Vol. 28 No. 2 Summer 2023 1 Editorial Team: Kirsty Behan Alan Edmiston Peter Jarrett Alison Roulstone Les Staves Nicky White Letters and other material for the attention of the Editorial Team to be sent by email to: edmiston01@btinternet.com ©The Mathematical Association The copyright for all material printed in Equals is held by the Mathematical Association Advertising enquiries: Charlotte Dyason charlotted@media-shed.co.uk D: 020 3137 9119 M: 077 1349 5481 Media Shed, The Old Courthouse, 58 High Street, Maidstone, Kent ME14 1SY Published by the 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 From the archive – Number as a quantity 4 The conference has given all at Equals much food for thought and one effect has been an exploration of the treasures in our archive. In the light of what Louise Langford writes in this edition please tell us what you make of this piece from a few years back. Would it be feasible to replace formal maths exams with ongoing assessment? 7 In this piece Mark Pepper asks the perennial question about the place and role of formal exams. We will be discussing this at the 2023 conference so please get in touch if you have something to say. Number Stacks – 3 years on 10 James Aylott has taken the time to share how far Number Stacks have come in the three years since he first shared his idea with us. Representations of number - Securing a visual structure to transfer knowledge 14 Louise Langford has taken the time to share how her work with the RIWG for SHaW Maths Hub has been developing their thinking around ‘representations of number.’ Inclusion with Technology at LEO Academy Trust 18 Just before Easter Janet Goring heard Natasha Dolling speak and was impressed with how they are using technology, within the LEO Academy Trust, to create an inclusive maths classroom. Please let us know what you are doing and what you think about Natasha’s work. How can cute puppies help teach Maths? 21 Hema Tasker attended the 2022 Equals conference and was inspired to make some changes to his practice. In this piece he reflects upon his attempts to support his pupils to become more independent. No one left behind 23 I recently had the pleasure of meeting Tracey Roberts and when I heard about what they are doing to support her learners’ at Key Stage 4 I invited her to share her story.
Vol. 28 No. 2 Summer 2023 2 Editors’ Page This edition of Equals reflects our commitment to access and inclusion and the articles that we have been kindly sent clearly illustrate this as they range from early number sense to formal examinations at the end of Key Stage 4. Please do get in touch if you have something to share as did James Aylott in 2019 and it is lovely to be able to catch up with him to see just how his idea has grown and developed. Our first conference was a pivotal moment for us and much has changed in terms of our editorial board and the number of colleagues who are coming forward to share their work as a consequence. This is great news and bodes well the 2023 conference which I can reveal will take place at the Tudor Grange Academy Kingshurst, near Birmingham, on Friday 6th October. Details of how to register for this event can be found on page 3 and please do book early as places are limited and last year we sold old two weeks prior to the event. Equals is growing, as you can see from the announcements below, and we would like you to be part of this. Please get in touch if you would like to partner with us in any way at all. Opportunities to get involved with us Equals now has two working groups that are open to all those interested in SEND: The metacognition discussion group This group has developed as a consequence of the 2022 conference and affords all colleagues who have an interest in metacognition at any level to share experiences and support each other. The next meeting of this group will take place in June and if you would like to join us please let me know: edmiston01@btinternet.com The curriculum advisory group (CAG) This group formed as a result of the seminar on the Roots of Maths given by Les Staves last term. The CAG reflects the fact that many schools are working on their SEND provision and following his seminar six colleagues joined forces to collaborate and support each other as they seek to update and renew their curriculum. The next meeting of this group will take place in June and if you would like to join us please let me know: edmiston01@ btinternet.com We are also very pleased to announce that, due to popular demand, Les Staves has offered two more seminars on early number: Tuesday 9th May @ 4.00pm – Developing number sense Tuesday 6th June @ 4.00pm – Counting
Vol. 28 No. 2 Summer 2023 3 EQUALS WEBINAR SERIES Presented by Les Staves The roots of number sense 9th May 2023 4-5pm The roots of counting 6th June 2023 4-5pm FREE for MA members, non members £6 www.m-a.org.uk/online-webinar SAVE THE DATE Following on from the huge success of last year’s event ‘Success Across all Settings - SAS: Who Cares Wins’ where topics including maths anxiety, dyscalulia and metacognition were discussed. The Equals team are pleased to announce the 2nd Equals Conference 6th October 2023 Tudor Grange Academy Kingshurst Register your interest www.m-a.org.uk/equals-conference
Vol. 28 No. 2 Summer 2023 4 Number as a Quantity Part 1 In the first of two articles, Stewart Fowlie argues that children’s progress in early numbers should be understood in its complexity. There are counting, measurement and meaningful addition and subtraction. But there are also pitfalls. A series of activities may help to avoid these, all based on handling NUMBER as a QUANTITY. When a child begins to turn counting into adding, he learns that for example 3 + 2 = 5. His teacher may think he is adding two numbers together, but he is not: he is looking at three objects and two objects and realising that together they make a collection of five objects. He may use a thumb and two fingers to represent the three objects, and two more fingers to represent the two objects. If he has met a number line, he may think of moving 3 steps and then 2 steps, and develop the idea that all the steps must be the same length. Of course the points