Vol. 28 No. 3 Autumn 2023 1 Editorial Team: Kirsty Behan Alan Edmiston Peter Jarrett Alison Roulstone Les Staves Nicky White Letters and other material for the attention of the Editorial Team to be sent by email to: edmiston01@btinternet.com ©The Mathematical Association The copyright for all material printed in Equals is held by the Mathematical Association Advertising enquiries: Charlotte Dyason charlotted@media-shed.co.uk D: 020 3137 9119 M: 077 1349 5481 Media Shed, The Old Courthouse, 58 High Street, Maidstone, Kent ME14 1SY Published by the The Mathematical Association, Charnwood Building, Holywell Park, Loughborough University Science and Enterprise Park, Leicestershire, LE11 3AQ Tel: 0116 221 0013 Fax: 0116 212 2835 (All publishing and subscription enquiries to be addressed here.) Designed by Nicole Lane The views expressed in this journal are not necessarily those of The Mathematical Association. The inclusion of any advertisements in this journal does not imply any endorsement by The Mathematical Association. Editors’ Page 2 Harry Hewitt Award 3 Access for All – the 2023 Equals Conference 4 Ray Gibbons Memorial Prize 11 At Equals we are so pleased to be able to recognise all the work that Kat Adams has done by awarding her the Ray Gibbons Memorial Prize for 2023. Teaching for Mastery and SEND 14 Jo Cayley is someone who has only recently made the move from mainstream to special schools. Such a change of perspective is important as it helps you to, which she does so very well, reflect upon the real differences and to focus instead upon what all children have in common. Compulsory Maths up to 18 19 Mark Pepper has done it again – in this very thoughtful piece he reflects upon the state of maths for those coming to the end of formal education in his usual thought provoking manner. Number, numbers, numerals and SQUIGGLES! 23 Inspired by one of the Les Staves seminars Tandi Clausen-May put pen to paper to articulate her thinking on how confusing the acquisition of number facts can be for many children. By taking a child’s perspective she suggests some helpful ideas that we could all adopt with those at the earliest stages of mathematical thinking. Approaches to multiplication and division 28 In this reflective piece Gill Knight tracks the development of the FunKey time tables cards through her collaboration with Maggie Steel. Like all good ideas it began with a simple question: “So many children (and adults) struggle to secure fluent recall of multiplication facts. Why is this and how can adapt our teaching approaches to help them?” From the archive 31 We are going back 13 years for this edition’s archive feature. I picked this interesting piece by Margaret Haseler as it contains so much that both resonated with me and is of relevance today in the time of CPA. Please get in touch to let me know your thoughts on what Margaret has to say.
Vol. 28 No. 3 Autumn 2023 2 Editors’ Page This edition of Equals is earlier than usual because our second conference is fast approaching and will soon be upon us! The 6th October promises to be a significant SEND event and one not to be missed. The line-up of speakers is exceptional and the programme has been constructed to enable the day to meet your needs. The conference includes a wide range of inputs from experts and it will showcase of the brilliant work currently taking place across the UK. The presenters have been carefully chosen to appeal to as wide as range as possible and also to address the theme of the conference which is access for all. The conference programme is outlined on pages 5 - 10 and please book now to avoid disappointment: Tickets on sale for Access for All I am also looking beyond the conference and two events are on the horizon and I mention these for they are significant SEND developments and at Equals we feel you need to be aware of how the future is shaping up. 1. The Maths Hubs and SEND For the past two years I have had the great privilege and pleasure of working as part of the SEND Research and Innovation Work Groups run by the Maths Hubs. This has given me access to so many wonderful schools, teachers and SEND experts. It is clear that a real support community is developing as a consequence of this focus by the NCETM. The SEND work is developing and maturing as it builds upon this foundation and is being translated into a form that is accessible for all. The coming year looks as if it’s going to be the best twelve months yet and we will keep you updated on developments in future issues of Equals. 2. AMIE and the future of Equals For those of you who do not know the maths subject associations are amalgamating to form a new organisation called AMIE (Association for Mathematics in Education). Currently we are the only maths subject association SEND-focused voice and so consequently are very excited about what the future holds and how Equals will develop within this new and exciting association. I suggest you watch this space and get in touch if you would like to be involved as we seek to professionally represent and support all teachers of mathematics. Research and Innovation Work Groups (RIWGs) are maths-specific professional development projects funded by the NCETM (National Centre for Excellence in the Teaching of Mathematics) and developed and led as an integral part of the Maths Hubs Programme. Through cultures of research, innovation and collaboration, they rigorously explore new approaches to better
Vol. 28 No. 3 Autumn 2023 3 understand how children and young people learn maths. The outcomes of these RIWGs influence Maths Hub professional development projects at a national level and have an impact on individual teachers. This piece is from an educator leading a RIWG on behalf of a Maths Hub, and is entirely the work of the author. The Harry Hewitt Memorial Award This prize is awarded to any pupil who has overcome barriers with mathematics and is now making real progress. Do you have a pupil who has struggled but is growing in confidence as they engage with mathematics? The please why don’t you celebrate their success with us? We are offering a prize of a £25 book token to the award winner and the opportunity to have their work published in Equals. To nominate a child simply choose a piece of work that both you and they consider successful and send it to Equals. Please include: • the original piece of work, photograph or photocopy, • an explanation of work and its context and a description of the barriers which they have overcome or is in the process of overcoming, • the child’s age, school and the context of the class in which they learn and, if possible, some comments from the pupil, and staff, about what they are pleased with about this piece of work and/or the learning it shows. Please do take the time to honour one of your children as it is a lovely way to mark and celebrate their success.
