Vol. 26 No. 2 Summer 2021 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 Supporting the Assessment of SEND 3 Dan Blundred and Amy Vigus kindly offered to provide an insight into what they are doing at Ormiston Meridian Academy in Stoke. Moving forward 8 Natalie Kerslake has a real passion for helping her colleagues to better understand and support children with dyscalculia. Here she shares her views on the way forward at her school. Maths in a time of Covid 9 Lucy Rycroft-Smith - a long-term friend and supporter of Equals got in touch to share some lessons learned during the past 12 months! A manifesto for SEND 12 During an on-line training course forty people were consulted on their views on what should be done now to support pupils with SEND. In this piece Alan Edmiston shared what was said. Ideal Maths Policy 13 In a very prescient piece Mark Pepper ponders some answers to the question: In an ideal world in which there were no restraints or restrictions, what would consist of an efficient maths teaching policy aimed at the creation of effective teaching and learning strategies? Don’t imply that or How children jump to the wrong conclusion 19 From her treasure trove of teaching wisdom Fiona Allen reflects upon the impact of the words teachers use upon the thinking of their students. Sticking points in maths teaching and learning – 24 global issues, local solutions From a global perspective Pete Jarrett uses this article to make a very personal appeal for a curriculum based upon the four C’s: Critical thinking, Communication, Collaboration and Creativity. Book review: ‘How We Learn’ by Stanislas Dehaene 28 reviewed by Alan Edmiston Alan Edmiston shares his views on a book that seeks to provide clear advice for those who wish their teaching to be informed by the latest knowledge from neuroscience.
Vol. 26 No. 2 Summer 2021 2 Editors’ Page Welcome to the Summer 2021 edition of Equals – we hope you enjoy the articles and find them a source of inspiration and support. This edition can be split into two as the first half deals with the impact of the past, difficult and challenging, twelve months upon all of us. The focus here is upon the sharing of the changes people are making to support their learners. The second contains an interesting range of pieces that should help anyone who wishes to develop an inclusive classroom where all children reach out to fulfil their potential. If any of these articles resonate with you then please let us know as we would be delighted to share what you are doing. Also if you need any advice or support then get in touch as we have a range of expertise and experience in supporting school ready and waiting to help. “Having a sharp, well-defined repertoire of approaches will help ensure high expectations for all are maintained, next steps are well-informed and pupils with SEND thrive.” I have recently had cause to read and study the recent SEND publication from the EEF including the five key recommendations for mainstream schools. In addition to this I also managed to catch a blog on the same matter by Kirsten Mould whose closing advice is given above. Rather than comment upon what she wrote or even the report itself I want to share the main questions that occurred to me as I read Kirsten’s blog post. Kirsten’s comment My questions Scaffolding is a metaphor for temporary support and refers to the role of the teacher and assessment. How do people think they do this? What about peer support is that part of scaffolding? What is effective assessment and what do schools use to do this? Explicit instruction is vital. What is it and how do Rosenshine’s Principles of Instruction apply to mathematics and SEND? Cognitive and metacognitive strategies are important. What theories underpin these and do the adults in the room model their thinking? High - quality teaching for all. What is high quality teaching? What are high expectations in mathematics for those who can be five or more years behind their peers? Although I agree with every statement on the left I am not really sure of their true meaning and if a colleague in my school thinks they mean the same to her as they mean to me. It struck me that after teaching for 30 years I now have a professional radar that switches on when I hear
Vol. 26 No. 2 Summer 2021 3 phrases like those above. There are so many of these in use in daily educational life that they can end up meaning all things to all people or taken to justify a particular perspective. Now in the days of corporate educational multi-academy trusts I feel such words are said so often that they can become a sort of tribal mantra and with that alarm bells should begin to ring. Before the first lock down a colleague shared some really interesting data that I will repeat here. When comparing lesson observation judgements an agreement of 0.6 was found between HMI’s trained in lesson observation. The point of this for me was if HMI’s have such a difference of opinion then what about the rest of those who observe? Now I know what I think a good lesson should look like but would my colleague above agree. I wonder how much time is spent actually unpicking the meaning of phrases like those used in the EEF report and how it could apply to mathematics and SEND. With this is mind I will end this editorial with a series of questions and I look forward to receiving your responses and debating this further in the next edition. 1. Do you use and implement the guidance in the EEF report? If you do which bits are the most helpful to you? 2. How do you view scaffolding? What does it look like in your classroom? 3. What assessments do you use with your SEND learners and how do they inform what happens in the classroom? 4. Is explicit instruction part of your repertoire of approaches? 5. What does high quality teaching look like to you and what words would you use to describe it? 6. What cognitive and metacognitive strategies are common in your school? Do you even talk about this aspect of pedagogy? Dan Blundred (Director of KS3 Achievement and Head of KS3 Maths) and Amy Vigus (SENDCo) kindly offered to provide an insight into what they are doing at Ormiston Meridian Academy in Stoke. Supporting the success of SEND students post lockdown There are a number of things that happen in education that take us by surprise, but nothing can compare to the effects and impact of the Covid 19 pandemic. Every single student in every single educational setting will have been affected by the disruption to their education over the last twelve months, and there is no doubt that the task of trying to reduce the impact of lost learning time is monumental. Perhaps when it comes to our SEND learners the impact in many cases has been greater and the task of supporting them to achieve the best possible outcomes presents us with even
