Motivating for Maths Mastery

Using elements of play, independent and self-directed learning, all within a game-based framework, Bebington High has found a way to inspire students who are behind in maths. Christine Edge-Sayer shows how her school has successfully engaged and supported their most reluctant students.

Gamification or ‘game-based learning’ put simply, means using elements of play to help children learn. This can include point-scoring, competing against other pupils, rules and rewards for success. While hardly a new concept, the constant evolution of the online world means that game-based learning is taking on new and increasingly innovative approaches to help engage pupils in learning.

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      string(5912) "colleagues in secondary who are specialists. Mastery might actually be a way for many teachers to improve their own maths. In Shanghai, maths at both primary and secondary is taught by mathematics specialists who teach one to three 35 minute lessons daily and spend the rest of their time planning collaboratively, marking pupils’ work and engaging in teaching research. Interestingly, these teachers also hold regular interventions in small groups to address areas of difficulty and provide pupils with extra practise on the same day.Unbelievably, Shanghai teachers are also expected to complete 240 hours of further training every five years. Compare this to UK teachers who might not even get any formal CPD now thanks to strangled budgets.  The situation is different in Singapore where specialist maths teaching is the norm only in upper primary and secondary. Their teachers are also ‘entitled’ to a whopping 100 hours of professional development every year.Maths mastery has been seductive and there are many evangelists singing its praises but what’s the evidence? Does it work? Can it really be a magic bullet? Yes, within high performing countries, disadvantaged pupils do as well as their peers (PISA, 2014). The mastery approach is credited with boosting students in China, Hong Kong and Singapore to the top of the OECD’s Pisa rankings. What works well in one part of the world is worth investigating however, we need to exercise caution against assuming a simple causal relationship between the systems in high performing jurisdictions and their outcomes. Outcomes are the result of more than the education system and as Pasi Sahlberg says in his Foreword to Teach Like Finland by Timothy D. Walker, “…I think it is impossible to transfer education systems from one place to another. Education systems are complex cultural, organic entities like plants or trees that grow well only in their home soil and climate.” What works in Shanghai might work in Swindon but the parental attitudes and wider cultural values at work are going to have an impact. Still, this hasn’t stopped Collins Learning from producing ‘Real Shanghai Mathematics’ a primary mathematics programme for Years 1-6 which will be directly translated from the Chinese with no revisions except a currency change from renminbi symbols to British pound signs.It would be too far-fetched to say that mastery has been a huge con and we have fallen for it hook, line and sinker. Mastery comes in all shapes and sizes and the ideas around it have been interpreted and developed in different ways – context is king and it depends which wind you are standing in. There is no special curriculum which is ‘The Maths Mastery Curriculum’ but ‘mastery’ is used so freely and generically that it is easy to see why there might be a misunderstanding that there is only one maths mastery approach. Over the last few years there have been a number of  ‘providers’ and publishers eager to put their own spin on maths mastery as a way to teach maths. Inspired by Singapore and Shanghai, they have evolved their own philosophies and ‘sold’ mastery to teachers as an engaging and accessible style of maths teaching full of promise. Put simply, whilst the general principles might be similar, the mastery approach can vary which means schools can get quite a different version of it depending on what and who they ‘buy into’.  Many favour the Singapore brand of mastery to the Shanghai approach because the Singaporean system is more closely aligned to the English curriculum whereas the Shanghai method relies on specialist maths teachers. Whether or not introducing East Asian teaching methods into western schools is a good idea is something that has been independently evaluated for the Education Endowment Foundation by  Dr John Jerrim at the Institute of Education, University of London and Professor Anna Vignoles at the University of Cambridge. They evaluated the ARK ‘Maths Mastery’ programme, a Singaporean inspired ‘mastery’ teaching approach adopted by a selection of England’s primary and secondary schools and endorsed by government ministers. This branded variant of the mastery model is made up of three parts: Deep understanding, through a curriculum that combines universal high expectations with spending more time on fewer topics and heavy emphasis on problem-solving.Integrated professional development through workshops, visits, coaching and mentoring and ‘access to exclusive online teaching and learning materials, including lesson guides for each week’.Teacher collaboration – primary schools are allocated a geographical cluster of 4-6 schools while secondary schools attend a ‘national collaboration event’. There is also an online dimension.The EEF report and reached four key conclusions:‘On average, pupils in schools adopting Mathematics Mastery made a small amount more progress than pupils in schools that did not. The effect detected was statistically significant, which means that it is likely that that improvement was caused by the programme.’‘It is unclear whether the programme had a different impact on pupils eligible for free school meals, or on pupils with higher or lower attainment.’‘Given the low per-pupil cost, Mathematics Mastery may represent a cost-effective change for schools to consider.’‘The evaluations assessed the impact of the programme in its first year of adoption. It would be worthwhile to track the medium and long-term impact of the"
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Leadership

Maths Mastery – Is It Just Another Fad?

