1

(The Role of) Mental Imagery(The Role of) Mental Imageryin the Teaching and Learningin the Teaching and Learning

of Mathematicsof Mathematics

John MasonJohn Mason

Tower HamletsTower Hamlets

June 08June 08

3

OutlineOutline

Some tasks (paration)Some tasks (paration) Some reflection (post-Some reflection (post-

paration)paration) Some pre-parationSome pre-paration

5

Backwards ArithmeticBackwards Arithmetic

When I call out a number, you When I call out a number, you must write it down from right must write it down from right to left, with the units digit on to left, with the units digit on the leftthe left

Useful Strategy:impose an unusual constraint

so as to force yourself tobring internalised procedures

to the surface

6

TulipsTulips

7

KitesKites

8

aaa

30361635542428402863483642364232455681497254635160119905

24206478979654CopperPlate CopperPlate

CalculationsCalculations

aaa

30361635542428402863483642364232455681497254635160119905

79654242064789

9

What’s Being Done?What’s Being Done?

aa

562548254795478683983983983948 79 86

Does ‘it’ always work?Does ‘it’ always work? What is the it?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)

11

ConstructionConstruction

12

ConstructionConstruction

13

Raise your hand when you Raise your hand when you can seecan see

Something which is 2/5 of somethingSomething which is 2/5 of something Something which is 3/5 of somethingSomething which is 3/5 of something Something which is 2/3 of somethingSomething which is 2/3 of something

– What others can you see?What others can you see? Something which is 1/3 of 3/5 of something Something which is 1/3 of 3/5 of something Something which is 3/5 of 1/3 of somethingSomething which is 3/5 of 1/3 of something Something which is 2/5 of 5/2 of somethingSomething which is 2/5 of 5/2 of something Something which is 1 ÷ 2/5 of somethingSomething which is 1 ÷ 2/5 of something

14

What fractions can you What fractions can you ‘see’?‘see’?

What relationships between What relationships between fractions can you see?fractions can you see?

15

Using Mental ImageryUsing Mental Imagery

When planning a lessonWhen planning a lesson When reflecting on a lessonWhen reflecting on a lesson When introducing a lesson or a taskWhen introducing a lesson or a task Getting learners to use their Getting learners to use their

imageryimagery– AnticipatingAnticipating– PredictingPredicting– Going beyond the physically possibleGoing beyond the physically possible