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East midlands primary mathematics conference Teaching and the curriculum Jane Jones HMI, National lead for Mathematics 10 February 2015

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Aims In the context of the new primary mathematics curriculum, to explore: characteristics of effective teaching and learning how these might be developed, particularly in relation to mastery.

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Shapes activity: squares How many squares of different sizes can you draw on the dotty paper, (with vertices placed on the dots)? You might want to draw one on each mini grid. How do you know for certain that each shape is a square?

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Shapes activity: squares (continued) Have you got them all?

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What does Ofsted say about mathematics teaching?

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Findings ( Mathematics: made to measure ) The best teaching develops conceptual understanding alongside pupils’ fluent recall of knowledge and confidence in problem solving. In highly effective practice, teachers get ‘inside pupils’ heads’. They find out how pupils think by observing pupils closely, listening carefully to what they say, and asking questions to probe and extend their understanding, then adapting teaching accordingly. Too much teaching concentrates on the acquisition of disparate skills that enable pupils to pass tests and examinations but do not equip them for the next stage of education, work and life.

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The National Curriculum for mathematics aims to ensure that 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 the ability to recall and apply knowledge rapidly and accurately 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.

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Characteristics of a Mastery curriculum An expectation that all pupils can and will achieve. The large majority of pupils progress through the curriculum content at the same pace. Differentiation emphasises deep knowledge and individual support/intervention. Teaching is underpinned by methodical curriculum design, with units of work that focus in depth on key topics. Lessons and resources are crafted carefully to foster deep conceptual and procedural knowledge. Practice and consolidation play a central role. Well-designed variation builds fluency and understanding of underlying mathematical concepts in tandem. Teachers use precise questioning to check conceptual and procedural knowledge. They assess in lessons to identify who requires intervention so that all pupils keep up.

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Extracts from Ofsted’s grade descriptor for good teaching ( Jan 2015 handbook ) Teachers have high expectations. They plan and teach lessons that deepen pupils’ knowledge and understanding… Teachers listen to, carefully observe and skilfully question pupils during lessons in order to reshape tasks and explanations to improve learning. Teachers create a positive climate for learning in lessons … Teachers assess pupils’ learning and progress regularly and accurately … Effective teaching strategies, including well-targeted support and intervention, are matched closely to most pupils’ needs, including those most and least able, so that pupils learn well in lessons.

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Inspecting mathematics teaching ( Summary of para 55, Jan 2015 handbook ) How well does the school identify and tackle inconsistency in the quality of mathematics teaching? How well does teaching: foster mathematical understanding of new concepts and methods, including teachers’ explanations and the way they require pupils to think and reason mathematically ensure that pupils acquire mathematical knowledge and enable them to recall it rapidly and apply it fluently and accurately, including when calculating efficiently use resources and approaches to enable pupils in the class to understand and master the mathematics learnt develop depth of understanding & next-stage readiness enable pupils to solve a variety of mathematical problems

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Pause for thought … … in what ways are Ofsted’s findings, inspection grade descriptors, and guidance on inspecting mathematics, the NC’s aims, and the characteristics of Mastery: aligned in conflict? Discuss!

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The National Curriculum: expectations The NC states: The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage.

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Pause for thought … … what implications does the expectation, ‘the majority of pupils will move through the programmes of study at broadly the same pace’, have for lesson planning and teaching? Does this imply a re-interpretation of what ‘differentiation’ means? Discuss!

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The National Curriculum: differentiation? The NC states: Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.

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Developing fluency … … what does good consolidation look like?

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Practice makes perfect? Compare these two multiplication exercises. Which supports the development of fluency better? Why?

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Pause for thought … … in your school, how is the formula for the area of a rectangle taught?

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Problems and puzzles Problems do not have to be set in real-life contexts. Providing a range of puzzles and other problems helps pupils to reason strategically to: find possible ways into solving a problem sequence an unfolding solution to a problem use recording to help their thinking about the next step. It is particularly important that teachers and teaching assistants stress such reasoning, rather than just checking whether the final answer is correct. All pupils need to learn how to solve problems from the earliest age – the EYFS early learning goals also include problem solving.

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Common weaknesses in teaching problem solving Pupils are expected to acquire problem-solving skills without them being made explicit. Lesson objectives and planning tend to focus on content rather than specific problem-solving skills. Teachers/TAs are too quick to prompt pupils, focusing on getting ‘the answer’ – pupils need to build their confidence and skills in solving problems, so that they can apply them naturally in other situations. When problems are set, teachers do not use them well enough to discuss with pupils alternative approaches and why one is more elegant than another. Problems for high attainers involve harder numbers rather than more demanding reasoning and problem-solving skills.

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Problem solving: nrich.maths.orgnrich.maths.org The nrich website is a good source for problems. It includes printable resources, notes for teachers and solutions written by pupils. Each problem has been mapped against the new NC.

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20cm 5cm Improving problems … … what is the area of this rectangle? Adapt this question to encourage pupils to think harder about how to solve it, and to develop better their problem-solving skills and conceptual understanding of area of a rectangle.

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Development of Maths Capabilities and Confidence in Primary School F-RR118.pdf F-RR118.pdf Developing reasoning … … research by Terezinha Nunes (2009) identified the ability to reason mathematically as the most important factor in a pupil’s success in mathematics.

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Reasoning Reasoning is integral to the development of conceptual understanding and problem-solving skills. Of the three National Curriculum aims, it is the least well developed currently. Not all classrooms have a positive ethos that encourages learning from mistakes. Tasks are not used well enough to develop reasoning. Talk often focuses on the ‘how’ rather than the ‘why’, ‘why not’, and ‘what if’ in: teachers’ explanations and questions pupils’ responses.

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NCETM progression maps The National Centre for Excellence in the Teaching of Mathematics (NCETM) has produced progression maps for different strands of mathematics within the NC at KS1-3 It has added questions to each section within the strands to encourage discussion and reasoning. These include: Such questions are useful to encourage all pupils to think and reason, but also good for challenging the more able.

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