Vol. 28 No. 3 Autumn 2023 4 For programme information visit https://www.m-a.org.uk/events/?id=294
Vol. 28 No. 3 Autumn 2023 5 Conference Session Outlines For more information visit https://www.m-a.org.uk/events/?id=294 Opening Plenary Les Staves. The roots of maths for children with very special needs The presentation will discuss why it is important to have a curriculum that will encourage relevant math’s to grow for all children, including those who are at the earliest levels of learning. It will review the nature of relevant content and approaches to teaching and learning. Its starting points may relate to children who have not yet reached numeracy, or are well behind age related expectations. But I hope it will provoke consideration if its messages are relevant at later levels. Closing Plenary Kinga Morsanyi. Dyscalculia: What it is and what to do about it Dyscalculia (specific learning difficulty in mathematics) is a condition that affects about 6% of the school-age population. Although dyscalculia is equally prevalent as dyslexia, and it can seriously affect people’s life chances, it is neglected by both educational professionals and policy-makers. Currently, in the UK (and in many other countries), a diagnosis of dyscalculia is almost non-existent, and educational support and official recognition is lacking. In this talk, I will present information about current conceptualisations of dyscalculia, how it can be identified, and how it can be discriminated from other conditions. I will also introduce a new screening tool to identify pupils with mathematics difficulties in the classroom, and make some recommendations for best practice.
Vol. 28 No. 3 Autumn 2023 6 Short bio: Dr Kinga Morsanyi is a Senior Lecturer in Mathematical Cognition at Loughborough University. She has broad research interests, which lie at the intersection of research into mathematical cognition, and reasoning and decision making. Currently, one of her main research interests is in dyscalculia. Dr Morsanyi is researching the cognitive profile of individuals with dyscalculia, the demographic risk- and protective factors, typical symptoms (including problems in everyday settings), and co-morbidity with other developmental conditions. She is also leading the development of an app (Numeralis), which will comprise a standardized screening instrument for dyscalculia and tasks to assess the broader cognitive profile of learners. Dr Morsanyi is associate editor or editorial board member of several academic journals, member of the UK Young Academy, and advisory board member of the Dyscalculia Network. Alison Roulstone. Practical strategies and interventions to support learners with difficulties in mathematics. i-CAN Maths Workshop Session Tracey Roberts. No One Left Behind i-CAN Maths is a maths magnetic counting frame that uses a Hungarian number frame (double five domino pattern). i-CAN Maths can help learners build good number sense using the maths mastery approach. Designed to be used one-to-one, in small groups and with the whole class i-CAN Maths is fully flexible and each session can be tailored to meet learners’ specific needs. i-CAN Maths encourages learners to visualize number, builds subitizing skills, develops mathematical reasoning skills and mathematical fluency. Finding the right course for learners is often tricky, even trickier for pupils who are often disengaged or demotivated with education. The session will explore an alternative course for KS4 pupils to enable a positive and successful experience within maths. The qualification in particular will support learners with identified skills gaps in Maths and can be used to support progression or even run alongside GCSE. Come and see what we have learnt from delivering the course at both Level 1 and 2 along with the changes in pupils’ perception to maths following their success, and how we make this work within our curriculum.
Vol. 28 No. 3 Autumn 2023 7 Pete Jarrett. The transdiagnostic model of neurodiversity and the maths learner How can the conceptualisation of neurodiversity across multiple dimensions that breaks away traditional models of diagnostic siloes help us to understand barriers to mathematics in the classroom? There is a lot of debate about whether categories such as dyscalculia, dyslexia or ADHD adequately describe what we see as teachers. This presentation will take recent academic discussion as a start point for understanding the complexity of cognitive processes, for example, the role language development plays in the development of mathematical understanding, and how we can begin to understand the impact environmental and behavioural variables can magnify difficulties. Janet Goring. Using the EEF’s Five-a-day principle to increase access for learners with special needs in the Mathematics classroom. This session will use the EEF’S: “Five-a-day” approach to discuss how we can support all learners to make progress and feel more confident, with a focus on pupils with special needs. Each of the five strands will be explored through case studies of teachers successfully implementing strategies in their settings. Participants will also be encouraged to discuss and share their own experiences and develop a one-page plan to take away and implement. The Equals Curriculum discussion - Thinking Aloud about Curricula Kat Adams (Rocklands School), Angela Miller (Forest Oak School) and Les Staves This is a practical session that will allow participants to share their curriculum stories and solutions. If you are seeking to, or are currently developing, your special school maths curriculum then this is the session for you. Angela and Kat will lead the session and share what they have done respectively over the past 12 months in their schools. Please join them and come ready to share what you do and to support each other.
