Mastery Unlocked Session 4: Fluency

The 5 big ideas Give participants an overview of the session: Defining fluency & making the case for it What number facts do children need to know? How do I get children fluent in these facts? The role of memorisation in learning

Fluency is more than memorisation Fluency rests on a well-built mathematical foundation with three parts: an understanding of the meaning of the operations and their relationships to each other -- for example, the inverse relationship between multiplication and division; the knowledge of a large repertoire of number relationships, including the addition and multiplication "facts" as well as other relationships, such as how 4 × 5 is related to 4 × 50; and a thorough understanding of the base ten number system, how numbers are structured in this system, and how the place value system of numbers behaves in different operations -- for example, that 24 + 10 = 34 or 24 × 10 = 240. (Russell 2000) Jo Boaler – Fluency without fear: research evidence on the best ways to learn maths facts Jan 28th 2015 Tall and Gray 1994 – students between 7 and 13 put into groups by their teachers as either low, mid or high achievers in maths. The researchers found that the high achieving students used number sense whilst the low achieving students didn’t. Eg with 19 + 7 high schievers would change it to 20 + 6 When low achieving children are given subtraction calculations eg 21 – 16 they try to count back rather than changing if to 20 – 15 or counting on.

Why Facts and Procedures? In teaching procedural and factual knowledge, ensure the students get to automaticity. Explain to students that automaticity with procedures and facts is important because it frees their minds to think about concepts. It’s vital that children get to automaticity with their mathematical facts and procedures because it lessens the cognitive load, allowing them to apply their mathematical understanding to solve problems. Daniel Willingham Is it true that some people just cant do math?

Fluency involves: 18 + 14 = + 15 32 – 27 = 18 + 14 = + 15 32 – 27 = = 25 x 9 42 ÷ 3 = 2.4 – 0.7 = Discuss the knowledge & strategies which make you fluent answering these questions. Memorisation isn’t enough to solve these problems – to have flexibility you need to understand the structure of the maths and number. Relies upon the relationship between 14 and 15 Need to understand concept of difference and number bonds to 10 (counting on from 27 to 32) Multiplying by near multiples Partitioning 42 into 30 and 12 to divide by 4 (flexibility with partitioning and recognising multiples) Partitioning 0.7 into 0.4 and 0.3 in order to subtract from 2.4

What does mastery of number facts mean? Recall facts out of sequence: 3 + 4 = 7 Use the fact to derive other facts: 8 + 2 = 10, so 8 + 3 = 11 Apply laws and principles – inverse relationships, commutative and associative laws: 3 + 5 = 5 + 3 7 + 5 + 3 = 7 + 3 + 5 = 10 + 5

Number sense v memorisation 18 x 5 Jo Boaler video https://www.youcubed.org/what-is-number-sense/ Efficient multiplication relies on a knowledge of times tables but we need to change the attention away from memorising towards developing strategies and number sense.

Number talks Ruth Parker and Kathy Richardson Pose an abstract problem eg 18 x 5 and ask the children to solve it mentally. Teacher collects different methods and the class looks at why they work.

www.insidemathematics.org

+ 1 2 3 4 5 6 7 8 9 10 Y1 facts Y2 facts Adding 1 Bonds to 10 Bridging/ compensating Y1 facts Y2 facts Adding 2 Adding 0 Doubles Near doubles + 1 2 3 4 5 6 7 8 9 10 0 + 0 0 + 1 0 + 2 0 + 3 0 + 4 0 + 5 0 + 6 0 + 7 0 + 8 0 + 9 0 + 10 1 + 0 1 + 1 1 + 2 1 + 3 1 + 4 1 + 5 1 + 6 1 + 7 1 + 8 1 + 9 1 + 10 2 + 0 2 + 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 + 8 2 + 9 2 + 10 3 + 0 3 + 1 3 + 2 3 + 3 3 + 4 3 + 5 3 + 6 3 + 7 3 + 8 3 + 9 3 + 10 4 + 0 4 + 1 4 + 2 4 + 3 4 + 4 4 + 5 4 + 6 4 + 7 4 + 8 4 + 9 4 + 10 5 + 0 5 + 1 5 + 2 5 + 3 5 + 4 5 + 5 5 + 6 5 + 7 5 + 8 5 + 9 5 + 10 6 + 0 6 + 1 6 + 2 6 + 3 6 + 4 6 + 5 6 + 6 6 + 7 6 + 8 6 + 9 6 + 10 7 + 0 7 + 1 7 + 2 7 + 3 7 + 4 7 + 5 7 + 6 7 + 7 7 + 8 7 + 9 7 + 10 8 + 0 8 + 1 8 + 2 8 + 3 8 + 4 8 + 5 8 + 6 8 + 7 8 + 8 8 + 9 8 + 10 9 + 0 9 + 1 9 + 2 9 + 3 9 + 4 9 + 5 9 + 6 9 + 7 9 + 8 9 + 9 9 + 10 10 + 0 10 + 1 10 + 2 10 + 3 10 + 4 10 + 5 10 + 6 10 + 7 10 + 8 10 + 9 10 + 10 This is a grid showing the addition facts that children should be fluent with by the end of Year 2. Take a few moments to look at it and think about the order in which we can help them to become fluent. By fluent we mean either instantly recalling or having a very quick mental strategy to get to the answer. They have been grouped into “families” of similar facts. The white boxes ie the ones without fact families have been minimised.

