Collaborative Approaches in Mathematics Education

1. Working with Colleagueson Mathematicsand 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 • 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 could be varied? What’s The Difference? What then would be the difference? What then would be the difference? First, add one to each First, add one to the larger and subtract one from the smaller Investigative & Thematic Mathematics

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

7. Grid Movement ((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 x2 ? ÷2 7 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 -3 +3 Investigative & Thematic Mathematics

8. 6 7 2 1 5 9 8 3 4 Sum( – Sum( ) = 0 ) Magic Square Reasoning What other configurationslike thisgive one sumequal to another? Try to describethem in words Investigative & Thematic Mathematics

9. Sum( ) – Sum( ) = 0 More Magic Square Reasoning 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

11. Number Spirals 43 45 46 47 48 49 50 44 49 42 21 22 23 24 25 26 25 41 20 10 27 9 40 19 9 1 1 7 6 5 4 3 2 8 11 28 39 18 12 29 4 38 17 16 15 14 13 30 16 35 34 33 36 37 36 32 31 Investigative & Thematic Mathematics

12. 43 45 46 47 48 49 50 44 42 21 22 23 24 25 26 41 20 10 27 40 19 1 8 9 6 7 4 3 2 5 11 28 39 18 12 29 38 17 16 15 14 13 30 35 34 33 81 64 37 36 32 31 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 & UndoingFreedom & 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 – ArticulatingTasks – 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 togenerate that behaviour for themselves

21. Inner World of imagery enactive iconic symbolic Worldof Symbols Classroom actions Material World Getting-a-sense-of Articulating Manipulating Worlds of Experience

22. Sources of Support Mcs.open.ac.uk/jhm3 • 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