Vol. 27 No. 3 Autumn 2022 1 Editorial Team: Kirsty Behan Carol Buxton Alan Edmiston Peter Jarrett Louise Needham 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 Mathematical Association, 259 London Road, Leicester LE2 3BE 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 The 2023 Equals Conference 3 SAS – Success across all settings. SEND and the Math’s Hubs 11 Alan Edmiston highlights the SEND work that is currently taking place among the English Maths Hubs. Keep it simple 12 Lou Hoskyns-Staples shares her experience of supporting special school teachers to research their own practice in the context of Teaching for Mastery. Playing the game 20 Rebekah Gear and Matt Woodford discuss evidence from a recent study, whereby they invited pupils from a primary school with SEND to co-contribute to the design of their mathematical learning experiences. They discuss how inviting pupils to contribute to an intervention design can have an impact on how they experience mathematics that can support their self-esteem. How does Understanding Happen 23 In how does understanding happen Les Stave takes the time to share his thoughts upon the nature of number sense. In the process he asks the right questions to support the profession to think about how early number work should be shaped around the developmental needs of children.
Vol. 27 No. 3 Autumn 2022 2 Editors’ Page The SEND work taking place in many of the forty English Maths Hubs is beginning to bear fruit and this edition seeks to witness to this. One interesting outcome is the realisation of just how valuable it is for teachers and schools to network and support each other. With this in mind an online system of communication has been set up just for special schools and alternative providers to facilitate such critical networking. If you work in a special school you can contact your local Maths Hub to find out how you can use something called ‘basecamp’ to communicate, collaborate and share resources with colleagues who work in other special schools around the country. There are fewer pieces in this edition of Equals for the simple reason is that all of our efforts are focused upon our first ever conference. This marks a milestone for us and you can see on page 4 details of the very exciting programme we have been able to put together. I would like to give thanks in advance to all of the presenters for giving their time voluntarily to share their knowledge on the 25th November at Parliament Hill School. Through the efforts of Kirsty Behan, lead practitioner at Parliament Hill, the school have kindly donated the venue to us for the day – this is wonderful and I doubt that without such an offer the conference would actually take place at all. So a big thanks must go to Sarah Creasey the Headteacher and all of the leadership team – thank you so very much for all of your help and support in getting our first conference off to such a wonderful start. The Conference reflects the way Equals has evolved in its support for teachers. Last year we were able to offer a very well received series of SEND seminars, the aim of which was to give teachers access to the people we thought would be able to help them deal with the range of needs within their classrooms. It was then felt that it was time for a face to face event and the notion of the SAS – success across all settings Conference was born. The conference will see over 100 people from across the UK gather together under one roof and as such it affords us a wonderful opportunity for us to take stock. All participants will be asked about their views on what Equals should be doing more of to support SEND children in mathematics. We would also like to ask you the same thing. Please let us know what you think by adding your views to the following google form: https://forms. gle/o6KLh3WggutWha9X9 Just a quick glance at the programme will reveal just why we are so excited - the strength and depth of the sessions and presenters reflects the commitment of Equals to give teachers access to the best support and advice.
Vol. 27 No. 3 Autumn 2022 3 The 2023 Equals Conference SAS – Success across all settings We have included both the programme and the session details so you can get a sense for how special the day is going to be. All of the session presentations will be grouped together as a conference proceedings book and the coming editions of Equals will share as many of the outcomes from the 25th November as possible. The conference on the 25th November is simply a first chapter in how Equals will seek to support teachers of mathematics. There is far more we could do and so much more to come but before we do so we need to get a feel for what you think should be our priorities in this role. To help us do this could you take the time to use the following google form to share your views: https://forms.gle/o6KLh3WggutWha9X9 Advertisement
Vol. 27 No. 3 Autumn 2022 4 MA Equals Conference: SAS who cares wins! Friday 25th November Opening 9.00 – 10.00 Welcome, Refreshments and exhibition Plenary 10.00 – 10.45 Steve Chinn - Thinking about dyscalculia Break 10.45 – 11.15 Refreshments and exhibition A B C Session 1 11.15 – 12.15 Amy How What the Heck is a Rekenrek? Pete Jarrett Designing interventions for maths learning difficulties Lou Hoskins – Staples Keeping it simple Lunch 12.15 – 13.00 Lunch and exhibition Session 2 13.00 – 14.00 Alan Edmiston and Kirstin Mulholland Exploring the role of Metacognition Les Staves The roots of maths Alison Roulstone What do we know about Developmental Dyscalculia? Break 14.00 – 14.30 Refreshments and exhibition Session 3 14.30 – 15.30 Rachael Addy Maths Anxiety Anne Haberfield Castle Schools’ journey to a wonderful Maths Curriculum Janet Goring Games for Maths, Motivation, Metacognition and More Plenary 15.30 – 16.15 Margaret Brown - Changing Perceptions of Maths