representing 1, 2, 3…on a number line don’t need to be the same distance apart just as, on a chart showing successive stations on a railway, all that matters is that the names of the stations are in the correct order. The teacher is thinking ahead to learning to measure lengths with a ruler when the numbers must be equidistant. When a syllabus asks for adding bonds up to ten, what the child is really expected to know is what sorts of things it is meaningful to put together. Thus you can put 3 boys and 2 boys together and get 5 boys, but if you put 3 boys and 2 girls together you get 5 children. If you put 3 boys and 2 books together all you get five of is things beginning with “B”. In my experience children know that you cannot take 5 boys from 3 boys (though they may think 3 - 5 is 2), but will have never thought explicitly that whereas you can add 2 girls to 3 boys, you can’t subtract 2 girls from 3 boys! From the archive – Number as a quantity The conference has given all at Equals much food for thought and one effect has been an exploration of the treasures in our archive. In the light of what Louise Langford writes in this edition please tell us what you make of this piece from a few years back.
Vol. 28 No. 2 Summer 2023 5 Never add without a context I would like to suggest that children, as they move on from counting, should never simply be asked to add two numbers together without a context. If they are, they will always use the same model or procedure, and may not develop a full understanding of the concept of addition. For example, all children realise quite quickly that if you are asked to add a larger number to a smaller number you will get the same answer if you add the smaller to the larger (and that is easier to do), but it may not occur to them for example that to add 9 they could add 10 and subtract 1. What I would suggest is that they be invited to solve a problem which involves adding numbers of specific things together, if the teacher wants to ask a dozen questions to practice adding it can be done by giving a table with three columns headed perhaps “number of children in room” “ number of children who come in” “number of children there now”. number of number of number of children in children who children room come in there now 3 5 ? 4 7 ? --- --- -- 2 ? 9 8 ? 3 ----- ---- ---- ? 6 12 ? 11 12 ------ ----- ---- ? 10 3 * First there might be a dozen examples where a value was given in each of the first two columns, an answer to be written in the last column. Then perhaps a dozen where a value was given in each of the first and third columns; then a dozen where a value was given in each of the second and third columns, and finally a dozen with the three types mixed up. Before these questions are attempted, an appropriate number of children in the class should come to the front, to be joined by the number representing those who come in. This is to make sure the set-up is understood and be visualised. Thereafter any still having difficulty should be encouraged to use cards or counters to represent children. Incidentally, understanding will be enhanced if a few examples are included which are impossible to answer, when a cross should be written instead of an answer. Here an impossible situation would arise if there were fewer children in the room after some have come in than there were before, or if more have come in than there are then in the room: in the former case, it would mean that some had really left the room, in the latter, there is no reasonable explanation. No word or sign to do with adding or subtracting is needed at this stage. A similar exercise with the columns headed “Number of children in room” “Number of children who leave” and “Number of boys remaining” could be presented. The children will see this as quite a different situation. Notice that it never becomes apparent in this example why, if, for example, 8 - 3 is 5, then 8 - 5 is 3. Climbing up and down as a model for adding and subtracting The situation of climbing upstairs might be used: the first column would be the number of steps first climbed, the second a further number of steps climbed, and the third how many have been
Vol. 28 No. 2 Summer 2023 6 climbed altogether. A further activity could be based on climbing up and then climbing down. Another might be based on the game “Pass the parcel” with the players arranged in a circle perhaps of five, so that passing the parcel 3 places and then 4 would be the same as passing it 2 places. One might even have a postman delivering letters to houses labelled 1, 3, 5, 7. . . along a street drawn at the top of the page. He delivers a letter to the 3rd house and then to the 4th house after that - what are the numbers of the houses he has delivered to? For the teacher always to use the same model would be as restricting as specifying none and thereby allowing the child to use the same model every time. Examples involving as wide a range of situations as possible should be considered. After meeting several different situations of this sort, the children should have in their minds the concepts of what we call adding and subtracting, realising what the different situations they have met have in common. After these tasks have been completed successfully there can be meaningful discussion about how everyone got their answers. Only then it will be appropriate to give a name to something everyone knows how to do. As a follow up, and to give a reason for introducing the +, - and = signs, the use of calculators could be invited to check answers. Where they had to be put in one of the first two columns this should be done by verifying that the number in the first column acted on appropriately by the number in the second column gave the number in the third column. Here are a few more suggestions for topics: • Having a few pennies and either being given a few more, or spending a few, and finding how much one then has. • Change of 5p, 10p, 20p, 50p, £1 when article of different price is bought. • Going up or down so many floors in a lift. • Length of a line if one end is at say 6 on a ruler and the other end is at 9. (This is equivalent to how many steps it is from one number to another on a number line). • Time now (only in exact hours) and what time it will be after so many hours. • Weight of a birthday card and an envelope in grammes, or of a suitcase and its contents in kilogrammes. • Temperature going up or down so many degrees. What we want children to know and understand first is not so much a series of facts like 5 + 3 = 8 but rather to recognise ren to know and whether or not such a not so much a fact is meaningful in a + 3 = 8 but rather given situation. There er or not such a are situations where a given situation. some properties of number apply but not others. For example if people waiting in a queue are given numbered cards, adding two people’s numbers doesn’t mean anything, but subtracting the number of the person now being served from What we want child understand first is series of facts like 5 to recognise wheth fact is meaningful in
