Primary PD Lead Support Programme

Introduction to:

The focus and demands of

the new curriculum

The Aims of The New Curriculum

The National Curriculum for mathematics aims to ensure all pupils:

become fluent in the fundamentals of mathematics so that they are efficient in using and selecting the appropriate written algorithms and mental methods, underpinned by mathematical concepts

can solve problems by applying their mathematics to a variety of problems with increasing sophistication, including in unfamiliar contexts and to model real-life scenarios

can reason mathematically by following a line of enquiry and develop and present a justification, argument or proof using mathematical language.

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The focus and demands of the new curriculum

The new National Curriculum is likely to have • An increased focus on arithmetic proficiency• A greater emphasis on learning number facts• Higher expectations in fractions, decimals and

percentages• Less emphasis on data handling

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Mathematical Proficiency

• Mathematical proficiency requires a focus on core knowledge and procedural fluency so that pupils can carry out mathematical procedures flexibly, accurately, consistently, efficiently, and appropriately. Procedures and understanding are developed in tandem.

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Arithmetic Proficiency: achieving fluency in calculating with understanding

... an appreciation of number and number operations, which enables mental calculations and written procedures to be performed efficiently, fluently and accurately.

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BALANCE

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Procedural Fluency

Conceptual Understanding

INTEGRATION

Is this image preferable?

Procedural Fluency

Conceptual Understanding

Developing conceptual understanding

The teacher’s job is to organise and provide the sorts of experiences which enable pupils to construct and develop their own understanding of mathematics, rather than simply communicate the ways in which they themselves understand the subject.(1989 National Curriculum, Non- Statutory Guidance, page C2)

Is this how teachers see their role?

Ofsted: Made to Measure

“Schools should:

• tackle in-school inconsistency of teaching• increase the emphasis on problem solving across the

mathematics curriculum • develop the expertise of staff:

• in choosing teaching approaches and activities that foster pupils’ deeper understanding,

• in checking and probing pupils’ understanding during the lesson,

• in understanding the progression in strands of mathematics over time, so that they know the key knowledge and skills that underpin each stage of learning

• ensuring policies and guidance are backed up by professional development for staff “

(Source Ofsted 2012  - Mathematics: made to measure)

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Flexibility

● 25 + 42

● 25 + 27

● 25 + 49

● 1462 + 2567

● 24 50

● 24 4

● 24 17

● 1424 x 17

● 123 3

● 325 25

● 408 17

● 67 - 45

● 67 - 59

● 3241 - 2167

● 672 - 364

What aspects of conceptual understanding and procedural fluency are required for each of these calculations?

Improving children’s Arithmetic Proficiency

Findings from Ofsted 2011:• Practical, hands-on experiences of using, comparing and

calculating with numbers and quantities … are of crucial importance in establishing the best mathematical start …

• Understanding of place value, fluency in mental methods, and good recall of number facts … are considered by the schools to be essential precursors for learning traditional vertical algorithms (methods)

• Subtraction is generally introduced alongside its inverse operation, addition, and division alongside its inverse, multiplication

• High-quality teaching secures pupils’ understanding of structure and relationships in number …

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Other findings:• Skills in calculation are strengthened through solving

a wide range of problems.• Schools recognised the importance of moving

towards more efficient methods• Schools in the good practice survey had clear,

coherent calculation policies• Schools recognise the importance of good subject

knowledge and subject-specific teaching skills and seek to enhance these aspects of subject expertise.

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Improving children’s Arithmetic Proficiency

(Source Ofsted 2011  - Good Practice in Primary Mathematics)

Primary PD Lead Support Programme

What can we learn from other

countries?

Problem solving, reasoning and procedural fluency’

“...there is a wider consensus amongst mathematics educators that conceptual understanding, procedural and factual fluency and the ability to apply knowledge to solve problems are all important and mutually reinforce each other.”

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DfE RR178, 2011

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Singapore mathematics framework

Procedural fluency and conceptual understanding

• Singapore Maths does not rely on rote learning of facts and procedures without the underlying understanding required to use them effectively. The Ministry of Education (2006) explicitly states: “Although students should become competent in the various mathematical skills, over-emphasising procedural skills without understanding the underlying mathematical principles should be avoided”;

• Improving pupils’ attitudes is likely to raise achievement – links between success, confidence, enjoyment;

• High expectations;• Concrete, pictorial, abstract (CPA);• Visual problem solving strategies;• Focus on number and calculation from the beginning.

