1. 1 Noticing: the key to teaching, learning and research John Mason Modena & Napoli 2007

2. 2 It is only after you come to know the surface of things that you venture to see what is underneath; But the surface of things is inexhaustible (Italo Calvino) He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast. Practice always rests on good theory. (Leonardo Da Vinci)

3. 3 Say What You See

4. 4

5. 5 CopperPlate Multiplication

6. 6 Giacomo Candido 1871-1941 What does it say? [x 2 + y 2 + (x + y) 2 ] 2 2[x 4 + y 4 + (x + y) 4 ] = +

7. 7 Reflections  What did you do with your attention? –Hold wholes (gazing)? –Discern details? –Recognise relationships? –Perceive properties? –Reason on the basis of properties?

8. 8 What Can a Teacher Do for Learners?  Direct attention  Display mathematical behaviour  Reinforce behaviour  Get out of the way  Stress (and consequently ignore)  Amplify & Edit (what learners say and do)

9. 9 In Order to Direct Learner Attention  I need to know what I attend to myself –So, that I can direct learners attention  I need to know how I am attending –What can I do to prompt learners to attend to the same things in similar ways? Noticing

10. 10 What does ‘develop/research teaching’ mean?  Assemble different strategies & tactics, different actions to initiate  Watch out for opportunities to use them  Observe what happens Noticing

11. 11 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

12. 12 Number Spiral 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 1 4 9 16 25 49 36

13. 13 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 64 81

14. 14 Tunja Display (1) 2x2 - 2 - 2 = 1x1 - 1 3x2 - 3 - 2 = 2x1 - 1 4x2 - 4 - 2 = 3x1 - 1 5x2 - 5 - 2 = 4x1 - 1 … 3x3 - 3 - 3 = 2x2 - 1 4x3 - 4 - 3 = 3x2 - 1 5x3 - 5 - 3 = 4x2 - 1 … Generalise! Run Backwards 1x2 - 1 - 2 = 0x3 - 1 0x2 - 0 - 2 = (-1)x3 - 1 (-1)x2 - (-1) - 2 = (-2)x3 - 1 2x3 - 2 - 3 = 1x2 - 1 1x3 - 1 - 3 = 0x2 - 1 0x3 - 0 - 3 = (-1)x2 - 1 … … … (-1)x3 - (-1) - 3 = (-2)x2 - 1 … … … … … ……

15. 15 Tunja Display (2) 4x3x2 - 2x3 - 4x2 = 4x3 - 2 4x4x2 - 2x4 - 4x2 = 6x3 - 2 4x5x2 - 2x5 - 4x2 = 8x3 - 2 4x6x2 - 2x6 - 4x2 = 10x3 - 2 … 4x3x3 - 2x3 - 4x3 = 4x5 - 2 4x4x3 - 2x4 - 4x3 = 6x5 - 2 4x5x3 - 2x5 - 4x3 = 8x5 - 2 4x6x3 - 2x6 - 4x3 = 10x5 - 2 … Generalise! Run Backwards 4x2x2 - 2x2 - 4x2 = 2x3 - 2 4x1x2 - 2x1 - 4x2 = 0x3 - 2 4x0x2 - 2x0 - 4x2 = (-2)x3 - 2 4x(-1)x2 - 2x(-1) - 4x2 = (-4)x3 - 2 …

16. 16 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 5  and another  Write down one which you think no-one else here will write down.

17. 17 Remainders of the Day (2)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

18. 18 Remainders of the Day (3)  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4  … … … …

19. 19 Remainders of the Day (4)  Write down a number which is 1 more than a multiple of 2  and 1 more than a multiple of 3  and 1 more than a multiple of 4  … … … …

20. 20 Magic Square Reasoning 519 2 4 6 83 7 –= 0Sum( ) Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

21. 21 More Magic Square Reasoning –= 0Sum( )Sum( )

22. 22 Powers  Specialising & Generalising  Conjecturing & Convincing  Imagining & Expressing  Ordering & Classifying  Distinguishing & Connecting  Assenting & Asserting

23. 23 Themes  Doing & Undoing  Invariance Amidst Change  Freedom & Constraint  Extending & Restricting Meaning

24. 24 Attention  Gazing at wholes  Discerning details  Recognising relationships  Perceiving properties  Reasoning on the basis of properties

25. 25 Observation  A name or label immediately shapes what is seen and what is recalled (Frederick Bartlett 1932)  Observation is theory laden (George Hanson 1958)  We want our theories to be as fact laden as our facts are theory laden ( Nelson Goodman 1978)

26. 26 What are the significant products of research?  Transformations of the ‘being’ of the researchers  Increased sensitivity to notice what was previously not noticed  Refined vocabulary for discussing, discerning and analysing  Awareness which informs future choices of action  Self  Others

27. 27 Accounts-of & Accounting-for Accounts-of: brief-but-vivid accounts Reduce-remove theorising, judgement, excuses, evaluations, justifications I cannot evaluate your analysis if I cannot distinguish it from the data itself

28. 28 Reporting Data “They couldn’t …” “They can’t” “They didn’t display evidence of …” “They don’t display evidence of …” Accounting ForAccount of “I didn’t detect evidence of …” “I don’t detect evidence of …”

29. 29 Precision Conjecture The more precisely the data is specified, the more we learn about the researcher ’ s sensitivities to notice

30. 30 Protases ` I cannot change others; I can work at changing myself ` To express is to over stress ` One thing we do not seem to learn from experience, is that we do not often learn from experience alone

31. 31 Natural Epistemology  Noticing – Marking – Recording  Conjecturing  Resonance seeking –with own experience –with others  Validity found in –use by local community of practice –own future practices informed Avoid the teaching of speculators, whose judgements are not confirmed by experience. (Leonardo Da Vinci)

32. 32 Specting Interspective Extraspective IntrospectiveIntrospectiveIntrospectiveIntrospective Intraspective

33. 33 Essence of Discipline of Noticing  Systematic Reflection –Past (accounts-of not accounting-for)  Preparing & Noticing –For Future & Present  Recognising Choices –Could-have & Could-be (not should have or should be)  Validating –for Self & with Others

34. 34 Recognising Choices Distinguishing Choices Accumulating Alternatives Identifying & labelling Validating with Others Describing Moments Refining Exercises Systematic Reflection Keeping Accounts Seeking Threads Preparing & Noticing Imagining Possibilities Noticing Possibilities

35. 35 Interwoven Worlds Own world of experience Trying ReflectingSeekingresonance with others Colleague's world of experience World of observations & theories Recognising Possibilities Expressing Preparing

36. 36 The universe is a mirror in which we can contemplate only what we have learned to know about ourselves (Italo Calvino)

37. 37  Developing Thinking in Algebra (Sage 2005)  Researching Your Own Practice using the Discipline of Noticing (Routledge 2002)  Fundamental Constructs in Mathematucs Education (Routledge 2005)  John Mason  j.h.mason @ open.ac.uk  http://mcs.open.ac.uk/jhm3