MATHEMATICS PROGRAMME PRIMARY ADVANTAGE MATHEMATICS PROGRAMME KS1 Parents Workshop – Mathematics Elizabeth Winterton, Jane Woolley and Catherine Thomas A MODEL OF BEST PRACTICE

The Fundamentals – Year 1

The Fundamentals – Year 2

Experience and the National Curriculum Procedural Fluency Conceptual Understanding Primary Advantage Maths Programme Cover what conceptual understanding and procedural fluency is. What does this mean for how we teach?

Key Concepts - CPA The concrete-pictorial-abstract approach, based on research by psychologist Jerome Bruner, suggests that there are three steps (or representations) necessary for pupils to develop understanding of a concept. Reinforcement is achieved by going back and forth between these representations.

Concrete Pictorial Abstract 3 + 4 = 7

Addition – Mental Methods 5 + 31 32 + 8 60 + 60 32 + 19 15 + 16 27 + 36 + 13 Did you: Count on from the largest number? Re-order the numbers? Partition the numbers into tens and ones? Bridge through 10 and multiples of 10? Add 9, 11, etc. by adding a multiple of 10 and compensating? Use near doubles? Use knowledge of number facts? These are all strategies that children need to be aware of when carrying out addition calculations.

Models for Addition Combining two sets of objects (aggregation) Tom had two sweets and John had three sweets: how many did they have altogether? Adding onto a set (augmentation) Tom had two sweets and bought two more. How many sweets does he have now? 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

Counting on with a bead bar/number line Counting on with straws Models for Addition Counting on with a bead bar/number line + 7 5 12 +5 10 +2 Counting on with straws + =

Models for addition 25 + 47 You may wish to dispense with the slides 55 – 62 initially and invite participants to think how the use of Dienes can be used to model how you might add two larger numbers and how this might be recorded. You may then wish to show these slides afterwards.

Models for Addition 25 + 47

Models for Addition 25 + 47

Remember to use the terms regroup and rename. Your turn: Use the deines to complete these sums: 17 + 14 22 + 19 16 + 16 172 + 50 T O Remember to use the terms regroup and rename.

20 + 5 10 4 30 9 20 + 7 10 5 40 2 Expanded Method Models for Addition Ask participants regrouped and renamed

Subtraction – Mental Methods 95 - 86 32 - 8 60 - 30 30 - 14 25 - 9 27 - 18 Did you: Count up from the smallest number? Round the numbers? Partition the numbers into tens and ones? Adjust the place value? Subtract 9, 11, etc. by subtracting a multiple of 10 and compensating? Use near halves? Use knowledge of number facts? These are all strategies that children need to be aware of when carrying out subtraction calculations.

Models for Subtraction Removing items from a set (reduction or take-away) 12 - 4 - 5 - 1 - 2 - 3 = 7 Comparing two sets (comparison or difference) N.B. Important to have materials (Dienes, PV counters, etc.) on tables for participants to use throughout these sessions on the 4 operations. For this session on subtraction you may wish to dispense with slides 50 to 54 initially and work directly with the materials laid out on tables. You may wish then to show slides after some hands-on activity. Start by saying that there is a difference between teaching children to take away and teaching the concept of subtraction. Children need to understand what subtraction means in all its different forms. The first part of this session is an attempt to survey some of the important models and images for subtraction: [N.B. It is important to be aware of where number and number operations are being thought of in a cardinal or an ordinal sense for the different models] Reduction or take-away – one set being reduced in size by removing objects. Could use a bead bar or bead string here to show how removing 5 (say) from either the left or the right of 12 can be used to think of either count back or difference on a number line. Comparison or difference – When two sets are of a different size, it makes sense to ask ‘which one is bigger?’ and ‘by how much?’. By lining them up and counting the ‘surplus’ you can find the difference. Partitioned set – one image that can be seen as representing 2 additions (5 + 7 = 12 and 7 + 5 = 12) and 2 subtractions (12 – 5 = 7 and 12 – 7 = 5) Counting back on a bead bar or number line – the bead bar provides a useful bridge from the cardinal sense (of both number and of the operation) and the ordinal. Finding the difference on a number line. N.B. Having numbers lines marked with the tens makes it possible to use knowledge of number bonds to 10 and place value to be more efficient in both counting back and finding the difference. Seeing one set as partitioned Seeing 12 as made up of 5 and 7

Models for Subtraction Counting back on a number line - 5 12 7 10 -3 -2 Finding the difference on a number line 7 5 10 12 5 2 - 5 =

Models for Subtraction 72 - 47 You may wish to dispense with the slides 67 – 75 initially and invite participants to think how the use of Dienes can be used to model how you might subtract one number from another when the numbers are large and how this might be recorded. You may then wish to show these slides afterwards.

Models for Subtraction 72 - 47

Models for Subtraction 72 - 47 This is now “Sixty- twelve” 7 12 6

Models for Subtraction 72 - 47

Models for Subtraction 72 - 47

Models for Subtraction 72 - 47 = 25

Remember to use the terms regroup and rename. Your turn: Use the deines to complete these sums: 17 - 14 22 - 9 32 - 16 172 - 61 32 – 16 = 16 T O 1 6 Remember to use the terms regroup and rename.

Models for Subtraction Expanded Method 20 + 5 10 4 - 1 20 + 3 10 5 - 10 30 Ask participants 10 8

Lots of the ‘same thing’ Models for Multiplication Lots of the ‘same thing’ Bead Bar Number Line Talk through the different models of multiplication outlined on this and the next slide, emphasising the importance of the bead bar in linking the cardinal and ordinal images of number. 3 6 9 12 Fingers “3” “6” “9” “12”

Multiplication is commutative Models for Multiplication How can it be represented? Multiplication is commutative 4 x 3 3 x 4 Four groups of 3 Three groups of 4

Models for Division The power of the place value counters for larger numbers 12 ÷ 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 When there is a need to use the place value counters to represent larger numbers the removing of groups of 3 counters can be used to model dividing by 3 because removing 3 counters and placing them in an array is equivalent to sharing between three. In this way larger number can be divided efficiently and this awareness is an important pre-cursor to the division algorithm introduced in the next slides. 4 3 1 2

Number bonds within 10 Number bonds to 10 Number bonds to 20 The Facts Number bonds within 10 Number bonds to 10 Number bonds to 20 Doubles and halves to 20 Two times tables Five times tables Ten times tables Look, cover, write and check I say, you say Five minutes whenever you get the chance

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