Vol. 26 No. 2 Summer 2021 4 more challenges. Working alongside our SENCo Amy Vigus, we have put together a brief overview of the approach that we have taken at Ormiston Meridian Academy. First of all, we think it is important to consider how students have been affected by the pandemic; in doing this it has helped us to identify the key areas where students may need further support and intervention. The most obvious area that has been impacted on from a school perspective is academic success and progress. Although many schools went to great lengths to provide online learning through the production of online resources and in some cases remote teaching, this could in no way replicate the learning experience that students would get in the classroom. No longer will I take for granted the opportunity to question and probe the students to build on their understanding of a key concept, or the chance to produce a pictorial representation of a particular maths problem that was proving troublesome. All these things have a significant impact in the classroom and were more difficult to achieve whilst we were delivering remotely. What we must not underestimate though, is the social and emotional impact of lockdown. We are social creatures, and we learn so much by watching, listening to, and interacting with other people particularly when we are of school age. Many students have missed almost twelve months of social interaction with their peers, their teachers and even their extended families. Often their only connection to other people was through social media, which has been demonstrated by the explosion of platforms such as TikTok, and this in itself will present us with challenges as we start to return to normality and the ‘real’ world. These missed experiences as a result of the school closures have meant that SEND students have been largely impacted and school staff are going to have to work closely alongside parents/carers to ensure effective strategies are implemented quickly and effectively to reduce the long term impact the pandemic may have on these students. Schools reopening does not come without its added complications and new things to consider. This alongside the anxieties that children may be facing only begins to touch the surface of the job schools have on their hands to ensure the smooth transition back into ‘normal’ school life. That said, it’s also a time of excitement. For us, the opportunity to have our students in front of us again is extremely valuable and, I am sure, something that will never be taken for granted again. Schools have a unique opportunity to play a big part in children’s lives right now and help them adjust to the new ‘normal’. To support our students and in particular our SEND students we have taken the following steps: • Prioritise mental health and wellbeing - With an ever-growing focus on mental health within schools, the impact of a global pandemic is only going to increase the pressures we face to ensure the appropriate measures are put into place to support the varying mental health issues presented by students. This should be at the forefront of every school when No longer will I take for granted the opportunity to question and probe the students to build on their understanding of a key concept
Vol. 26 No. 2 Summer 2021 5 considering the return of all students, however the support available may vary from student to student. For some, the return to school will be long overdue and once the initial ‘first day back’ nerves are out of the way, normality will resume. For others, the ability to ‘bounce back’ and respond quickly to changes is not a luxury, and it is these students who will need more careful planning and attention to enable them to fully immerse themselves back into everyday school life. During this time, there is not going to be a one size fits all approach which schools can undertake to support SEND students; instead, schools should aim to collaboratively work together with all stakeholders to establish the most effective actions required for that individual child. • Effective communication - Communication during this period of uncertainty is of paramount importance. As a SENCo’s, we play a vital role in ensuring the individual needs of the child are shared with relevant members of staff. Doing this allows staff to take into account the strategies and reasonable adjustments suggested for a particular child when planning subsequent lessons. Throughout both lockdowns, we were in constant communication with parents/ carers, students and staff to guarantee the remote learning being delivered still catered for our most vulnerable students. This included sending guides to both parents and teachers to explain how to set coloured overlays for students with dyslexia on their devices and programmes which would support with reading work aloud. With the return to school, this communication has shifted towards the sharing of individualised strategies tailored to the needs of students as we slowly adapt to the new norm. This level of rigour around our SEND students and provision needs to continue now we have returned. • Review the support on offer - For a considerable proportion of our SEND students, it was clear that their primary needs had changed and as a result, so had their learning requirements. Some students appeared to have regressed in confidence and ability so additional emphasis has been spent on developing the child’s self-esteem or subject specific interventions have been offered. For others, adapting to the rules and routines of a school environment has been challenging. We are extremely fortunate to have a base within our school which solely caters for students with social, emotional and mental health difficulties. Within this provision, students can access bespoke time tables, art therapy, counselling sessions and our school therapy dog (just to name a few). This has been a saving grace for our SEND students and has meant that the support we have been able to offer goes beyond re-establishing There is not going to be a one size fits all approach which schools can undertake to support SEND students. Some students appeared to have regressed in confidence and ability.
