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**Manipulatives, Mastery and Calculation**

Good evening and welcome to manipulatives, mastery and calculation.

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**Aim of Workshop To understand what is a manipulative**

To understand the mastery approach To consider a progressive calculation policy To know ways you can use manipulatives in your own setting Over the next 30 minutes I will share my experience of exploring manipulatives and the impact it has had since January last year. How I have used different manipulatives in my own class and school alongside research to support the use of manipulatives in mathematics across the 3-19 age range. I will then go on to talk about the mastery curriculum. How it links with manipulatives and pictorial representation and what it looks like in school. At the end there will time for questions. I will do my best to answer and if I can’t hopefully I can signpost you in the right direction.

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What is a Manipulative? Manipulatives are all practical bits of equipment that children can pick up and manipulate to help them get to grips with the very abstract notions of numbers, the relationships between them and the ways in which they work in the number system. Some examples include Multilink cubes, Dienes apparatus, counters, place value counters, bead strings, Cuisenaire rods, sticks divided into 10 equal sections and also those that use numerals such as place value cards, hundred squares, digit cards, dice, dominoes and so on. Examples on your table. Later we will look at how you can use them to support understanding and rapid progress in class

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**Manipulatives Bead strings Cuisenaire Rods Numicon Dienes**

Place value counters Place value arrow cards These are examples of manipulatives we use in our school across all year groups. Later I will go in to more detail about how I used place value counters for teaching division and Cuisenaire rods for teaching fractions.

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**Why a Mastery Curriculum?**

In 2012 the OECD’s PISA study found that Shanghai and Singapore topped the table, with students scoring the equivalent of nearly 3 years of schooling above most other OECD counties. The top performing countries all use a mastery curriculum approach. The findings highlight that our children’s current achievements in mathematics are not the best they can be. Organisation for Economic Co-operation and development. Organisation founded in 1961 to stimulate economic growth and world trade to provide a platform to share policies, experiences, seek answers to common problems and share good practice. OECD tested more than 510,000 students in 65 different countries and economies with a focus on mathematics. Several others Asian countries were also top performing as well as 3 European neighbours – Liechtenstein, Switzerland and the Netherlands.

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**Aim of a mastery approach?**

A mastery approach is about closing the gap between our highest and lowest achievers, also about raising achievement for all. The aim is to teach every concept or skill in a way that promotes understanding and problem solving so it is not a collection of memorised techniques but a coherent body of interconnected knowledge that can be flexibly applied to solve problems in unfamiliar contexts. Mastery approach places problem solving at the heart of mathematics for every child. Three core aims for success are 1) Conceptual understanding2) Mathematical thinking 3) Language & communication. To achieve conceptual understanding children need to represent concepts using objects and pictures and then make connections between the different representations. Mastery and manipulatives go together hand in hand.

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**Achieving a mastery approach**

Deepening conceptual understanding through the use of physical and pictorial representations (Very effective in countries such as Singapore & Netherlands) Developing pupils’ communication, through explicitly teaching pupils’ to discuss mathematics through grammatically correct full sentences with accurate vocabulary (A key priority in Asian countries) Encouraging pupils to think like mathematicians through giving them opportunities to seek patterns and rules, and ask and answer questions The three dimensions to a mastery approach are to deepen conceptual understanding and not just procedural understanding through the use of physical and pictorial representations. Develop pupils communication about maths using correct vocabulary and reasoned arguments. I predict this because….. I know this because…. Finally to encourage children to think like mathematicians by providing lots of opportunities to seek out pattern and rules and time to ask and answer questions.

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**What is a Mastery Curriculum?**

A cumulative curriculum, with sufficient time for every child to access age appropriate concepts and skills. Involves supporting and challenging pupils through depth. Involves purposeful planning that considers the use of different manipulatives and representations. All children working on the same learning intention at the same time. Teachers who teach mastery expect every single child to succeed. Dweck (1999) wrote that “Teachers adopting a mastery approach believe that every child mindset is more important than prior attainment in determining the progress they will make. In mathematics you know you’ve mastered something when you can apply it to a totally new problem in an unfamiliar situation. Whilst an hour’s lesson might be sufficient to say you have learnt something new, mastering is much longer term investment. Adopting a whole school mastery approach needs commitment, shared expectations, high quality training and resourcing.

