New for the 2014 National Curriculum The key to success in KS3 mathematics Why nurturing confidence raises achievement The thinking behind 10 key principles of www.pearsonschools.co.uk/progresswithks3maths
Why nurturing confidence raises achievement Nurture students confidence so that they can work independently, take risks and persevere, and they will experience success. If students do believe they are no good at mathematics, they are likely to give up before they have really tried. That s why research shows a strong link between confidence and achievement in mathematics. And that s why nurturing confidence is at the heart of our new Key Stage 3 Mathematics course. The thinking behind In preparing our new KS3 Maths course, leading mathematics education researchers and Key Stage 3 teachers across a range of schools: l closely examined the new National Curriculum for Mathematics (including Key Stage 2) l appraised the best performing countries in mathematics l reviewed some recent and most commonly cited research papers in mathematics education* l audited the most respected and well-used mathematics resources, both print and digital. As a result we ve developed a brand-new approach to Key Stage 3 Mathematics, designed to nurture confidence and raise achievement. The 10 Key Principles of l Fluency l Mathematical Reasoning l Multiplicative Reasoning l Problem Solving l Progression l Concrete Pictorial Abstract (CPA) l Relevance l Modelling l Reflection l Linking This guide explains each of these principles in more detail, describing how they raise confidence and raise achievement and how they appear throughout the course. 2
An innovative structure for Each Unit is organised as follows: Master Check Strengthen Extend Test 1 Students are helped to master fundamental knowledge and skills over a series of lessons. 3 They decide on their personalised route through the rest of the Unit: 4 Finally, students do a test to determine their progression across the Unit. 2 Before moving on with the rest of the Unit, they check their understanding in a short formative assessment, and give an indication of their confidence level. In areas where they have yet to develop a solid understanding and/or they do not feel confident, they can choose to strengthen their learning; In areas where they performed well in the assessment and also feel confident, they can choose to extend their learning. Each Unit is supported by a range of learning materials designed to promote confidence, by encouraging students to visualise concepts, discuss their understanding and reflect on their learning. How this structure nurtures confidence and raises achievement l Students who do not master the topic in the first few lessons do not carry on regardless. Instead, they have the opportunity to revisit key concepts, explained in a different way, and at a slower pace. l Students who do master the topic in the first few lessons do not simply do more of the same. Instead, they are challenged by increasing the breadth and depth of their understanding. 3
Fluency When certain mathematics facts are so familiar that their recall or use is automatic, this is called fluency. How fluency nurtures confidence and raises achievement Psychologists believe we only have a limited amount of brain power to apply to mathematics learning and problem-solving at any one time. This means that if we can rely on the speedy retrieval of known facts to help us, we have spare brain power to confidently engage in the trickier elements of a problem. For this reason, good mathematical fluency is known to enhance students mathematics achievement.* This is also why developing fluency is now a key feature of the new National Curriculum for Mathematics and will play an important part in progressing to GCSE Mathematics. How fluency appears in Lessons begin with questions for students to develop their mathematical fluency in the facts and skills they will soon be using. Students also have the opportunity to practise their fluency in questions that require them to use previously learnt knowledge. ActiveTeach Presentation provides a digital front-ofclass resource for building and embedding fluency, which students consolidate through online homework in ActiveCourse. There is also teacher support on how to build fluency in each lesson in ActiveTeach Planning. Taken from: KS3 Maths ActiveCourse 4
Mathematical reasoning When describing why mathematical knowledge, skills or methods are used, this is called mathematical reasoning.. How reasoning nurtures confidence and raises achievement Students may be able to state what they know and show what they can do in mathematics, but it is not until they can reason why, that they truly demonstrate their understanding. Only then can they describe why they chose a particular approach, why that approach may be better than another, and why their final solution made sense. In essence, only then can they be confident in their answers. Therefore, it is no surprise that research shows a strong relationship between students reasoning ability and their mathematical achievement.* As a result, reasoning mathematically is now a key feature of the new National Curriculum for Mathematics and will play an important part in progressing to GCSE Mathematics. How reasoning mathematically appears in Through the course, lessons often include prompts that encourage students to explain their reasoning. Sometimes these appear at the end of a problem. At other times, these appear as a statement for discussion, where students are asked to give a reason why the statement is correct or incorrect. 5
