Remainders 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
-
22Four Odd Sums
23Slope Reading
24Cutting 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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