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Mathematics Mastery head teachers, senior leaders & mathematics mastery school leads Introduce yourself to a person on your table who is not from your school. Introduce this person to the rest of your table. Name, role, and three facts. Pioneer partner schools Manchester (1) Birmingham (4) Peterborough (1) Northants (1) London (20) Suffolk (6) Essex (2) Reading (2) Portsmouth (1) Kent (1) Sussex (2) “Mathematics Mastery looks like it has all of the attributes that we are seeking to transform mathematical learning and success in our school.” A belief and a frustration ARK Schools wanted a new maths curriculum to ensure that their aspirations for every child’s mathematics success becomes reality, through significantly raising standards. • Success in mathematics for every child • Close the attainment gap The connections Best practice – national and international Research findings and evidence ARK Schools Mathematics Mastery Curricular principles • Fewer topics in greater depth • Mastery for all pupils • Number sense and place value come first • Problem solving is central Feedback from 2012-13 Problem solving and investigations give pupils the opportunity to demonstrate an in-depth understanding of the topic. Since teaching in a mastery style, I have really had to think about my questioning which has improved my subject knowledge. Agenda 10.00am What's really possible? 11:15am Break 11.30am What does ‘mastery’ look like in the classroom? 12.30pm Lunch 1.00pm Mastery in your school 2:45pm Action planning 3:30pm Questions and answers 4:00 Close Why are we here? Why have you and your school chosen to join the Mathematics Mastery community? What do you want/need to know by the end of today? Why are we here? “We know that no child is limited by their background and that by working hard all children can become excellent mathematicians. ” Research shows: • The gap at age 10 between our strongest and weakest maths performers is one of the widest in TIMSS - with fewer of our pupils overall reaching the very highest levels • The 10% not reaching the expected level at age 7 becomes 20% by age 11 and, in 2012, almost 40% did not gain grade C at GCSE • Girls are less likely than boys to study maths beyond 16 and less confident about their ability overall • Lower income pupils are falling behind in maths International Trends 2009 PISA Nationally, what are we doing well? What are we not doing so well? Maths is not a measuring tool “Mathematics education should be so much more than just passing exams and Mathematics Mastery will help us achieve this. We want every child to not just pass GCSE mathematics but pass with top grades and to leave our school with a love of mathematics. ” Our shared vision • Every school leaver to achieve a strong foundation in mathematics, with no child left behind • A significant proportion of pupils to be in a position to choose to study A-level and degree level science, technology, engineering and mathematics-related subjects What is necessary to make this vision a reality? Who are our students and what are they capable of? “Many of our students come from disadvantaged backgrounds and arrive at secondary school not fully equipped mathematically and so there is a necessity to close the attainment gap.” “We have a key stage two score on entry that is below the national average. This means that in order for our young people to achieve A and A* grades at GCSE mathematics we have to work harder for them and develop a mathematics curriculum that is beyond outstanding.” A/A* B C Above expectations Meeting expectations Below expectations Maths achievement in your school What is your expectation of your students’ achievement in mathematics? • • • Who will study maths-related subjects at degree level? Who will study A-level maths? How will they do? How will students achieve at 16? Consider your new intake of Year 7 students. Collaborative Community “One of the most exciting things about the programme is the idea that we will be able to talk to colleagues in different schools about what we are doing. We would hope to learn from others and to share the ideas we have that we find work well. ” Shared curriculum framework Online • • • • • Task banks Assessments Training Videos Blogs Lesson observation tools Collaborative cluster workshops Mathematics Mastery Launch training • Teachers • Leaders In-school development visits Our approach You say: Conceptual “The mathematics team is firmly committed to a problem solving understanding approach which will equip our students for later life.” Mathematical problem solving Mathematical thinking Language and communication Our approach: problem solving What does it mean to teach through problem solving? What does it mean to teach for problem solving? Problem solving – you say: “Our evidence shows that students are performing markedly worse on problem solving questions in GCSE exams in comparison to procedural questions.” “We promote problem solving within the mathematics department through our use of starters and as part of the differentiated extension tasks. However, deeper problem solving occurs more in the higher sets. Mathematics Mastery will enable us to bring the culture of problem solving in the mixed ability, lower sets and intervention sets.” Mastery for all Represent Mathematical