Vol. 27 No. 3 Autumn 2022 5 Thinking about dyscalculia • What do the definitions tell us about dyscalculia? • What are the consequences for pupils learning maths? • What might a diagnostic protocol look like? • What classroom interventions might help? Steve has many accolades to his name. Exec committees of UAU and BUSF. 64-65 PhD in applied physics 1967; Head of maths and science Faculty, Kings of Wessex School, Cheddar; Lifetime achievement awards from BDA, and Dyslexia Action; Yes award winning author (More Trouble with Maths. A diagnostic manual for maths LD and Dyscalculia) SASC approved; Founder Mark College which won ISA award and DfE Beacon School status; Ran first UK PG Diploma course for ‘Maths and Dyslexia’; Head of Chautauqua Academy Baltimore USA; Author many books and published research papers Teacher training in 30* countries inc Singapore Dept of Ed; First Chair BDA Dyscalculia Committee; Chair of 3rd BDA International Conference; Co-founder and Chait of CReSTeD Steve has also completed NYC marathon! 1A Amy How – What the Heck is a Rekenrek? Have you heard about this versatile, visual, concrete manipulative? Perhaps you are curious… or even sceptical? This is your chance to have a go at a few hands-on tasks. You will be amazed at how this tool can be the mess-free answer for children developing deeper understanding of number sense while naturally engaging in rich mathematical talk. Join in on this introductory rekenrek workshop and hopefully you too will be singing the praises of this simple tool. Amy is a Canadian who moved to the UK to pursue her dream of spreading the word globally on using the rekenrek. She has been a Primary teacher for 17 years in Canada as well as at an international school in the Netherlands. She has a master’s degree in Education and has taught Bachelor of Education maths courses at Acadia University. Amy is an author, presenter, primary maths specialist, SLE and university professor. She has an infectious passion and enthusiasm for teaching and will keep you engaged and entertained throughout the presentation. Opening Plenary – 10.00 -11.00 Keynote Steve Chinn Session 1 11.20 – 12.20
Vol. 27 No. 3 Autumn 2022 6 2A Pete Jarret - Designing interventions for maths learning difficulties This session will investigate the design principles that underpin effective intervention. In other words – what works well for people that struggle to understand number. The similarity with more familiar ‘mastery’ approaches will be clear. Looking at building a picture of prior learning, and moving through tangible access to mathematical understanding, we will draw on the extensive evidence base to support learners to offer a straightforward approach to intervention. Pete has over 30 years’ experience in the education sector and works as a maths and SEND consultant alongside running an ed tech Company. As someone with dyslexia and ADD, Pete is well aware of how education can be challenging for some people and needs an update for the 21stcentury. Pete was a founding member and current chair of the BDA Dyscalculia and Maths Learning Difficulties Committee and was instrumental in developing a definition for dyscalculia. 1C Lou Hoskin-Staples - Keeping it simple Lou Hoskyns-Staples shares her experience of supporting special schoolteachers to research their own practice in the context of Teaching for Mastery. In the spring term of 2022 Lou worked with eight teachers in the Central Maths Hub region as part of an NCETM Research and Innovation Work Group: Teaching for Mastery in the context of Special Schools and Alternative Provision. Her role was two fold: to provide external continuing professional development (CPD) and to support the teachers to design small-scale projects that could support change in their schools. During the session Lou would love to talk to others about their experiences of engaging in practitioner research. In 2012-14, she worked for the DfE as a mathematics education researcher and in that role began her journey towards understanding the specific learning needs of SEND pupils.
Vol. 27 No. 3 Autumn 2022 7 2B Les Staves - The roots of maths The session will delve into the relevance and roots of mathematics for children with very special needs. It will explore what Ofsted have described as the ‘cultural capital’ they need, and the skills and knowledge teachers need to help them engage in developing. It will range from sensory beginnings of learning through to number sense and the beginnings of becoming numerate. Les Staves worked as a teacher with children with very special needs in the UK since 1973. He was Head teacher at an outstanding special school, before becoming an independent consultant, trainer and author. He has worked internationally training teachers and working on curriculum development for special schools, government bodies, and SEN organisations. Though Les is proud of his reputation for curriculum development and writing he is most proud of his reputation as an inspirational teacher and trainer. His school courses are firmly based in practical experience and knowledge of children’s development, and engaging ways of teaching. His aim is always to provoke your thinking and motivate your teaching, he always hopes to both entertain you and touch your teaching heart. 2A Alan Edmiston and Kirstin Mulholland – Exploring the role of Metacognition In this session Kirstin and Alan are seeking to work with schools who are keen to explore the role that metacognition has in the learning of all children. They will share their understanding of the meaning of the term metacognition and some of the strategies they have used in schools. They are keen to partner with schools to further develop our understanding of metacognition and how it applies to SEND children. In this respect this session will start the ball rolling with a metacognition support group that will share its findings through Equals over the coming months. Dr Kirstin Mulholland currently works as the Education Endowment Foundation’s Maths Content Specialist. Alongside this role, she is an Assistant Professor in Education at Northumbria University, as well as a former primary school teacher, maths lead, and Specialist Leader of Education. Alan Edmiston is the current editor of Equals and also works as a Research and Innovation Lead for the Maths Hubs supporting the development of their SEND work. Originally a science teacher Alan began to explore mathematical thinking in children over 25 years ago. He is close to the end of his PhD looking at small group talk in secondary science lessons. Session 2 13.05 – 14.05