Vol. 26 No. 2 Summer 2021 6 academic success, but also addresses the social and emotional needs faced by an increasing number of students. • Focus on reading and literacy – even the most ardent mathematicians amongst us have to admit that reading fluently and having a good understanding of key vocabulary plays a huge part in being successful in maths. At Ormiston Meridian we have made reading and literacy one of our whole school priorities and it is being delivered and supported through every faculty. In maths we are introducing a number of comprehension style activities linked to maths as well as using the Count On Words resources that have been developed by Rachael Lethbridge, SENDCo and Assistant Principal at the Mary Hare School. The count on words resources aim to develop student’s maths vocabulary therefore increasing access to mathematical learning and student’s ability to achieve in the subject. The success of this programme is built on exposing students to new words, repeating words so that they are committed to long term memory and varying the words in all of their forms and meanings to further develop understanding. You can download these resources for free by visiting the Mary Hare website and selecting the downloadable resources section in the centre of excellence. • Using manipulatives and visual resources – the concrete, pictorial and abstract delivery of maths is at the very foundations of teaching for mastery. Using manipulatives is difficult at the moment but if it is possible to create class sets and keep them in class bubbles then using them is really helpful in putting in place strong foundations to key mathematical concepts. If you don’t have access to concrete resources, then Mathsbot has a variety of tools that can be projected and used to demonstrate the use of manipulatives. We have also moved to White Rose Maths in KS3 and have found their pictorial representations really helpful. • Find the gaps – one of the most useful things that we can do is find the gaps that students have in their learning and adapt our curriculum and teaching to ensure these gaps are addressed so that future learning is built on strong foundations. In maths we are very fortunate that there are a number of diagnostic tools out there which provide us with question level analysis. Diagnostic Questions, developed by Craig Barton, is something that we use, and the great thing is it’s free! There are lots of pre-prepared question sets and you can also develop your own. We also use Learning by Questions. Although there is a cost to this, the wealth of questions and the level of feedback has enabled us to track and monitor students really closely without additional workload for the class teacher. Learning by Questions also links very closely with the White Rose Maths schemes of learning. Both of these resources provide low stakes assessment opportunities in a fun and engaging way and produce Reading fluently and having a good understanding of key vocabulary plays a huge part in being successful in maths.
Vol. 26 No. 2 Summer 2021 7 valuable data for students and teachers. • Manage transition points – no matter what setting you are in there will be key transition points. For us, we are focused on the move from KS2 to KS3 and KS3 to KS4. Now, more than ever, it is important to gather as much information as possible but this year there are additional things we want to know about so that we can offer every child the support they need to be successful. In addition to the standard information around attainment and levels of support, finding out who accessed remote learning and who was affected directly by the pandemic might be two things to consider so we can direct the right support. Getting our feeder schools to outline which areas haven’t been covered will also be useful in supporting us to shape our curriculum as we move forward. • Curriculum design - Within our academy, faculties have been asked to reflect on the curriculum offered and respond to the necessary challenges now presented for a lot of our students. Faculty provision maps have been used to highlight how individual subject areas are embedding and implementing quality first teaching, whilst also demonstrating how the curriculum has been adapted for SEND students as a result of the missed learning. The power of a coherently planned curriculum cannot be ignored, and this factor alone will support greatly with the overall experience and success of SEND students post lockdown. This is by no means a complete list of what we can do to support our students, and in particular our SEND and most vulnerable students, at what has been one of the most challenging periods for education in our country. There will be numerous things that we were doing before lockdown that we need to continue and that any changes that we make must be student focused and support their mental health and well-being as well as their academic progress. It is also worth pointing out that there have been some positives to come out of this period of remote learning. In some cases, students, and in particular SEND students, have demonstrated immense resilience and independence not seen in the classroom at school, and for some, this way of learning has allowed students to flourish and develop skills they potentially would not be able to within the school environment because of the anxieties, distractions, and difficulties they face within school. The question educators should be considering is how to capture and harness these skills to ensure they are not lost when we return to face-to-face teaching. It is also worth pointing out that there have been some positives to come out of this period of remote learning. Finding out who accessed remote learning and who was affected directly by the pandemic might be two things to consider.
