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. 2

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 3

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. 4

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. 5

BALANCE 6 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 Mathematics: made to measure) 9

Flexibility ● ● ● ● ●24  50 ●24  4 ●24  17 ●1424 x 17 ●123  3 ●325  25 ●408  17 ● ● ● ● 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 … 11

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. 12 Improving children’s Arithmetic Proficiency (Source Ofsted Good Practice in Primary Mathematics)Source Ofsted 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.” 14 DfE RR178, 2011

15 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. 16

TIMSS 17 HIGHER performance compared with England Participants performing at a significantly higher level than in England SIMILAR performance compared with England Participants performing at a similar level to England (not statistically different) LOWER performance compared with England Participants 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 Korea Hong Kong Chinese Taipei Japan Northern Ireland [North Carolina, US] [554] Belgium (Flemish) Finland [Florida, US] ENGLAND Russian Federation United States Netherlands Denmark [545] [Quebec, Canada] Portugal Germany Ireland, Rep of [Ontario, Canada] Australia Austria Italy [Alberta, Canada] Sweden Kazakhstan Norway New Zealand Spain [518] [507] Table 1.1 TIMSS 2011 performance groups: mathematics at ages 9-10 Source: 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?   Pupil forgot to ‘put a nought’ here because there already was one here

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 x

See NCETM Mathemapedia – for example Counting and calculating in the Early YearsCounting and calculating in the Early Years 23 Addition

57 Models for addition Combining two sets of objects (aggregation) Context: Combined price of items Adding on to a set (augmentation) 12 Context: Increase in price Increase in temperature Age in x years

More than single digits?

26

27

Compacted leading to 28

Compacted leading to

30 Subtraction

12 Models for subtraction Removing items from a set (reduction or take-away) = 7 Issue: 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 31

Models for subtraction Counting back on a number line Finding the difference on a number line Issue: Number line helps to stop ‘counting all’. Also: Knowledge of place value and number bonds can support more efficient calculating Issue: Useful when two numbers are ‘close together’, use of number bonds and place value can help

More than single digits? 33 This is now “Sixty- twelve”

Compacted leading 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? 35

Primary PD Lead Support Programme Towards written algorithms: Multiplication and division

37 Multiplication

Models for multiplication Bead Bar Number Line Fingers “6”“6”“9”“9” “12” “3”“3” Lots of the ‘same thing’

Models for multiplication 39 Scaling 3 times as tall

Models for multiplication 40 3  44  3

An image for 7  8 = 56 41

An image for 7  8 = 56 42

More than single digits? 18 x

Progressing towards the standard algorithm

 ? 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

An image for 56  7 How many 7s can I see?

How many 7s can I see?

How many 7s can I see?

How many 7s can I see?

120  The power of the place value: counters for larger numbers

1200  3.. and for 100s

54 Now try one yourself 138 divided by 6 Use the counters

HundredsTensOnes

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. 56