1. 1 Teaching as The Directing of Attention and Learning as the Transformation of Attention 2007 John Mason Stowarzyszenia Nauczycieli Matematyki Muszyna

2. 2 Outline: teaching using attention  Directing Attention Towards: –stressing, focusing –confronting  Directing Attention Away: –de-emphasising through exploring (integration through subordination)  Structure of Attention

3. 3 Directing Attention Towards

4. 4 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 difference would that make to the differences? Directing Attention Towards

5. s are in... 13 one fifth? s are in... 14 How many... are in...? How many s are in 12 1 ? 2 ? 3 ? 4 ? blah ? oop ?... ? What are you doing? one third? 23 ? s are in... 23 Directing Attention Towards5

6. fraction which divides into both an integral number of times. Find a … 1415 10 21 and and another find the largest. 6 Directing Attention Towards

7. Find a... fraction which both divide into an integral number of times. 14151021 and and another find the smallest. Directing Attention Towards7

8. 8 More doesn’t always go with More and Less doesn’t always go with Less  Draw a (non-square) rectangle –Now adjust it so that the area goes up while the perimeter  goes up  stays the same  goes down Adjust = cut &/or paste Adjust = alter rectangle Area Perimeter UpSameDown Up Same Down Dina Tirosh 2007

9. 9 Directing Attention Away (ignoring, de-emphasising; exploring)  In how many different ways can a unit fraction be written as the difference of two unit fractions?  In how many different ways can you decompose a given rectilinear region into non-overlapping rectangles? Directing Attention Away

10. 10 Learner Choice: Breadth & Depth Exercises 3y = 12 – 2x x = 3 2x + 3y = 12 7x – y = 19 2x + 3y = 12 21x – 3y = 57 2x + 3y = 12 23x = 69 3y = 13 – 2x x = 5 2x + 3y = 13 9x – 6y = 39 2x + 3y = 13 11x – 3y = 52 2x + 3y = 13 13x = 65 2y = 14 + 5x x = -2 -5x + 2y = 14 8x - y = -18 -5x + 2y = 14 16x – 2y = -36 -5x + 2y = 14 11x = -22 5y = 14 + 3x x = 2 -3x + 5y = 14 4x + y = 12 -3x + 5y = 14 20x + 5y = 60 -3x + 5y = 14 23x = 46 (Dave Herring 2007)

11. 11 Using Attention

12. True or False What do you do to decide? What might 8yr olds do to decide? Marta Mollina (Granada) 12 + 11 = 11 + 12 51 + 51 = 50 + 52 27 + 48 - 48 = 27 257 - 34 = 257 - 30 - 4 137 - 137 = 5 12 Directing Attention Towards

13. 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 5/2 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; 2/5 ÷ 3/5 of something? 3/4 ÷ 5/4 of something? something which is 2/5 of something; 3/5 of something; 5/2 of something; 5/3 of something; 2/5 of 5/2 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; 2/5 ÷ 3/5 of something? 3/4 ÷ 5/4 of something? 13 Pat Thompson (2004) Directing Attention Towards

14. Ride & Tie Which graph makes sense, and why? What is the same, and what different about... 14 Directing Attention Towards

15. Structures of Attention Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties 15

16. Structures of Attention Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties 16

17. 17 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 Directing Attention Towards

18. 18 More Magic Square Reasoning –= 0Sum( )Sum( ) Directing Attention Towards

19. Questions ★ Why is directing attention important? Need for confluence between learner and teacher attention Being aware of the structure and focus of your own attention is necessary if you are to direct the attention of learners appropriately Being aware of the importance of attention can alert you to reasons for what learners say and do ★ Why is directing attention important? Need for confluence between learner and teacher attention Being aware of the structure and focus of your own attention is necessary if you are to direct the attention of learners appropriately Being aware of the importance of attention can alert you to reasons for what learners say and do 19

20. Summary ★ ★ How is attention directed? Towards (stressing, focusing) Towards (stressing, focusing) Away (ignoring, de-emphasising) Away (ignoring, de-emphasising) Strategies for Directing Attention Say What You See Watch What You Do In How Many Different Ways Can You...? Show me that you see... What is the same, and what is different about … ? Making Use of the Structure of Attention ★ ★ How is attention directed? Towards (stressing, focusing) Towards (stressing, focusing) Away (ignoring, de-emphasising) Away (ignoring, de-emphasising) Strategies for Directing Attention Say What You See Watch What You Do In How Many Different Ways Can You...? Show me that you see... What is the same, and what is different about … ? Making Use of the Structure of Attention