Vol. 26 No. 2 Summer 2021 8 Natalie Kerslake has a real passion for helping her colleagues to better understand and support children with dyscalculia. Here she shares her views on the way forward at her school. Moving forward After starting my first Teaching Assistant post back in 2011, I taught a learner with dyscalculia. Dyscalculia was something that I did not know much about at the time and wanted to know more about it. As I looked into this area, I found that not much was known about this, in comparison to other Specific Learning Differences. Therefore, when I came to do my MA Education dissertation, specialising in SEND, there was only one area that I wanted to look into. My MA Ed SEND dissertation, in 2015, looked at ‘How can Teachers and TA’s effectively support children with dyscalculia in one primary school’. The study consisted of a questionnaire, plus interviews with the special educational needs co-ordinator (SENCO), maths leader, and a class teacher of a learner with possible dyscalculia. One of the main findings from my study found that more awareness and training needs to be provided to aid them to support learners with dyscalculia effectively. I therefore wanted to use the findings from my dissertation to raise awareness of dyscalculia. As part of my Continuing Professional Development over the last year, I wanted to investigate the current knowledge and understanding of dyscalculia in the school that I work in. I invited all Teaching Assistants to complete a questionnaire to identify their current knowledge and understanding. From these questionnaires, three main areas were identified which were: more knowledge about dyscalculia; how to identify and assess these learners; along with strategies to use. This formed the basis of CPD training led by me in January 2021 to all TAs within the school. I hope to share this dyscalculia training with more schools in the Trust, to raise further awareness of dyscalculia. I have been fortunate to disseminate the findings of my MA Ed research and my subsequent work around raising awareness of dyscalculia further within the Dynamo Maths SENCO newsletter, Dyscalculia Blog, Equals Online and speaking at the recent Network ED East Midlands conference. I am due to speak at the virtual Mathematical Association conference in April, with my session focusing on the findings from my MA Education research, along with looking at what dyscalculia is, how learners with dyscalculia are identified, and strategies to support these learners effectively. I have just completed my Level 3 course in Dyscalculia and Maths Difficulties with the SEND Group and will be progressing onto the Level 5 More awareness and training needs to be provided to aid them to support learners with dyscalculia effectively.
Vol. 26 No. 2 Summer 2021 9 course in the near future. This would give me the opportunity to further develop my knowledge and skills in this area, through the planning and teaching of sessions for individual learners in school. Over the last year, during the COVID 19 pandemic, I have been supporting children with SEND in many ways, from supporting their learning in the classroom, through intervention and catch-up groups, along with remote learning and phone calls to families. All of this has helped to ensure children with SEND and their families are supported, in light of the global pandemic, both within school and at home. I look forward to continuing to raise awareness of dyscalculia in the future through my various projects – a big thank you has to go to Equals Online, for allowing me to share my work around dyscalculia with a wider audience. Lucy Rycroft-Smith - a long-term friend and supporter of Equals got in touch to share some lessons learned during the past 12 months! Maths in a time of Covid We are yet to determined exactly how 2020-21 will be remembered, but we know that issues of access, equity and identity in mathematics education have been disrupted during this period of lockdowns, virtual education and self-isolation, revealing that which may have been previously invisible and rendering less important that which may have been pivotal in the before-times. One example of the former might be the type of internet access, hardware and access to quiet space pupils have in which to study. We have learned to value the physical space of schools in different ways, even as we understand that they are not safe from infection, and this has felt complicated. An example of the latter: many schools have realised that the format and timing of work handed in is less important than previously thought; perhaps we have begun to make peace with messier, less constrained work, and with it begun to expose mathematical thinking in new ways. Below, I look at some of these issues in more detail, considering the question: in an ideal world, what lessons might we have learned from lockdown learning? Lesson 1: Students need well-designed tasks that give them early confidence In person, we can reassure our students that the work set for them is at the right level, that they are capable of accessing it with effort, Many schools have realised that the format and timing of work handed in is less important than previously thought.
Vol. 26 No. 2 Summer 2021 10 and that they can trust us to match them with suitable challenges. In short, that they can do it. Cheerleading matters, and especially in mathematics, where persistence is more than half the battle. When students are not present in the same physical location as the teacher, we see that they succumb to their inner saboteur faster and more frequently. Task choice is key, and disrupting spaces helps us remember that a well-structured task helps to minimise the need for this cheerleading by giving early success and interest in exploring the problem. We have also had the chance to consider the role of feedback and the importance of when and how it is given. Lesson 2: Let ‘em wear PE kit all day COVID has taught us that, contrary to the ‘Gospel of Schooling’, there are indeed greater goods than the correct uniform at the correct time. To minimise changing and infection spread, many schools have adopted a ‘PE kit all day on PE days’ policy. Similarly, when students are learning virtually, almost no-one would have dreamt of mandating what they wear to lessons (and a few schools hit the headline for supposedly attempting to do so). We have, in effect, conducted a large-scale experiment on the question: does uniform help students concentrate? And the results are in: comfort appears to have no particular detrimental effect, at least on the students themselves. What might this mean for mathematics? Maybe we can stop sweating the small stuff; maybe schools aren’t the place for extreme ‘broken window’ policies. If the goal for our students is deeper, more sense-making mathematical thinking, it may be time to stop telling them to tuck their shirt in constantly while they’re doing it - we’re just needlessly interrupting the good stuff. Lesson 3: Relationships matter If you had told most school-aged children they would be getting months and months off school before last year (alright, not exactly), many of them might have cheered and lobbed their rubbers in the air with joy. But life teaches us once again to be careful what you wish for. After the first few weeks, the shine wore off, and many of our students began to actually miss school and – surely not – their teachers. Students with particular needs around structure and boundaries had ample opportunity to see how carefully those things were provided for them throughout the school day and campus. A wide range of responses to a wide variety of lockdown teaching structures is likely to have deepened inequity in many ways – we all suddenly feel a painful longing for what we used to call banality, and now call consistency. But we have learned to miss one another, something we may not have previously thought possible. Students at risk from missing learning have often realised that they didn’t know what they had till it Contrary to the ‘Gospel of Schooling’, there are indeed greater goods than the correct uniform at the correct time. Students at risk from missing learning have often realised that they didn’t know what they had till it was gone.
