1. Anne Watson & John Mason
Patterns & Powers
making use of learners’ powers to seek, generate and represent patterns in mathematics
Anne Watson & John Mason
We start with and from mathematics and mind
Convinced that everyone has particular powers and cultural ways of seeing which must be called upon and drawn out to produce learning
EME Viana do Castelo 2006

2. 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

3. Clapping Counts How many claps were there? How did you do that?
Dum Dum Da Da Da Da Dum Da Da Da Da Dum Da Dum Dum

4. Remember this:
Here sight combined with sound might actually make it easier to internalise and hence remember
Anne’s account (14 years)
Bird Song

5. 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?
10% of 23 is 2.3
20% of 23 is 0.23

6. Tabled 1 x 7 = 2 x 7 = 3 x 7 = 4 x 7 = 5 x 7 = 6 x 7 = 7 x 7 = 8 x 7 =
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 =
A class was asked to complete this table provided for them on a printed sheet.
How do you think most children went about it?
Make 23 and n come up separately
WITH AND ACROSS THE GRAIN
With systematic sequential based on cultural practice
Across is Template
23 x 7 =
n x 7 =

7. Partial Tables 4 8 3 14 + 3 7 1 6 21 9 8 4 x 3 7 1 6 21 9 8 4 2 9 3 3 7 1 6 21 9 8 4
How few entries are required in order to be able to determine all the entries?
Create your own pair of addition and multiplication ‘tables’ with the matching entries.
With the Grain: churning out the entries using some simple reasoning
Across the Grain: focusing on the relationships, especially when constructing your own
Power of constraints when constructing examples.

8. Remainders of the Day (1)
Write down a number which, when you subtract 1 from it, is divisible by 5
and another
Write down one which you think no-one else here will write down.
Pattern generating, seeking pattern, even reject
5n + 1
Creative powers

9. Tunja Sequences
-1 x -1 – 1 =
-2 x 0
0 x 0 – 1 =
-1 x 1
With the Grain
1 x 1 – 1 =
0 x 2
2 x 2 – 1 =
1 x 3
3 x 3 – 1 =
2 x 4
Across the Grain
4 x 4 – 1 =
3 x 5

10. Diamond Multiplication
Is it correct?
How did you make sense if this ?
This is what children are up against in the classroom everyday.
You know what you know and look for it?

11. Mistaken Patterns To find 10% you divide by 10
Adding makes things bigger,
Subtracting makes things …
Multiplying makes things …
Dividing makes things …

12. Rounding Counts
Out loud, we will count together from zero in steps of 0,5
HOWEVER we will all say the number rounded up to the nearest whole number!
0,7
OWN LANGUAGE! 0.5 is rounded up every time.
Layout to draw attention
0.7 count

13. Patterns & Powers
Pattern 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’
Pattern seeing, seeking, generating, imposing, rejecting, representing

14. Using Systematic Variation= = = … = = = = = … = =
Example of how these ideas apply to ordinary lessons in school
‘systematic’ here means: we offer mathematical situations which are symbolically, visually, linguistically systematic – not conceptually systematic in the sense which Klein rejected.

15. Anticipating Generalising
Differences
Anticipating Generalising
If n = pqr then 1/n = 1/pr(q-r) - 1/pq(q-r) so there are as many ways as n can be factored as pqr with q>r
Rehearsing
Checking
Organising

16. 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
Refer back to slides
7: why three?
8: how many did you need?

17. 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

18. 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: mcs.open.ac.uk/jhm3