7 Nov 2017

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      string(6007) "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. 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Inclusion

Mathematics Mastery

20 Jun 2017

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      string(5734) "‘It sometimes happens that during an arithmetic lesson, when the very atmosphere seems heavy with tedium, a butterfly will flutter into the classroom from out of doors. Then the urchins will all crane their necks and follow its flight with curiosity, as if they saw before them something strange and new and not simply a butterfly.’So wrote Chekov in his short story Champagne. It is an uncomfortable question to ask: whether the perception of maths lessons has changed all that much in the 130 years since Chekov’s story was written. Many people who are passionate about mathematics and about teaching argue that it is a creative discipline, but I would suggest that Chekov’s description persists in the imaginations of many. There are also fears that an increased focus on arithmetic skills, and a push for ‘mastery’ teaching—taking slower steps through learning, perhaps with children sitting in rows facing the front—might actually be rewinding the clock to those lessons heavy with tedium. To communicate just what we mean when we say that mathematics is a creative discipline requires more than just showing the pretty pictures of the Mandelbrot fractal or the tessellations and paradoxes of Escher’s etchings. If anything, these only confirm the sense that mathematics is, at its core, incomprehensible to most of us. Even where the message has been communicated in part, there is an entrenched cultural norm: mathematics is something a very few unusually clever people can do, for whom it might indeed be creative though abstruse and abstract, but that most people cannot do, and for whom it is alienating, frightening or just plain dull. A student who has success in maths and shows an interest in the subject is not usually described as creative. Logical, perhaps. Clever, bright or intelligent maybe, but creative is hardly ever heard. The student who paints portraits or writes poetry with a good grasp of rhythm and metaphor, or strums a guitar soulfully—this student is deemed creative. In this short article I have picked out some of the ways in which doing and learning maths require such facets of creativity as intuition, revelation and the construction of meaning in the world. Rather than showing how these are present in the higher mathematics of academia, I have used examples that will be familiar to any teacher or pupil from primary school.Story and making meaning‘A class of 25 children needs to be in groups of 4 for their tennis lesson.’This is a pretty standard setup for a word problem in a lesson on division with remainders and will likely result in the calculation: 25 ÷ 4 = 6 remainder 1What does the 6 mean? That there are 6 groups. What about the 25? That there are 25 children in the class. What does that 1 mean? The remainder? It means that there is one child left out, on his own, quite probably feeling awkward and ashamed.That the numbers and symbols of an equation have meaning can be a surprise for many pupils, who have grown used to the implied sense that arithmetic is a sort of game with some specific rules to follow but with little physical reality outside of the maths tests on a Friday.In fact, we can use mathematics to describe the world. Now, just as we might use a sentence to describe the world, let us write some mathematics instead:25 = 6 x 4 + 1We can say that the 25 is the whole group and the 6 x 4 are our groups happily playing tennis, and our +1 is scuffing his shoes on the outside feeling excluded and tearful. And so, the ‘solution’ becomes the beginning of a journey instead of the end. The teacher is unlikely to leave the +1 on his own, and might ask him to join one of the other groups. Then the mathematics that describes this next step is:25 = 5 x 4 + 5Now, though, the +5 is a rather crowded group in which one of the pupils is likely to get a bloody nose from a carelessly swung racket, and so a different solution is needed. This time, a kind pupil has noticed the lonely +1, and so offers to join him, and the description of the world becomes:25 = 5 x 4 + 3 + 2Unhappy with this, one from a group of four complains that it’s not fair, as the +2 have a whole court to themselves, so the teacher seeks the ‘fairest’ solution, and sends the complainer of the 4 to join the +2, and now we write:25 = 4 x 4 + 3 x 3The different narratives can be described in words based on the problem context, enacted using equipment such as counters, and as calculations. The act of doing mathematics is not then about grinding through a process, but instead becomes a way of triangulating meaning from a context, a representation and a symbolic description.Elegance and empathyAs well as considering different end results and their implications, the class can also compare methods, and learn to see things from each other’s point of view. Taking Polya’s dictum that it is sometimes better to solve one problem in five different ways than to solve five problems in one way, taking time to push the boundaries of problem solving and look for alternative methods is an important way to share creativity with the class. Pupils who are quick with number and procedural fluency may solve a problem first, but the first method may not be the most elegant. Take the case of finding the area of a compound rectilinear shape. A first reaction for many pupils is to seek to split the shape into two rectangles and calculate the dimensions of each. However, the pupil who has a more creative, perhaps thoughtful eye, or who is willing to think a little differently sees the negative"
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Creative Teaching & Learning

