Download Developing mathematical thinking in the curriculum 2008

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 ) = x2 - x - 2 = x(x - 1) -2
( x – 3 ) ( x + 1 ) = x2 - 2x - 3 = x(x - 2) - 3
( x – 4 ) ( x + 1 ) = x2 - 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
– 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
2.8
38
-34
9028
.0058
and 34
and 3.4
and 44
and -28
and 9034
and .0064
Extend methods
• Transformation
• What else can I do with this?
Find a number half way between:
28 and 34
and
and
and
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
sin2x + cos2x
2 sin2x + 2 cos2x
3 sin2x + 3 cos2x
4 sin2x + 4 cos2x
exsin2x + excos2x
cosx sin2x + cos3x
=
=
=
=
1
2
3
4
= ex
= cosx
Hide this structure
• Substituting expressions for numbers and
letters
• Shift from specific equations and identities
to recognising them when they are
obscure
2x + 1
3x – 3
2x – 5
x+1
-x – 5
x–3
3x + 3
3x – 1
-2x + 1
-x + 2
x+2
x-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
9
4√3
4√4
√32 8
2√8
2√9
(√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
[email protected]
www.education.ox.ac.uk/people/academics
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
8th Annual Institute of Mathematics Pedagogy
July 28th to 31st
Cuddesdon near Oxford
[email protected]
John Mason, Malcolm Swan, Anne Watson