the new national curriculum for mathematics: a personal view anne watson ironbridge sept 2014
TRANSCRIPT
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The new national curriculum for mathematics: a personal view
Anne WatsonIronbridgeSept 2014
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Key features
• Primary year by year: only KS is statutory• Split KS2• Secondary is two lists of content: find your own
connections• “Secondary ready”: !• Sketchy guidance for primary; no guidance for
secondary• No levels• Hidden strands
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Three integrated aims– fluency • increasingly complex problems• conceptual understanding• recall and apply
– reasoning• line of enquiry, • conjecturing relationships• argument, justification or proof
– solving problems • nonroutine problems • simpler steps• persevere
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Schedule• All maintained schools in England must start
teaching programmes of study for mathematics as follows:– KS 1, 2 & 3 from Sept 2014 except yrs 2 and 6– year 10 pupils, new PoS from Sept 2015– year 11 pupils, new PoS from Sept 2016
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• TES website• @educationgovuk• Films on DfE website
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What the talking heads say:
• freedom in teaching methods• opportunities for collaboration: (a little)
central support from NCETM and new hubs• build on strengths and initiatives (outside
Strategy)
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What I say ....• Challenges– formal algorithms– Roman numerals– problem-solving and reasoning– all children learning harder content– time, knowledge and risk
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What are key ideas that make a difference?
• Not formal algorithms but .....– multiplicative reasoning– developing algebra– mathematical reasoning
• Vertical planning for coherent learning• Personal freedom v. whole school approach
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Multiplicative reasoning
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What if yellow is 5 and blue is 9?
What if yellow is 10?
What if blue is 27?
What if yellow is 1?
What if blue is 1?
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What if yellow is 10?
What if blue is 27?
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9 41
5 5
18 41
10 5
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What if yellow is 5 and blue is 9?
What if yellow is 10? What if blue is 27?
What if yellow is 1?
What if blue is 1?
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Connections
• Measuring• Factors• Highest/any common
factor• Lowest/any common
multiple
• Division as a fraction• Remainders• Remainders as
fractions • Scaling by fractions• Ratio
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MR Year 1
• counting• comparing quantities • halving and quartering• measures• fractions ?
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MR Year 2
• objects, materials and contexts for grouping and sharing; discrete and continuous quantities
• relate equal sharing to fractions of stuff and measures
• commutativity, e.g. arrays• language of multiplication and division• multiplication facts
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MR Year 3
• • correspondence: n objects are connected to m objects
• measuring and scaling in context with integers
• connect tenths, decimal measures, division by 10
• connect unit fractions and division by integers
• fractions as parts of ...• simple scales in pictograms and bar
charts• multiplication facts
3 hats and 4 coats, how many different outfits?
12 sweets shared equally between 4 children 4 cakes shared between 8 children
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MR Year 4• x by 0 and 1; ÷ by 1• distributive & associative laws• derived facts & mental methods• correspondence (unfamiliar fractions e.g. 7 cakes
between 4 children)• decimals, fractions as different ways to express both
number and proportion• multiplication and arrays
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MR Year 5
• fraction answers in division• x and ÷ as inverses• powers of 10 in scale drawings• rates (so much per so much)• % proportions• multiplication by fractions is ‘fractions of’• scaling by simple fractions, < and > 1
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MR Year 6
problems involving:–unequal sharing and grouping (ratio)–ratio comparing quantities, sizes, scale
drawings–percentages–scale factors of similar shapes
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MR at KS3• ratio and scale factor: convert, mix, measure, scale,
compare • express algebraically and graphically • when to use multiplicative reasoning• % change, simple interest, repeated growth • fixed product and fixed ratio problems: graphical and
algebraic representations • compound units: speed, unit pricing and density
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MR at KS4• compare quantities: links to similarity (trig ratios)• convert: speed, rates of pay, prices, density, pressure• inverse proportionality• gradient of a straight line graph as rate of change• direct and inverse proportion graphs• instantaneous and average rate of change;
numerical, algebraic and graphical contexts• growth and decay
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Threads of development• measure• continuous number• division and fractions• scaling• proportion• representations• ‘per’• rate of change• integer multiplication facts, derived facts, factors ....
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Algebraic reasoning
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AR Year 6
• 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
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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)
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missing numbers, lengths, coordinates and angles
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formulae in mathematics and science
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arithmetical rules
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generalisations of number patterns
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number puzzles
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Find the hidden algebra
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AR at KS3: number threads
• conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
• understand the relation between operations and their inverses and identify the inverse of a given operation where this exists
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AR at KS3 geometry threads
• apply properties of angles at a point, on a straight line, vertically opposite etc.
• use relationships between alternate and corresponding angles
• use the sum of angles in a triangle to deduce the angle sum in any polygon
• apply angle facts ... to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
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AR @ KS1&2 (guidance)• 9 + 7 = 16, 7 = 16 – 9; 4 × 5 = 20, 20 ÷ 5 = 4 etc.• meaning of the equals sign• equal expressions: 39 × 7 = 30 × 7 + 9 × 7• check calculations using inverses• commutativity: a x b = b x a • associativity: a x (b x c) = (a x b) x c• distributivity: a(b + c) = ab + ac• expressions e.g. perimeter of rectangle = 2(a + b) • missing number problems, e.g. angle sum; shape properties: if a
+ b = 180 then 180 – b = a; • coordinates for shapes, e.g. (a, b) & (a+d, b+d) opposite vertices
of a square. • number sequences, term-to-term• explore order of operations
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AR at KS 3 and 4
• Needs sorting and planning .....
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The good news: hard work but worth it
• Room for a ‘thinking’ curriculum• Need for working together across years and
phases to develop coherent development
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• further questions: [email protected]