1. 1 Attention to Attention in the Teaching and Learning of Mathematics John Mason Open University & University of Oxford Flötur Selfoss Sept 2008

2. 2 Say What You Saw

3. 3 One Sum Diagrams 1 1 (1- ) 2 Anticipating, not waiting 1- 2

4. 4 Reading a Diagram: Seeing As … x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

5. 5 Reasoning from Diagrams … … has a long tradition!

6. 6 CopperPlate Calculations

7. 7 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

8. 8   The calculation comes from an Arabic manuscript Hindu Reckoning written by Kushyar ibn-Lebban about 1000 C.E. (quoted in NCTM 1969 p133)

9. 9 Treviso & Pacioli Calculations 934 3 6 0 9 2 7 1 2 1 6 0 3 0 4 0 9 1 2 4 1 3 293 6 7 2 Treviso and Pacioli Multiplications Historical Topics for the Mathematics Classroom, NCTM p134. 2 7 0 9 1 2 0 9 0 3 0 4 3 6 1 2 1 6 4 1 3 276 2 9 3 93 4

10. 10 Word Problems In 26 years I shall be twice as old as I was 19 years ago. How old am I? + 26 40 ?=?2( - 19) ?26 ? 19 ? =

11. 11 Mid-Point  Where can the midpoint of the segment joining two points each on a separate circle, get to?

12. 12 Scaling P Q M Imagine a circle C. Imagine also a point P. Now join P to a point Q on C. Now let M be the mid point of PQ. What is the locus of M as Q moves around the circle?

13. 13 Additive & Multiplicative Perspectives  What is the relation between the numbers of squares of the two colours?  Difference of 2, one is 2 more: additive  Ratio of 3 to 5; one is five thirds the other etc.: multiplicative

14. 14 Raise your hand when you can see  Something which is 2/5 of something  Something which is 3/5 of something  Something which is 2/3 of something –What others can you see?  Something which is 1/3 of 3/5 of something  Something which is 3/5 of 1/3 of something  Something which is 2/5 of 5/2 of something  Something which is 1 ÷ 2/5 of something

15. 15 What fractions can you ‘see’?  What relationships between fractions can you see?

16. 16 Two-bit Perimeters 2a+2b What perimeters are possible using only 2 bits of information? a b

17. 17 Two-bit Perimeters 4a+2b What perimeters are possible using only 2 bits of information? a b

18. 18 Two-bit Perimeters 6a+2b What perimeters are possible using only 2 bits of information? a b

19. 19 Two-bit Perimeters 6a+4b What perimeters are possible using only 2 bits of information? a b