Vol. 27 No. 3 Autumn 2022 8 2C Alison Roulstone - What do we know about Developmental Dyscalculia? Specific learning difficulties in mathematics, also known as Developmental Dyscalculia (DD), are estimated to affect about 1 in 20 children in the UK. This means that there is at least one child with dyscalculia in most classrooms. Consequently, education professionals need a solid understanding of dyscalculia to know how to best identify and support children at risk. Currently, there is limited research that explores educators’ awareness and understanding of DD. Therefore, the purpose of this session is to share my findings from a recent study that explores educators’ awareness of DD. This interactive session will involve exploring 24 specific statements about DD, which were previously validated by thirty academic research experts. The “what can we learn about DD?” workshop aims to be an informative presentation for anyone wanting to strengthen their understanding of specific learning difficulties in mathematics. Alison Roulstone is a qualified primary schoolteacher and Special Educational Needs Co-ordinator (SENDCo) with seventeen years of teaching and leadership experience in mainstream schools across the East Midlands, UK. Having taught Mathematics to children from the early years all the way up to Key Stage 2, Alison has a good understanding of the primary maths curriculum and age-appropriate skills progression. Currently a PhD student within the Centre for Mathematical Cognition, based at the University of Loughborough, her research project focuses on understanding and raising awareness of dyscalculia amongst educational practitioners, and developing a screening instrument to identify children who are at risk of dyscalculia. Presently, she is collaborating with researchers from the UK, Vietnam, South Africa, Italy, and Germany to better understand what teachers know about dyscalculia. 3A Dr Rachael Addy – Maths Anxiety It is estimated 2 to 6% of mainstream school students report higher levels of maths anxiety, yet many educators lack awareness and/or understanding of how to identify and support these students. In a novel approach to maths anxiety research, observations of classroom behaviours exhibited by students with higher levels of maths anxiety were carried out, aiming to provide a new technique to further understanding and better support and prepare both pupils and educators. This session will explore maths Session 3 14.30 – 15.30
Vol. 27 No. 3 Autumn 2022 9 sound knowledge and understanding of each child’s needs. In mathematics this means shaping teaching around the way different pupils learn and taking care to nurture the unique talents of every pupil. Castle have developed a coherent curriculum based on a range of compelling learning experiences, which excite and inspire our learners. In this session Anne, the acting Headteacher, will share the journey they have taken to produce a truly wonderful mathematics curriculum. Anne Haberfield is Acting Head Teacher at Castle School a Special School in Cambridge. Having taught in both Mainstream and Special schools she has extensive experience in curriculum development and design. In particular using innovative ways of developing and delivering a curriculum to hard-to-reach pupils. Anne has been working with Cambridge Maths hub for eight year and her previous work groups “Maths in Stories” and Maths and SEND have received outstanding feedback . The school have been actively exploring how Teaching for Mastery can be utilised within their setting and have developed some innovative practice. She is delighted to be leading the current innovation work group TfM in the context of special schools within Cambridge Maths Hub and excited about being able to shape national developments with the NCETM. 3B Anne Haberfield – Castle Schools’ journey to a wonderful Maths Curriculum At Castle School the teachers seek to make the child’s learning stimulating, creative, fun, and successful whilst setting high expectations for every child. It is delivered through high quality teaching based on anxiety, findings from these observations and discuss the future of the research. An interactive session considering how students experiencing higher levels of maths anxiety may behave in the classroom and why. The session will combine a PowerPoint presentation with online polls and questions that participants may interact with on their personal devices and which shape discussion. Rachael is a Lecturer at the University of Lincoln, specialising in mathematics and statistics. With a background in inclusive education and additional needs, Rachael is furthering her knowledge by studying for a Doctor of Education at the University of Derby, with research into identifying classroom behaviours exhibited by students with higher levels of maths anxiety. Having taught for many years in a Community Hospital School, Rachael has experienced teaching students with significant interruptions to learning and is passionate about ensuring opportunities for students requiring additional support or alternative paths through education.
