1 Essential Mathematics: Core Awarenesses & Threshold Concpets Core Awarenesses & Threshold Concpets John Mason NCETM London Nov 2011 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 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) … 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:

3 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

4 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 objects 3 colours  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 … You only appreciate / under-stand / over-lie when you can place something in a more general context.

5 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?

6 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)

7 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?

8 Thinking Algebraically a b a+ba+ba+ba+b a+2b 2a+b2a+b2a+b2a+b a+3b 3a+b3a+b3a+b3a+b 3b-3a (3b-3a) = 3a+b 12a = 8b So a/b = 2/3 For an overall square 4a + 4b = 2a + 5b So 2a = b For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1)

9 3:2 2:3 2:3 a b a+ba+ba+ba+b a+2b a+3b 2(a+3b) 3 2a+3b 2(2a+3b) 3 2a+b+2a+b+2a+b+2a+b+ 3 2(a+3b) 3 +b–a 2( 2a+b+2a+b+2a+b+2a+b+ 2(2a+3b) 3 ) 3 ) 2a+b+2a+b+2a+b+2a+b+ 3 2( 3 = 2(a+3b) 3 +b–a -a+b–a -a+b–a -a+b–a -a a b = 9 32

10 More Formations

11 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 –Key developmental understandings –Conceptual Analysis –Historical-Genetic Analysis –Necessary Shifts –Canonical Images –(Core) Awarenesses

12 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

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