Mathematically Powerful Task Design Anne Watson John Mason

Mathematically Powerful Task Design Anne Watson & John Mason Matematikbiennalen 2008 Stockholm 1

Teaching Context • All learners generalise all the time • It is the teacher’s role to organise learners’ experience • It is the learners’ role to make (mathematical) sense of their experience 2

What do you know about …? 143 1. 43 Factors Place value Position on number line Approximation … … Take two sub-domains that are often seen as separate; Choose an object in one which with the slightest of alterations becomes an object in the other; Aim: to prompt richer connections between the subdomains 3

Find a number half way between: 28 and 34 2. 8 and 3. 4 38 and 44 -34 and -28 9028 and 9034. 0058 and. 0064 Provide a context in which ‘questions’ vary a little, but suitable methods may vary a lot 4

Reflective questions • What generalisations are available to be made? • What expectations (conjectures) and surprises are possible? • What conceptual understandings might be induced? 5

Find a number half way between: 28 and 34 1 4 and 1 2 3 8 and 3 4 2 5 and 4 7 a b and x y Seems like the previous type of task, but this time the teaching intention is to reject limited methods and provide ‘need’ for a general method 6

Reflective questions • What generalisations are available? • What habits might be developed (practised)? • What extensions of experience might be induced? 7

Construct a … … pair of straight lines whose x-intercepts differ by 2 … pair of straight lines whose y-intercepts differ by 2 … pair of straight lines whose slopes differ by 2 … pair of straight lines satisfying all of the above! Ask learners to construct an object according to constraints, chosen to encourage them to focus on particular aspects of the object 8

Reflective questions • What generalisations are available? • What habits might be developed (practised)? • What concepts might be thought about? 9

Variation theory (Marton) • What variables are controlled? What variables are allowed to vary? How do they vary? • What can be perceived? What do we see as being the same and being different? • If all varies, little can be perceived beyond the immediate • If all is the same, attention is drawn nowhere 10

An Interpretation of Variation Theory 4 pens plus 5 pencils cost £ 2. 60 4 pens plus 2 pencils cost £ 2. 00 5 oranges plus 3 apples cost £ 2. 36 5 oranges plus 1 apple cost £ 2. 12 8 stamps plus 5 envelopes cost £ 3. 90 8 stamps plus 4 envelopes cost £ 3. 60 11

An Interpretation of Variation Theory 4 pens plus 5 pencils cost £ 2. 60 2 pens plus 1 pencil cost £ 1. 00 4 pens plus 5 pencils cost £ 2. 60 4 pens plus 2 pencils cost £ 2. 00 Start from questions which are easy to solve using ad hoc methods, then vary in controlled ways so learners can adapt their methods to develop more general methods, and to understand conventional methods 12

Reflective questions • What generalisations are available? • What habits might be developed (practised)? • What concepts might be thought about? 13

More or Less Draw a scalene triangle Pay attention to its altitude and its area altitude more 14 more same more alt more area more alt same area same Same alt more area less more alt less area same alt less area less alt same area less alt less area

Summary of Task Design Questions • What varies? What is invariant? • What generalisations are available to be made? • What generalisations are prerequisites of the experience? • What expectations (conjectures) and surprises are likely? • What habits might be developed (practised)? • What conceptual understandings might be induced? 15

v Watson & Mason (1998): Questions and prompts for mathematical thinking. (ATM: Derby) v Bills, Watson & Mason (2004): Thinkers. (ATM: Derby) v Watson & Mason (2005): Mathematics as a constructive activity. (Erlbaum: Mahwah, NJ) 16