problem solving Generalise Communicate Potential barrier 1: language and communication Represent Mathematical problem solving Generalise Communicate Mastering mathematical language Mathematics Mastery lessons provide opportunities for pupils to communicate and develop mathematical language through: • Sharing essential vocabulary at the beginning of every lesson and insisting on its use throughout • Modelling clear sentence structures using mathematical language • Insisting on correct use of language – “I know what you’re trying to say” as start not end • Talk Tasks • Continuous questioning in all segments which give a further opportunity to assess understanding through pupil explanations Potential barrier 2: reasoning Represent Mathematical problem solving Generalise Communicate Mastering mathematical thinking “Mathematics can be terrific fun; knowing that you can enjoy it is psychologically and intellectually empowering.” (Watson, 2006) We believe that pupils should: • Explore, wonder, question and conjecture • Compare, classify, sort • Experiment, play with possibilities, modify an aspect and see what happens • Make theories and predictions and act purposefully to see what happens, generalise Mathematical thinking – you say “By focusing on fewer topics whilst increasing their skills as independent learners (which fits fantastically with our whole school policy of collaborative learning) we will increase the confidence of a large majority of our students in their key mathematical skills.” Potential barrier 3: conceptual understanding Represent Mathematical problem solving Generalise Communicate What are manipulatives? Bead strings Bar models Dienes blocks Fraction towers 100 grids Conceptual understanding Number lines Cuisenaire rods Mathematical problem solving Multilink cubes Mathematical thinking Shapes Language and communication Let’s do some maths... Problem solving using bar models! • Pupils draw a visual representation of a word problem. • Taught early on in the programme, using concrete and pictorial representations, in the context of the four operations. • Pupils are then expected to use models for fractions, decimals, percentages, algebra, pie charts.... Solving problems with unknowns John gives his brother three marbles. Now his brother has three times as many marbles as John. Altogether they now have sixteen marbles. How many marbles did John have at the start? ? 3 John 16 John’s brother Ben scored 1,866 Abe scored 2,177 Conceptual understanding – you say “It is essential that all of our teachers aim for all our students to clearly understand a mathematical concept rather than simply learning the process.” “Our aim is to teach for understanding, but realistically this is not happening in all classes all the time.” “I feel that the use of concrete manipulatives and a constant focus on problem solving will mean that students are much more able to understand mathematical concepts.” Lesson structure New learning Do Now Talk task Independent task Develop learning Ofsted outstanding: • Planning is astute • Time is used very well • Every opportunity is used to successfully develop crucial skills (inc. literacy and numeracy) • Lessons proceed without interruption • Appropriate independent learning tasks are set • Pupils are resilient, confident and independent • Well judged and often imaginative teaching strategies are used Plenary YOU DON’T ACHIEVE MASTERY BY CLIMBING...YOU ACHIEVE MASTERY THROUGH DEPTH Generalising Modifying Comparing MATHEMATICAL THINKING Curriculum with problem solving at the heart Maths learning in your school What is consistent across the department? What happens in every lesson? What does ‘students’ work’ look like? How are students supported to: • use language to reason and communicate with accuracy? • represent mathematical concepts and techniques? • make connections within mathematics? • make connections beyond mathematics? • think mathematically and solve problems? Using data and evidence Fine grain detailed data analysis on a question level and by national curriculum sub-levels are essential to ensuring that every student is successful The big picture is what’s important – the focus should be on the best way to teach the students, and the best way to teach the concept or technique, with their long term success in mind ‘Big picture’ data can tell us… 1) What the essential concepts and techniques are for students to succeed at A-level and beyond. 2) What the essential concepts and techniques are for students who might otherwise fall behind. 3) That these are the same! 4) The ‘habits of mind’ that students need to succeed a) in maths b) in applying their maths Classroom data 1) In lessons: student responses, student engagement 2) Work scrutiny 3) Student voice 4) Student tracking 5) Case studies – progress of students in challenging circumstances 6) Views of parents 7) Discussion with staff and senior leaders Work scrutiny Classroom data can tell us… 1) Which students are at risk of underachieving 2) Which concepts and techniques need re-teaching/teaching differently next time round 3) Which students need additional challenge 4) Which pedagogic approaches were particularly effective ‘Exam’ data tells us… 1) Areas where the classroom evidence might be misleading 2) How students might perform in high stakes external exams Assessment Pre- and post-module