Putting research into practice

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Putting research into practice

• Anne Watson
• Oxfordshire
• Feb 2008

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Some key issues from research

• What mathematics?
• International comparisons who does well in what
  tests?
• Its the maths!
• What is difficult about maths?
• Collaboration of a particular kind

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Your first task
6 cm
3 cm
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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)

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What mathematics?

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

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Your second task

• What is the height of each of the 11 steps?

250 cm
495 cm
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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

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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

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Its 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

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Its 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

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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

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So what is so difficult about maths?

• Inherent difficulties
• Notations
• Changes of meaning
• Limitations of images

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So what is so difficult about maths?

• Concept images are individually constructed

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So what is so difficult about maths?

• Some images are better than others

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Importance of tasks

• Decay of tasks
• Freudenthal Its not the way that you do it
  its whether you get stuck in the way that you do
  it
• Task activity what learners actually get to
  do

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Your fourth 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

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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?

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Collaboration

• Currently collaboration is good - networking
  encouraged
• Need to discuss teaching in detail
• Differences in images and meaning make a
  difference to learning

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Awkward questions

• When does collaboration gloss over the detail?
• When does teachers mathematical subject
  knowledge get glossed over?

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Summary

• Teaching harder maths
• Mathematical coherence in lessons
• Mathematical rationale explicit
• Helping students do complex tasks rather than
  simplifying the tasks
• Good choice of examples
• Collaboration which focuses on mathematical
  details