Multiplicative reasoning When deliberately using multiplication (or division) to compare quantities and work out the value of one based on the values of others, this is called multiplicative reasoning. How multiplicative reasoning nurtures confidence and raises achievement Knowing if, when and why relationships are multiplicative is essential to the understanding of many Key Stage 3 concepts, among them ratio, proportion, percentages, area, volume, sequences and functions. This is why multiplicative reasoning is now a key feature of the new National Curriculum for Mathematics and will play an important part in progressing to GCSE Mathematics. Research shows that students gain confidence in multiplicative reasoning when they actually see multiplicative connections illustrated in diagrams, such as bar models. This, in turn, deepens their understanding of what is happening to one quantity as the other changes. Therefore, it helps them choose the correct calculations to do to solve problems in mathematical and real-world contexts.* How multiplicative reasoning appears in Lessons addressing concepts dependent on multiplicative reasoning include pictorial representations to support students understanding. Typically multiplicative relations will be illustrated in bar models or tables, or on number lines or graphs in ActiveCourse, ActiveTeach Presentation and in the Student Books. 6
Problem-solving When decisions have to be made about the steps to take to tackle a mathematical task, this is called mathematics problem-solving. How problem-solving nurtures confidence and raises achievement There is strong evidence to suggest that when students are taught strategies for mathematics problem-solving they become increasingly willing to have a go and persevere at tasks. This is because as they experience success, they gain confidence in their ability, and so become ever-more willing to tackle unfamiliar problems using such strategies. As a result, these students are better equipped to succeed in tests and exams and consequently in future study and employment. This is why mathematical problemsolving is central to the curriculum of high-performing jurisdictions.* Problem-solving is now a key feature of the new National Curriculum for Mathematics and will play an important part in progressing to GCSE Mathematics. How problem-solving appears in Lessons not only include problem-solving tasks, but also offer strategy hints, like draw a diagram, try working backwards, or first of all use easier numbers. Once students have met a range of these strategies, hints are removed and students are simply asked to tackle the problem, choosing a strategy for themselves. Problem-solving isn t just questions in contexts. ActiveTeach Presentation presents innovative problem-solving videos, where mathematical problems are shown in a real-world context and the process of solving them is deconstructed for students step-by-step. ActiveCourse offers plenty of problem-solving practice, with each question having an interactive worked example that breaks the problem down into simple steps. 7
Progression When the order of mathematics teaching is carefully planned and mathematical understanding is carefully scaffolded, this secures progression in students learning through KS3 and on to KS4 and beyond. How progression nurtures confidence and raises achievement Success in mathematics is dependent on offering students just the right amount of challenge, at just the right moment. This means ensuring all the necessary prior knowledge, skills and understanding are in place, and gradually building to enable students to progress. Research suggests that when students are made explicitly aware of this progression, not just topic-by-topic, but lesson-by-lesson, then their confidence and performance improve.* How progression appears in Students follow a Pi (Tier 1), Theta (Tier 2) or Delta (Tier 3) route. Within each route, we have worked with teachers to achieve the best possible order for teaching topics, as well as the most appropriate progression routes through the Units themselves (mastering first, then choosing to strengthen or extend). There is smooth progression in lessons, both in the introduction of new concepts, the comprehensive coverage of fluency, problem-solving and reasoning and the pace of the questions. ActiveTeach Presentation provides adaptive multiple-choice exercises an innovative classroom test that adapts to the students success and provides harder questions when they are ready. In ActiveCourse, progression is measured and tracked by ability and by level of confidence. Taken from: KS3 Maths ActiveCourse 8
