Aims of the Session To build understanding of mathematics and it’s development throughout KS2 To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods) To enhance subject knowledge of the pedagogical approaches to teaching mathematics

The Aims of The New Curriculum The final version of National Curriculum for mathematics aims to ensure all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop 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. These aims come from research carried out by the DfE into high performing jurisdictions https://www.education.gov.uk/publications/standard/publicationDetail/Page1/DFE-RR178

Mathematical Proficiency (NCETM) 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. This slide provides a NCETM definition of mathematical proficiency which is central to the New Curriculum 3

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. This is the NCETM’s agreed working definition of a sub-set of mathematical proficiency. 4

Mathematics Learning Bruner – children need to experience a mix of three different modes of learning: Enactive, Iconic and Symbolic. Dienes – children learn mathematics by means of direct interaction with their environment – variability principles Use 2+3=5 to show three stages. Joins a set of two objects with a set of three to determine 5 altogether – ENACTIVE A series of pictures – picture of two objects joined with a picture of 3 ICONIC Recorded as 2+3=5 SYMBOLIC Some people know this as CPA In Singapore they refer to concrete, pictorial, abstract – CPA – the same thing. Example – planning seating for a wedding can’t be done abstractly, some scribble and some put names on slips and move them. Modes can be translated eg; draw a picture of that calculation. ENACTIVE void if solely worksheets, textbooks and whiteboard. DIENES – variability principles Perceptual Variability principle – a variety of embodiments eg. Dienes, money but the same concept

Derek Haylock and Anne Cockburn 2008 A mathematical concept can be thought of as a network of connections between symbols, language, concrete experiences and pictures Derek Haylock and Anne Cockburn 2008 Linking the whole ideas of concrete materials – symbols – arranging cards with numbers on them, pressing buttons on calculators, etc language – discussing choices, reading, explaining, interpreting and pictures – drawing number lines, jottings, set diagrams etc

How do young children learn number skills? Nominal, ordinal and cardinal aspects to number Nominal – number as a name eg telephone number Ordinal – the order numbers come in eg 4 is always between 3 and 5,fourth, fifth Cardinal – when you finish counting that is the number of the set

Why is counting important? Numbers Count Teacher PD Y1 D3 S4 1st experience of number and maths Counting backwards is seen as difficult Why is counting important? Supports understanding Of the number system A tool for calculation strategies Counting is a child’s first experience of number and maths Learning to count can support understanding of the number system It’s one tool for building up calculation strategies You have three aspects to understanding of number in the early years – a misconception here often lies beneath later barriers to learning. There is the nominal aspect – a label, or name – the Number 3 bus! Ordinal aspect of number – the order of numbers and is why counting is really important – talk about the bikes Cardinal aspect – numbers in a set The bikes represent all three, brings in the concrete experience and early formation of a picture of a number track. COUNTING RACES – arbitrary starting number and difference, forwards backwards, extend decimals and fractions Counting stick 8

Where does it all start? 10 9 8 7 6 5 4 3 2 1 Counting is really important in the early years and remains so throughout – you can do lots of this. Two aspects to counting: Counting along a track – one to one correspondence Counting and realising that the final number represents how many you have (don’t have to recount ultimately!) At home, counting steps etc, objects, tracks in playgrounds, board games Which is bigger? How do you know? The questioning means you are asking your child to analyse their thinking.

From Counting to Calculation Explain that you are going to put up the next slide for a few moments only and they are going to say how many stars are there How many stars? How did you see them – how did you count them? Number talk – different ways of adding – gives permission to children to break numbers up and use them in different ways. Do 5 +7 Number talk Use what you know

How many stars? How did you see them – how did you count them? Take feedback from several people – draw diagrams of their patterns 334 2s Number talk – different ways of adding – gives permission to children to break numbers up and use them in different ways. Do 5 +7 How did they do it? Break the 7, break the 5, move one across? Not largest number in your head and count in ones Growing, throwing and bunny ears – Valuable at KS1 as well as KS2 Use what you know

Models of Addition There are 12 girls and 3 boys. How many children altogether? The chocolate bar was 12p last week, but today the price went up by 3p. What is the price now? These are both addition word problems – but they are different – how? Talk in pairs. Two models of addition: Union of two sets – Aggregation – cardinal aspect Counting on or increasing – Augmentation – ordinal aspect – Bonds to 10/100 - could be either – how are you teaching it? This is the link where the two models connect – starting from the first number and counting on by the second.

