Developing mathematical thinking in the core curriculum Anne Watson East London Maths Forum June 2008

What is specifically mathematical thinking? Tasks for you to do, within core curriculum, to illustrate key features of secondary mathematical thinking and to provide generic task types

2,4,6,8 … 5,7,9,11 … 9,11,13,15 …

2,4,6,8 … 2,5,8,11 … 2,23,44,65 …

2,4,6,8 … 3,6,9,12 … 4,8,12,16 …

Start the same; then are different Expectations and assumptions about pattern Application of number sense and experience Looking for similarities with other mathematical structures Testing a repertoire of operations Background generalisations

… ( x – 2 ) ( x + 1 ) = x 2 - x - 2 = x(x - 1) -2 ( x – 3 ) ( x + 1 ) = x 2 - 2x - 3 = x(x - 2) - 3 ( x – 4 ) ( x + 1 ) = x 2 - 3x - 4 = x(x – 3) - 4

‘With’ to ‘across’ the grain Why equivalent? Shifts from going ‘with the grain’ to making relationships ‘across the grain’ Shifts from ‘filling in’ to operating and transforming Shifts from answers to relationships

Construct a triangle which has a height of two and a height of one –and another Can you make one which has a height of three and a height of two and a height of one?

And another... Make an unusual one Making an example of a familiar object with given constraints Upsetting ‘standard’ examples Exploring a class of objects, beyond the obvious Exploring scope

Given that the sum of internal angles of a polygon is 2(n-2) x 90°, what does a 2.5 sided regular polygon look like?

Work backwards using properties Move on from whole numbers Application of formula Counter-intuitive ideas to form new concepts Construction Shift from discrete to continuous

Similar structures Similar structures; analogous reasoning Powerful (?) images Multi-purpose images –offer tools –offer meaning

Find a number half way between: 28 and and and and and and.0064

Extend methods Transformation What else can I do with this?

Find a number half way between: 28 and 34 and

Need new methods Beyond ad hoc methods Beyond visual models Fraction as mathematical structure Shifts in what is triggered when you see familiar objects

Plot y = 2x y = 3x + 2 y = 5x – 3 y = -4x - 5 Find the slope between (4, 5) and (7, 10) (4, 5) and (6, 10) (4, 5) and (5, 10) (4, 5) and (4, 10) (4, 5) and (3, 10)

Exercise as object Control variation so that concept emerges Shift from separate answers to raw material for conceptual understanding

sin 2 x + cos 2 x = 1 2 sin 2 x + 2 cos 2 x= 2 3 sin 2 x + 3 cos 2 x = 3 4 sin 2 x + 4 cos 2 x = 4 e x sin 2 x + e x cos 2 x = e x cosx sin 2 x + cos 3 x = cosx

Hide this structure Substituting expressions for numbers and letters Shift from specific equations and identities to recognising them when they are obscure

2x + 13x – 32x – 5 x + 1-x – 5x – 3 3x + 33x – 1-2x + 1 -x + 2x + 2x - 2

What varies? Sorting in different ways draws attention to all the things that can vary Sorting in several ways encourages focus on properties instead of visual similarities

Put these in increasing order: 6√2 4√3 √ √8 2√9 9 4√4 (√3) 2 & some of your own

Ordering Beyond obvious Beyond visual impact Gives need for method Continuity Creativity …

Arguing about Anne says that when a percentage goes down, the actual number goes down - Is this always, sometimes or never true? John says that when you square a number, the result is always bigger than the number you started with - Is this always, sometimes or never true?

Task types Start the same, then be different With and across the grain Why equivalent?..and another Make an unusual one Work back from properties Move on from whole numbers Similar structures Hide this structure Extend methods Need new method Exercise as object What varies? Order Argue

Thankyou for thinking

Anne Watson: Raising Achievement in Secondary Mathematics, Open University Press “Thinkers” & “Questions and Prompts for Mathematical Thinking” are available from Association of Teachers of Mathematics 8 th Annual Institute of Mathematics Pedagogy July 28 th to 31 st Cuddesdon near Oxford John Mason, Malcolm Swan, Anne Watson