Vol. 28 No. 2 Summer 2023 7 your number tells you how many people have to be served before it’s your turn. If you came 5th in a race for 10 people, how many people were slower than you? has answer 5, but the number faster than you is not 5 but 4. There are also many examples when subtracting doesn’t give the correct answer when you think it should.For example: when you are in your 7th year, are you 7 years old? How many birthdays have you had? If your 7th birthday was in 2003, in which year were you born? Teachers’ awareness of the non-examples of addition and subtraction would help children realise the appropriateness of the operations. At what stage children could tackle and recognise non-examples is another issue! Would it be feasible to replace formal maths exams with ongoing assessment? In this piece Mark Pepper asks the perennial question about the place and role of formal exams. We will be discussing this at the 2023 conference so please get in touch if you have something to say. A rising groundswell of support has recently emerged for radical reforms to be made to the educational system. This view has been expressed by a wide range of organisations as diverse as the National Education Union, the Kenneth Baker Think Tank, The Times Education Commission, the publishers Pearson and the Tony Blair Institute for Global Change. The consensus of these groups is that GCSE and A Level exams should be abolished and different teaching strategies should be introduced. Whilst the recommendations could be applied to all subjects, their relevance to maths will exclusively be considered here. The Main Recommendations A new assessment procedure should be introduced following the abolition of some of the formal exams. These would be replaced by a system of continuous assessment. There should be a major change to the methods of teaching such that active learning could take place as opposed to the current emphasis these groups is on teacher instruction vel exams should followed by a series ifferent teaching of written questions. be introduced. A specific suggestion would involve the use of the 4 Cs – critical thinking, creativity, communication and collaborative problem-solving. The consensus of that GCSE and A Le be abolished and d strategies should
Vol. 28 No. 2 Summer 2023 8 Students should develop a sufficiently high level of autonomy to equip them to be able to devise their own problem-solving strategies so that they would be able to overcome unforeseen problems in their present and future adult lives. Consequences of the proposed reforms Whilst these reforms would nominally apply to the 16-18 year old cohort they would also have a significant influence on education within the secondary sector as teaching would no longer be directed towards the current key objective of securing good GCSE results. The possibility of extending this approach to the primary sector could also be considered. If such a change did not take place there would be an abrupt change in the transition from primary to secondary education. It would surely be preferable to have a cohesive approach in which there would be continuity of teaching and assessment procedure from primary to secondary to further education. Key policies that could be incorporated into a reformed maths curriculum Primary sector If these changes were to be introduced in the primary sector then National Curriculum Tests (NCTs) would need to be abolished. The effect of this would result in a profound change in classroom teaching. The current emphasis on the key objective of obtaining good NCT results would no longer apply. Hence there would be no temptation for teachers to ‘teach to the test.’ Instead a teaching approach that made extensive use of the 4Cs could be used. The changes could include the reintroduction of the mental maths starter to open every maths lesson. This would provide opportunities to reinforce number fact knowledge as well as providing opportunities for the teacher and the class to consider different methods both of computation and of effective problem solving strategies. A regular investigation could be introduced as well as a weekly session of maths games to reinforce skills that had been learnt and to make maths lessons enjoyable. These initiatives would be reminiscent of the reforms that took place in the period immediately succeeding the publication of the Cockcroft Report (1982). I taught in primary schools at that time and the pupils showed enjoyment of maths and a great deal of effective learning took place. A major departure from the current system of teacher instruction followed by the setting of a written exercise in which the pupils are required to answer a series of questions that all consisted of the application of a single taught skill would need to be discarded. Instead the pupils could be encouraged to consider a range of strategies of problem solving and to devise their own methods in response to the questions. Some useful aspects of current classroom practise could be retained such as those designed to reinforce number fact recall. Hence a blend of activities could be used to develop both number It would surely be preferable to have a cohesive approach in which there would be continuity of teaching and assessment procedure from primary to secondary to further education.