Concrete Pictorial Abstract (CPA) When students learn mathematics with objects, then pictures, followed by notation, this is called a concrete-pictorial-abstract (CPA) approach. How CPA nurtures confidence and raises achievement Sometimes students may start with the concrete manipulation of objects (be they blocks, sticks, dice); at other times they may begin with pictorial representations (whether moving animations, diagrams, charts, graphs, or bar models). They can then move onto abstract ideas expressed in symbols, numbers and letters. Bar models are rectangular bars that have been proven to help students visualise problems: Question: Pen and Dave hire a limousine together at a cost of 287. Pen pays 6 times more than Dave. How much does Dave pay? Answer: Pen 287 Dave Taken from KS3 Maths Student Book Theta 1 Dave pays 287 7 = 41 CPA is a common approach to teaching mathematics in high-performing jurisdictions, and research shows that it gives students the time and space for conceptual understanding to develop and confidence to grow.* How CPA appears in The Pi (Tier 1) route leads wherever possible with concrete representations of problems. In the Theta (Tier 2) route, the emphasis is more on pictorial representations, with students encouraged towards abstract understanding. Then, in the Delta (Tier 3) route, the focus is on abstract representations of mathematics, supported by some pictorial ideas. The CPA approach is supported by ActiveTeach Presentation, a visually rich resource including hundreds of videos and digital animations to help students visualise mathematical concepts. In ActiveTeach Planning, a series of teacher videos provide lesson ideas for the new and trickier national curriculum topics, with the ideas arranged along concrete-pictorial-abstract progression. 9
Relevance When mathematics is used and applied in the real world, then its relevance is demonstrated. How relevance nurtures confidence and raises achievement The purpose of learning mathematics becomes clear to students when they are faced with real (rather than contrived) problems that they can relate to, such as: How many toilets are needed at the Glastonbury Festival? Is it cheaper to buy an ipad in the UK, France or the USA? How can you scale a small painting to make it street art on the side of a building? Research demonstrates that making students aware of the need for mathematics in these sorts of real-life situations increases motivation. At the same time, it gives everyone an opportunity to access mathematical problem-solving, and so increases confidence.* How relevance appears Lessons start with a real-life problem for students to Explore, loosely at first, then fully at the end of the lesson (having learned the necessary mathematical knowledge and skills). This shows students how much they have learnt and how it may be useful. Lessons are also dotted with further questions labelled real, which use teen-friendly, STEM or financial contexts. At appropriate points, whole lessons related to STEM or Finance are integrated in the course. These deliver learning objectives through a single theme each time, and are fully supported in ActiveTeach Planning. 10
Modelling When an attempt is made to understand and describe a real-world situation in mathematical terms, this is called mathematical modelling. How modelling nurtures confidence and raises achievement Give students opportunities to model and they begin to understand how mathematics can provide insights into important real-world situations. This is known to have a motivational effect. There is also evidence that modelling deepens students understanding of the concepts they bring to bear in devising, testing and evaluating a model. As well as improving their performance in mathematics, this also improves how they do in mathsrelated fields, such as STEM and finance, beyond school.* Therefore, modelling realistic situations is now a key feature of the new National Curriculum for Mathematics and will play an important part in progressing to GCSE Mathematics. How modelling appears in Lessons include problems labelled modelling, often drawn from STEM or finance. These mostly ask students to test a model in some simple way and then give reasons for whether it is a good model or not. 11
Reflection (Metacognition) When thinking about the processes and mental strategies involved when doing mathematics, this is called reflection or metacognition. How reflection nurtures confidence and raises achievement As they do mathematics, if students reflect on what they are doing, how they are doing it, and why they are taking a particular course of action, then they gain valuable insights into the way that they learn. If, afterwards, they are also encouraged to consider the understanding they gained, what they found easy or difficult, the mistakes that they made, and the merits of different approaches, they can confidently adjust how they do things in the future. Research demonstrates that students who regularly reflect in this manner demonstrate greater perseverance and success at solving mathematics problems.* How reflection appears in Lessons and Units end with Reflect questions that ask students to examine their thinking and understanding, emphasising the important role of reflection in learning mathematics. Adaptive multiple choice quizzes in ActiveTeach Presentation provide an innovative in-class resource enabling teachers to capture students confidence levels. The quizzes ask students to reflect on their level of certainty when giving an answer: are they just guessing, feeling doubtful or confident? This gives much greater insight into students results at the end of each quiz. ActiveCourse allows students to log their confidence level for each question, and also feed back any problems they have to the teacher through the interactive workspace or comments field. Guidance is given to teachers on using reflection in the classroom in ActiveTeach Planning for each lesson. Taken from: KS3 Maths ActiveCourse 12