Vol. 27 No. 3 Autumn 2022 10 3C Janet Goring - Games for Maths, Motivation, Metacognition and More The concept of using games in maths is not new as a way of making maths “fun” but how can we make the most of games to increase student enjoyment, increase their mathematical thinking and gain insight into their learning to inform assessment for learning. This session will consider the research into games and use a small number of games to demonstrate how these can be adapted to different ages and abilities. Prepare to play! Janet Goring has taught for over twenty years as a class teacher, maths subject leader, Advanced Skills teacer and currently as a specialist teacher for pupils with specific learning difficulties. She manages a locala authority team of 11 specialist teachers working with students from KS1-4 with literacy and/or maths difficulties. She has a Masters in Specific Learning Difficulties. Her dissertation researched how playing a board game can increase number sense, following a career-long interest in using games to develop understanding and increase enjoyment and an ever-expanding dice collection. Janet is an accredited PD lead for the NCETM and has led SEND work groups for the last four years in her local Maths Hub. She is also on the Dyscalculia committee of the British Dyslexia Association. Changing Perceptions of Maths To improve young people’s experiences of and perceptions of maths, and to stop them and their teachers feeling anxious about maths lessons and tests, we need a sea change in the maths curriculum, and in assessment methods. I will describe some initiatives to change perceptions but basically this needs a large-scale national initiative. Margaret is an Emeritus Professor of Mathematics Education at King’s College London. She has taught in primary and secondary schools, trained maths teachers and directed research and development projects concerned with learning, teaching and assessing mathematics at all ages and levels. She is an ex-President of the Mathematical Association and has been a member of several government advisory committees. Currently she is President of The Maths Anxiety Trust and Deputy Chair of MathsWorldUK. Closing Plenary – 15.30 – 16.30 Keynote – Margaret Brown OBE
Vol. 27 No. 3 Autumn 2022 11 Alan Edmiston highlights the SEND work that is currently taking place among the English Maths Hubs. SEND and the Maths Hubs Every year the NCETM fund the Maths Hubs to run a number of research and innovation work groups termed RIWG’s. The work groups seek to provide a local context for teachers across England to explore areas of interest. Since 2021 there have been two SEND focused RIWG in operation: • Supporting SEND in mainstream schools and, • Exploring the Teaching for Mastery (TfM) in special schools and alternative provision. Work groups usually last for an academic year and are co-ordinated by a lead and all work groups in a strand are overseen by a national lead. The outcomes and findings from each set of work group’s then influence the subsequent work of the NCETM through the Maths Hubs. I have had the great pleasure of being involved in both groups at all levels and this year I am leading the strand and it looks like we are going to have a very interesting time. The plan is that the outcomes from the work that take place this year will be shared and disseminated more widely by the NCETM. How this looks will be the focus of the Summer 23 edition of Equals and there will be many more submissions from this work in future editions. This edition contains two illustrations of how the groups work. Firstly I am including an appeal for help from one of the work group leads, Max Hogg. Max is running a group under the Characteristics of SEND banner and this clearly highlights how this way of working facilitates teachers to improve and develop their practice. An appeal by Max Hogg Are you leading or working in a school that has seen a significant increase in SEMH needs since the pandemic? Have you found an effective way to intervene to support those students to get back on track in maths? Or are you searching for a way to support those students? If you answered yes to any of these questions, we would love to hear from you. The Boolean Maths Hub (based in Bristol, covering Bristol and Somerset) is convening a work group this year to explore how to intervene effectively to support students who feel anxious about maths, and are struggling to engage in lessons as a result. We are particularly interested in visiting schools that have intervened effectively with a vulnerable group of students, where the school has measurable impact on their progress in maths or where the professionals involved judge that the intervention has been effective over time.
Vol. 27 No. 3 Autumn 2022 12 The group could be those with SEND, PP, P/LAC, refugees, those who struggled to engage with maths during remote learning etc. We are also keen to share the findings from the work group towards the end of this academic year. If you think your school might be a source of good practice for us to reflect on and share, or if you would like to hear the conclusions of the work group, please contact the work group lead Max Hogg on mhogg@bristolcathedral.org.uk. Many thanks. Keeping it simple Lou Hoskyns-Staples shares her experience of supporting special school teachers to research their own practice in the context of Teaching for Mastery. Context In the spring term of 2022 I worked with eight teachers in the Central Maths Hub region as part of an NCETM Research and Innovation Work Group: Teaching for Mastery in the context of Special Schools and Alternative Provision. My role was twofold: to provide external continuing professional development (CPD) and to support the teachers to design small-scale projects that could support change in their schools. Introduction The teachers, who came from a range of special schools and alternative provision settings, developed their understanding of Teaching for Mastery, planned and executed a simple practitioner research cycle (Plan-Do-Review), focusing on a single pedagogic concept, and gathered evidence to support wider change in their schools. From the beginning of the project, we discussed the need for a final ‘output’ that could be used by others to learn about the work we did together. This article forms part of that output. Each teacher provided a simple write up of their evidence to share with school colleagues, members of the Work Group and nationally with the NCETM. To support the teachers to develop their projects we read the EEF guidance report Improving Mathematics in Key Stages Two and Three (Education Endowment Foundation, 2017) and throughout the project we used the EEF school improvement cycle (see Figure 1, blue labels added). This gave credence to the projects, which helped gain support from school leadership teams (SLT).