Vol. 28 No. 2 Summer 2023 9 fact recall and number sense. The National Curriculum could be retained as it consists of a useful reference point for the maths skills that need to be covered. It would be imperative to reinstate AT 1 Using and Applying which featured in earlier versions of the National Curriculum. Secondary sector The abolition of GCSE exams would lead to a marked change in both the content of maths lessons as well as the teaching strategies. With teachers freed from the pressure of obtaining good GCSE grades from their students, they would be in a position to promote creative skills and increase levels of autonomy. This could be achieved by setting open-ended questions in which the students would be encouraged to formulate their own problem-solving strategies. This could be achieved on an individual basis or through working collaboratively with other students and with the teacher. This approach is commonly referred to as number sense which is an alternative strategy to number fact recall which includes the learning by rote of multiplication tables. These approaches need not be mutually exclusive as they as they can complement each other and could be used cohesively. Whilst such strategies should greatly enhance the understanding of mathematical concepts, their introduction should not affect the teaching and learning of maths skills in accordance with the National Curriculum. The occasional use of a mental maths starter and a regular weekly investigation could also take place. Further education sector One of the key objectives at this stage of the students’ education should be to help to equip them with the necessary skills to deal effectively with the challenges that arise in their day-to-day lives. Such challenges could occur in their future working lives and so Functional Skills could play a prominent role in maths lessons. This would help to forge links between the maths skills that had been acquired and its application to various vocations. For assessment purposes Functional Skills written exams could be replaced by a system of continuous assessment conducted by the teacher. Similarly it would be essential to provide regular Financial Literacy lessons. The content of such lessons could include the costs of interest charges when taking out a loan/ mortgage, understanding of contracts such as those associated with mobile phones and a good understanding of charges levied on credit cards. Assessment The provision of an effective means of assessment presents the most challenging aspect of a reformed system in which formal exams had been abolished. Current system of assessment with use of exam results The use of formal exams to assess students’ mathematical attainment is a deeply flawed system as it exclusively provides information relating to levels of success within rote learning and number fact recall whilst failing to provide any With teachers freed from the pressure of obtaining good GCSE grades from their students, they would be in a position to promote creative skills and increase levels of autonomy.
Vol. 28 No. 2 Summer 2023 10 information regarding such qualities as autonomy, creativity and problem-solving ability. This system does, however, have a significant benefit in that it is easy to administer. This is due to the fact that the vast majority of questions are closed and usually require a single word answer. A mark can then be gained for a correct answer. In some of the questions a further mark is available if the student demonstrates that a correct method has been used. In the case of NCTs this mark will only be awarded if the student has used the “formal method”. This means that students who demonstrate initiative and formulate their own response to a question will not gain the additional mark whilst students who use a taught algorithm will receive the extra mark as they will be deemed to have used the formal method! Thus from a purely administrative perspective it is an efficient system. A system of assessment not based on exam results The great strength of such an assessment system is that it can assess qualities such as levels of autonomy, creativity and problem-solving ability. As these are intangible qualities it would represent a complex undertaking for teachers. A system would need to be developed that produced maximum useful information whilst keeping the administrative workload to a minimum in which records would need to be concise. These could be maintained on a half-termly basis and could consist of discussion with each student on a one-to-one basis. The teacher could also review recent coursework with the student. This could help the teacher to identify any misconceptions and then take action to rectify these. In the case of students who had displayed relatively poor levels of number fact recall, a check could be made on recent progress within this. The reaction of inspectors The response of inspectors would be crucial in determining the success or otherwise of the enterprise. If they respected the teachers’ professional judgement and accepted their findings it should operate smoothly. If, on the other hand, they demanded ‘evidence’ on the progress of every student to justify the teachers’ judgements then the massive administrative workload would make the system untenable. Number Stacks – 3 years on James Aylott has taken the time to share how far Number Stacks have come in the three years since he first shared his idea with us. Back in the summer of 2019, I was invited to write an article for Equals about Number Stacks, a new maths intervention resource I had created and launched earlier that year. You can read the full article in Volume 24 Issue 3 but for those who haven’t heard about Number Stacks, I’ll try and summarise it briefly for you below.
