1 Working with Colleagues on Mathematics and on Mathematics Education John Mason SWMA Sept 2007

2 Outline  Working on mathematics together  Ways of working with colleagues Reminder: My way of working is experiential: What you get from today will depend on what you notice happening inside you, and how you relate that to what you do

3 Themes & Concerns  Investigative & Thematic Mathematics  Seeking consistency between –ways of working with learners on mathematics, and and –ways of working on teaching and learning of mathematics with colleagues.

4 What?  A spokesman for Thames water said that since 40% of their water was lost in broken pipes, they would need to build their new reservoir 40% bigger than previously planned.  The BBC reported that in parts of Gloucestershire there are 100 slugs per square foot.  Clerk working for an auctioneer: with inflation running at 3%, shouldn’t we raise our commission in line with it? Prospect; New Scientist; Maths Gazette; …

5 What’s The Difference? What could be varied? –= First, add one to each First, add one to the larger and subtract one from the smaller What then would be the difference? Investigative & Thematic Mathematics

6 Grid Sums In how many different ways can you work out a value for the square with a ‘?’ only using addition? 7 ? ? To move to the right one cell you add 3. To move up one cell you add 2. Using exactly two subtractions?

7 Grid Movement 7 ? +3-3 x2 ÷2 ((7+3)x2)+3 is a path from 7 to ‘?’. What expression represents the reverse of this path? What values can ‘?’ have: - if only + and x are used - if exactly one - and one ÷ are used, with as many + & x as necessary What about other cells? Does any cell have 0? -7? Does any other cell have 7? Characterise ALL the possible values that can appear in a cell Investigative & Thematic Mathematics

8 Magic Square Reasoning –= 0Sum( ) Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? Investigative & Thematic Mathematics

9 More Magic Square Reasoning –= 0Sum( )Sum( ) Investigative & Thematic Mathematics

10 Raise Your Hand When You See … Something which is 2/5 of something; 3/5 of something; 5/2 of something; 5/3 of something; 2/5 of 5/3 of something; 3/5 of 5/3 of something; 5/2 of 2/5 of something; 5/3 of 3/5 of something; 1 ÷ 2/5 of something; 1 ÷ 3/5 of something

Number Spirals Investigative & Thematic Mathematics

Extended Number Spirals

13 Varying & Generalising  What are the dimensions of possible variation?  What is the range of permissible change within each dimension of variation?  You only understand more if you extend the example space or the scope of generality

14 Investigative & Thematic Mathematics  What blocks colleagues from teaching mathematics investigatively?  What constitutes teaching investigatively?  Very often the mathematics arising from ‘themes’ is trivial and does not advance learners’ mathematical thinking  How can contact with mathematical structure and concepts arise from thematic work?

15 Teaching Maths Investigatively  Phenomenal Mathematics  Mathematical Themes –Invariance in the midst of change –Doing & Undoing Freedom & Constraint  Prompting learners to use their mathematical powers –Imagining & Expressing –Specialising & Generalising –Conjecturing & Convincing –Stressing & Ignoring –Ordering & Classifying

16 Teaching Maths Effectively  Conjecturing atmosphere  Raising mathematical questions  Making sense of phenomena with mathematics  Making sense of mathematics –Senses accessed through Doing – Talking - Recording Manipulating – Getting-a-sense-of – Articulating Tasks – Activity – Interaction – Reflection Manipulating – Getting-a-sense-of – Articulating Tasks – Activity – Interaction – Reflection

17 Phenomenal Mathematics  Material world phenomena  Virtual world phenomena  Dead birds & Easter eggs (Janet Ainley)

18 Getting the Most out of Themes  Being aware of and exploiting pervasive mathematical themes –Doing & Undoing –Invariance in the midst of Change –Freedom & Constraint –Extending & Restricting

19 Desire  What do you most want to ‘tell’ colleagues, or to have colleagues ‘appreciate and understand’ about teaching?  How do you go about raising this as an issue with them?  What is the difference between seeing teaching as –implementing some approach –focusing/stressing some aspects Ways of working with colleagues

20 Didactic Tension The more clearly I indicate the behaviour sought from learners, the less likely they are to generate that behaviour for themselves

21 Worlds of Experience Material World World of Symbol s Inner World of imagery enactiveiconicsymbolic Manipulating Getting-a-sense-ofArticulating Classroom actions

22 Sources of Support  Mathemapedia (NCETM)  Colleagues (NCETM)  Reading and Writing (journals) Questions & Prompts for Mathematical Thinking (Primary & secondary versions) (ATM) Thinkers (ATM) Malcolm Swan; Susan Wall; Afzal Ahmed Mcs.open.ac.uk/jhm3