1. 1 (The Role of) Mental Imagery in the Teaching and Learning of Mathematics John Mason Tower Hamlets June 08

2. 2 Conjectures  Mental imagery is perhaps THE fundamental power possessed by humans –It is the basis for planning and for carrying out plans; –It is the root of expectation and the mechanism for motivation –It is the basis for algebra and geometry

3. 3 Outline  Some tasks (paration)  Some reflection (post-paration)  Some pre-paration

4. 4 Imagine …  A straight line …

5. 5 Backwards Arithmetic  When I call out a number, you must write it down from right to left, with the units digit on the left Useful Strategy: impose an unusual constraint so as to force yourself to bring internalised procedures to the surface

6. 6 Tulips

7. 7 Kites

8. 8 CopperPlate Calculations

9. 9 What’s Being Done?  Does ‘it’ always work?  What is the it? The calculation comes from an Arabic manuscript Hindu Reckoning written by Kushyar ibn-Lebban about 1000 C.E. (quoted in NCTM 1969 p133)

10. 10 Layout 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

11. 11 Construction

12. 12 Construction

13. 13 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

14. 14 What fractions can you ‘see’?  What relationships between fractions can you see?

15. 15 Using Mental Imagery  When planning a lesson  When reflecting on a lesson  When introducing a lesson or a task  Getting learners to use their imagery –Anticipating –Predicting –Going beyond the physically possible