Vol. 28 No. 3 Autumn 2023 8 The Equals Assessment discussion - Thinking Aloud about Assessment Matt Welborn + Neil Barker Matt and Neil will lead this session that will focus upon Assessment of maths within different schools to explore what other practitioners are doing for assessment, what is working well, and what areas can we focus on to develop. If you are asking: ‘How do we assess, what do we assess our pupils against and what methods of assessment do we use to both record and track progress?’ then this is the session for you. Matt and Neil will share how they assess in their own settings, some of the benefits and areas to develop before opening the room up to discussion to explore how other schools use assessment. We hope you will leave with ideas to develop assessment of SEND learners, and share what best practices we all use. Natasha Dolling. How LEO Academy Trust is using technology to improve inclusion for children with SEND. At LEO Academy Trust, we are excited to be at the forefront of using technology to improve inclusion. We believe that technology can be a powerful tool for ensuring that all children have the opportunity to succeed. Technology is being used to help children with SEND access learning, work independently, and overcome barriers to learning. I am excited to explore the future of technology in education. I believe that technology has the potential to revolutionise education and make it more inclusive for all children. I am excited to see how LEO Academy Trust and other schools continue to use technology to improve inclusion.
Vol. 28 No. 3 Autumn 2023 9 Louise Langford. Representations of number- Securing a visual structure to transfer knowledge This will be a practical session based on my article in the Equals Magazine summer 23 and is appropriate to delegates teaching all ages and stages, as well as those involved in professional development work. We will look at a variety of dot patterns to study developing a secure representation of numbers, based on effective practice for teaching those with Mathematical Learning Difficulties and Dyscalculia. We will explore ‘cluster’ recognition, building on subitising and the idea of efficiently composing and decomposing number, discuss the importance of linking this to linear representations to connect understanding of the amount, its magnitude and number order, then focus on known representations of ten and how these expose different mathematical structures to broaden and deepen thinking, as well as enable learners to communicate their mathematical thinking more readily. Lara Lalemi. Creative Tuition Ltd. Beyond the Western Canon: Enriching Secondary School Mathematics Education with Diverse Mathematical Knowledge In recent years, there has been a growing recognition of the importance of incorporating diverse perspectives and knowledge into educational curricula, especially in mathematics. This talk highlights how traditional mathematics education often overlooks the rich and diverse contributions of global cultures and knowledges, leading to missed learning opportunities for school students to engage with the subject fully. The talk discusses the impact of colonialism on mathematics education and advocates for using an anti-colonial framework to critically examine and enrich the curriculum. By integrating captivating examples from ancient civilizations and indigenous cultures, teachers can encourage school students to gain a deeper appreciation for the universality and interconnectedness of mathematics across societies and with themselves in daily life. Through the incorporation of diverse mathematical examples, teachers can show students alternative ways problem-solving methods and different perspectives, fostering critical thinking and creativity. Moreover, exposure to the accomplishments of mathematicians from non-Western cultures can empower students from diverse backgrounds, allowing them to see themselves as active participants in the field of mathematics.
Vol. 28 No. 3 Autumn 2023 10 Magdalene Lake. The vicious cycle of poor reasoning: Putting the horse back before the cart to address inequalities and improve maths attainment for everyone. Reasoning is fundamental to good maths attainment for everyone, and yet, we often think that pupils must have certain mathematical skills before trying any reasoning, which leads to fewer opportunities for those with lower attainment to develop their thinking, and a vicious cycle ensues. In this session, we will look at who is at risk of this vicious cycle, why reasoning will improve number sense and calculation, and how to improve reasoning when number skills are low. Rob Jennings. The Dyscalculia Network The Dyscalculia Network have just signed a deal for a new maths assessment with Jessica Kingsley publishing. It is due out next year and is entitled … ‘The Maths and Dyscalculia Assessment’ and is an assessment tool aimed at formulating a focused teaching intervention plan. Come along to this session and find out more!
Vol. 28 No. 3 Autumn 2023 11 Ray Gibbons Memorial Prize At Equals we are so pleased to be able to recognise all the work that Kat has done by awarding her the Ray Gibbons Memorial Prize for 2023. Kat Adams, Deputy Headteacher at Rocklands School, was nominated for the Ray Gibbons Memorial Prize by Jane Elliker her Headteacher. The statement below by Jane should leave you in no doubt how strong a candidate she was and is. We are happy that Kat will be co-leading a workshop at this years conference. “Kat began her role at Rocklands in September 2022 having previously been a deputy in a mainstream setting. The leadership of Maths at Rocklands had not been effective and I decided to appoint her as the new Maths lead. OFSTED had paid us a visit in July 2022 and had given the following action point: ‘At times in mathematics, staff move on learning too quickly for some pupils. They introduce new information before pupils are fluent in what they are studying or before their learning has been consolidated. Leaders should ensure that all staff have the knowledge and skills they need to deliver the mathematics curriculum consistently well so that all pupils have strong mathematical knowledge’. Kat’s knowledge was strong in the National Curriculum but she knew she needed to commence research in what we call our pre-formal classes and our more complex learners. She began by spending time in classes and observing how the children learn. She named this the ‘Explore’ phase and carried out book trawls, resource audits, learning walks and had conversations with staff and gained pupil voice around mathematical teaching and learning. This is what she discovered that she needed to action: • A clear curriculum for pre-numerate children, where the foundational knowledge of ‘roots of mathematics’ should be taught and embedded. • The need to ensure staff were not moving on to quickly and allowed time to ‘plug the gaps’. Earning needed to be revisited regularly. Kat panned to introduce ROCK time (Recall Of Calculation Knowledge). • Interventions needed to be delivered y trained staff and identified through teaching and learning and formal assessments.
