Advisory Committee on Mathematics Education Working algebraically 5-19 Anne Watson South West, 2012

Advisory Committee on Mathematics Education What is algebra? What are the pre-algebraic experiences appropriate for primary children?

Advisory Committee on Mathematics Education Hypothetical but still a valuable exercise

Advisory Committee on Mathematics Education Sources Draft curriculum ACME synthesis of responses from mathematics education community Research (e.g. nuffieldfoundation.org.uk) Glimpse of possible KS4 content

Advisory Committee on Mathematics Education Where are we going with algebra for everyone (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

Advisory Committee on Mathematics Education To summarise Generalising Notation Equivalent expressions Solving equations (finding values of variables) Expressing real and mathematical situations algebraically (recognising additive, multiplicative and exponential relations) Relating features of graphs to properties of functions and situations (e.g. gradient of straight line) New relations from old

Advisory Committee on Mathematics Education Key ideas Generalise relationships Notation Equivalent expressions Solve equations Express situations Relate representations New from old

Advisory Committee on Mathematics Education Additive reasoning a + b = cc = a + b b + a = cc = b + a c – a = bb = c - a c – b = aa = c - b

Advisory Committee on Mathematics Education Multiplicative reasoning a = bc bc = a a = cb cb = a b = a a = b c c = a a = c b

Advisory Committee on Mathematics Education Draft primary national curriculum structure Aims Statements: programme of study (left: statutory) Notes and guidance: to support pedagogy(right: non-statutory but....)

Advisory Committee on Mathematics Education Explicit statements about algebra POS year 6 Pupils should be taught to: –solve linear missing number problems, including those involving decimals and fractions, and find pairs of number that satisfy number sentences involving two unknowns [280] –use simple formulae expressed in words [281] –generate and describe linear number sequences including those involving negative and decimal numbers, and proper fractions e.g. 1.4, 1.1, 0.8. [282] Notes and Guidance year 6 –Ensure pupils write some known arithmetical rules algebraically, such as a + b = b + a, and known relations such as p = 4s for the perimeter of a square. They should also interpret word problems as statements about number and record as a mathematical statement. [283] –Pupils should also write missing number problems algebraically; for example, 2x – 4 = 8 therefore 2x = 12 therefore x = 6 or finding missing lengths in perimeters and missing angles at a point. Pupils should also find possible solutions for equations with two unknown variables, for example x + y = 5 includes solutions x = 1 and y = 4, x = 2 and y = 3. [284]

Advisory Committee on Mathematics Education Your immediate thoughts/concerns?

Advisory Committee on Mathematics Education My immediate thoughts/concerns solving linear equations with non-integer solutions is hard and algebraic methods take time to learn why are sequences here? word formulae have occurred throughout the rest of the document, so what is extra here and where is the progression? does this build on what children already know?

Advisory Committee on Mathematics Education Searching for hidden pre-algebra using the key ideas Generalise relationships Notation Equivalent expressions Solve equations Express situations Relate representations New from old

Advisory Committee on Mathematics Education Searching for hidden algebra in the primary draft curriculum, yrs 1-2 number bonds in the three forms (e.g = 16; 16 – 7 = 9; 16 – 9 = 7) using related facts to perform calculations (e.g. using = 10, = 3 and 10 – 3 = 7 to calculate = 100, = 30 and 100 – 30 = 70) check calculations, including by adding to check subtraction and adding numbers in a different order to check addition; for example = =

Advisory Committee on Mathematics Education Hidden in years 3-4 commutativity (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240) and multiplication and division facts (e.g. using 3 x 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3 to calculate 30 x 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3). count in multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 50, 100 and 1000 from any given number complete number sequences derive facts, for example 300 x 2 = 600 into 600 ÷ 3 = 200 use the distributive law to derive facts, for example, 30 x x 7= 39 x 7. count forwards and backwards with positive and negative whole numbers through zero write and use pairs of coordinates, e.g. (2, 5) practise recognising line symmetry in a variety of diagrams one or more lengths have to be deduced using properties of the shape.

Advisory Committee on Mathematics Education Hidden in years 5-6 calculation of perimeter of composite shapes order of operations include the use of brackets, for example: x 3 = 5 and (2 + 1) x 3 = 9. relationship between unit fractions and division to work backwards; for example, if ¼ of a length is 36cm then the whole length is 36 x 4 = 144cm. 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

Advisory Committee on Mathematics Education What else do you currently teach that feeds in to algebra?

Advisory Committee on Mathematics Education ACME comments (what is ACME?) PoS does not provide a coherent progression towards formal algebra Procedural approach discourages thinking before acting Expectations of algebraic thinking could be even more challenging if they were based on reasoning about relations between quantities, such as patterns, structure, equivalence, commutativity, distributivity, and associativity, and models and representations of these Early introduction of formal algebra before secondary school can lead to poor understanding without a good foundation Algebra in primary connects what is known about number relations in arithmetic to general expression of those relations, including unknown quantities and variables. Operations need to be thoroughly understood in order to make this connection. The only generalising currently explicit in the PoS relates to pattern sequences and not other forms of relations A coherent developmental strand for algebra should be made explicit, making clear the connections between knowledge of number, mental methods, generalizing, and representing relations between quantities and unknowns