Download Trondheim Generalisation

Generalisation:
Fostering & Supporting
Algebraic Thinking
John Mason
Trondheim
Oct 2007
1
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!
2
What’s The Difference?
–
=
First, add one to each
First,
add one to the larger and
subtract one from the smaller
3
What then
would be
the difference?
What could
be varied?
Think Of A Number (Thoan)
intrigues adolescents
 Displays power over numbers
 Introduces a device for dealing with as-yetunknown numbers

4
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?
+1
+2
+3
4
5
+6
Powers
Imagining
& Expressing
Specialising & Generalising
Conjecturing & Convincing
Classifying & Characterising
Fixing & Changing
Stressing & Ignoring
Attending & Intending
6
Pattern Continuation
…
…
7
Experiencing Generalisation
with the grain:
enactive
generalisation
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Going
Going
across the
grain:
cognitive
generalisation
Pleasure
in use of powers;
disposition: affective generalisation
(Helen Drury)
8
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
9
CopperPlate
Multiplication
10
796
7964455
64789
64789
30
2420
361635
54242840
4236423245
28634836
497254
5681
63
5160119905
Paper Folding
Shape?
Shape?
11
What Would Happen If …?
The
tap wasn’t turned off
It never rained
The power went off
A nearby stream flooded
You kept on cutting a piece of paper in
half
…
12
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 a+b
more than it, is a perfect square, of the number
squared plus a+b times the number plus ab/2
squared.
13
Perforations
How many holes
for a sheet of
r rows and c columns
of stamps?
14
If someone claimed
there were 228 perforations
in a sheet,
how could you check?
15
Consecutive Sums
Say What You See
16
Say What You See
Worlds of Experience
Inner
World of
imagery
Material
World
enactive
17
iconic
World of
Symbol
s
symbolic
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.
18
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?

19
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
…

20
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
…

21
Four Odd Sums
22
Slope Reading
c
b
23
a
d
a
c
a
a+c
c
if Š then Š
Š
b b+d d
b d
Cutting Chocolate Bars
In
how few cuts can
you separate the bar
into its pieces?
You can only cut
one piece at a time!
24
43
44
45
46
47
48
49
42
21
22
23
24
25
26
41
20
7
8
99
10
27
40
19
6
1
2
11
28
39
18
5
4
3
12
29
38
17
16
15
14
13
30
37
36
35
34
33
32
31
50
64
36
37
38
39
40
41
42
43
44
35
14
15
16
17
18
19
20
45
34
13
2
3
4
21
46
33
12
11
10
1
5
22
47
32
31
30
9
8
7
6
23
48
29
28
27
26
25
24
49
50
81