assessments Termly holistic assessments Expectations What does success look like? Year 7 Term Autumn Spring Summer Evidence of success Attainment at end of term Fewer topics, greater depth “Our students have a brief knowledge of a large range of topics but their foundation skills crumble as we approach more complex topics.” “By focusing on fewer topics whilst increasing their skills as independent learners (which fits fantastically with our whole school policy of collaborative learning) we will increase the confidence of a large majority of our students in their key mathematical skills.” “We want to close the achievement gap across all year groups so that it is not necessary to put all of our resources and effort into just the Y11 groups each year and so that maths can be truly understood and embraced by all year groups prior to their GCSE year.” Year 7 Knowledge, understanding and skills by week Weeks Integer place value Round and estimate Half terms Multiple, LCM Add and subtract Word problems Multiplication Multiplication of decimals Rectangle and triangle area Estimate measures Read scales Draw, measure and name angles Angle types Fractions as numbers, fractions as operators Decimal numbers Decimal place value Factor, HCF Division Mean average Triangles Quadrilaterals (Investigation) Multiplicative relationships with fractions Fraction of a quantity Multiply and divide fractions Equivalent fractions Compare and order fractions Order of operations Symbolic notation Interpret pie charts Convert fractions, decimals and percentages Find perimeter (Investigation) Substitute and simplify Percentage of a quantity (Investigations) Half term 1 Half term 2 Half term 3 Half term 4 Half term 5 Half term 6 Number sense Multiplication & division Angle and line properties Fractions Algebraic representation Percentages & pie charts Place value Fractions, decimals and percentages Addition and subtraction Perimeter Multiplication and division Area Using scales Year 7 KEY Half term topic Big idea Substantial new knowledge mastered Angle and line properties Calculating with fractions Algebraic notation Support and challenge Conceptual understanding Mathematical problem solving Mathematical thinking Language and communication Low threshold, high ceiling Pupils compare the area and perimeter of rectangles. Which is bigger? Which is smaller? 8cm 2cm 4cm 4cm Can/does this task involve modifying? How could the teacher prompt this? Can/does this task involve generalising? How could the teacher prompt this? Summary What does Mastery mean? • Assumption is that what has been learnt really has been learnt – this doesn’t happen after one lesson however! • Achieving mastery through ‘less explicit’ repetition • Mastery occurs as a result of repetition through using skills continuously across genres • Taking time to explore, clarify, practice and apply, not memorise • Valuing concrete and pictorial, not rushing to the abstract Supporting, monitoring and evaluating What to look for in a lesson • evidence of C-P-A • use of language structures • great questioning • 100% involvement • AfL Looking at a lesson What is 0.27 plus 1.009? It’s one point two seven nine. That means it has nine thousandths. Good. How many tenths are there in 0.27? There are two tenths, in 0.27 and two tenths in the answer. Actually there are twelve tenths aren’t there? Yes. There are. What is 1.27 plus 1.009? Supporting, monitoring and evaluating Conceptual understanding Mathematical problem solving Mathematical thinking Language and communication Looking at a lesson I know you can already do this, but you need to show this using manipulatives. But why, sir? We’re a top set. It’s important. Show me which is bigger, 0.2 multiplied by £25 or 250% of £2. How many different ways can you find of showing this. Supporting, monitoring and evaluating Conceptual understanding Mathematical problem solving Mathematical thinking Language and communication Looking at a lesson What is a fraction? It’s a number less than one. Good. Any examples? It’s like three quarters, the top number is divided by the bottom number. Brilliant. Division. Anyone else? It’s part of a whole. Excellent! Supporting, monitoring and evaluating Conceptual understanding Mathematical problem solving Mathematical thinking Language and communication What are you looking for? New learning Do Now Talk task Independent task Develop learning Plenary Ofsted: Made to measure (2011) Evidence from successful schools •Pupil collaboration and discussion of work •Mixture of group tasks, exploratory activities and independent tasks •Focus on concepts, not on teaching rules •All pupils tackled a wide variety of problems •Use of hands on resources and visual images •Consistent approaches and use of visual images and models •Importance of good teacher subject-knowledge and subjectspecific skills •Collaborative discussion of tasks amongst teachers Do the maths I can't stress enough how vital it is for teachers to complete the tasks before teaching them. By doing this I have found it far easier to anticipate what my students might do or where they may struggle (particularly with the open problems) so I can plan scaffolding carefully. On the occasions I haven't done this I have ended up having very 'teacher led' lessons which were not as effective. Emily Hudsmith, Head of Maths Charter Academy