How to get students to work mathematically

The new secondary school curriculum demands that KS3 students be capable at “working mathematically”. But what does this entail? In this article, Barbara Miller breaks down the underlying concepts into helpful summaries.

The new curriculum considers “working mathematically” to be achieved when students can solve problems, reason mathematically and are mathematically fluent. But interpreting those objectives on a year-by-year basis has been left inadequately vague. In this issue’s ‘Strategies for the new curriculum’, our Maths expert, Barbara Miller, examines the key curriculum objectives relating to working mathematically and shares a variety of approaches for teaching them in KS3.

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      string(5998) "England has lagged behind the performance of others, as measured by international studies such as the Trends in International Maths and Science Study (TIMSS) and the Programme for International Student Assessment (PISA).  Results from the TIMSS report in 2016 for eight year olds showed that while England has risen to its highest point in these rankings for 20 years for mathematics, the gap between the lowest performing and the highest performing children remains too wide. There is also a widening gap between the performance of boys and girls. Performance in the PISA tests for 15 year olds show that while pupils in the United Kingdom achieve above the OECD average for science and reading, performance in mathematics is around average, and has remained stable since 2006. This article explores how the Mathematics Mastery programme, developed to overcome the difficulties faced by many children in developing a secure understanding of mathematical concepts, can improve outcomes in mathematics while supporting lower attainers and reducing the attainment gap.Drawing on the evidence from high-performing jurisdictions such as Singapore and Shanghai, which use the mastery approach to teaching maths, Mathematics Mastery began in 30 pilot schools in 2012 and by the end of 2016, the partnership had grown to 264 partner primary schools. Mathematics Mastery, founded by Dr Helen Drury, is a not-for-profit organisation; partner schools pay an annual subscription to gain access to their resources.What is Mathematics Mastery?Mathematics Mastery is based around the key principles of mathematical thinking, language and communication, and conceptual understanding, underpinned by problem-solving.  Referred to as the dimensions of depth, they are explained more fully in the following diagram:   What is mastery teaching?Mastery teaching has become the new buzzword in education. People often tell me that they are already ‘doing’ maths mastery in their schools. In further discussion it seems that they are using commercially available schemes, textbooks as well as resources from the NCETM’s Maths Hubs (National Centre for Excellence in the Teaching of Mathematics), sometimes with great success – usually less so.  So what do we understand mastery teaching to be? Mastery teaching has moved on from the approach first proposed by Benjamin Bloom in 1968, when it was referred to as ‘learning for mastery’. Then a rather narrow view of mastery meant that children were expected to attain a certain level before moving on to learn new material. Children who did not reach the standard were given additional tuition in order to get there and could not move on until they had acquired the concepts. What would this have indicated to children with additional learning needs? At Mathematics Mastery, our definition of mastery is that pupils learn to apply what they have learned to a totally new and unfamiliar situation. We don’t hold pupils back if they have not ‘got’ something (although we would suggest that pupils with difficulties are given additional support outside of the daily mathematics lesson to keep with their peers), nor do we advance pupils into new concepts beyond what others are learning.What does Mathematics Mastery look like in the classroom?High expectations for allWe believe that all pupils should have access to high quality teaching.  Pupils in Mathematics Mastery schools are not set or streamed.  We don’t label children as SEND, talk about the ‘bottom group’ or ‘lower ability’. These are banned words. Instead, all pupils quickly learn from their mistakes, understand that challenge and hard work are exciting and worthwhile, and learn to persevere with their learning.  A teacher told me that almost immediately after starting the approach, pupils in her class developed more confidence in mathematics and that the differences between higher and lower attaining children were not as obvious as they had been before using the programme. Daily interventions are used to help targeted children to keep up with the learning rather than catch up after gaps have widened. Pupils quickly begin to participate in all parts of the maths lesson and to enjoy their learning. 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Mathematics