Vol. 27 No. 3 Autumn 2022 13 We had four collaborative online meetings during the year, and I also met individual teachers to support their implementation of their research cycles. The project took place over a very limited time period (January to April) but teachers continued to gather evidence until the end of June. Pupil and teacher evidence was gathered by each teacher throughout the implementation stage. Working collaboratively not only supported the teachers to develop their projects through the sharing and refining of ideas but also meant that they were more confident in giving themselves ‘permission’ to try out something new. Knowing that at the ‘review’ stage, questions and plans could be adapted for a further cycle, and that this was a normal part of researching practice, also reinforced their confidence to ‘have a go’. Ongoing Covid-19 caused high pupil and staff absence, which inhibited progress in some projects and not all reached a conclusion during the time available. School pupils are referred to with pseudonyms. Developing the practitioner research cycle Background reading Although background reading was not a core part of the project beyond the EEF (2017), participants and I shared interesting articles that we thought would support others. During the project Rebekah Gear published an article in Impact discussing how the Five Big Ideas of Mastery can influence curriculum design more generally (Gear, 2022). For two projects, subitising formed part of the focus and we used an article from Equals (Langford, 2019) and the activities provided by Alan Edmiston as part of the thinking for the projects. Finding the right question Finding good questions for the research projects was the hardest part: ideas needed to become reality; questions needed to be precise, measurable and simple enough to be carried out in the time available. The teachers worked Figure 1. An evidence-informed school improvement cycle (adapted from EEF, page 4) STEP 1 Decide what you want to achieve Identify school priorities using internal data and professional judgement. Question STEP 2 Identifying possible solutions External evidence from this guidance and elsewhere can be used to inform choices. STEP 5 Securing and spreading change Mobilise the knowledge and use the findings to inform the work of the school to grow or stop the intervention. STEP 3 Giving the idea the best chance of success Applying the ingredients of effective implementation. STEP 4 Did it work? Evaluate the impact of your decisions and identify potential improvements for the future. Plan Do Review Output
Vol. 27 No. 3 Autumn 2022 14 collaboratively, supporting each other to develop and refine their ideas. This process took longer than I had anticipated, everyone wanted to right all the wrongs in their schools, but the effort at this stage was worth it and enabled the teachers to have clear ideas making the planning much simpler. ‘Keep it simple’ became a repeated catch phrase in our discussions. To help with a narrow focus, we explored all the skills that underpinned the learning of a concept by using ‘knowledge packages’ (Ma, 1999). Exploring an apparently simple arithmetic operation, in this case subtraction with regrouping, see Figure 2, was a useful tool for demonstrating complexity and considering the preceding steps required for learners. Subtraction with regrouping of large numbers The composition of numbers within 100 The composition of 10 The rate of composing a higher value unit Subtraction without regrouping Addition without carrying Addition and subtraction within 10 Addition and subtraction as inverse operations Addition and subtraction within 20 Composing and decomposing a higher value unit Subtraction and regrouping of numbers between 20 and 100 Where does this learning lead? Figure 2. Adapted from Ma (1999, p. 19)
Vol. 27 No. 3 Autumn 2022 15 Planning With clear questions in mind, the planning stage was relatively easy. The teachers considered who to work with in school and which pupils would provide the strongest evidence. Pupil evidence was gathered at three points during the project to provide a baseline, monitoring and a final measure of impact. The projects Eight teachers began the Work Group and of those eight, three are shared here. Razia’s project Razia is a secondary mathematics teacher and the director of cognition and learning/ mathematics at the Wilson Stuart School and the Education Impact Academy Trust. The Trust includes Mayfield and Queensbury whose mathematics leads also engaged in the Work Group. The pupils in the Wilson Stuart School are MLD learners. Razia’s initial idea was to develop mathematical vocabulary through reasoning, but she was struggling to work out how this could be implemented. Pupils’ explanations of their reasoning were not strong and they found it hard to articulate coherent thought processes, which was inhibiting their ability to reason. After discussion, Razia amended her question to, ‘How to develop reasoning through the use of mathematical vocabulary?’ Razia initially used a word board to support the use of vocabulary, but to increase the immediacy of the language created laminated words that the pupils could use. Razia used video to capture pupil data which could be shared with school colleagues to demonstrate impact. Razia began with three focus pupils but high absence meant that there was insufficient data for one pupil. Steve’s journey Steve is in year 10; he has developmental delay and struggles to make connections or see patterns in mathematics. He is working well below age-related expectations (ARE). Initially, Steve struggled to explain his understanding and most of his answers were guessed. Steve found it difficult to explain his thinking and lacked confidence when sharing ideas in class. Razia challenged Steve to explain his thinking rather than give an answer and encouraged him to use the word board. He needed prompting and during the project has had a significant amount of learning support. Gradually, Steve showed an increase in using the word board; however, this was not consistent. With a persistent teaching approach focusing on the word board and additional laminated vocabulary Steve had developed the confidence to explain his reasoning to his peers and uses correct mathematical vocabulary. David’s journey David is in year 7; he finds reading challenging and his speech is not always clear. He needs constant reassurance when completing tasks. Pupils’ explanations of their reasoning were not strong and they found it hard to articulate coherent thought processes.