Progression in teaching addition facts. Group A: Year 1 Adding 1 (e.g. 7 + 1 and 1 + 7) Doubles of numbers to 5 (e.g. 4 + 4) Adding 2 (e.g. 4 + 2 and 2 + 4) Number bonds to 10 (e.g. 8 + 2 and 2 + 8) Adding 10 to a number (e.g. 5 + 10 and 10 + 5) Adding 0 to a number (e.g. 3 + 0 and 0 + 3) The ones without a family: 5 + 3, 3 + 5, 6 + 3, 3 + 6 Knowing these facts by the end of Year 1 will mean children will know 87 of the 121 addition facts in the grid. (Could also teach ‘5 and a bit’ facts as a family) Don’t just memorise – you need to connect facts together to help children learn them.

Know or derive a quick strategy (not counting): Progression in teaching addition facts: Group B: Year 2 (those that bridge 10) Know or derive a quick strategy (not counting): Doubles: 7 + 7 Near doubles: 8 + 9 = 8 + 8 + 1 Bridging: 8 + 9 = 8 + 2 + 7 Compensation 8 + 9 = 8 + 10 – 1  

Mid attaining Y3 Know Count Strategy Middle attainer (start of Y3)

High attaining Y4 Higher attainer (start of Y4) Aim to get rid of counting as a strategy.

The Impact of Factual & Procedural Fluency “In the course of this year, I have come to realise that the procedural gap in my Y6 class was much bigger than the conceptual gap.” (Tom, Cohort 1 Mastery Specialist) Look at the lower attaining children in your class – it is likely that the procedural gap is bigger than conceptual one. Just because you’re teaching in upper KS2, don’t assume that all the children in your class are fluent in their number facts and procedural fluency. You may have to go back and revisit in order that they become fluent and lessen the cognitive load which could be preventing them for applying their mathematical knowledge and solving problems. Jo Boaler – “Low achievers are often low achievers not because they know less, but because they don’t use numbers flexibly.”

Some Reflections Outcomes are stronger where there is a combination of recall and strategies The route to memorisation is recognising relationships and making connections Need to move away as quickly as we can from counting strategies The best way to develop fluency with numbers is to work with numbers in different ways not to blindly memorise without number sense. Don’t let children switch off from maths because they can’t memorise facts Fluency is not only about speed and memorisation.

Activities to develop fluency Make it fun! If the activities are meaningful children will begin to commit maths facts to heart at the same time as developing number sense Don’t think that timed testing is the only way to achieve rapid recall

Snap it Children make a train of blocks of a specified length. On the signal “snap” they break their train and hid one part behind their back. The other children have to work out how many blocks are hidden.

Competitive aim – stop your partner from going Strike it Out - nrich 6 + 4 = 10 10 take away 9 makes 1 1 add 17 is 18 18…… Competitive aim – stop your partner from going Collaborative aim – cross off as many as possible This is a game called, “strike it out,” on nrich. Run through the instructions and get the participants to play in pairs. Think about the skills you’re using when you’re playing. There are lots of ways of making learning number facts fun.

What whole numbers between 0 and 12 do the shapes represent?

Solution: Purple square = 2 Yellow semicircle = 8 Orange oval = 4 Blue rectangle = 3 Red circle = 12 Green triangle = 6 Green star = 9 Purple star = 5 Blue hexagon = 10 Red triangle = 0 Yellow diamond = 1

The 5 big ideas Give participants an overview of the session: Defining fluency & making the case for it What number facts do children need to know? How do I get children fluent in these facts? The role of memorisation in learning

Strike it out - nrich