Mathematics Mastery

20 Jun 2017

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      string(5787) "Global research and good practice consistently delivers the same message: learning is greatly affected by what learners perceive they can or cannot do. What’s more, it is also shaped by what their parents, carers and teachers perceive they can do. Mindset – the thinking that determines our beliefs and behaviours – therefore has a fundamental impact on teaching and learning.Mastery methods in maths focus on the distinction between ‘fixed’ and ‘growth’ mindsets1. Those with a fixed mindset believe that their basic qualities (e.g. intelligence, talent and ability to learn) are pre-wired or fixed traits: ‘If you have a talent for maths, you’ll succeed at it. If not, too bad!’. By contrast, those with a growth mindset believe that hard work, effort and commitment drive success and that ‘smart’ is not something you are or are not, but something you become. In short, everyone can do maths!A growth mindset needs to be actively nurtured and developed. The methods used when teaching for mastery in maths focus on the same dispositions we use with a ‘growth’ mindset. Here are some key strategies for fostering healthy growth mindsets in your maths classroom.1. Remember that mistakes are GREATIt’s important to see mistakes as opportunities for learning. When presenting maths problems to a class, you could suggest incorrect answers as well as correct ones. You can then use these as opportunities to discuss children’s reasoning and encourage them to explain to each other what they think and why they think it. At its peak, in a growth mindset maths classroom, children love getting answers wrong as it’s an opportunity to learn and discuss. Mistakes are valuable opportunities to re-think and understand more deeply. Learning is richer when children and teachers alike focus on spotting and sharing mistakes as well as solutions. Encourage children to spot their own mistakes and the mistakes of others, discuss them and find a solution together.2. Use the right languageThe language we use around learners has a profound effect on their mindsets. Make a habit of using growth phrases, such as, ‘Everyone can!’, ‘Mistakes can help you learn’ and ‘Just try for a little longer’. The king of them all is just one little word, ‘yet’… I can’t solve this…yet!’ Encourage parents and carers to use the right language too and celebrate children’s efforts and not the results. When children talk purposefully together about maths, barriers of fear and anxiety are broken down. Children should be encouraged to use full sentences when reasoning, explaining or discussing maths and key vocabulary should be used precisely. Don’t feel you need to dilute language for children – most children will embrace the challenge of learning new maths words. 3. There is no one right answer or representationTry to choose tasks where maths is shown in different ways and there is no one right representation or answer. This is often called variation and is a key feature of intelligent practice. For instance, don’t always show a quantity as apples – it could be people, shapes, things on the table or images on the whiteboard. It’s also important to show that there are many ways of getting to an answer in maths and while some may be more efficient ways than others, it’s critical that children see and understand each others’ methods and learn from one another. 4. Make growth mindset relevant to children’s interests You could draw on pupils’ role models and interests when trying to convey the importance of ‘mindset’ in learning. For example, why not explain how musicians, sports personalities or actors have succeeded by working hard, practising and not just relying on being naturally gifted or talented. You can then explain how this mindset can be applied to the maths classroom and when solving maths problems. Talk about your learning preferences as a class, reinforcing how together you have the skills to tackle any problem. 5. Be a growth mindset role model yourselfIt can be helpful for pupils to understand that their teachers find things difficult at times too. Don’t be afraid to get things wrong – show pupils how you might have to try hard to succeed and allow them to help you find a solution. Exploring maths together in this way could be reflected in the way you currently structure your maths lessons. Do lessons give all children the opportunity to access the same tasks and achieve the same learning goals? Low-threshold, high-ceiling tasks are great for getting all children working together and using growth mindset can help you model successful strategies.6. Show that struggle brings successDo you celebrate when a pupil has got to an answer through struggle? Make sure you are rewarding such efforts as it encourages others to persevere when they are struggling too.  Creating opportunities for success through a step by small step approach enables children to enjoy the experience of accomplishment. It can be best to avoid ability grouping and encourage every child to answer questions and explain or demonstrate their methods to others. Praise is a great motivator, and by focusing on praising effort and learning rather than success, children will be more willing to try harder, take risks and persist for longer. 7. Don’t just give it lip-serviceIt’s very easy to have a growth mindset display or to hold a one-off assembly on how to use a growth mindset, but how do you make it deeper? How do you make is consistent? How do you give"
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Growth Mindset