Vol. 26 No. 2 Summer 2021 11 was gone. Where previously, they couldn’t think of anything worse than Mrs Smith for maths, they’ve now found it. Our collective imaginations have deepened. Oscar Wilde once said, “The only thing worse than being talked about is not being talked about.” Perhaps some of our most challenging students have realised that the only thing worse than a maths lesson is not having a maths lesson? Either way, there is a softness to us now, a kind of present-nostalgia that means we are all unusually attentive to relationships this year, and this can only be a positive influence on our mathematical practices together. Lesson 4: Bring the mathematics to the learner, not the other way around Rochelle Gutierrez wrote, “The assumption is that certain people will gain from having mathematics in their lives, as opposed to the field of mathematics will gain from having these people in its field.” Dispersing our students physically helps us remember the diversity of where they come from mentally too. With many of the minutiae of classroom life removed, the focus on being present and participating in mathematical dialogue has also reminded us of the most urgent priority of maths learning: you show up. And when our students showed up, so did their cats, their toddler siblings, their noisy boilers, and their neighbour’s motorbikes: in other words, we remembered they were humans with lives, bringing with them a social and cultural richness that they couldn’t discard even if they tried. It was always there in the classroom; it was just quieter (usually). What of this will bleed into the mathematics classrooms of the future? How might we remember the importance of those pets and siblings and hobbies to our student’s whole selves when they are invisible again, and how might we incorporate them into lessons to make our students think and smile and feel part of a whole again? Lesson 5: Space and safety Perhaps the greatest lesson of COVID disruption has been the important reminder that - ironically - the constraint of spaces we inhabit in schools has often made our students feel safe. Watching them try to learn (or switch off completely) from afar has reminded us that students can be hard to reach, but at least when physically present we can wrap the comfort of routine and environment around them – not as a straitjacket, but as a weighted blanket. When we return to classrooms (and many students and teachers never left) we will need a period of adjustment to the loss of control, the cold blankness that we all may have felt when that blanket was lifted – and we all will need to relearn how to be a member of a community that practises mathematics together again in new and different ways. Perhaps we can take the opportunity to question what value we gain from that community, and how we share the space of mathematics thinking together. Perhaps some of our most challenging students have realised that the only thing worse than a maths lesson is not having a maths lesson? We can wrap the comfort of routine and environment around them – not as a straitjacket, but as a weighted blanket.
Vol. 26 No. 2 Summer 2021 12 During an on-line training course forty people were consulted on their views on what should be done now to support pupils with SEND. In this piece Alan Edmiston shared what was said. A manifesto for SEND I am currently acting as a SEND work group lead for two Hubs in the Midlands. In January, during the feedback at the start of one of our meetings, the discussion turned to the end of lockdown and what that might look like for pupils with SEND. The discussions became more focused as we began to share just what needs to be done to nurture our pupils after a year like no other. The following gives you a flavour of the feelings and comments from the 40 colleagues who took part in this discussion. I think their views on how the mathematics classroom could be a basis for a manifesto for SEND that can be used to shape and guide provision to help us overcome the damage that the past 12 months has done. At this stage I am simply going to leave their words as they were said rather than paraphrase or merge them. • We must focus on their emotional well-being and reduce their anxiety • We must change our scheme of learning as they will never catch up. • Catch up is the most unhelpful phrase being used right now. It could be used in a very accusing way that will cause even more damage. • Maths clubs can now help. They will reduce stigma of catch up and focus upon maths in a supportive, safe and familial setting. • It’s not about catching up – we need to change/readjust what we teach and move from where they are not where they should have been last year. • They are too used to not going to school and their computers and isolation are the new normal. This raises massive questions for those of us who view learning as a social process. Computers make school redundant for some pupils – they can get all the feedback they need so why should they come to us. • It’s now about pastoral support - they need to feel safe before they can learn. We need to enhance and boost the pastoral system to complement the teaching that will be taking place. • What does progress mean now? This is a very real challenge for teachers now. • Funding is needed to meet the need as we cannot go back to how it was and pretend the treadmill can start at the same speed!
