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Anne Watson South West 2013 KEY IDEAS IN THE NATIONAL CURRICULUM: WHAT’S X GOT TO DO WITH IT? WHAT IS ALGEBRA? What are the pre-algebraic experiences appropriate for primary children? HEALTH WARNING 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 PRIMARY NATIONAL CURRICULUM STRUCTURE 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 SOURCES FOR TODAY Draft curriculum ACME synthesis of responses from mathematics education community Research (e.g. nuffieldfoundation.org.uk) Possible GCSE content Mathematics ACME COMMENTS (WHAT IS ACME?) 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. WHERE ARE WE GOING WITH ALGEBRA FOR EVERYONE? FROM KS4: 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 TRANSFORMED THINKING ABOUT ALGEBRA 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 KEY IDEAS Generalise relationships Equivalent expressions Solve equations Express situations Relate representations New from old Notation EXPLICIT STATEMENTS ABOUT ALGEBRA YR 6 (HEALTH WARNING) 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 NON-STATUTORY GUIDANCE YR 6 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). YOUR IMMEDIATE THOUGHTS AND CONCERNS? 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). MY IMMEDIATE THOUGHTS/CONCERNS 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! SEARCHING FOR HIDDEN PRE-ALGEBRA USING THE KEY IDEAS Generalise relationships Equivalent expressions Solve equations Express situations Relate representations New from old Notation SEARCHING FOR HIDDEN ALGEBRA IN THE PRIMARY DRAFT CURRICULUM, YRS 1-2 Year 1 counting as enumerating objects patterns in the number system repeating patterns number bonds in several forms add or subtract zero. Generalise Equivalence Solve Express Representations New from old Notation Year 2 add to check subtraction (inverse) add numbers in a different order (associativity) inverse relations to develop multiplicative reasoning ENUMERATION 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 c c= ab = ba = 2( ) = 2( - 1) + 2 etc. 2 2 DIFFERENT KINDS OF PATTERN Repeating a, b, b, a, b, b, ...... (3n+1)th square is red Continuing (arithmetic, linear ...) 1, 4, 7, 10 .... Spatial (nth term is 3n+1) ADDITIVE REASONING a+b=c b+a=c c–a=b c–b=a Generalise Equivalence Solve Express Representations New from old Notation c=a+b c=b+a b=c- a a=c- b MULTIPLICATIVE REASONING a = bc a = cb b=a c c=a b bc = a cb = a a=b c a=c b Generalise Equivalence Solve Express Representations New from old Notation HIDDEN IN YEARS 3-4 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 + 9 × 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 HIDDEN IN YEARS 5-6 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 WHAT ELSE DO YOU CURRENTLY TEACH THAT FEEDS IN TO ALGEBRA?