Vol. 27 No. 3 Autumn 2022 16 Like Steve, he was encouraged to explain his thinking rather than giving an answer. The word board proved less effective as a form of support due to his limited reading ability. Razia consistently modelled the language she wanted Steve to use and he was given a significant amount of learning support to aid his reasoning skills. The focus was reduced to the four operations but David’s responses remained inconsistent. Targeted laminated vocabulary and a focus on explaining his thinking has developed David’s confidence to share with the group even though he is not always correct. Figure 3 is taken from the final piece of video evidence and demonstrates a marked increase in confidence. Overall impact and next steps The pupils showed a marked increase in confidence when sharing their reasoning and are no longer afraid to make mistakes. Pupils now explain their mathematical reasoning rather than giving a monosyllabic answer. The laminated vocabulary has become part of classroom practice and is available for all pupils to use. A next step is to use symbols to support pupils with low reading ability. The project is being expanded to more classes where the teachers are confident to apply a different approach and Razia is creating a video bank as additional CPD. Rebecca’s projects Rebecca is a relatively new mathematics lead at Mayfield School in Birmingham. The school had undergone a period of transition and teachers were not always open to further change. Rebecca had previously led mathematics in a mainstream primary school and had a clear understanding of how to develop numerosity and number sense. The school is in an economically deprived area of Birmingham and is split over two sites. The first site has predominantly ASC learners and the second has a mixture of SLD and PMLD learners. Rebecca chose a different focus for each site. Her two projects have enabled her to work with teachers on each site and to gain advocates for a change in practice. Throughout the Work Group, Rebecca shared inspiring examples of pupil progress demonstrating the impact of core representations and subitising. Figure 3 is taken from a short video clip shared by Rebecca in which Nat demonstrates how he knows there are 12 counters. Nat is able to quickly see the four missing counters below the blue counters and moves the red counters to fill the tens frame. During his explanation he uses lovely mathematical language as he explains his thinking, ‘these combine together there’ tapping the empty four, ‘but these two … not go there’ tapping the remaining two counters. Figure 3. David confidently counts backwards at the front of the class.
Vol. 27 No. 3 Autumn 2022 17 ASC – Simplifying work trays and using key representations The focus pupil, Dwayne, is 14 years old, non-verbal and is operating at pre-key stage standard 1. For the two previous years, he has been working towards recognising numbers 1–5. Pupils in the school use work trays and normally these have a series of mathematical activities cycling through number, shape and space, and geometry. Rebecca chose to keep the same concept for a period of two weeks and to focus on using the key representation of a five frame. First observation Dwayne showed that he was able to identify numerals 1, 2 and 3 if placed in numerical order and some evidence of him being able to match two dots to the numeral 2 with support. Second observation Dwayne is becoming increasingly fluent in his ability to match representations of quantities to three with minimal support. Third observation Dwayne is able to complete the activity without prompting or adult support. With prompting he is attempting quantities up to five. Impact Dwayne’s teacher was very impressed with his progress. In a matter of weeks Dwayne has made significantly faster progress than over the last two years. SLD – Developing subitising Gab is in year 9 and has global development delay and is operating at pre-key stage standard 2. Historically there has been a strong reliance on counting in the school rather than developing an approach based around number sense. Rebecca chose to clear images and five frames to support subitising. First observation Gab is able to give an adult one item from a group, but unable to provide two when asked. He could continually repeat giving more items, but each time labelled them as ‘one’. Second observation The adult working with Gab used consistent language to support subitising, ‘I can see two; what do you see?’ alongside subitising pictures and a five frame. When playing subitising games on the whiteboard, Gab needed less support to identify groups of two. Figure 4. Nat demonstrating how he knows there are 12 counters.
Vol. 27 No. 3 Autumn 2022 18 Third observation Gab can now make groups of two or three items with minimal prompting. Impact Gab’s teacher has noticed an improvement in his confidence and his ability to join in with mathematics activities with less resistance. Overall impact and next steps Rebecca is excited about the progress that these pupils have made in eight weeks and alongside a newly structured curriculum and more effective assessment systems, focusing on learning rather than targets, she is looking forward to leading CPD, providing support and helping with interventions. Rebecca now has some strong advocates in her schools to support future change. Kay’s project Kay is the mathematics teacher at New Ways School. The pupils are of secondary age and are ADHD and ASD learners. Kay was not initially trained as a mathematics specialist and found the mathematics input in the Work Group hugely valuable. Some of Kay’s pupils were too afraid to even enter the mathematics classroom, making it difficult to find an appropriate focus for the project. The discussion in the Work Group enabled Kay to find a creative approach through looking for ‘hidden’ mathematics opportunities across the school in cooking, games, art, sport and shopping, depending on the interests of her focus pupils. Kay began the project with three focus pupils but due to a deterioration in the circumstances of one pupil, Kay was unable to gather sufficient evidence. Riley Riley is in year 11 and is due to sit Functional Skills Level 1. Riley has missed a significant amount of school due to an eating disorder and is petrified of mathematics. Riley has poor written skills and finds sitting still in any lesson difficult; he loves art, especially graffiti. Riley was observed over ten lessons. Initial observations Riley used a range of behaviours to avoid engaging with mathematics including pacing, forgetting to bring stationery, needing the lavatory and a negative conversation from a previous lesson. He needed scrap paper for drawing and a mini-board so that he can erase his work and had a constant need for reassurance. Kay read and scribed for him to reduce the level of anxiety. Ongoing observations Kay gave Riley the space to go through his process of entering the mathematics classroom and only worked with him one-to-one. His art book was used as a reward system. Art and manipulatives supported Riley in his interpretation of mathematics and with positive reinforcement Riley was able to sit for longer than 20 minutes exceeding his EHCP expectations. Riley was particularly proud when Kay informed his mum of his progress and this became a routine at home The discussion in the Work Group enabled Kay to find a creative approach through looking for ‘hidden’ mathematics opportunities.