Vol. 26 No. 2 Summer 2021 13 • We need a plan to re-engage and to build a learning community. • A new curriculum is necessary for the impact lag will be in the system for years to come. • All years are struggling and we are having real issues in Year 10 with anxiety. We could do with the exams being held a year later as many go to college to resit anyway. • We cannot recap and revisit because some have made no progress at all. This is very different and new, we need to get a sense of where they are at and where we would like them to be. • We have a terms window to repair some damage so let’s treat the Summer term differently. If we don’t it will be a massive intervention with the issues that come with that and the problems will be perpetuated. • We need to consolidate for the rest of the year and now try not to do too much that is new. It’s about recovery and recuperation just like any illness. • It’s the pupils with poor social skills who worry me for they don’t want to or will not be able to come back till they feel safe. • I think we should go back twelve months and start building again – we need to know what foundations they are standing on so we can secure learning next year. What would you say? Do you agree? What do you think? What is your school putting in place to support learners and their learning rather than cover the curriculum – please get in touch let us know so we can celebrate your good work. In a very prescient piece Mark Pepper ponders some answers to the question: In an ideal world in which there were no restraints or restrictions, what would consist of an efficient maths teaching policy aimed at the creation of effective teaching and learning strategies? Ideal Maths Policy Part 1 At the outset it is helpful to consider the policy initiatives that have been highly influential in the changing provision of maths teaching in the classroom. Perhaps the most significant of these took place with the introduction of the National Curriculum (NC) in 1989. It is helpful to consider the difference between classroom teaching of maths Pre- National
Vol. 26 No. 2 Summer 2021 14 Curriculum and Post - National Curriculum. In the former case it was commonly claimed that what took place in the classroom was entirely in accordance with the individual teacher’s discretion. With the initial introduction of the NC a widespread reservation of many teachers was that their autonomy would be eroded. In theory it may be felt that if we returned to the Pre-NC position then teachers would be able to teach the content and use teaching strategies that they considered to be appropriate. There are, however, difficulties with this approach. In the former system there were doubtless many teachers who would have covered an appropriate curriculum and used effective teaching strategies. Nevertheless there had been misgivings generated by the suspicion that some teachers taught a restricted curriculum in accordance with their own strengths. For this reason it would be unwise to revert to the pre N.C. position. How an effective alternative policy could be produced? Let us imagine a utopian fantasy world in which the maths community is empowered to produce an effective teaching and learning maths policy that is completely insulated against external controls. At the outset it would be necessary to establish a collective forum charged with the responsibility of producing the policy. This group could include experienced maths teachers, well qualified and experienced maths advisors and commentators as well as organisations such as the Advisory Committee of Mathematics Education (A.C.M.E.). Encouragement could also be provided for teachers and other interested parties to send suggestions to the appointed group. Once this committee had created a written policy then this could be disseminated to the teaching community and to the general public. Organisations and individual teachers would then have the opportunity to react to this and to make alternative suggestions. Once this policy had eventually been refined and had received widespread acceptance then teachers would be expected to adhere closely to its key principles. There would, however, be flexibility for individual teachers to present the policy in a manner in accordance with their style of teaching. Consideration of key policy initiatives over the past 40 years One of the first tasks of the appointed group could consist of a review of the key policy initiatives of the past 40 years or so. Consideration could then be made of the policies that could be included and those that could be discarded. There is a high likelihood that the main policies that would be reviewed would consist of the National Curriculum (1989) with revisions in 1991, 1995 and 2014, the Cockcroft Report (1982) and the National Numeracy Strategy (1999). There had been misgivings generated by the suspicion that some teachers taught a restricted curriculum in accordance with their own strengths.