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**CLF Calculation Policy**

A policy that has been updated to mirror the new national curriculum expectations. It was designed in collaboration by different maths experts. It is a progressive approach to teaching mathematics across a school. Each school can then supplement strategies and resources at their own discretion. Calculation Policy as a hand out on tables or in packs. Bring up PDF file

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**Plan for Implementation**

Maths Leaders from each school will meet to discuss the implementation of the policy, including misconceptions, hesitations and learning implications. The next step will be to raise people’s understanding of the pedagogical choices teachers have and reasoning behind certain aspects such as using a blank number line and teaching grouping and not sharing. Over the next few terms our maths leads will work together to discuss the policy and consider all the implications before taking it in to school. As a whole federation and as individual schools we then plan to support staff with their own understanding before fully implementing it.

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**Using Manipulatives in Class**

Fractions: Division: Since January 2014, the use of different manipulatives has supported all children in my class in many areas of teaching and learning in mathematics. However, I have chosen to share my experience of fractions and division because I believe it is these two concepts that manipulatives have had biggest impact on. I have taught these two concepts many times before but never as successfully as over the last year! It is on reflection that I realise it is the manipulatives and not the cohort of children or that my teaching has dramatically improved.

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**Using Manipulatives in Class**

Cuisenaire rods for fractions Last year I had a mixed 3 / 4 class (children aged 7 -9). During the autumn term I spent two weeks teaching fractions before I had introduced any manipulatives. It wasn’t that successful! On reflection children hadn’t mastered a new concept but I still moved them on and because it was too abstract, numbers on a page with the occasional picture to represent. So after attending a training course and having more confidence to have explore further I tried using the Cuisenaire rods for the first time. This is what I did! Children in my class could read and write fractions but I don’t believe many of them understood what the numbers on the page represented.

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**The pink rod is one. What are the other rods worth?**

Have a go! The first activity with Cuisenaire rods was to work in mixed ability groups to explore fractions. They were told the pink was worth one, and from that they had to work out the value of the other coloured rods. Everyone was eager to manipulate the rods and started comparing the sizes.

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From children’s questioning and reasoning to each other they then started adding fractions to make equivalent fractions. Pit Stops were then used to address misconceptions over how to write mixed fractions then later another pit stop to introduce the term improper fractions. Children with SEN were able to access the activity by comparing and recognising fractions. The children led the learning with the teacher stepping every now and again to address misconception or extend the learning. The children then identified the next steps for their learning. Within one lesson children were independently ordering fractions, comparing fractions, working out equivalent fractions, relating fractions to decimals and referring to terms such as improper fractions and mixed fractions.

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Mathematical talk “The bottom number is the denominator. That tells you how many equal parts it is divided up in to. The top number is the numerator, the fraction part” Hajra “6/4 is equivalent to 1 2/4 and 1 ½” Holly “When the numerator matches the denominator that means it’s one whole” Arafath “2 ¼ that’s a mixed fraction” Pitro “ If the white rod is ¼ the dark green rod is six lots of a ¼, that means it is 6/4” Ellie

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**Recording the Learning**

At the beginning the children's talk was written by the teacher. Quickly it moved on to the children recording their own thoughts, questions, observations. From reflecting on my practice, I now I ensure every group member has a different colour pen or pile of post it notes to record their thoughts, observations or questions about the maths. This way I can monitor that every child has contributed and it can be used as evidence of progress within the lesson and over the unit of work. Photographs are also used to record learning

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**Further Learning using Cuisenaire Rods**

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**Using Manipulatives in Class**

Place Value Counters for Division Teach explicitly vocabulary for the unit then build in opportunities and time for conversations about the maths using the correct vocabulary. Calculate division mentally. At first the children needed to use jottings to support their thinking. This was over several lessons until all children had a clear understanding. Children were challenged through applying the skills to real life word problems. Design a series of lessons with a clear learning intention for each. First I found out from the children what they knew about division - what it was, vocabulary, misconceptions and what they could do. At the beginning we would also discuss as a class the relevance of division in the real world. I do this with every unit and at first the class found this very hard but now they are being to recognise maths is all around us. Then I planned a series of lessons with the main aim of mastering division by a single digit. The unit of work took in to account the aims of fluency, reasoning and solving problems. Mastering a new skill is a long term aim not one of days or even weeks. (Times table knowledge, making links to division, zero as a place holder, correct vocabulary and explicitly teaching reasoning.) I know that…..so that means….. I started by teaching explicitly the language involved with division. This was modelled and reinforced daily to embed words and structured sentences through pupil talk in pairs and small groups. After reflecting on my previous practice I now see the importance of children working in pairs and small groups some of the time. When using whiteboards children now work in pairs verbalising what is happening. When manipulatives are involved again children start by working in pairs. One to manipulate the concrete, one to record the abstract, both verbalising what is happening making links between the two.

