Anne Watson South West 2013

 What are the pre-algebraic experiences appropriate for primary children?

 The Secretary of State and Ministers for Education are legally entitled to make changes without consultation and without giving reasons  My role on the panel and here

 Aims: fluency, reasoning, problem-solving  Statements: programme of study (statutory list of content: mainly things to do)  Notes and guidance: to support pedagogy and progression (non-statutory)  Two-year chunks

 Draft curriculum  ACME synthesis of responses from mathematics education community  Research (e.g. nuffieldfoundation.org.uk)  Possible GCSE content  Mathematics

 Expectations of algebraic thinking could be based on reasoning about relations between quantities, such as patterns, structure, equivalence, commutativity, distributivity, and associativity  Early introduction of formal algebra can lead to poor understanding without a good foundation  Algebra connects what is known about number relations in arithmetic to general expression of those relations, including unknown quantities and variables.

 arithmetic sequences (nth term)  algebraic manipulation including expanding products, factorisation and simplification of expressions  solving linear and quadratic equations in one variable  application of algebra to real world problems  solving simultaneous linear equations and linear inequalities  gradients  properties of quadratic functions  using functions and graphs in real world situations  transformation of functions

 Generalising relations between quantities  Equivalence: different expressions meaning the same thing  Solving equations (finding particular values of variables for particular states)  Expressing real and mathematical situations algebraically (recognising additive, multiplicative and exponential relations)  Relating features of graphs to situations (e.g. gradient of straight line)  New relations from old  Standard notation

 Generalise relationships  Equivalent expressions  Solve equations  Express situations  Relate representations  New from old  Notation

Programme of study:  express missing number problems algebraically  use simple formulae expressed in words  generate and describe linear number sequences  find pairs of numbers that satisfy number sentences involving two unknowns.  enumerate all possibilities of combinations of two variables

 Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:  missing numbers, lengths, coordinates and angles  formulae in mathematics and science  arithmetical rules (e.g. a + b = b + a)  generalisations of number patterns  number puzzles (e.g. what two numbers can add up to).

Programme of study:  express missing number problems algebraically  use simple formulae expressed in words  generate and describe linear number sequences  find pairs of numbers that satisfy number sentences involving two unknowns.  enumerate all possibilities of combinations of two variables Notes and guidance: Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:  missing numbers, lengths, coordinates and angles  formulae in mathematics and science  arithmetical rules (e.g. a + b = b + a)  generalisations of number patterns  number puzzles (e.g. what two numbers can add up to).

 How can this build on what children already know?  missing number problems  simple formulae expressed in words  linear number sequences  number sentences involving two unknowns  combinations of two variables  What do you do already? Year 6 is too late!

 Generalise relationships  Equivalent expressions  Solve equations  Express situations  Relate representations  New from old  Notation

 Year 1  counting as enumerating objects  patterns in the number system  repeating patterns  number bonds in several forms  add or subtract zero.  Year 2  add to check subtraction (inverse)  add numbers in a different order (associativity)  inverse relations to develop multiplicative reasoning Generalise Equivalence Solve Express Representations New from old Notation

 12 = 3 lots of 4  12 = 4 lots of 3  12 = two groups of 6  12 = 6 pairs  12 = 2 lots of 5 plus two extra  c= ab = ba = 2( ) = 2( - 1) + 2 etc.

DIFFERENT KINDS OF PATTERN a, b, b, a, b, b, (3n+1)th square is red Repeating Continuing (arithmetic, linear...) Spatial 1, 4, 7, (nth term is 3n+1)

a + b = cc = a + b b + a = cc = b + a c – a = bb = c - a c – b = aa = c - b Generalise Equivalence Solve Express Representations New from old Notation

a = bc bc = a a = cb cb = a b = a a = b c c = a a = c b Generalise Equivalence Solve Express Representations New from old Notation

 Year 3  mental methods  commutativity and associativity  Year 4  write statements about the equality of expressions (e.g. use the distributive law 39 × 7 = 30 × × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4))  write and use pairs of coordinates, e.g. (2, 5)  one or more lengths have to be deduced using properties of the shape Generalise Equivalence Solve Express Representations New from old Notation

 perimeter of composite shapes  order of operations  relate unit fractions and division.  derive unknown angles and lengths from known measurements.  use all four quadrants, including the use of negative numbers  quadrilaterals specified by coordinates in the four quadrants Generalise Equivalence Solve Express Representations New from old Notation