The New Primary Curriculum for mathematics: what does it mean to you? 17.10.14 Caroline Clissold: ccmi@Hotmail.co.uk

The National Curriculum for Mathematics aims to ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions These three aims should not be lost in the detail of the programmes of study. The new draft seeks to strengthen these aims

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on. Stage 1: emerging Stage 2: expected/achieving: it is at this stage that teachers often accelerate through new content Stage 3: exceeding – great depth and complexity of problem solving

Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils’ conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used.

EYFS handbook: Key aspects of effective learning characteristics include children: • being willing to have a go; • being involved and concentrating; • having their own ideas; • choosing ways to do things; • finding new ways; • enjoying achieving what they set out to do. These develop the children’s ability to reason and problem solve – these characteristics must be developed throughout a child’s education.

Something to consider…… Divide the requirements into these sections to promote reasoning: Number sense Additive reasoning Multiplicative reasoning Geometric reasoning Build two or three week units that are based on these. Include measures and statistics within them and not separately – they are real life applications of the mathematics we teach. This will allow more time for teaching, practising, consolidating and deepening children’s understanding of the fundamentals of mathematics.

Mastery curriculum Some points to consider: Differentiation is achieved by emphasising deep knowledge and through individual support and intervention - not in the content taught. Questioning and scaffolding vary, different problems to solve, higher attainers given complex problems which deepen their knowledge of the same content. Misconceptions dealt with immediately. Fluency comes from deep knowledge and practice. Ability to recall facts and manipulate them to work out other facts is important. Concrete and pictorial representations help build procedural and conceptual understanding. Practice and consolidation within different contexts, e.g. time, money, length. There is a focus on the development of deep structural knowledge and the ability to make connections.

Where it all begins…… Counting: Stable order principle One to one principle Cardinal principle Order irrelevance principle Abstraction principle Gelman and Gallistel

= = The equals sign the same as the same as equivalent equivalent Not the answer to a calculation! Not the answer to a calculation! equal equal balance balance

What is Place Value? 2 3 4 5 (Ross 1989) Positional Base 10 Multiplicative Additive 1000 100 10 1 2 3 4 5 (Ross 1989)

Does this represent understanding of place value? JULIA ANGHILERI1, MEINDERT BEISHUIZEN2 and KEES VAN PUTTEN (2001) FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES: MIND THE GAP! Educational Studies in Mathematics 49: 149–170, 2002.

1 2 3 1 0 2 0 1 0 0 2 0 0 1 0 0 2 0 3

1000 100 10 1 . 10th 100th

Using Place Value Charts 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 Activity: Give out place value charts and counters Demo activities involving partitioning and combining. 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000

Place Value and Decimals 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000 50000 60000 70000 80000 90000 Each row is 10 times bigger/smaller than the row above/below. Move up and down each row multiplying and dividing by 10 and multiples of 10.

Calculators for place value

A sledgehammer to crack a nut 1 0 0 0 - 9 9 7 3 1 9 97 x 100 00 000 9700 16 - 9 7 1 7 5 56 0 8

Well known mental calculation strategies Partition and recombine Doubles and near doubles Use number pairs to 10 and 100 Adding near multiples of ten and adjusting Using patterns of similar calculations Using known number facts Bridging though ten, hundred, tenth Use relationships between operations Counting on x4 by doubling and doubling again x5 by x10 and halving x20 by x10 and doubling

Resources to support the memorisation of number facts This resource is based on the slavonic abacus and represents quantity through a more concrete and visual representation, rather than just abstract number symbols Establishing 10 as a visual image of two groups of 5 These cards can be helpful for children developing mental images of number bonds

Resources to support the memorisation of number facts How many are there? How many more to make 10? Seeing number and quantity without counting.

Resources to support the memorisation of number facts Could make reference to work on the Slavonic Abacus by Tandi Clausen-May (MT 168 – “Spatial Arithmetic” http://www.atm.org.uk/journal/archive/mt168files/ATM-MT168-42-44.pdf) Also look at flip flops as representing an equation How many are there? How many more to make 10? Seeing number and quantity without counting.

Tens Frames Notice how the numbers are arranged in Numicon shapes, drawing out the odd and even properties. Flip flops!