Vol. 27 No. 3 Autumn 2022 19 time. Riley’s confidence increased throughout the project and he was able to complete all eight components of the ELC mathematics papers. Pat Pat is in year 10 and suffers from a range of attention disorders with poor recall and understanding. She is operating at lower key stage 2 levels in both mathematics and English. Pat needs to work one-to-one to minimise distraction and avoidance techniques. Initial observations Pat panics in lessons without a calculator, will restart an entire piece of work if there is a single error, tries to dictate what she will learn, will leave the classroom if faced with anything new and reacts poorly to any change in routine. Pat suffers from attachment disorder. Ongoing observations Pat built a strong relationship with Kay, alleviating the behaviours associated with her attachment disorder. Pat worked well with manipulatives and open tasks and although she still dislikes praise is able to accept positive reinforcement. She likes to copy out work that has been completed and shows this to the headteacher and her parents. She still worries about things she cannot control but can now stay in the classroom for an entire lesson. Overall findings and next steps Removing pupils from group scenarios reduced their anxiety and the developed confidence. The use of the ‘hidden’ maths in other subjects increased engagement and demonstrated that mathematics is also a ‘life skill’. Previous history means that direct praise is difficult so Kay provides positive reinforcement in other ways such as rewards and sharing success with parents. Conclusions The teachers involved in the project are confident to initiate curriculum and pedagogic changes in their schools. Working collaboratively improved each of the projects and the discussions around Teaching for Mastery gave confidence to try out new techniques. The teachers benefited from having the ‘permission’ to step back from their normal teaching routines and try something new, always bearing in mind the need to keep it simple. On many occasions we were all astounded by how much progress individual pupils made in the limited timeframe of the project. I very much enjoyed working with these teachers and we all learnt a huge amount about their pupils and their schools. References Education Endowment Foundation (2017) Improving Mathematics in Key Stages Two and Three A self-assessment guide. Available at: https://educationendowmentfoundation. org.uk/public/files/Presentations/Publications/Maths/5660_ EEF_-_Maths_Guidance_RAG_v5.pdf. Gear, R. (2022) ‘A curriculum masterclass: Inspired by lessons we have learned from mathematics’, Impact, (14), pp. 59–61. Langford, L. (2019) ‘Starting with dots!’, Equals, 3, pp. 3–8. Ma, L. (1999) Knowing and Teaching Elementary Mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Abingdon: Routledge. On many occasions we were all astounded by how much progress individual pupils made in the limited timeframe of the project.
Vol. 27 No. 3 Autumn 2022 20 Playing the game Rebekah Gear and Matt Woodford discuss evidence from a recent study, whereby they invited pupils from a primary school with SEND to co-contribute to the design of their mathematical learning experiences. They discuss how inviting pupils to contribute to an intervention design can have an impact on how they experience mathematics that can support their self-esteem. Our study was informed by some earlier research we carried out that indicated the rehearsal of declarative knowledge can have negative effects on pupils’ self-esteem. This was despite the recommendations which emerged from the recent Ofsted review (2021) which advocated that both systematic and instructional approaches to retaining new learning, can enhance the progress, attainment and self-esteem of pupils with SEND. However, after we observed a pupil working on a task designed to help him retrieve information, he stated: “It just made me sad, because I knew I was getting some wrong on the test, and it made me panic!” We concluded that what we as practitioners may consider as a low-stakes quiz (EEF 2021) is rarely seen that way for our pupils. Alongside this, we observed other signs of stress in pupils including an outright refusal to complete the task and it was apparent that this current approach had impacted their self-esteem. Therefore, we set up a piece of practitioner research designed to investigate the experience of pupils with SEND when asked to create mathematical activities that they felt would support their learning and retention of multiplication tables. We saw this as an opportunity to invite pupils to become co-researchers and contributors to our research practices, in the expectation that it would help us understand how they best like to learn. Our study took place in a mainstream school with a group of 7 boys aged between 9 and 10 who had been identified as having moderate learning difficulties. Design 1: Playing cards The first design suggested involved each pupil in a pair selecting a set of 10 playing cards from a pack and turning over two at a time in order to create a calculation. From this starting point the pupils came up with two variations to create calculations. Firstly, to allow a focus on a specific multiplication table, pupils suggested selecting a static multiplier and turning over the other cards to reveal the multiplicand (see Figure 1). In a second variation they suggested both the multiplier and the multiplicand could be changed at the We observed other signs of stress in pupils including an outright refusal to complete the task.