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TIMSS

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HIGHER performance compared with EnglandParticipants performing at a significantly higher level than in England

SIMILAR performance compared with EnglandParticipants performing at a similar level to England (not statistically different)

LOWER performance compared with EnglandParticipants performing at a significantly lower level than England

6 countries [and 1 benchmarking participant], with their scale scores

6 other countries [and 1 benchmarking participant], with their scale scores

37 countries [and 5 benchmarking participants] including…With their scale scores

Singapore KoreaHong KongChinese Taipei JapanNorthern Ireland[North Carolina, US]

606605602591585562[554] 

Belgium (Flemish)Finland[Florida, US]ENGLANDRussian FederationUnited StatesNetherlandsDenmark

549545[545]542542541540537

[Quebec, Canada]PortugalGermanyIreland, Rep of[Ontario, Canada]AustraliaAustriaItaly[Alberta, Canada]SwedenKazakhstanNorwayNew ZealandSpain

533532528527[518]516508508[507]504501495486482

Table 1.1 TIMSS 2011 performance groups: mathematics at ages 9-10Source: Exhibit 1.3 international mathematics report

Primary PD Lead Support Programme

Towards written algorithms: Addition and subtraction

“Dependence on facts and procedures alone cannot ensure competence with arithmetic at the required level; competence also depends on identifying, understanding and acting on the underlying relations, equating and estimating quantities.”

ACME Report on Primary Arithmetic, Dec. 2010

Now what was the next step?

3 5+ 4 7 9 1

2

4 5- 3 7 1 2

3 5 3915

248 251240 496

Pupil forgot to ‘put a nought’ here because there already was one here.

1 82162

The need for algorithms

“Civilization advances by extending the number of important operations we can perform without thinking.”

Alfred North Whitehead

A sledgehammer to crack a nut

1 0 0 0- 7 9 9 3

10 11

99

16- 9

7

0 1 97x 100

00000

97009700

7 5 65

0 80

See NCETM Mathemapedia – for example Counting and calculating in the Early Years

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Addition

5 7

Models for additionCombining two sets of objects (aggregation)

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Context:• Combined price of

items

Adding on to a set (augmentation)

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Context:• Increase in price• Increase in temperature• Age in x years

More than single digits?

25

25 + 47

26

27

Compacted

leading to

28

Compacted

29

5

7

2

4leading to

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Subtraction

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Models for subtraction

Removing items from a set (reduction or take-away)

- 1- 2- 3- 4- 5 = 7Issue:Relies on ‘counting all’, again.

Comparing two sets (comparison or difference)Issue:Useful when two numbers are ‘close together’, where ‘take-away’ image can be cumbersome

Seeing one set as partitioned

Seeing 12 as made up of 5 and 7

Issue:Helps to see the related calculations; 5+7=12, 7+5=12, 12-7 = 5 and 12-5=7 as all in the same diagram

N.B.When this is done on a bead bar, there are links with both counting back and difference on a number line

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Models for subtraction

Counting back on a number line

Finding the difference on a number line

12 0

- 5

7

Issue:Number line helps to stop ‘counting all’.

Also:Knowledge of place value and number bonds can support more efficient calculating

12 0 5

7 Issue:Useful when two numbers are ‘close together’, use of number bonds and place value can help.

5 2 10

10 -2-3

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72 - 47

More than single digits?

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This is now “Sixty-twelve”

7 126

Compacted

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2

7

7

4

7 2

4 7leading to

Task

Explore some subtraction calculations using the different manipulatives.

• How well do the manipulatives help you to solve the calculation problems?

• How well do the manipulatives help to move pupils towards written methods?

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Primary PD Lead Support ProgrammeTowards written algorithms: Multiplication and division

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Multiplication

Models for multiplication

Bead Bar

Number Line

Fingers“6” “9” “12”“3”

0 3 6 9 12

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Lots of the ‘same thing’

Models for multiplication

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Scaling

3 timesas tall

Models for multiplication

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3 4 4 3

An image for 7 8 = 56

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An image for 7 8 = 56

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More than single digits? 18 x13

18

13

1810 8

13

3

10100 80

243043

Progressing towards the standard algorithm

1 0 8

1 0

3

1 0 0 8 0

3 0 2 4

44

10 8

10

3

100 80

30 24

1 8

1 3

5 4

1 8 0

2 3 4

?

When?

Whether?

How?

Division

An image for 56 7

Either:

• How many 7s can I see? (grouping)

Or:

• If I put these into 7 groups how many in each group? (sharing)

The array is an image for division too

5 67

8

An image for 56 7

How many 7s

can I see?

5 67

8

How many 7s

can I see?

5 67

8

How many 7s

can I see?

5 67

8

How many 7s

can I see?

5 67

8

120 3

52

40

1203

The power of the place value:counters for larger numbers

1200 3

.. and for 100s

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400

12003

54

Now try one yourself

138 divided by 6

Use the counters

2 01 3 8

Hundreds Tens Ones

610

55

31

1386

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Task

Explore some division calculations using the different manipulatives.

• How well do the manipulatives help you to solve the calculation problems?

• How well do the manipulatives help to move pupils towards written methods?

• Reflect on your own practice about how a written method for division can be taught.

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