Vol. 26 No. 2 Summer 2021 15 The National Curriculum It would be helpful to retain this as it consists of a useful record of the content required for the teaching and learning of Number and Algebra, Shape, Space and Measure and Data Handling. It would be essential to include a Using and Applying Attainment Target. The inclusion of a mental maths component would also be helpful. The Cockcroft Report The Cockcroft Report was highly influential in the classroom teaching that took place in the immediate years following its publication. The central recommendations are encapsulated in the pivotal Para 243: Exposition by the teacher Discussion between teacher and pupils and between pupils themselves Appropriate practical work Problem solving including the application of mathematics to everyday situations Investigational work. A consequence of Cockcroft was that problem solving and investigations took place in classrooms on a regular basis. Additionally discussion both within groups of students and between the teacher and student groups were regular features of maths lessons and this helped to promote collaborative learning. A consistent theme in the Report is that pupils should consider mathematics to be an enjoyable subject. Para 347 applies this to the primary sector: The challenge for the teacher is to present mathematics in a way which continues to be interesting and enjoyable. As a result of this regular maths games sessions became a widespread feature in many primary classrooms. Aspects of the Cockcroft Report that could be included in a new policy The putative committee could consider including the following components of the Cockcroft Report: 1. The widespread use of investigations and problem solving activities in all maths classrooms. 2. The use of a teaching style that incorporated a collaborative approach with the widespread use of group discussion. 3. A strong commitment to make maths lessons interesting and enjoyable. This would include the regular use of maths games in KS1 and KS2 classrooms. The National Numeracy Strategy The NNS introduced some useful initiatives. In particular it revolutionised the structure of maths lessons with the introduction of the three part Daily Mathematics Lesson that consisted of a mental maths starter, a main teaching component and a plenary. The first of these was crucial as it consisted of a mandatory requirement for every maths lesson to open with mental maths activities. This provided a useful opportunity for discussion
Vol. 26 No. 2 Summer 2021 16 of mental maths computation as well as providing opportunities for the teacher to correct any misconceptions. Furthermore it was useful as a means of informal assessment of the progress of individual members of the class. Other aspects of the Strategy were less welcome such as the inflexible requirement that teachers adhered to a strict timetable to deliver the prescriptive components of the National Curriculum. The consequence of this was that teachers were not empowered to use their professional judgement to decide when a class had gained sufficient understanding of a concept to enable them to move on to a new topic. Aspects of the National Numeracy Strategy that could be included in a new policy It is probable that the committee would be in favour of the re-introduction of the mental maths starter in every maths lesson. It is unlikely that they would wish to retain any other aspects of the NNS. Assessment KS1 and KS2 National tests (NTs) Statutory assessment linked to the mathematics NC was initiated with the introduction of Standard Assessment Tasks ( SATs) in 1992. These were later renamed National Tests ( NTs). This involved a significant change from the performance of tasks to answering test questions with the use of pen and paper (with the exception of a mental maths component which required an oral response). A further change of name took place in 2014 when the tests were renamed National Curriculum Tests. This change of name was accompanied by a significant change of approach. This was evidenced by the withdrawal of a mental maths paper, the withdrawal of the option to use a calculator in any part of the test and a significant increase in the number of questions involving long multiplication and division. Furthermore the marking scheme was amended such that an additional mark was made available for use of the Formal Method. The effect of all of these changes has been that primary teachers have been placed under considerable pressure to use a formal approach to teaching with an emphasis on the learning of number facts and the application of taught algorithms. The effect of this has been to reduce opportunities to encourage creative thinking, problem solving and the use of enjoyable activities such as maths games. Assessment procedure that could be used in a future policy The first question that would need to be considered would we be whether to retain a formal test. An alternative would consist of teachers making informal ongoing assessments on each pupil. If it was decided that a formal test should also be in place then this could apply to pupils in Teachers were not empowered to use their professional judgement to decide when a class had gained sufficient understanding of a concept.
Vol. 26 No. 2 Summer 2021 17 Year 6 only. A return to the format used in the early versions of the NTs could be considered. Hence there would be a mental maths component and a calculator could be used throughout one of the papers. The questions could consist of a mixture of those involving formal skills and those that were open ended such that a problem solving approach would be required. A mark scheme could be devised in which marks would be made available for the use of diverse methods of calculation and for the use of efficient problem solving techniques. It could be emphasised that any efficient method of calculation would be accepted. The extra mark for use of the formal method would be abolished. KS3 and KS4 The Mathematics GCSE There appears to be a consensus that the most recent changes to the GCSE syllabus has consisted of some movement towards a greater proportion of questions that would require a problem solving approach. Nevertheless there are aspects of the GCSE that are unsatisfactory such as the enforced resits for candidates who do not achieve Level 4. Indeed it has been suggested that the GCSE is not an appropriate course for lower achievers. The Smith Review (2017) makes a compelling case for the introduction of a new course to