Ten Tips For Using A Growth Mindset In Maths

2 Jul 2018

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      string(4931) "engagement with their children’s learning. A further advantage of the program is that it is online, so can be used at home as well as at school. We would therefore like to use it more consistently for homework, which would have the dual benefit of saving us time on marking!A step that we’ve already made is to purchase additional licenses for our mainstream classes. We’ve already started talks with the maths department about using the resource as an intervention tool. We are excited to continue using game-based learning to raise standards at Bebington High!The benefits of game-based learning: 1. Supports less-confident learners For many less confident learners, traditional classroom practices can often fail to engage children in advancing their learning. In subjects sometimes perceived as ‘scarier’, such as Maths, it’s not uncommon to encounter pupils who simply shut down when faced with a tricky question they aren’t immediately able to answer. Often this doesn’t mean that a child isn’t capable of discovering the answer, merely that a fear of maths is preventing them from attempting to solve the problem. Game-based learning offers a non-threatening and supportive environment where students can explore their skills, without feeling like they are failing—helping students to feel secure in their ability and motivated to improve.2. Encourages mastery Getting students to use game-based learning can be a useful way of prompting them to revisit concepts until they improve and have really understood the methods. Achieving mastery is the key to being able to answer any question thrown at students in an exam. Surveys have found that learners generally recall just 10 per cent of what they read and 20 per cent of what they hear. If there are visuals accompanying an oral presentation, the number rises to 30 per cent, and if they observe someone carrying out an action while explaining it, 50 per cent. But learners remember 90 per cent ‘if they do the job themselves’, even if only as a simulation. Learners’ favourite gamification techniques include the ability to progress to different levels, receiving scores and playing through an avatar—all often found in high-quality digital education resources. 3. Allows autonomyThe vast majority of game-based resources are learner-led, meaning that students must progress in order to complete a task. All teachers are familiar with the benefits of a learner-led approach rather than teacher-based practice. Students who may not consider themselves ‘good at school’ have the chance to see that progression is something everyone can achieve—not just the perceived high-flyers. Similarly, student-led learning provides greater opportunity for students to make mistakes and try again, seeing that one-time failure doesn’t brand them as a failure for life. Empowering students through independent learning will have a huge impact on their skills and ultimately give them more confidence when it comes to tackling exams. Unlike exams, games give constant feedback and keep the stakes low.4. Reinforces classroom practice The majority of online resources are accessible from home, as well as school, and can be accessed on multiple or different devices. This not only saves time for teachers in setting and marking homework, but also provides a more seamless integration between work completed at home and at school. If a child is struggling with a particular concept and there isn’t time to address this fully in class, homework can be set to encourage a student to practise this skill at home. A good online tool will allow a teacher to view progress and if a student is still struggling, this can be picked up on back in school. 5. Builds on children’s existing hobbiesLet’s not forget that students are playing games when they are out of school. 57 per cent of the UK’s population are ‘gamers’ in their spare time, with the largest single age/demographic being 15–24-year-old males making up 16 per cent of all players.So if students are voluntarily investing countless hours in developing gaming skills outside of school, it has to be sensible to show them how the experience they gain through extended practice and strategies such as ‘risk-taking’, reasoning and problem-solving can be harnessed and used successfully in the classroom too.By playing an educational ‘game’, students spend more time actively learning and reinforcing their knowledge, which can provide for a proper flipped learning experience. By incentivising this effort, we support that human desire to self-improve—and just as importantly, allow students to demonstrate that improvement to themselves and others.Christine Edge-Sayer is Assistant SENCO and Head of Resourced Base Provision at Bebington High. "
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Creative Teaching & Learning