Vol. 28 No. 3 Autumn 2023 12 • Staff required CPD around the ‘small steps’ of learning and the roots of maths. Including the importance of the sensory curriculum. • The scheme that was previously followed was White Rose and although this was appropriate for those working on National Curriculum levels it needed to be reviewed for the pre-formal learners. Kat met regularly with me to share her plans and then she created an action plan over the next three terms. She delivered to SLT her ‘Prepare phase’. This included the following: • Leading on Department meetings, • Research – Les Staves, Flo Longhorn and Karen Mcguigan, • Working with the Maths Hub, • Collaboratively working with other SEND schools, • Updating the Maths curriculum aims and vision, rewriting school Maths and calculation policies, • The calculation policy included specific language that is taught using concreate concepts of the 4 operations as this had not been consistent. Kat has spent the Spring term attending a range of CPD sessions with the Maths Hub and continuing working with other schools to develop the pre-formal departments. This inspired content modification ensured the curriculum is sequenced with mathematical content and knowledge and skills that build upon each other. Kat spent time in our Puffins class (first pre-formal class) observing ‘maths exposure’ sessions based on a fluid rolling programme. She created a sensory activities document based on sensory processing following all of her research suggested by Les Staves and is currently piloting papers for 2D and 3D shapes. Kat has written and delivered very bespoke training to all three of departments pre, semi and formal – sharing the clear ethos, expectations and rationale behind all the changes. Following feedback staff have felt both ‘inspired’ and ‘empowered’ in their delivery of maths. Their understanding of how to deliver maths in small steps and realisation that ‘maths is all around’ in the curriculum has changed the thinking of our teachers. This is all down to Kat and her work. Kat has a clear plan for her next steps: • Kat will be sharing her work at the Equals conference in the Autumn term, • Following introducing ROCK time for ten minutes three times a week, analysing the impact of this, • Further work with our MAT and other SE school, • Further work with the Maths Hub, also applying to be part of the development programme ‘Mastering Number’ in September 2023, • Creating knowledge organisers and memory cards to support the achieve of recall of calculation facts, Staff have felt both ‘inspired’ and ‘empowered’ in their delivery of maths
Vol. 28 No. 3 Autumn 2023 13 • Finally introduce mathematical apps such as Numbots and TTRockstars. Our school improvement advisor came to visit in March and reported the following: ‘The School has undertaken a substantial piece of work to improve the mathematics curriculum. This has been very effectively and ably led by the Deputy Headteacher. The management and organisation of the Mathematics review has been very thorough and based on the most up to date research. The result is a detailed framework and rationale for the teaching of mathematics has been produced. This work has been backed up with focussed training for all staff setting out expectations for the teaching of mathematics at Rocklands. There are also several advice and guidance documents that will support staff with their planning and delivery of mathematics lessons. Discussions with a sample of staff I talked with during the day demonstrated that they were enthusiastic about the changes and thought it improved the quality of mathematics teaching across the School’. So, to conclude, Kat has had a huge impact on Rocklands in her development of the Maths. She has based all of these changes on the research she has completed learning about sensory approaches and the roots of maths. Kat has updated policies, has a clear rationale, has produced clear, small steps of learning in Maths with specific appropriate language to be used within the steps of mathematical knowledge. Her delivery of six hours training was outstanding sharing her work, practical activities and getting the staff themselves to have a go at activities. I am extremely proud of all the Kat has undertaken in Maths to ensure the best possible outcomes for our learners. I am delighted she is not only working and sharing this within our MAT, but Nationally too, working with highly respected experts in the field Comments from a recent peer review from two Trust schools. There was a sense of excitement around maths from the children and a real buzz. Children use investigative skills to explore concepts. The children were all excited and through discussion, it was identified that children who had previously found maths difficult or less engaging, were excited to be actively participating. Quote from a recent School Improvement Audit following interviews with staff. I picked up on my last visit that staff were supportive of the development of the Mathematics curriculum. I was interested as to why they had such strong feelings. Their view was they liked the way this was managed. It felt like they were part of it and their individual needs were considered during the process. They particularly likes developing an understanding of how the Mathematics changes affected teaching in other departments. In simple terms it was not a top-down model. Kat has had a huge impact on Rocklands in her development of the Maths. She has based all of these changes on the research she has completed learning about sensory approaches and the roots of maths.