Vol. 28 No. 2 Summer 2023 11 Number Stacks was created as I struggled to find a maths intervention that did everything I wanted when I was maths lead at my former primary school. It has physical manipulatives at its heart in the form of stackable place-value counters and these are combined with online video tutorials that guide learners through using the counters to develop understanding and confidence in 69 Key Skills of Number & Calculation. The resource kits and videos are accompanied by Initial Assessments that can be used to discover gaps in understanding and establish starting points; printable Fluency Activities that are great for checking a child’s understanding at the end of each key skill; and opportunities to revisit and practice previously learnt skills with our Fluency Question Generator (a database of over 900 questions that can be filtered by age & category) and games ideas. When used together, all these elements provide a complete maths support program that can be used by any adult, regardless of their teaching experience, to help children improve their confidence and understanding of mathematical concepts. What’s more, the fact that all the teaching is done through the video tutorials means the need for training, planning and preparation is virtually eliminated, so all available time (and we know how precious that can be!) is dedicated to working with the child. So what has happened in the 3 years since the last article? Well, it’s all been a bit of a whirlwind but I’m delighted to say that Number Stacks has proved to be a great success and it’s lovely to be invited back to write for Equals again to say a bit more about how its helped learners of all abilities, but particularly those who have struggled with maths in the past, to gain a new found love of maths! As 2019 drew to a close, none of us knew what was about to hit us the following year and before we knew it, the country was forced into lockdown due to Covid-19. As a small business still in its infancy, this was a worrying time, but it also presented us with an unexpected opportunity to test out the theory of our resources. As schools were closed to all pupils whose parents weren’t considered to be Key Workers, the vast majority of children were expected to learn at home and parents were expected to become their stand-in teachers! Thanks to online recommendations from schools who had already been using Number Stacks, and the power of social media, we sent out hundreds of resource kits in the following months and were fortunate to hear from many users how using them alongside our video tutorials had really helped their child enjoy their maths sessions again, rather than simply completing online activities or worksheets.
Vol. 28 No. 2 Summer 2023 12 One particular example that stood out was a young lady with Down Syndrome who had hit a real barrier with maths. Her mother decided that lockdown was the perfect time to try something new and after consulting with the school, she decided to go ahead and buy a Number Stacks resource kit to use at home. The progress made in a short space of time was phenomenal and they were soon ordering extra resources that could be used in school so they could continue the new approach that was working so well at home. What was even better was that they would make videos to show off her newfound skills which they would share online via her Facebook page. It was truly heart-warming to see these videos and know that Number Stacks had provided the ‘lightbulb’ moment which allowed a child to completely change their attitude towards maths! My own children didn’t get away with it during lockdown either and we took the same approach in sharing our Number Stacks journey online. You can see many of Jess & Holly’s videos on our YouTube channel and a particular video of Jess finding fractions of amounts got over 99 thousand views, which was crazy but really helped people see the power of using manipulatives! As schools started back, our increased online presence and the allocation of ‘Catch-Up’ funding meant that many more schools decided to give Number Stacks a go. As they started using it, the quick impact on pupil understanding led them to recommend it to others and the snowball effect began. We were also fortunate that some schools were kind enough to share their progress data, allowing us to create some case studies. Highlights include KS2 pupils from Pleasant Street Primary in Liverpool progressing by an average of 14 Key skills in just over two terms, and at Tockwith Primary Academy in North Yorkshire, Year Four pupils improved their standardised assessment scores by over 10 points in just a term. You can view the full case studies on the Reviews page of our website, but a couple of the accompanying staff & pupil comments are below: ‘I like Number Stacks. It makes Maths easy because we have counters. It helps me with my maths because I remember all the things I have done in Number Stacks when we do Maths in class.’ (Y4 Pupil) ‘Number Stacks has had a big impact on the pupils at Tockwith. As well as increasing confidence, children accessing the programme have made accelerated progress during the year.’ (Maths Leader) Informal feedback from other schools has included
Vol. 28 No. 2 Summer 2023 13 the following positive aspects of Number Stacks: • Pupils love the hands-on resources and the visual nature of the videos. • The fact that the videos are split into short sections means most of the sessions are spent ‘doing’ rather than ‘listening’ which helps those with shorter attention spans. • The consistency of physical resources and visual representations allows pupils to make links between concepts that they previously found difficult. • The counters make maths accessible as they ease the cognitive load and don’t rely on children memorising number facts which they might find difficult, which in turn reduces anxiety. • Older pupils enjoy using the stacking counters and don’t relate them to something they used in KS1. Recommendations and word-of-mouth are our main forms of marketing and the fact that we receive enquiries from new schools on a daily basis is testament to the impact that schools are seeing on their pupils. New additions to the resources include the Puzzle Box Challenges, in which pupils solve mathematical problems to gain padlock codes