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

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

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

Remember this: 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 Here sight combined with sound might actually make it easier to internalise and hence remember Anne’s account (14 years) Bird Song

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

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 =

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.

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

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

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?

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

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 0 1 1 2 3 4

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

Using Systematic Variation 17 - 9 = 27 - 9 = 37 - 9 = … 107 - 9 = 257 - 9 = 117 - 99 = 127 - 99 = 237 - 99 = … 1007 - 99 = 3257 - 99 = 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.

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

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?

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

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