Vol. 28 No. 3 Autumn 2023 14 Teaching for Mastery and SEND Just like Kat Jo Cayley is someone who has only recently made the move from mainstream to special schools. Such a change of perspective is important as it helps you to, which she does so very well, reflect upon the real differences and to focus instead upon what all children have in common. I have recently made the move from a mainstream primary to a special school after teaching in mainstream for 25 years. As a Teaching for Mastery specialist, my first question was ‘How can Teaching for Mastery be adapted for a special needs setting?’ Having used and applied the NCETM five big ideas over the course of my Teaching for Mastery journey; working in special education, a key question we must ask is how the five big ideas of teaching for mastery apply to children with high levels of special needs? Many children in special school settings are learning mathematics at levels which are earlier than National Curriculum descriptors, some are even pre-numeric and non-verbal, and therefore the five big ideas as they currently stand may not serve our purpose. Nevertheless, the mathematical world is part of their practical and social lives, and we should therefore aim to help them develop the fundamental roots of mathematical learning that is essential to their lives. Within our special settings there are a wide range of children with very different needs. Working in a special school can allow teachers to focus on each student as an individual with needs that need addressing. Mainstream teaching can allow us to make a positive impact on many children, but within special schools we can have a deep impact on individual learners. A key aim of Teaching for Mastery is to be inclusive for all pupils. In whole class teaching, the use of one curriculum that works for all is encouraged, with everybody studying the same topic and being provided with support and challenge as needed. High expectations can still be applied within special education and children within the same class may be able to access the same curriculum to a certain extent but the use of individual differentiation may be needed and children are more likely to learn at different rates, with some needing more support and repetition than others. Within our school, we wonder if we need one curriculum for all, one curriculum for each class or one curriculum for each learner. In reality, it is often the latter but working together with others on the same mathematical task is an important part of mathematical learning that we try to enable whenever possible. In some classes, groups of children can be taught together, in others it is more individualised. The Ofsted framework says ‘teachers present subject matter clearly…they respond and adapt their teaching as necessary, without unnecessarily elaborate or
Vol. 28 No. 3 Autumn 2023 15 differentiated approaches’ but also ‘the provider has the same academic, technical or vocational ambitions for almost all learners. Where this is not practical – for example, for some learners with high levels of SEND – its curriculum is designed to be ambitious and to meet their needs.’ (2022) The mastery principle suggests that pupils should broadly move through the curriculum at the same pace. As far as possible, pupils should stay together on the same topic with necessary differentiation such as removing barriers and providing targeted support, but without the need for many different levels for each step. ‘Low floor, high ceiling tasks’ support this approach to differentiation together with reasoning and problem solving questions that will challenge all pupils and allow all children to access the maths at their own level. This approach works well within special needs, with children being able to work together on the same task at different levels, and we have found many useful activities from websites such as Nrich, I see maths and Youcubed as well as engaging children with creative activities and maths games. Many of the teaching strategies we advocate for all pupils are particularly useful for pupils with SEND. For example, using concrete and pictorial representations. The use of concrete and pictorial representations to develop strategies to deepen and embed understanding is important for all learners, particularly those with special educational needs. For many pupils, the CPA approach is a ‘way in’ to a topic whilst also it can be challenge for pupils to find an alternative representation to the ones they already have. It is essential for teachers to model the use of manipulatives and make it clear that the resources are accessible for all to use. Children with special educational needs may rely on the representations and may need support to make the links between the concrete and abstract. Sometimes, the use of too many representations may be confusing for children and it may be that one representation is more useful than another. In my class, Unifix cubes have been particularly useful in teaching the numbers to 20 as they are easy to handle and manipulate. Grouped in 5s or 10s helps the children to use these benchmark numbers and relate to the fingers on their hands. The rekenrek has also been useful and we have been successfully following the Mastering Number programme in many of our primary and secondary classes. Teaching for Mastery promotes multiple opportunities to look at topics again in new contexts. This enables teachers to support
Vol. 28 No. 3 Autumn 2023 16 students who have struggled with a topic to spend more time reconsidering and developing their understanding. Retrieval tasks are ideal to assess what has been learnt and what might need further work or intervention. Children with special educational needs may need more opportunities for revisiting and retrieval to embed the learning. Planning for misconceptions by pre-empting examples of where pupils could go wrong, challenging the pupils to spot, explain and rectify errors is an essential part of Teaching for Mastery. Pupils’ responses to these prompts helps teachers to identify and tackle misunderstandings early on rather than let these incorrect ideas become established in pupils’ minds. Thinking about what pupils need to have covered before in order to access a step and assessing these are important. Incorporating revisiting these before a topic is taught, helps support learning throughout the unit and ensuring connections are made. Children with SEND will often