Addition How would you solve these? 25 + 42 25 + 27 25 + 49 127 + 113 145 +127 3537 + 2946

45 + 77 25 25

45 + 77 26 26

45 + 77 4 5 + 7 7 1 2 2 1 27 27

Subtraction How would you solve these? 67 - 45 67 - 59 178 - 99 300 - 150 468 - 237 3241 - 2167

182 - 147 . 29

182 - 147 30

182 - 147 1 812 - 1 4 7 3 5 31

Multiplication How would you solve these? 24 x 50 52 x 4 26 x 15

Models for multiplication Arrays lead to the grid method and then the short written method 30 3 90 8 24 10 1 See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction https://www.ncetm.org.uk/resources/38453 38 x 3 1 1 4 2 What’s different about all these models? What is the same?

Division How would you solve these? 123÷ 3 165÷ 10 325÷ 25 408÷ 17 728÷ 5 623÷ 24

Models for division Using manipulatives to develop a conceptual understanding of the short written method Develop a conceptual understanding of the short written method 135 ÷ 3 We can’t make any groups of 3 hundred with the 1 hundred we have. Exchange the 1 hundred for 10 tens. 100 10 1 1 3 5 3 10 1 10 1 See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction https://www.ncetm.org.uk/resources/38453 What’s different about these models? What is the same? 1 1

Models for division Using manipulatives to develop a conceptual understanding of the short written method We now have 13 tens 10 10 10 1 3 1 13 5 10 10 10 1 10 10 10 1 See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction https://www.ncetm.org.uk/resources/38453 10 10 1 10 10 1 What’s different about these models? What is the same?

Models for division Using manipulatives to develop a conceptual understanding of the short written method 4 We can make 4 groups of three 10s, leaving one 10 10 1 3 1 13 5 1 10 10 10 10 1 See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction https://www.ncetm.org.uk/resources/38453 10 10 10 10 1 10 What’s different about these models? What is the same? 10 10 10 1

Models for division Using manipulatives to develop a conceptual understanding of the short written method We need to exchange the one 10 for ten 1s 1 1 1 4 1 1 1 3 1 13 15 1 1 1 10 10 10 10 1 1 1 See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction https://www.ncetm.org.uk/resources/38453 10 10 10 10 1 1 1 What’s different about all these models? What is the same? 10 10 10 10

Models for division Using manipulatives to develop a conceptual understanding of the short written method 4 5 We can make 5 groups of three 1s, giving an answer of 45 3 1 13 15 10 10Z 10 10 1 1 1 1 1 See Primary Magazine Essentials maths to share link – click on ‘calculation’ Issues 23 and 24 cover structures for addition and subtraction https://www.ncetm.org.uk/resources/38453 10 10 10 10 1 1 1 1 1 10 10 10 10 1 1 1 1 1 What’s different about all these models? What is the same?

The bar model (Singapore Bar) This has been extremely successful in helping children to make sense of problems in Singapore and Japan. It is increasingly being used in the UK. ‘It helps me see the story of the problem’ ‘I can have a go at any problem now’ David spent 2/5 of his money on a book. The book cost £10. How much money did he start off with? This also illustrates a part whole relationship Concrete to abstract In Singapore there would be a progression from concrete objects, in this case footballs, to pictures of objects to spaces on the bar(s) to represents objects Refer back to the structures of addition explored in residential 1 In this model aggregation can be seen in the two sets of balls coming together and being combined in the one strip There is also the potential to sow the seeds and link to proportional reason in seeing 3 balls as a proportion of the whole The importance of making connections in mathematics! What if the book cost….. £20? £6? £5? £10

Findings from Ofsted Good Practice in Primary Mathematics, 2011

Peter has 4 books. Harry has five times as many books as Peter Peter has 4 books. Harry has five times as many books as Peter. How many more books has Harry? Peter’s books Harry’s books

Sam had 5 times as many marbles as Tom Sam had 5 times as many marbles as Tom. If Sam gives 26 marbles to Tom, the two friends will have exactly the same amount. How many marbles do they have altogether? A computer game was reduced in a sale by 20% and it now costs £48. What was the original price? A gardener plants tulip bulbs in a flower bed. She plants 3 red bulbs for every 4 white bulbs. She plants 60 red bulbs. How many white bulbs does she plant?

Generalisation This can then help the children solve, for example, missing number problems: 45 + ? = 93, ? – 62 = 13, 146 - ? = 79, ? + 82 = 147 Generalising the structure of the mathematics to any context

KS2 2012 Ask participants to solve this problem and ask how hard Y6 children found this? The majority of children found this difficult – even those at L5! Explain that we will explore representations that might help children to more easily access the structure of the mathematics and solve the problem.

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