Transcript Patterns & Powers Portugal 06

Patterns & Powers
making use of learners’
powers
to seek, generate and
represent patterns in
mathematics
Anne Watson & John Mason
EME
Viana do Castelo
2006
1
Transcendence
To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.
William Blake
Auguries of Innocence
2
Clapping Counts
How
many claps were there?
How did you do that?
3
Remember this:
101101100101001101
4
Conjecture
Every
child who gets to school has
already displayed the power to detect,
express, and make use of patterns
Questions
Am
I making best use of children’s
powers?
Do my students sometimes get confused
by using them inappropriately?
5
Tabled
1x7=
2x7=
3x7=
4x7=
5x7=
6x7=
7x7=
8x7=
9x7=
23 x 7 =
nx7=
6
7x1=
7x2=
7x3=
7x4=
7x5=
7x6=
7x8=
7x9=
7÷1=
14 ÷ 2 =
21 ÷ 3 =
28 ÷ 4 =
35 ÷ 5 =
42 ÷ 6 =
49 ÷ 7 =
56 ÷ 8 =
63 ÷ 9 =
7÷7=
14 ÷ 7 =
21 ÷ 7 =
28 ÷ 7 =
35 ÷ 7 =
42 ÷ 7 =
56 ÷ 7 =
63 ÷ 7 =
Partial Tables
+ 3 4 7 1
8
4
9
8
3 6
2
14
1
x 3 2 7 1
7 1
3
8
8
4
4
9
9
9
2 6
6
2
2
3
1
1
Create your own How few entries are required
in order to be able to
pair of addition
and multiplication determine all the entries?
7
‘tables’ with the
matching entries.
Remainders of the Day (1)
Write
down a number which, when
you subtract 1 from it, is divisible by
5
and another
and another
Write down one which you think
no-one else here will write down.
8
Tunja Sequences
-1 x -1 – 1 = -2 x 0
0 x 0 – 1 = -1 x 1
1x1–1=
2x2–1=
3x3–1=
4x4–1=
9
With
the
Grain
0x2
1x3
2 x 4 Across the Grain
3x5
Diamond
Multiplication
Is it
correct?
10
796
7964455
64789
64789
30
2420
361635
54242840
4236423245
28634836
497254
5681
63
5160119905
Mistaken Patterns
To find 10% you divide by 10
To find 20% you divide by …
Adding makes things bigger,
Subtracting makes things …
Multiplying makes things …
Dividing makes things …
11
Rounding Counts
Out
loud, we will count together from
zero in steps of 0,5
0,7
HOWEVER we will all say the number
rounded up to the nearest whole
number!
12
Patterns & Powers
Pattern
13
seeing, seeking, generating,
imposing, rejecting, representing
Natural propensity to detect and use
patterns
Value of directing learners’ attention
to structural mathematical patterns
Developing disposition to see
patterns outside classroom
mathematically
Importance of pattern seeking and
generating rather than only pattern
‘seeing’
Using Systematic Variation
14
17 - 9 =
117 - 99 =
27 - 9 =
127 - 99 =
37 - 9 =
237 - 99 =
…
…
107 - 9 =
1007 - 99 =
257 - 9 =
3257 - 99 =
Differences
15
1  1 1
1 1 1
7 6 42
2 1 2
1 11
1 1 1  1 1  11
3 2 6
8 7 56 6 24 4 8
Anticipating
1  1 1  1  1
Generalising
4 3 12 2 4
Rehearsing
1 1 1
5 4 20
Checking
1  1  1  1 1 1 1  1  1
Organising
6 5 30 2 3 3 6 4 12
Questions for Further Study
How
do we decide on the amount of
variation displayed before getting
learners to articulate their awareness
of pattern?
How do we decide how much
systematic variation to offer learners
in order to prompt pattern
seeking-generating-representing
16
Transcendence
To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.
William Blake
Auguries of Innocence
17
Some Resources
Thinkers: a collection of activities to provoke
mathematical thinking (ATM, UK)
Primary Questions & Prompts (ATM, UK)
Supporting Mathematical Thinking (Fulton, UK)
Mathematics as a Constructive Activity:
learners constructing their own examples
(Erlbaum, US)
Structured Variation Grids:
18mcs.open.ac.uk/jhm3