Download Essential Mathematics

Promoting Mathematical Thinking
Essential Mathematics:
Core Awarenesses
& Threshold Concpets
John Mason
NCETM London
Nov 2011
The Open University
Maths Dept
1
University of Oxford
Dept of Education
Vocabulary
Essential (essence) mathematical
concepts/understandings/knowledge/appreciation
Key Developmental Understandings (Simon, Tzur)
Conceptial Analysis (von Glasersfeld, Thompson)
Historical-Genetic Analysis (Schmittau)
Necessary Shifts (Watson)
Canonical Images (Tahta)
(Core) Awarenesses (Gattegno)
Purposes:
… that observers (researchers, teachers) can impose a coherent and
potentially useful organization on their experience of students’ actions
(including verbalizations) and make distinctions among students’
abilities to engage with particular mathematics
(Simon 2006 p360).
2
Number
Order (ordinals)
Quantity (cardinals)
Naming of numbers (base ten)
Putting things in and taking things out of ‘bags’
Scaling
Numbers as actions on objects
Relationships between actions
3
Bag Constructions (1)
Here there are three bags. If you
compare any two of them, there is
exactly one colour for which the
difference in the numbers of that
colour in the two bags is exactly 1.
You only appreciate / under-stand / over-lie when you
can place something in a more general context.
Can the number of objects be
reduced?
Can the number of colours be
reduced?
What about four bags?
What about ‘exactly two colours’ for
which the difference …
4
17 objects
3 colours
Bag Constructions (2)
Here there are 3 bags and two
objects.
The symbol [0,1,2;3] records the
fact that the bags contain 0, 1 and 2
objects respectively, and there are
3 bags altogether
Given a sequence like [2,4,5,5;6] or
[1,1,3,3;6] how can you tell if there
is a corresponding set of bags?
In how many different ways can you
put k objects in b bags?
5
Arithmetic
Can’t learn arithmetic without thinking ‘algebraically’ (ie in
generalities)
Addition commutative, associative
Multiplication commutative, associative
Multiplication distributes over addition
Acknowledging your ignorance, denoting it,
and then expressing what you do know
(Mary Boole)
6
Square Development
Idling sketching one day, I produced the following rough
diagram. Everything that looks square is meant to be …
Can squares be packed into a rectangle in this way?
Is it possible for the outer rectangle to be a square?
7
Thinking Algebraically
3b-3a
a+3b
3a+b
a b
2a+b
a+b
a+2b
3(3b-3a) = 3a+b
12a = 8b
So a/b = 2/3
For an overall square
4a + 4b = 2a + 5b
So 2a = b
3
9
7
8
For n squares upper left
n(3b - 3a) = 3a + b
So 3a(n + 1) = b(3n - 1)
1
1
2 3
5
8
2a+b+2(2a+3b)
3
3:2
2(a+3b)
3
+b–a
2( 2a+b+2(2a+3b)
)
3
3
a
b
a+b
2a+3b
a+3b
2:3
2(a+3b)
3
a+2b
2:3
2(2a+3b)
3
2( 2a+b+2(2a+3b)
)
3
= 2(a+3b) +b–a -a
3
3
9
a = 9
b
32
More Formations
10
Conjectures about New National Curriculum
In addition to pedagogical strategies and didactic tactics
…
No matter how it is stated and whatever it stresses (and
consequently ignores) …
What we as CPD providers need to promote
are the essential (essence) mathematical concepts
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Key developmental understandings
Conceptual Analysis
Historical-Genetic Analysis
Necessary Shifts
Canonical Images
(Core) Awarenesses
Geometry
Actions on points, lines, circles, …
Relations between components of diagrams
Relations between actions on diagrams
Isosceles Triangles (equal angles iff equal sides)
– Steph Prestage & Pat Perks –> circle theorems
Translations: orientation & relative angles and lengths
preserved
Rotations: orientation; relative angles & lengths preserved
Reflections: relative angles & lengths preserved
Scaling: angles preserved; ratios of lengths preserved;
result independent of centre of scaling
Shears: ratios of lengths in parallel directions preserved
12
Reflexive Turn
What struck you that you might want to work on for
yourself?
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13
Multiplicity of vocabulary?
Difficulty of being precise / locating essence?
Use of tasks to focus attention on key ideas?
…