need support to see these links and have them made explicit to them. When teaching, we need to consider the key aim or aims of a lesson and be aware of cognitive overload. Fluency is an important part of maths, but some children struggle to recall facts so support such as a prompt sheet may help children to be able to access the mathematics and not get bogged down with the calculations. For example, if the focus of a lesson is understanding that the area of a rectangle is found by multiplying the length by the width, then providing pupils with times-tables grids or calculators will help them to focus on this rather than struggling with the mechanics of the calculations if these are an obstacle. Pre-teaching; going through some key ideas with pupils in short targeted sessions before the topic is taught, enables them to have a head start and be prepared for what’s coming up often hugely increasing confidence and participation at the start of a topic. Providing additional individualised support after lessons through targeted interventions can help to close gaps and/ or deepen understanding. It may be necessary to look back at previous steps to support this. The Ofsted maths review (2021) says, ‘Pupils with SEND benefit hugely from explicit, systematic instruction and systematic rehearsal of declarative and procedural knowledge. The benefits of these approaches extend beyond enhanced academic attainment and proficiency. The relationship between cognitive ability and academic attainment, including in numeracy, is in fact bidirectional. Therefore, educational outcomes for pupils with SEND are likely to improve if teachers use systematic instruction and rehearsal to help pupils learn planned content. This approach is particularly useful for pupils with moderate learning difficulties who have slower cognitive processing speed. Systematic approaches increase the amount of content considered per
Vol. 28 No. 3 Autumn 2023 17 unit of time. Systematic curricular approaches give pupils with SEND and disadvantaged pupils a better chance of success, of keeping up and therefore of feeling included. Playing to pupils’ strengths: the powerful declarative memory systems of pupils with autism Many pupils with autism have ‘normal to above average algorithmic thinking ability’ but can struggle with reasoning and problem-solving because of: • language processing deficits • difficulties in classifying problems by type • lack of knowledge of strategies • the use of ‘inefficient and overly complex procedures’ for calculation Teachers can fill these gaps in knowledge with systematic curriculums, teaching approaches and rehearsal. For example, teaching efficient algorithms to pupils with autism speeds up their calculations. They then have more time to learn strategies for solving classes of problem. However, research also shows that the unique organisation and powerful declarative memory systems of many people with autism help them study, and develop proficiency in, the subject. Potentially, a powerful declarative memory system can take on a compensatory role; thus many pupils with autism might benefit from a deliberate focus on memorisation of core facts and methods. Leaders should therefore consider ways to give autistic pupils more time to study core content so that they can close gaps in learning through deliberate memorisation. Leaders should also make sure pupils’ lesson time is used efficiently and effectively. Based on the above, high-quality maths education may have the following features: • New content draws on and makes links with the content that pupils have previously acquired. • Curriculum progression is by intelligent design rather than by choice or chance. • Rehearsal sequences align with curriculum sequences. Pupils who are more likely to struggle or who are at risk of falling behind are given more time to complete tasks, rather than different tasks or curriculums, so that they can commit core facts and methods to long-term memory. The ATM / Maths Association response to the Ofsted Review has some useful practical ideas to support the statements. Adapting Teaching for Mastery for SEND has been a challenge, but the more I have thought about and understood the five big ideas, the more I have been able to apply these to the individual children I work with. Here is our school’s Maths Curriculum, which is still work in progress: Curriculum Aims: The Intent - The Why We strive to make maths link to real life and be relevant, engaging and fun. Maths can be found in every part of day-to-day life and is a fundamental skill that we need to support our pupils to develop in preparation for adult life. Curriculum Content: The What
Vol. 28 No. 3 Autumn 2023 18 We teach the fundamental skills needed for future life. Our delivery is based on the Maths mastery approach and closely mirrors the National Curriculum expectations. There is a strong emphasis on developing number skills as well as a focus on : - Measurement — weight, capacity, length and height - Time - Money - Shape - Pattern - Statistics - Maths skills are generalised and extended within all curriculum activities. Curriculum Delivery: The Implementation – The How In primary and secondary semi-formal classes, maths skills are taught though the context of familiar stories, where abstract maths ideas are introduced in a fun and ‘imagined’ way so that children are encouraged to think freely and learn creatively. Interventions have a maths mastery approach though play. Maths is delivered in context and follows three stages. • Firstly, ‘Concrete’ using tactile (physical) objects that build on a child’s existing understanding of maths and makes ‘unfamiliar’ or abstract concepts such as ‘fractions’ to life by allowing children to interact, solve problems by experiencing and handling concrete objects. For example, using real fruit for the story ‘Handa’s Surprise’ • The ‘Pictorial’ stage is using visual representations of concrete objects used to model problems. Children are encouraged to make connections between the concrete objects and abstract concepts of math e.g. ‘fractions’ and by ‘seeing’ the pictorial image, helps children to visualise the abstract problem and makes it more accessible and understandable. • The final stage is ‘Symbolic’ stage. This is where children use abstract symbols such as ‘+.