Starting Point Before launching in to the expectations of KS2, the following are the new National Curriculum expectations for year 2: Discuss your thoughts, particularly with reference to year 3 expectations. Explain that year 2 addition is based on mental approaches. Eg if you add 24 and 32 mentally you automatically combine the tens first – mental strategy. But when in y3 they meet the algorithm, they need to start with the units. If they play this game in y2, they really understand the principle of exchange without having to direct them prematurely to the algorithm and they are well placed to enter y3

Reflections High expectations for end of year 2 – Is this where your children are at the start of year 3? Where is the reference to using "the number line"? Ensure that non- statutory guidance is being used Trading game to follow as transition

Trading Game - Addition H T U 4 30 Roll a dice and add that amount to the board. When you bust 10 swap for a ten rod.

Building the Journey Year 3 Addition and Subtraction up to 3 digits using formal methods Year 4 Addition and Subtraction up to 4 digits using formal methods (Solve simple measure and money problems involving decimals to 2 dp) Year 5 Addition and Subtraction more than 4 digits using formal methods (Solve problems involving number up to 3 dp) (They mentally add and subtract tenths, and one-digit whole numbers and tenths) (They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1)). All years groups also refer to "estimation", "inverse operations" to check, "problem solving"

Three-Digit Column Addition – Stage 1 Pupils will still, even if they say they don’t!, require concrete resources. This is likely to be Dienes (or equivalent) to model two digit addition. No re-grouping to take place. Pupils record concrete and abstract calculations together. T U T U 2 3 4 1 + Keep an eye out for the bright child who skims the reasoning which will cap his learning later on. Reward best use of resources and reasoning

Three-Digit Column Addition – Stage 2 Pupils will still, even if they say they don’t!, require concrete resources. This is likely to be Dienes (or equivalent) to model two digit addition. No re-grouping to take place. Pupils record concrete and abstract calculations together. T U T U 2 5 4 7 + Keep an eye out for the bright child who skims the reasoning which will cap his learning later on.

Three-Digit Column Addition – Stage 1 remodelled Pupils still require concrete resources. Some pupils will want to move away from Dienes, and handle resources less cumbersome (as they now have a feel for “size”) – e.g. place value counters (cheap to make, expensive to buy!) T U 10 1 T U 2 3 4 1 +

Three-Digit Column Addition – Stage 2 remodelled Pupils still require concrete resources. Pupils are now ready to tackle problems requiring “re-grouping”. There are different way pupils could effectively communicate their thoughts. In time pupils won’t need counters. T U 2 5 + 4 7 T U 10 10 1 1 1 1 1 10 10 10 10 1 1 1 1 1 1 1 T U 2 5 + 4 7

Three-Digit Column Addition – Stage 3 Pupils still require concrete resources. Pupils are now ready to tackle problems requiring “re-grouping”. There are different way Pupils could effectively communicate their thoughts. In time pupils won’t need counters. T U 2 5 T U + 4 7 1 2 10 10 1 1 1 1 1 6 10 10 10 10 10 1 1 1 1 1 1 1 T U 2 5 + 4 7 2 1

Three-Digit Column Addition – Stage 3 Pupils still require concrete resources. Pupils are now ready to tackle problems requiring “re-grouping”. There are different way pupils could effectively communicate their thoughts. In time pupils won’t need counters. T U 2 5 + 4 7 T U 1 2 10 10 1 6 10 7 2 10 10 10 10 1 T U 2 5 + 4 7 7 2 1