to try and open the ‘virtual puzzle box’. I’ve also started offering short online training sessions via Zoom to talk through how the elements of Number Stacks work together as despite no specific training being necessary, we found that some people want to make sure they’re on the right path and have the opportunity to ask questions. So, what’s next? Awareness and understanding of dyscalculia finally seem to be on the rise and we’ve heard from some specialist tutors that Number Stacks is already making a difference for pupils experiencing these difficulties. We’d like to explore this area further and have already made links with some specialist providers to help them support children and their families but would love to get more special schools on board. We’re also working with more schools on some new case studies and who knows, maybe a formal EEF trial might be something to consider soon! If you’re still reading, thanks for taking the time to share our journey and if you have any questions, or would like to find out more about Number Stacks, our website is www.numberstacks. co.uk and you can contact me at support@ numberstacks.co.uk James Aylott (creator of Number Stacks)
Vol. 28 No. 2 Summer 2023 14 Representations of number - Securing a visual structure to transfer knowledge Louise Langford has taken the time to share how her work with the RIWG for SHaW Maths Hub has been developing their thinking around ‘representations of number.’ This year’s Work Group key question is: How can the teaching of mathematics in mainstream be adapted to reduce the barriers to learning for children with Mathematical Learning Difficulties and Dyscalculia? As a part of the Research and Innovation Work Groups (RIWG), I have been privileged to work with a range of colleagues from both mainstream and specialist educational provision over the last five years, to develop understanding of the characteristics of SEND and how teaching can be adapted to enable all to access. This year, our Work Group has specifically focused on developing the effective use of well-chosen manipulatives and structured representations as part of the learning sequence, to enable all learners to communicate their mathematical thinking and successfully remove barriers to learning. To develop participants understanding of MLD and Dyscalculia, we initially looked at and discussed the main areas of difficulty for those struggling in maths. From videos of those with dyscalculia talking about their struggles, participants observations of focus pupils, sharing personal experience, to evidence-based research (NCTL, 2014), it became clear that difficulties with understanding basic number, magnitude, order and having the ability to confidently compose and decompose numbers was having a significant impact on pupils with SEND’s ability to work efficiently, effectively, and flexibly with numbers. We started by looking at a variety of dot patterns to explore developing a secure representation of number, based around effective pedagogy for teaching those with dyscalculia and MLD. We looked at the idea of ‘cluster’ recognition, which builds on the ability to subitise, that is to rapidly count or instantly recognise quantities (Clements, 1999), which supports the idea of efficiently composing and decomposing number. Alongside this, we used multi-sensory teaching techniques; by linking speech, with actions and visual images, memory is supported through connections being made in different areas of the brain and it is believed that by making learning more memorable, children are able to internalise key models, images and patterns to enable better recall (Goswami and Bryant, 2007 cited in Gifford and Rockliffe, 2012). We explored the use of dice dot patterns advocated by experts in dyscalculia (Butterworth and Yeo, 2004; Emerson and Babtie, 2014; Chinn, 2019) and investigated Mahesh Sharma’s (2019) technique of teaching through dice patterns linked to Cuisenaire, he skilfully demonstrates this in this video: Introducing a Number using Visual Cluster Cards and Cuisenaire Rods - YouTube. We also
Vol. 28 No. 2 Summer 2023 15 explored the use of the Hungarian Number Frame (Nrich, 2021); this often-familiar structured pattern enables pupils to ‘see’ numbers within numbers and therefore learn to partition and recombine in multiple ways, using structured visual images flexibly. The discussion in the Work Group around these structures highlighted the importance of linking the ‘cluster’ representation to a ‘liner’ representation to connect understanding of the amount, it’s magnitude and number order. Participants particularly noticed ‘the importance of pattern layout when embedding knowledge.’ When trialling these methods with pupils with SEND, participants were fascinated with how these enabled pupils of all ages, to demonstrate their mathematical ability, visualise and work more flexibly with number. One participant summed this up commenting that ‘with the pictorial representation the learner was much more able mathematically.’ We then focussed our attention on different representations of ten. This was again based on research around MLD and Dyscalculia, as it is suggested (Chinn, 2012 and Benz, 2014) that using structured visual images will support memory function, flexible partitioning of numbers and recall of core number facts. Over many years, educationalists (Van de Walle, 1988; Clements, 1999; Wright et al, 2008 and Benz, 2014) have argued that identifying quantities and being able to decompose a quantity into parts is an important mathematical skill that allows for the development of conceptual and procedural fluency. As already suggested, there is a correlation between low achievement in mathematics and problems identifying quantities, as well as recognising the structures in visual representations of number. Mulligan et al. (2010, cited in Benz, 2014:3) draw attention to the fact that visual resources need to be structured to allow pupils to subitise small amounts. Van de Walle (1988) and Wright et al. (2008) suggest that identifying five and ten as key reference points supports pupils with discovering relationships between numbers. Whilst Chinn (2019) goes further to suggest that 1, 2, 5 and 10 are the ‘key numbers’ pupils need, as they are the numbers used most in our monetary system (1p, 2p, 5p, 10p…), link to finger counting and are used to derive unknown facts (if I know 5x and 2x, I can work out 7x). The NCETM (2020, 2015) support this idea of using key facts to transfer knowledge, stating that the Tens Frame is a core representation in their Mathematics Guidance for KS1 and KS2 and recommend using specific structured images to teach numbers which support memory function and involve part/whole relationships. As a Dyscalculia accredited teacher and assessor, I have found developing a secure representation of five and ten to be a powerful tool. Based on Chinn’s idea of efficient teaching (Chinn, 2019), I explore how certain images, techniques and concepts are important pre-cursors to later ideas, linking the development of number sense to generalising and later mathematical concepts. This is of particular importance for pupils with barriers to learning but also secures deep understanding for