-,=.x,÷’ to solve problems. This stage requires adult modelling of mathematical symbols and concepts using numbers, names and notations. It is the final stage of maths mastery where fluency and reasoning of maths skills are related from stories and applied to different contexts and everyday life. Teaching and learning maths in stories combine all three of these stages. The following interventions craft a powerful mental connection children make between, concrete, pictorial and the abstract stages, whilst being attuned to the child’s own learning style. In secondary formal classes maths is taught through half termly units there is strong focus on practical application and problem solving. In sixth form maths is delivered through a series of projects that link to real life events. Maths is delivered through taught sessions, across the curriculum and in the community. There is a strong focus on money and time for example budgeting and shopping within cooking sessions. The school shop offers an opportunity for pupils to consolidate, generalise and extend their maths skills. Special Events: The Celebration In the Autumn term we have a half termly focus
Vol. 28 No. 3 Autumn 2023 19 on maths this includes special events such as the shopping street, maths week and a celebration of learning maths in the outdoor area. The Christmas Bazaar is organised and run by the students. Pupil Attainment and Progress: The Impact We have developed our own assessment system based on National Curriculum expectations. These are called the Rainbow and Gemstone levels. Progress is tracked within an app called Evidence for Learning. Pupils in KS4 and above follow accredited courses linked to Equals moving on, OCR life and living skills, Entry Level and functional skills. Compulsory maths up to 18 Enquiry Mark Pepper has done it again – in this very thoughtful piece he reflects upon the state of maths for those coming to the end of formal education in his usual thought provoking manner. On 7th February 2023 a select committee convened to contribute to the Compulsory maths to 18 Enquiry. The proceedings were televised and shown on the Parliament Channel. The participants included members of the main political parties, representatives of some of the teaching trade unions and of various maths organisations and researchers. One unexpected feature of the discussion was the high consensus on a number of key issues. There was virtual unanimous agreement that to compel students to repeatedly retake the maths GCSE exam was an inefficient policy and resulted in only approximately one in four of the students achieving a pass. There was also virtually total agreement amongst those that spoke that there should be a significant increase in problem-solving and creative activities and there was also widespread condemnation of “teaching to the test.” Some of the speakers heavily emphasised that the early years were of crucial importance in maths learning and recommended that there should be priority funding for this cohort. This was indirectly linked to the difficulties associated with “norm referencing” in which a pre-determined number of candidates are allocated to pass the GCSE exam. This system was criticised by some of the speakers on the grounds that it mislead lower-achieving students and their parents into the belief that with intense practise they had a reasonable chance of making sufficient improvement to be successful in the exam. It was pointed out that it has been statistically shown that to a marked extent students who achieve highly at 11 continue to achieve highly at 16.As only a finite number of candidates will pass then
Vol. 28 No. 3 Autumn 2023 20 the vast majority of the allocated number of passes will be taken by the consistently higher achieving students. With so few passes remaining the prospects of success for the rest of the candidates will be extremely limited. A concern was expressed regarding the use of teachers not qualified in maths taking maths lessons. A specific concern was that such teachers were liable to exhibit anxiety due to their feelings of inadequacy and this anxiety could be transmitted to their students. The point was raised by one of the speakers that maths should be taught in the primary sector in such a way as to make it an attractive subject for the pupils. There was general agreement with this approach amid a belief that this should have the effect of reducing levels of anxiety. Maths teaching in the further education sector Functional Skills There were conflicting views regarding Functional Skills. Some of the speakers had misgivings concerning the efficiency of this means of teaching maths. Others strongly disagreed and recommended its increased use. They did, however, make the proviso that it was essential to integrate it into the prospective vocation of the students. Financial Literacy There was strong support in favour of an increased use of Financial Literacy. In particular it was suggested that the link between Financial Literacy and everyday life should lead to increased motivation for the students to engage in maths lessons. It was felt that it was essential to prepare students to make informed decisions in their future adult lives regarding financial matters. This could involve issues such as budgeting, understanding interest rates for loans or mortgages and making decisions regarding mobile phone contracts. Maths Mastery The topic of Maths Mastery was raised at a late stage of the discussion. Relatively few speakers spoke about it but of those that did there was strong support for it with no-one expressing any opposition to it. Assessment There were mixed views regarding assessment. On the one hand the pressure on both students and teachers created by the insatiable quest for acceptable exam results was described as a “cliff edge” which was “bad for health”. On the other hand there were misgivings regarding the probable repercussions of replacing formal exams with ongoing assessment. This concern was articulated by one of the union representatives who predicted that it would lead to such an increase in teacher workload that it would have a strongly adverse effect on the retention of maths teachers. A very personal view of the policies that should be implemented In the previous edition of Equals Vol 28 No 2, I wrote a piece entitled Would it be feasible to replace formal maths exams with ongoing assessment? In which I made various suggestions It was felt that it was essential to prepare students to make informed decisions in their future adult lives regarding financial matters.