1 1 1 10 10 1 1 5 2 1 + 4 7 10

1 1 1 10 10 1 1 5 2 1 + 4 7 10

1 1 1 10 10 1 1 1 5 2 + 4 7 10

1 1 1 10 10 1 1 1 5 2 + 4 7 10

1 1 1 10 10 1 1 1 1 1 1 1 1 1 5 2 + 4 7 10 12

1 1 1 10 10 1 1 1 1 1 1 1 10 1 1 5 2 + 4 7 10 2 1

1 1 1 10 10 1 1 1 1 1 1 1 10 1 1 5 2 10 + 4 7 2 1

1 1 1 10 10 1 1 1 1 1 1 1 10 1 1 10 5 2 + 4 7 2 1

1 1 1 10 1 10 1 1 1 1 1 10 1 10 1 1 5 2 + 4 7 2 1

1 1 1 10 10 1 1 10 1 1 1 1 1 10 1 1 5 2 + 4 7 Show Video of introducing written column subtraction 7 2 1

Year 3 Essentially year 3 becomes a time when more "formal methods are introduced". This needs to take place as/when the individual pupil is ready: e.g. 34 + 21 35 + 19 U T U T 10 1 10 1 1 10 1 10 10 10 1 1 1 1 Delegates replicate the shown strategy 1

Addition with 3-digits (an end of Year Objective) Why is this a poor example? e.g. 345 + 126 283 + 142 364 + 159 U T H U T H U T H 100 10 1 100 10 1 100 10 1 100 10 1 100 10 1 100 10 1 10 1 100 10 1 100 10 1 10 10 1 10 1 1 10 10 First is a poor example because the units sum to 11 (cannot spot if the child has a misconception about tens and units ) Heirarchy is exchange in units, exchange in tens, exchange in both. 10 10 10 10

Some day-to-day pointers: If some pupils need refreshing (say on single digit addition or partitioning) how will you be using your TA effectively? Is your TA going to be up to speed on the different strategies that you will be employing? How/when is pupils' work marked and assessed? Can you reduce your workload here and at the same time give better feedback to pupils? If a pupil has "mastered" the process in the Autumn term, they need their knowledge extended in the spring/summer terms. Often problem solving is the key here. How to we ensure that work doesn't fall into the "boring" category? Initially pupils will be motivated by ticks on a page, but if the style of question become repetitive, there might be a tendency to "switch off"

Alternatively, set the addition into an application phase: Find the perimeter of this shape: 123 268 352 Show this pattern goes up by the same amount each time. Then find the next number in the pattern: 325, 462, 599, …

Play the role of the teacher: Mark the following questions. If they are right give a tick, if they are wrong, explain why you think the mistake has been made: Reflection: is column addition the most efficient way to tackle this question? 346 + 300 – 5 Where does planning allow for reflection? 2 9 5 2 9 5 + 3 4 6 + 3 4 6 9 1 1 3 2 9 6 4 1 1

Year 4 The strategies met in year 3, extend into year 4 - with addition and subtraction now with four digits. My suggestion would be: Although pupils might be able to naturally extend the method, revisit the kinaesthetic examples so they link their new objective to prior learning. If some pupils need longer working kinaesthetically than others - fine! Some very visual learners will even remember "counter colours" from the previous year, so ensure complete consistency between year groups.

Upto Four-Digit Addition H Th 1000 Can you scaffold a hierarchical set of questions for addition? Is the column addition approach always sensible? 100 10 1

Year 5 More than Four-Digit Addition 10000 1000 Can you scaffold a hierarchical set of questions for addition? Is the column addition approach always sensible? 100 10 1

Year 5 They mentally add and subtract tenths, and one-digit whole numbers and tenths) They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1) Note - we will include some work on decimals and percentages. Here the most important concept to introduce is 0.9 + 0.1 ≠ 0.10 Two helpful strategies: Bead bar – show counting/decimals http://www.mathsisfun.com/numbers/number-line-zoom.html

Aims of the Session To build understanding of mathematics and it’s development throughout KS2 To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods) To enhance subject knowledge of the pedagogical approaches to teaching mathematics

Where now? By the next meeting, I am going to trial/action... * Where next? * Progress in subtraction * Progress in multiplication * Progress in division and ratio.