Vol. 28 No. 2 Summer 2023 16 all pupils, regardless of their attainment levels. Consequently, we spent much time exploring, discussing, and reflecting on the different representations of ten and how these expose different mathematical concepts. Through a range of activities, that involved for example: exploring how to teach ‘near doubles’ from naturally occurring dice doubles, using structured representations to expose the commutative aspects of addition and multiplication, using known facts for ten to generalise other place value facts, participants have developed their practice, commenting that they now have a ‘deeper understanding of how children take in their mathematical learning.’ They have considered how a secure representation of number, specifically ten, can enable knowledge to be transferred e.g., bridging through ten to add or subtract using partitioning, or generalised e.g., if 7+3=10 then 70+30=100 and 0.7+0.3=1, and can support knowledge of place value, written calculation methods and ‘exchange.’ Furthermore, participants have noticed that using certain manipulatives enables pupils to ‘access deeper thinking skills,’ through ‘exploring the use of different manipulatives’ they can expose the maths, and this can ‘broaden and deepen thinking for abstract mathematical concepts.’ For example, finding the total for 5, 4 and 8. Which method would you use in each case? Near doubles, multiples of five, bridge through ten to name a few! This exploration, discussion and reflection all led to interesting conversations around representations of 10 and the importance of children having this to transfer knowledge but being more flexible in that representation. We discussed the importance of letting pupils choose which structure helps them, which one they can visualise and manipulate to support their mathematical thinking. Alongside this, participants acknowledged the importance of pupils having a strong, secure visual image of the maths and the need to ‘encourage the use of concrete resources or pictorial representation’ to enable learners to communicate their mathematical thinking. The participants are currently researching this further through school-based case study, focussing on using ‘clusters’ to consolidate learners understanding of number, developing the use of the Hungarian Number Frame linked to dice
Vol. 28 No. 2 Summer 2023 17 patterns and developing the use of the Tens Frame linked to understanding of place value facts. Reflecting on our work so far, reminds me of research by Benz (2014). Benz (2014) suggests that structuring quantities in a specific way can help inform pupils choices about how to decompose numbers in different ways but concluded that pupils do not automatically decompose numbers into parts referencing five and ten, suggesting that sometimes pupils decompose numbers based on what they know rather than the structure given. This brings me back to the participants findings from their observations and work with pupils in class, which has led us to conclude that it is very important to develop a secure representation of ten but in the words of a participant ‘representations should be fluid to allow children to choose the most effective methods for them.’ References: BENZ, C., 2014. Identifying quantities of representations- Children using structures to compose collections from parts or decompose collections into parts. In: C, BENZ and B, BRANDT eds. Early Mathematics Learning, London: Springer. pp.189-203 Butterworth, B and Yeo, D., 2004. Dyscalculia Guidance: Helping pupils with specific learning difficulties in maths. London: NFER-Nelson. CHINN, S.J., 2012. The trouble with maths: A practical guide to helping learners with numeracy difficulties. 2nd ed. Abingdon: Routledge. Chinn, S.J., 2019. Maths Learning Difficulties, Dyslexia and Dyscalculia, 2nd ed. London: JKP CLEMENTS D.H., 1999. Subitizing: What is it? Why teach it? Teaching Children Mathematics NCTM. 5 (7), pp. 400-405. Emerson, J and Babtie, P. (2014) The Dyscalculia Solution foreword by Butterworth. London: Bloomsbury Education. GIFFORD, S. and ROCKLIFFE, F., 2012. Mathematics difficulties: Does one approach fit all? Research in Mathematics Education. 14 (1), pp. 1–15. Nrich. 2021, Hungarian Number Picture Resource [online]. Available from: https://nrich.maths.org/content/id/8119/ Hungarian%20number%20pictures.pdf NATIONAL COLLEGE FOR TEACHING AND LEADERSHIP, 2014. Closing the gap with the new primary national curriculum. Nottingham: Carmel Education Trust. VAN DE WALLE, J.,1988. The early development of number relations. Arithmetic Teacher. 35, pp.15-21 WRIGHT, R, J., STANGER, G., STAFFORD, A, K. and MARTLAND, J., 2008. Teaching Number in the Classroom with 4-8 year olds. 2nd ed. London: Sage publications ltd Research and Innovation Work Groups (RIWGs) are maths-specific professional development projects funded by the NCETM (National Centre for Excellence in the Teaching of Mathematics) and developed and led as an integral part of the Maths Hubs Programme. Through cultures of research, innovation and collaboration, they rigorously explore new approaches to better understand how children and young people learn maths. The outcomes of these RIWGs influence Maths Hub professional development projects at a national level and have an impact on individual teachers. This article is from an educator leading a RIWG on behalf of a Maths Hub, and is entirely the work of the author.
