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Essential Mathematics:Essential Mathematics:Core AwarenessesCore Awarenesses

& Threshold Concpets& Threshold Concpets

John MasonJohn Mason

NCETM LondonNCETM London

Nov 2011Nov 2011

The Open UniversityThe Open UniversityMaths DeptMaths Dept University of OxfordUniversity of Oxford

Dept of EducationDept of Education

Promoting Mathematical ThinkingPromoting Mathematical Thinking

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VocabularyVocabulary

Essential (essence) mathematical Essential (essence) mathematical concepts/understandings/knowledge/appreciationconcepts/understandings/knowledge/appreciation

Key Developmental Understandings (Simon, Tzur)Key Developmental Understandings (Simon, Tzur) Conceptial Analysis (von Glasersfeld, Thompson)Conceptial Analysis (von Glasersfeld, Thompson) Historical-Genetic Analysis (Schmittau)Historical-Genetic Analysis (Schmittau) Necessary Shifts (Watson)Necessary Shifts (Watson) Canonical Images (Tahta)Canonical Images (Tahta) (Core) Awarenesses (Gattegno)(Core) Awarenesses (Gattegno)

… 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).

Purposes:

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NumberNumber

Order (ordinals)Order (ordinals) Quantity (cardinals)Quantity (cardinals) Naming of numbers (base ten)Naming of numbers (base ten) Putting things in and taking things out of Putting things in and taking things out of ‘bags’‘bags’

ScalingScaling Numbers as actions on objectsNumbers as actions on objects Relationships between actionsRelationships between actions

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Bag Constructions (1)Bag Constructions (1) Here there are three bags. Here there are three bags. If you compare any two of If you compare any two of them, there is exactly one them, there is exactly one colour for which the colour for which the difference in the numbers difference in the numbers of that colour in the two of that colour in the two bags is exactly 1.bags is exactly 1.

17 17 objectsobjects

3 3 colourscolours

Can the number of objects Can the number of objects be reduced?be reduced?

Can the number of colours Can the number of colours be reduced?be reduced?

What about four bags?What about four bags? What about ‘exactly two What about ‘exactly two colours’ for which the colours’ for which the difference …difference …

You only appreciate / under-stand / over-lie when you can place something in a more general context.

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Bag Constructions (2)Bag Constructions (2)

Here there are 3 bags and Here there are 3 bags and two objects.two objects.

The symbol [0,1,2;3] records The symbol [0,1,2;3] records the fact that the bags the fact that the bags contain 0, 1 and 2 objects contain 0, 1 and 2 objects respectively, and there are respectively, and there are 3 bags altogether3 bags altogether

Given a sequence like Given a sequence like [2,4,5,5;6] or [1,1,3,3;6] [2,4,5,5;6] or [1,1,3,3;6] how can you tell if there is how can you tell if there is a corresponding set of bags?a corresponding set of bags?

In how many different ways In how many different ways can you put can you put kk objects in objects in bb bags?bags?

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ArithmeticArithmetic

Can’t learn arithmetic without thinking Can’t learn arithmetic without thinking ‘algebraically’ (ie in generalities)‘algebraically’ (ie in generalities)

Addition commutative, associativeAddition commutative, associative Multiplication commutative, associativeMultiplication commutative, associative Multiplication distributes over additionMultiplication distributes over addition

Acknowledging your ignorance, Acknowledging your ignorance, denoting it, and then expressing denoting it, and then expressing what you do know (Mary what you do know (Mary Boole)Boole)

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Square DevelopmentSquare Development

Idling sketching one day, I produced the Idling sketching one day, I produced the following rough diagram. Everything that following rough diagram. Everything that looks square is meant to be …looks square is meant to be …

Can squares be packed into a rectangle in Can squares be packed into a rectangle in this way?this way?

Is it possible for the outer rectangle to Is it possible for the outer rectangle to be a square?be a square?

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Thinking AlgebraicallyThinking Algebraically

aa bb

aa++bb aa+2+2bb22aa++bb

aa+3+3bb33aa++bb

33bb-3-3aa

22 33

558877

991111

33

3(33(3bb-3-3aa) = 3) = 3aa++bb

1212aa = 8 = 8bb

So So aa//bb = 2/3 = 2/3

For an overall For an overall squaresquare

44aa + 4 + 4bb = 2 = 2aa + 5 + 5bb

So 2So 2aa = = bb

For For nn squares upper squares upper leftleft

nn(3(3bb - 3 - 3aa) = 3) = 3aa + + bb

So 3So 3aa((nn + 1) = + 1) = bb(3(3nn - - 1)1)

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3:23:2

2:32:32:32:3

aa

bbaa++bb

aa+2+2bbaa+3+3bb

2(2(aa+3+3bb)) 33

22aa+3+3bb

2(22(2aa+3+3bb))33

22aa++bb++2(22(2aa+3+3bb))33

2(2(aa+3+3bb)) 33++b–b–aa

22((22aa++bb++2(22(2aa+3+3

bb))33))

33

))22aa++bb++2(22(2aa+3+3bb))33

22((

33== 2(2(aa+3+3bb

)) 33 ++bb––aa - -aa

aabb ==

99

3232

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More FormationsMore Formations

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Conjectures about New National Conjectures about New National CurriculumCurriculum

In addition to pedagogical strategies and In addition to pedagogical strategies and didactic tactics …didactic tactics …

No matter how it is stated and whatever it No matter how it is stated and whatever it stresses (and consequently ignores) …stresses (and consequently ignores) …

What we as CPD providers need to promoteWhat we as CPD providers need to promoteare the essential (essence) mathematical are the essential (essence) mathematical conceptsconcepts– Key developmental understandingsKey developmental understandings– Conceptual AnalysisConceptual Analysis– Historical-Genetic AnalysisHistorical-Genetic Analysis– Necessary ShiftsNecessary Shifts– Canonical ImagesCanonical Images– (Core) Awarenesses (Core) Awarenesses

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GeometryGeometry Actions on points, lines, circles, …Actions on points, lines, circles, … Relations between components of diagramsRelations between components of diagramsRelations between actions on diagramsRelations between actions on diagrams

Isosceles Triangles (equal angles iff equal sides)Isosceles Triangles (equal angles iff equal sides)– Steph Prestage & Pat Perks –> circle theoremsSteph Prestage & Pat Perks –> circle theorems

Translations: orientation & relative angles and Translations: orientation & relative angles and lengths preservedlengths preserved

Rotations: orientation; relative angles & lengths Rotations: orientation; relative angles & lengths preservedpreserved

Reflections: relative angles & lengths preservedReflections: relative angles & lengths preserved Scaling: angles preserved; ratios of lengths Scaling: angles preserved; ratios of lengths preserved; result independent of centre of scalingpreserved; result independent of centre of scaling

Shears: ratios of lengths in parallel directions Shears: ratios of lengths in parallel directions preserved preserved

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Reflexive TurnReflexive Turn

What struck you that you might want to What struck you that you might want to work on for yourself?work on for yourself?– Multiplicity of vocabulary?Multiplicity of vocabulary?– Difficulty of being precise / locating essence?Difficulty of being precise / locating essence?– Use of tasks to focus attention on key ideas?Use of tasks to focus attention on key ideas?– ……