Motivating for Maths Mastery

8 Jul 2019

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Leadership Briefing 10.12(120)"
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      string(5779) "Leadership BriefingLeadership Briefing – www.teachingtimes.comReports Summaries – a weeklyservice of essential summaries,in-depth reports and criticalanalysis that’s new and relevantto education, linked to originalsource documents.EVALUATION OF THE MATHS TEACHEREXCHANGE: CHINA AND ENGLANDKeywordsEvaluation,programme,Maths TeacherExchange,China, England,primary schools,implementation,teachingSummaryThis interim DfE report on the first year of the maths teacher exchange programme betweenChina and England evaluates how primary schools in England are implementing learning fromthe exchange.The Mathematics Teacher Exchange is a highly innovative programme that aims to foster a radicalshift in primary mathematics teaching in England by learning from Shanghai mathematics education– a mastery approach to teaching and learning. The exchange is funded and managed by theDepartment for Education. The National Centre for Excellence in the Teaching of Mathematics(NCETM) lead the implementation of the exchange through the national network of Maths Hubs andare actively engaged in developing, refining and deepening school and teachers’ understanding ofShanghai teaching for mastery.The first year of the evaluation has focused on the implementation of the exchange and the waysin which the 48 participating schools – lead primary schools – have implemented learning from theexchange.Key Findings: ■ The visits of English teachers to Shanghai had a strong impact on teachers’ beliefs andmotivations towards teaching mathematics, although a minority (just over a fifth) of teachersstated that they would have preferred to spend more time in school than in university. ■ The visits of Chinese teachers to England were identified as particularly good opportunities forlearning by English teachers, who valued the chance to observe teaching and participate in post-lesson discussions in the English context. ■ Maths Hubs differed in their approach to implementing the exchange in two aspects: ■ Their orientation towards mastery. Existing mastery enthusiasts used the exchange toadd impetus to existing approaches within the hub. More recent converts became moreconvinced through participation in the exchange, but voiced concern that other teachers andschools may be harder to convince. ■ The steer and support provided. Some hubs acted as project managers providing logisticalsupport to lead primary schools, who then led on which aspects of the Shanghai approachwere adopted, and organised professional development. Other hubs took a more activestance, encouraging schools to adopt or experiment with particular aspects of mastery. ■ The schools varied in their attitudes and commitment towards Shanghai mastery approaches.Week Commencing 18th July 2016https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/536003/Mathematics_Teacher_Exchange_Interim_Report_FINAL_040716.pdfhttps://www.gov.uk/government/uploads/system/uploads/attachment_data/file/536003/Mathematics_Teacher_Exchange_Interim_Report_FINAL_040716.pdfLeadership BriefingLeadership Briefing – www.teachingtimes.comReports Summaries – a weeklyservice of essential summaries,in-depth reports and criticalanalysis that’s new and relevantto education, linked to originalsource documents.Of the 48 lead primary schools, seven were considered to be already committed to Shanghaimastery, whilst 36 were newly committed as a result of participating in the exchange. Fiveschools were more cautious in their outlook. ■ Teachers continued with aspects of Shanghai teaching practice with classes after the Shanghaiteachers’ visits. Patterns of implementation across other year groups were complex. In somecases, schools adopted particular practices such as same day intervention or changes in lessonstructure on a whole school basis. Other schools adopted a strategy of gradual implementation ofchange in multiple practices in specific year groups. ■ Overall, of the 48 schools, 38 were making medium to high levels of change. A medium level ofchange was found in 32 schools, comprising more extensive change in specific year groups oradoption of some Shanghai informed practices across multiple year groups. In six schools, ahigh level of change occurred with significant changes of multiple practices on a whole or nearlywhole school basis. Ten were making relatively smaller changes across the school in 2014/15,with changes restricted to continuing some Shanghai practices after the visit. Of these, fourexpressed a commitment to mastery and were planning changes for 2015/16, and in one, thelevel of change was smaller because their existing practices were already aligned with masteryapproaches. ■ Changes in both classroom practice and the organisation of maths in schools were found. Mostchange took place in classroom practices, with an overall increase in the amount of whole classinteractive teaching. 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Leadership

Evaluation Of The Maths Teacher Exchange: China And England

14 Jul 2016