Vol. 27 No. 3 Autumn 2022 21 same time. For this version of the game pupils suggested either a race or taking turns to provide opportunities for peer support. During our subsequent discussions with pupils, we asked why they had allowed a set of cards to be chosen rather than just use the whole pack. The pupils revealed thoughtful, meta-cognitive thinking in stating that the choice gave them more ownership over the calculations. For example, some explained that they only included certain card values, such as 3, 5, 2 to help them practice a particular multiplication table fact. Similarly, one pupil stated: “I knew I needed to test my memory of my 6’s, so I made sure all of the 6 cards were in the pack.” In terms of self-esteem, pupils valued the support provided by others, in assisting them in feeling successful. They were not upset by not being able to complete a calculation and this ‘failure’ no longer posed a threat to their self-esteem. Whilst playing the game we witnessed no signs of stress or refusal. There were numerous observations of how the pupils worked together and supported each other. Sometimes they would hold their fingers out so the partner could physically count in the multiplication table up to the number on the cards. One student stated: “I felt like I did well in the game and if I got really stuck, my partner helped me and that was okay.” Design 2: Dominos The second activity replicated a similar structure to the playing cards but used dominos. Again, pupils made decisions regarding the structure of the calculations. As a starting point for the game pupils would turn over two dominos (see Figure 2) and calculated 2 multiplied by 4 followed by 5 multiplied by 1. However, pupils also insisted on having the option of totalling the spots on each domino, for example in Figure 2 this would result in 7 multiplied by 5. Pupils made this choice based on their confidence with the respective multiplication table. Figure 1: Pupils playing the first variation of the card game they designed, using a static multiplier.
Vol. 27 No. 3 Autumn 2022 22 Similar to the card game, pupils spent time choosing which dominos they wanted to use. However, in the domino game pupils had the additional option of selecting which domino to play. We often observed one pupil in the pair purposefully selecting their domino to create a known or more comfortable calculation. For example, in Figure 2, the green domino of 4 and 1 was chosen first, before the second pupil deliberately selected their 2 and 5 domino. He explained that he felt comfortable because he could see the calculations 4 multiplied by 2 and 1 multiplied by 5. One pupil explained: “I liked that we could make them easier or harder for each other with the dominos, looking at all the spots of just the top or bottom part.” For us, this was evidence that these pupils were carefully considering the structures of the calculation itself and the value and position of the multiplier and multiplicand. In addition, we noted that the domino representation assisted the skill of subitising, and further supported their retrieval process. Our reflections on games and co-cognitive learning Throughout our discussions with the pupils, they described the benefits of using games to create a supportive learning environment. Pupils stated how they both learned from, and with, each other: “I liked playing with my partner and we both got chance to answer lots of questions, remembering and using our times table facts.” It was apparent to us, that these pupils not only rose to the challenge of being co-contributors to our research, but they curated experiences for themselves where they excelled and flourished. Success became a self-fulfilling prophecy; a stark contrast to the ‘low-stakes’ retrieval practice activity which had inspired this project initially. Our study evidenced the possibilities that the rehearsal and retrieval practice of declarative knowledge is not necessarily an individual exercise and may be better situated as a co-cognitive experience. This appeared to have the most positive implications for pupil’s self-esteem and how they viewed themselves: “I felt like I had proved to myself that I can do maths and I can learn my times tables when I saw how many questions I had answered right... it made me feel good about myself.” Throughout this study we noticed the delicate nature of self-esteem for pupils with SEND. Whilst Figure 2: Pupils playing their domino multiplication game. Success became a self-fulfilling prophecy; a stark contrast to the ‘low-stakes’ retrieval practice activity which had inspired this project initially.
Vol. 27 No. 3 Autumn 2022 23 we see the value of retrieval practice, we are reminded that for some pupils this is seldom seen as low-stakes. From our experience in this study giving pupils the responsibility to create games transferred ownership and led to a concomitant reduction in stakes. Furthermore, through pupil designed games we observed positive implications for pupils’ retrieval, metacognition and self-esteem. References: 1. Education Endowment Foundation [EEF]., 2021. Cognitive Science in the Classroom: Evidence and Practice Review. [Online]. Available at: https:// d2tic4wvo1iusb.cloudfront.net/documents/guidance/ Cognitive_Science_in_the_classroom_-_Evidence_and_ practice_review.pdf [Accessed 10 December 2021]. 2. Ofsted., 2021. Research review series: Mathematics. Manchester: Ofsted. Available at: https://www.gov.uk/ government/publications/research-review-series-mathematics [Accessed: 12 June 2021]. How does understanding number happen In how does understanding happen Les Stave takes the time to share his thoughts upon the nature of number sense. In the process he asks the right questions to support the profession to think about how early number work should be shaped around the developmental needs of children. Awareness is growing in schools about the importance of developing ‘Number Sense’ and teaching the skills of subitising that develop from it. It is at last gradually reaching our curriculum documents, but what is it? Where does it come from; how does it develop in our minds? These are questions for us to understand If we are to appreciate how to teach children to improve this natural skill which underpins mathematical learning. A sense of quantity. Pioneering work in 1992 by Karen Wynn on the early numerical cognition of infants discussed how in the first weeks of life infants could visually discriminate when there were quantitative differences between groups of objects. Their reactions indicated that they noticed changes and by five months they even showed some anticipation of what outcomes of a change should be. She dubbed this ‘number sense’ and noted it operated at two levels, exact and approximate number sense. With small groups babies have an ‘exact number sense’ they immediately notice differences between small groups of one to five things. It has been established that this sense is as automatic as discriminating colours. But there is a limitation and neither children, nor adults, can differentiate when groups are larger than five. So, for example we are not able to easily notice differences between random groups such as six and eight, or seven and 10 etc. For this we must use additional pattern thinking.
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