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**Division with no exchange or remainders**

Other examples 369 ÷ 3 = 488 ÷ 4 = Have a go! Tasks were differentiated by times tables. All children were applying the skill of division using manipulatives at their own level. Fluency with times tables is a focus for the school as a whole. The expectation was that children could explain what was happening at each stage using correct vocabulary. Again this was over several lessons.

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**Division with exchange but no remainders 618 ÷ 6 =**

The children were set this calculation as a problem to discuss before I started any direct teaching. The children worked in pairs and used manipulatives. Responses included… “Use the counter to show the dividend” “We can make one group of 6 with the hundreds but what do we do now?” “We could exchange the ten like we do in subtraction” “Zero is the place holder” Children worked independently to solve the calculation. I wrote post it notes as evidence which then get stuck in the back of their books. Have a go!

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**Division with exchange but no remainders 618 ÷ 6 =**

Daniel and Alhosna who are Year 4 working just below MARE could solve and explain how to solve.

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**Division with exchange but no remainders 832 ÷ 8 =**

“First I made 832 with my place value counters. I knew I had to make groups of 8 because 8 is the divisor, 832 is the dividend, the big number. With the hundreds, I had one group of 8. I wrote that to start the answer. I then looked at the tens, I only had 3 so I had to write zero as a place holder. I then exchanged 3 tens for 30 units making it 32. I then put the units into groups of 8. I checked with my times table knowledge. Each time I had four groups. The quotient or answer is 104.” Tyreke Year 4 (Working at ARE)

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1428 ÷7 = Children used the place value counters to illustrate their understanding of division with larger numbers Children explored more examples using the place value counters. They realised that some examples exchanging took a lot of time so automatically they started talk about the maths without the counters.

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**Moving away from manipulatives**

To begin my learning intention include using the manipulatives and reasoning. Can I use manipulatives to…. Can I use my reasoning skills to…. As children master a skill it is their decision when to move away from manipulatives, then on to pictorial representation and then the abstract. Challenge - Nrich website, how a question is presented. ? ÷ 8 = 208, using a different manipulative, working independently or a different person The unit is not complete and now we are moving on to division with remainders.

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**Division with Remainders**

446 ÷ 4 = Children first discussed in pairs if this calculation would have a remainder and how they knew. Children used their reasoning skills to estimate and predict even before they got out the place value counters. To begin children were asked if this calculation will have a remainder and how they knew. Children predicted what the quotient would be and explained why they thought this. By this stage most children are only using the place value counters to check their answers.

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Impact Children have become more enthusiastic about maths. It has supported all children with their speaking and listening as they feel empowered by their deeper understanding of number. Some of our reluctant speakers are now so engaged they want to talk about their maths all day long! With a high number of children with EAL and new arrivals to England we have found it has supported the children to demonstrate their understanding even when they can’t yet verbalise it. Using images and manipulatives has supported SEN children, those who are just below MARE and reluctant mathematicians. They are now more engaged, talk more about their learning and show their understanding more clearly. At first it did confuse the more able as they understood procedures but not necessary concepts.

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What the children say…. “The counters help me see what is happening” – Kelsie 1B – 2B “The place value counters help me to subtract when you have to take a ten from the tens column and then put it in the units. I like changing the ten in to units” Tyreke 2A – 3C “I get Dienes out and pick them up. I count in ones, the sticks in tens and the squares in hundreds. It helps me remember. I can add and take away big numbers now.”– Aiden P8 – 1A “When I have a tricky calculation I use the place value counters to help me get the numbers in my mind. It helps me remember” – Ellie 2A – 3A “They help me (Cuisenaire rods) add and subtract fractions because the rods are different colours and you can see. I liked finding things out by myself and then talking to my friends about it” – Sophie 2A – 3A

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My Favourite Quote “My Maths is on fire!” Tyreke

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Questions

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**Fractions using Cuisenaire rods**

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**Division using Place Value Counters (exchange but no remainders) 832 ÷ 8 =**

“First I made 832 with my place value counters. I knew I had to make groups of 8 because 8 is the divisor, 832 is the dividend, the big number. With the hundreds, I had one group of 8. I wrote that to start the answer. I then looked at the tens, I only had 3 so I had to write zero as a place holder. I then exchanged 3 tens for 30 units making it 32. I then put the units into groups of 8. I checked with my times table knowledge. Each time I had four groups. The quotient or answer is 104.” Tyreke Yr4 (Working at MARE)

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