Putting research into practice Anne Watson Oxfordshire Feb 2008

Some key findings from research What mathematics? International comparisons: who does well in what tests? It’s the maths! What is difficult about maths? Collaboration

Your first task 6 cm 3 cm

What mathematics? Five strands of mathematical competence: –conceptual understanding –procedural fluency and mathematical repertoire –strategic competence –adaptive reasoning –productive disposition (Kilpatrick et al. ‘Adding it up’)

What mathematics? Mathematical habits of mind (Cuoco, Goldenberg, Mark) What good mathematics students do (Krutetskii)

Your second task What is the height of each of the 11 steps? 495 cm 250 cm

International comparisons: what tests? TIMSS 2003 –Questions based on reproducing knowledge and techniques PISA/OECD 2006 –Questions revealing mathematical literacy: thinking and reasoning argumentation communication modelling problem posing and solving representation using symbolic, formal and technical language use of aids and tools

International comparisons: who does well in what tests? Korea, Hong Kong, Belgium, Sweden, Australia, Chinese Taipei, Netherlands, Estonia do better than UK in both Do better than us in TIMSS and worse in PISA: US, Lithuania, Slovak Republic, Russia, Latvia Do better than us in PISA but worse in TIMSS: New Zealand Do better than us in PISA and did not take part in TIMSS: Finland, Azerbaijan, Canada, Macao-China, Liechtenstein, Switzerland, Denmark, Ireland, Iceland, Czech Republic

It’s the maths What they did not have in common: –Lesson format –Task types –Calculator and ICT use –Amount of whole class interaction –Ways of arranging students –Number of examples –Techniques or problem-solving

It’s the maths Comparing 6 high-achieving nations - what they had in common: –Teaching harder maths –Mathematical coherence in lessons –Mathematical rationale explicit –Helping students do complex tasks rather than simplifying the tasks

Your third task Write down as many things as you can about the number 143 Write down as many things as you can about the number 1.43

So what is so difficult about maths? Inherent difficulties –Notations –Changes of meaning –Limitations of images

So what is so difficult about maths? Concept images are individually constructed

So what is so difficult about maths? Some images are better than others

Tasks Importance of tasks Affordances of examples

Importance of tasks Decay’ of tasks Freudenthal: It’s not the way that you do it; it’s whether you get stuck in the way that you do it Task – activity – what learners actually get to do

Your ….. task You have a collection of brightly coloured cardboard discs, a space in which to move 30 students easily, some rope, some white carpet tape, some compasses, some rulers, some pencils … outline a first lesson about loci

Loci What do you say? What do you indicate? What arm-waving do you do? What do you mark? What concept image of loci do your students get? Did you start by sharing your ideas of what locus is?

Collaboration Currently – collaboration is good: networking encouraged Need to discuss teaching in detail Differences in images and meaning do make a difference to learning