Reconsidering Maths as a Creative Discipline

8 Jul 2019

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      string(5658) "graphing proportional relationships.Visualisation continues to be an important tool for developing reasoning skills this year. In geometry, as pupils work with more complex shapes, they need to be able to move between 3D and 2D representations, for example in making sense of surface areas of prisms. Spend time helping pupils to understand the relationships between nets and 3D prisms. Invite pupils to identify the two sides on a net that join to make a given edge, challenge pupils to draw a net given just a cross section, including those formed from combined rectilinear shapes.In developing their skills of deductive reasoning, pupils should regularly break down mathematical situations like a complex calculation, a proportion problem, a formula for surface area, a geometric proof to demonstrate they understand the structure of the mathematical thinking. They should be able to think through and identify the steps (or possible steps) towards a solution or a proof. One approach is to ask pupils to annotate each stage of a complex problem with notes to highlight the key thinking. Be specific by modelling this approach for the class. Share good and interesting examples of pupils’ notes, then give pupils time to improve their own recording.Help to reinforce and embed new skills by inviting pupils to conjecture on the effects of a single change to a problem already explored. For example, inviting pupils to think through the effects on the answer to a volume of a cylinder problem if the radius was halved. Consider occasionally incorporating these kinds of ‘What if’ questions into work on a range of problems.Encourage pupils to engage in lines of enquiry to test out conjectures and hypotheses. Approaches like starting from a statement, for example All numbers can be expressed as an exact fraction and inviting pupils to discuss whether it is always, sometimes or never true. This is an effective approach for helping pupils to produce a convincing argument, particularly with statements that are sometimes true.Solve problemsYear 7Can solve and evaluate the outcome of problems including multi-step problemsPupils are expected to be able to analyse problems; to seek important information and clues when starting from any context, for example an equation to solve, a missing angle diagram, interpreting a statistical chart, solving a word problem. Be explicit about how to analyse different problems. Encourage pupils to look carefully at information provided and identify what they know.Avoid using a set of very similar problems as some pupils will just repeat the method without engaging with the mathematics.Invite pupils to consider the approximate size or likely answer to the problem. This kind of thinking helps pupils to engage with the problem. Give pupils the chance to compare a few suggested answers and the reasoning behind each.It is important to establish that the method and approach to solving a problem is mathematically more important than the answer. Give pairs or groups of pupils sets of problems and ask them to classify them, for example a set of missing angle problems, some that require use of the angle sum of a triangle and the others that don’t. Put most of the focus on method when going over a problem: the different ways pupils have approached it, how the first step was identified.Pupils need to have the confidence to try things out and not be put off when an approach doesn’t work. They will and should make mistakes and reveal misconceptions when problem solving. It is important to help pupils to value mistakes so they view them as a way to deepen understanding rather than evidence they can’t do something. Take feedback on mistakes made and corrected for a problem. Explore tips and advice to avoid common mistakes. Give pupils time to annotate their notes.Year 8Has begun to model situations and express results in a range of suitable formatsPupils should be able to adapt a given model to use for a related but different problem or situation, for example using the sample space for the equally likely outcomes when throwing two dice to produce a sample space that can be used for identifying outcomes when throwing a coin and a dice.When the mathematics requires pupils to generate information for analysis and interpretation, invite pupils to come up with their own ideas for organising and structuring the information. Use examples of pupils’ ideas for class discussion on advantages and disadvantages.Use real examples to engage pupils with business uses of modelling for formulae, for example generating monthly bills for mobile phone contracts. Involve pupils in investigating the constants (monthly charge) and the variables (charge for each photo text, charges for going above set limit on data, phone calls, texts) used to calculate mobile phone bills. Work on formulae and graphs to model and compare a few price plans for the same or different providers.Pupils should be aware of models that are used to help them make sense of a mathematical concept or idea, for example how the dimensions and area of a rectangle are used to model the distributive law in number and algebra, supporting pupils’ understanding of how to remove brackets in expressions.Provide contexts for pupils to extract information from a word problem and construct their own annotated diagram as a model for the problem. For example in line and angle problems, talk through the information orally and ask pupils to sketch and label their own diagram.Year 9Selects appropriate concepts, methods and techniques to apply to unfamiliar and"
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Creative Teaching & Learning

How to get students to work mathematically

20 Jun 2017

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