Vol. 28 No. 3 Autumn 2023 21 of policy changes that could be made in the teaching and assessment of maths. These can be summarised as: Implement a general approach that focused primarily on problem-solving and creativity and away from a heavy emphasis on rote learning and preparation for exams. Make maths lessons more attractive with a reintroduction of enjoyable activities such as the regular use of maths games in the primary sector. Reintroduce a session of mental maths in every maths lesson in the primary sector and on an occasional basis in the secondary and further education sectors. Increase the use of Financial Literacy. Abolish the formal exams of National Curriculum Tests, GCSEs and A levels. Further changes in response to points raised in the Review Maths Mastery Despite the emphatic support from those that spoke on this topic in the Review I am not in favour of the use of Maths Mastery. With its roots in Shanghai Maths I feel that there is too heavy a reliance on rote learning. Furthermore a policy of waiting for every student to show understanding of a particular concept before moving on to a new topic is likely to have an adverse effect on the learning opportunities of the rest of the class. This particularly applies to situations in which an able student is directed to support a classmate who is struggling to understand a concept. This policy no doubt affords considerable benefits for the student who is being supported but provides no learning opportunities for the student who provides the support. Functional Skills An increase in the use of Functional Skills would be a most welcome development. In my own past experience of teaching maths at a college of further education I found the use of Functional Skills to be a useful means of engaging students who otherwise showed a marked antipathy to the subject of maths. This can perhaps be attributed to the fact that many of them had a pragmatic attitude to life and had a strong incentive to gain the maths Functional Skills qualification to assist their quest for their vocational qualification whilst having no other interest in maths. Furthermore some of the content of the course, such as the use of measurement and the calculation of the area of a wall, had a direct relevance to their vocational aspiration. On these occasions they displayed a marked increase in their motivation to fully engage in the lesson. Retention/recruitment of teachers It is imperative that the issue of the retention and recruitment of teachers be addressed. This crisis can only be averted with the provision of a significant increase in pay and greatly improved working conditions for teachers. Otherwise the exodus of teachers is likely to greatly increase and there would be few incentives for new recruits to join the profession. The consequence of this is that there will be an increase in the use of unqualified staff taking maths lessons which will I found the use of Functional Skills to be a useful means of engaging students who otherwise showed a marked antipathy to the subject of maths.
Vol. 28 No. 3 Autumn 2023 22 have a deleterious effect both on the quality of teaching and could lead to heightened levels of anxiety. Prediction of the probable policy that will be introduced It will be interesting to see how much influence the Compulsory Maths to 18 Review will exert on the policy that will eventually be introduced. As it has been the norm within various fields for the Government to ignore the advice both of experienced professionals and researchers it seems unlikely that many of the recommendations made in the Review will be implemented. Recommended policies that are extremely likely to be implemented Compulsory resits of maths GCSE will almost certainly be abolished. Maths Mastery will continue to receive emphatic Government endorsement. The use of Financial Literacy is likely to be expanded. The use of Functional Skills is likely to be unchanged. Recommended policies that are unlikely to be implemented The recommendation for an increase in the use of problem solving over that of number fact recall is likely to go unheeded as the pressures for satisfactory exam results will almost certainly persist. For the same reason it is probable that there will be no directive to make maths a more enjoyable subject for pupils in the primary sector. The prospect of a reinstatement of regular sessions of maths games which was in widespread use in the 1980s and 90s can be considered non-existent. Instead the frantic pressure to produce acceptable National Curriculum Test results is likely to dominate primary maths lessons to the virtual exclusion of any other mathematical activities. It is highly unlikely that there will be any change in the assessment procedure with formal exams continuing to dominate. If this proves to be the case then all of the difficulties associated with high levels of stress for students and teachers will persist. It seems to be extremely unlikely that a significant improvement in the pay and working conditions of teachers will be provided in which case there will be no improvement in the situation regarding the recruitment and retention of teachers. It is truly ironic that if the plan for every learner to study maths up to the age of eighteen reaches fruition then a great many more maths teachers would be required and this will exacerbate the difficulties associated with the recruitment and retention of teachers. N.B. The views expressed here are those of the author and not necessarily shared by the rest of the editorial team of Equals It is probable that there will be no directive to make maths a more enjoyable subject for pupils in the primary sector.
Vol. 28 No. 3 Autumn 2023 23 Number, numbers, numerals and SQUIGGLES! Inspired by one of the Les Staves’ seminars Tandi Clausen-May takes a learner’s perspective to explain how, at any stage, numerals and symbols may create a barrier to the understanding of number facts and relationships. She suggests some alternative approaches to enable mathematical thinking and reasoning to develop. From Number to Squiggles The concept of Number. Quantities. A quantity of dots. I can see that. I can visualise it. That’s fine. So what about Numbers? Four dots. Four mugs, but only three plates. I can see the difference between the number of mugs and the number of plates. The difference between four and three. And having different words for the different quantities gives me a way to talk about the difference. Well, okay. I can see the point of that. But then we have Numerals. Squiggles. A way of representing number words. And that is where things get difficult. The squiggles are arbitrary. If you tell me that the numeral ‘4’ represents the number word four, or the representation , then I want to ask, ‘Why?’ Why does that particular squiggle, ‘4’, represent that particular number, four? Why can’t it represent six? How does the squiggle ‘4’ convey the fourness of four? The representation shows the concept of four clearly, and it is different from . It doesn’t matter which way I turn it, or or , it still shows the quantity ‘four’. But if I turn the squiggle ‘4’ on its side, , it doesn’t mean anything at all. Or, perhaps even worse, if I turn the squiggle ‘6’ through a half turn, it means something else – it means ‘9’, ‘nine’. And if I turn the squiggle ‘8’ through a quarter turn, it means… . Infinity! So… No. I don’t like the squiggles. They make me feel dizzy and sick, and their meaning changes depending on where they are. I can’t trust them. But if you can’t read and interpret the squiggles easily, then you can’t do maths, at least at school. It counts as ‘proper’ mathematics only if you can write it down or print it out. The problem is not with Number as such, not with the concept of ‘four’ or ‘six’ or ‘nine’, or even with infinity. The problem is with the numerals. The number concepts are reasonable and useful, but the numerals? They are a matter of rote learning, not of reasoning. 4 8
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