Linking When the connections between concepts within mathematics, or with other subjects, are realised, this is called linking. How linking nurtures confidence and raises achievement Students often perceive mathematics as a series of discrete topics and concepts. However, research shows that when students recognise the links between two concepts, their understanding of both of them deepens. This is because they have the opportunity to reinforce concepts previously mastered, and to use them again but in new contexts. Then, as their confidence in mathematics increases, research shows students become more willing to apply mathematical knowledge and skills in other subjects, such as science and geography.* How linking appears in Lessons begin with Warm-up questions where students are often asked to demonstrate their knowledge or skills in related concepts. Progression through the whole course, whether in ActiveCourse, ActiveTeach Presentation or the Student Books, is predicated on the idea of linking topics and concepts. Questions are structured to encourage students to use and apply prior knowledge and understanding as an integral part of their learning. The progression structure for KS3 Maths was developed in conjunction with a group of teachers from around the country, reviewers and our Series Editors. Throughout lessons, links are made to previously mastered topics, as well as contexts drawn from other subjects. Every lesson also has a list of cross-topic and subject links. 13
Series Editors Dr Naomi Norman Katherine Pate Progress with Our innovative Key Stage 3 Maths course embeds evidencebased approaches throughout our trusted suite of digital and print resources, to create confident and numerate students able to progress to KS4 and beyond. Pedagogy at the heart Our new course is built around a pedagogy based on leading mathematics educational research and best practice from teachers in the UK. The result is an innovative learning structure based around 10 key principles designed to nurture confidence and raise achievement. Progression to Key Stage 4 - In line with the new National Curriculum, there is a strong focus based on evidence on the pedagogical approaches for fluency, problem-solving, reasoning and progression to help prepare your students to advance. Nurturing confidence to raise achievement - Our course promotes confidence by encouraging students to understand and reflect on their learning. Stretch, challenge and support Differentiated materials catering for students of all abilities. Discover Sign up for your free evalua www.pearsonschools.co.u
confidence Learning beyond the classroom Focussing on breaking barriers to independent learning, ActiveCourse online homework is linked to every lesson to offer students extra practice, and a chance to reflect on their learning with our confidencechecker. Powerful reporting tools can be used to track student progression and confidence levels. Tried and tested front-of-class support Our enhanced online service ActiveTeach Presentation makes the Student Books available for display on your whiteboard along with an extensive range of videos and animations, to help your class progress their conceptual understanding at the right speed. Easy to plan and teach Our new digital Teacher Guides are full of comprehensive lesson plans and editable schemes of work that provide teacher support for the pedagogy and link to frontof-class ActiveTeach Presentation material and ActiveCourse homework activities. Practice to progress An extensive range of practice across topics and abilities. Student Books, write-in Progression Workbooks and ActiveCourse, ensure there is plenty of practice available in a variety of formats to suit both classroom and independent learning. now! tion and Student Book at: k/progresswithks3maths
Progress with confidence Our innovative Key Stage 3 Maths course embeds a modern pedagogical approach around our trusted suite of digital and print resources, to create confident and numerate students ready to progress further. Discover now! Sign up for free evaluation at: www.pearsonschools.co.uk/progresswithks3maths Pearson 11-19 Mathematics for Progression aims to create a new generation of numerate and confident young people who feel well equipped to progress through their Mathematics studies and beyond. The courses that build into Pearson s 11-19 Mathematics for Progression framework embed a deep and broad understanding of mathematics. They promote a can-do philosophy that prepares students in mathematical fluency, reasoning and problem-solving as well as specific key stage or qualification learning and teaching needs. These core principles - created to consistently track, build upon and promote progression - are delivered in the secondary Mathematics services at Pearson through evidence-based and best practice approaches. S876 S13MAT01513