cater for the needs of lower achievers. Methods of assessment at KS3 and KS4 that could be included in a future policy It is probable that the current GCSE syllabus with some modifications would be deemed to be appropriate for higher achievers. It would need to be edited with the removal of some of the components. These are likely to include the withdrawal of the requirement to memorise number facts related to Roman Numerals as well as multiplication tables up to x 12. It is likely that consideration would be made of the introduction of a new course of study and qualification in accordance with the needs of lower achievers. This could be heavily influenced by the recommendations made in the Smith Review for a new curriculum (Recommendation 5 (p62): In view of the low GCSE success rate… the Department of Education should review its 16-18 resit policy… Specifically, there should be fresh consideration of appropriate curricular and qualifications for these students and the extent to which current policy incentivises these to be offered. In response to this recommendation a Nuffield funded team at Mathematics Education Innovation (MEI) developed a new curriculum aimed at meeting the requirements of the resit students. Additionally the recommendations contained in the project Low Attainment in mathematics, an investigation of Year 9 students in England (Hodgen et al) (2019) could be considered. If a new course for lower achievers were to be introduced then students would not initially take the GCSE and so there would be no necessity It has been suggested that the GCSE is not an appropriate course for lower achievers
Vol. 26 No. 2 Summer 2021 18 for resits to take place. Nevertheless for those students who achieved highly in the new course then they would have the option of moving on to study for the GCSE. Entry Level The Entry Level system is unsatisfactory and in need of an overhaul. The main difficulty is that there is far too wide a gulf between Entry 3 and GCSE or Level 1 of a vocational course. The consequence of this is that many of the students who achieve Entry 3 are then put on a GCSE or Level 1 course and are unable to cope with the work. An additional difficulty with Entry Level exam papers is that questions are so predictable that there is a temptation for teachers to teach to the test and provide a maths diet exclusively consisting of repetitive exercises within a narrow range of skills whilst neglecting a problem solving approach that would encourage the development of logical reasoning. Aspects of Entry Level that could be included in a new policy The task of producing an effective policy to meet the needs of students currently categorised as being at Entry Level is a daunting prospect. An immediate issue that would need to be resolved consists of the difficulties caused by the gap between Entry 3 and GCSE. This could be alleviated by the introduction of a new course for lower achievers as suggested in the previous section. Entry level 3 students would then have the opportunity of progressing on to the new course instead of the GCSE. Students who achieve highly in the new course would then have the option of joining a GCSE group. The syllabus for Entry Level could be amended such that problem solving activities could be introduced to supplement the existing rote learning content. N.B. The views expressed here are those of the author and are not necessarily shared by members of the editorial team References Cockcroft W.H, (Chair) (1982) Mathematics Counts: Report of the committee of inquiry HMSO Mathematics in the National Curriculum (1989, 1991 and 2014) HMSO The National Numeracy Strategy: Framework for teaching mathematics from Reception to Year 6 (1999) QCA Smith A (2017) Review of post 16 mathematics Crown Copyright Mathematics Education Innovation (2020) A new mathematics GCSE curriculum for post 16 resit students Nuffield Foundation Jeremy Hodgen Margaret Brown and Robert Coe (2020) Low attainment in mathematics: an investigation of Year 9 students in England Funded by the Nuffield Foundation There is a temptation for teachers to teach to the test and provide a maths diet exclusively consisting of repetitive exercises.
Vol. 26 No. 2 Summer 2021 19 From her treasure trove of teaching wisdom Fiona Allen reflects upon the impact of the words teachers use upon the thinking of their students. Don’t Imply That! Or How children jump to the wrong conclusion Many things we say to children lead to misconceptions: • You always take the small number from the big number • The more digits there are, the bigger the number • To multiply by 10, simply add a 0 to the end of the number • To divide by 10, take off the last number and that becomes the remainder • When you multiply two numbers together, you always get a bigger number It is so easy to give children a rule, which works for them at a particular stage in their mathematical thinking but which will not work as they progress. Dr Malcolm Swan wrote: ‘Research has shown that teaching becomes more effective when common mistakes and misconceptions are systematically exposed, challenged and discussed.’ (Improving learning in mathematics: challenges and strategies, 2005) Recently, I have been thinking about children’s misconceptions and have started to think that there are also many things we show or tell children which cause them to infer a rule. = means work out the answer When I taught in a college, I was irritated by the number of A-level students who thought that an equals sign means ‘this is the next thing I thought of/am thinking about’. Reading more about the reasons for this, I found an interesting research paper. ‘Considering the equal sign as an operator places it in the same class of symbols as the addition, subtraction, multiplication and division signs instead of with other relational symbols such as the greater than (>) and less than (<) signs. This operational interpretation has been considered responsible for functional misconceptions among them, the one we term ‘running equal sign’ e.g., 2 + 3 = 5 * 2 = 10 – 2 = 8. Capraro et al. argue that children learn that = means ‘work out the answer’ in Reception classes, if they only meet questions of the type 2 + 3 = , 4 + 5 = 7 -1 = . The idea that = means ‘work out the answer’, is reinforced when children use a calculator, when, indeed, pressing the = sign tells the calculator to work out the answer! Capraro et al. write that to overcome the confusion, children
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