Vol. 28 No. 2 Summer 2023 18 Inclusion with Technology at LEO Academy Trust Just before Easter Janet Goring heard Natasha Dolling speak and was impressed with how they are using technology, within the LEO Academy Trust, to create an inclusive maths classroom. Please let us know what you are doing and what you think about Natasha’s work. At all LEO Academy Trust schools, inclusion is a fundamental principle of our Learning, Excellence, and Opportunity ethos. Every teacher is a teacher of SEND and every leader is a leader of SEND which is intrinsic to everything we do at LEO Academy Trust. The Trust’s vision is that Technology for Learning will enable children, staff and parents to enhance their teaching and learning. Effective use of technology for learning will be demonstrated when children are more engaged in their learning, can independently select the appropriate resources and as a result of utilising technology for learning, children’s outcomes and experiences are enriched and enhanced. The use of technology across the Trust has been of great benefit to including all children in learning,
Vol. 28 No. 2 Summer 2023 19 especially those with SEND. Using digital tools has increasingly become a powerful aid in maths lessons, both for teachers to better help those who would have previously been outside of a mainstream maths lesson keep up with their peers, and also for the children to have more autonomy over their maths learning and how they represent their understanding. One way we create more equitable learning environments is by having universally designed technology to all LEO pupils. Universal Design for Learning (UDL) is a framework that aims to remove barriers to learning, meaning it is inclusive for everyone. All LEO schools use ‘techquity’ which is about using technology specifically to help lower or close barriers to learning. ‘Techquity’ helps LEO give pupils the conditions they need to be successful. All pupils receive access to assistive technology resources in order to achieve success. Diving even deeper, let’s think about inclusive practice in terms of what tasks we are asking children to undertake in a maths lesson. How can we be flexible enough to allow children to have the freedom to respond to the tasks that we set in a way that suits them, builds on their strengths, and doesn’t undermine things that they find more challenging? Demonstrating understanding of a maths skill has often been confined to reading and writing, reading the question, writing numbers in a box etc. What about the children who struggle to access the maths because of the tools we are giving them? Using technology for learning can see maths questions being read to them, a choice of responses such as voice recording, screen
Vol. 28 No. 2 Summer 2023 20 recording, a choice of digital manipulatives from which to demonstrate understanding, but overall a much more interactive, dynamic and accessible maths experience from filling in a worksheet. This way of learning has opened up a whole new chapter of inclusive maths lessons and the benefit to children with SEND has been extensive. Some ways that using technology is improving inclusive practice at LEO Academy Trust: • Use of Nearpod, student paced lessons so children can go as fast as they like; • Teachers pre-recording explanations that children can watch back as many times as they need, including outside of the lessons; • Using tools such as Mote that allow children to verbally record answers; • Digital questions appearing alongside animated gifs of maths manipulatives- much more accessible that static images next to questions; • Use of AI to pinpoint exactly what a child is struggling with and start learning from that point; • Writing to text tool- children can write their answers using a stylus which technology then transforms to written numbers. Especially useful for children who struggle with number formation (as well as in EYFS); • Children screen recording their work as opposed to filling in a worksheet. The record of their learning might look like them narrating moving digital manipulatives on the screen. This method of independent work has hugely increased children with SEND moving away from TA support and being more independent; • … and many more! Throughout LEO Academy Trust, staff are able to point to this practice and say, “If it wasn’t for the technology, I wouldn’t be doing this.” When this gets extended to those with SEND and disabilities, there are children who would not be able to function in schools without technology and more
Vol. 28 No. 2 Summer 2023 21 importantly, there would be children with needs that were not able to flourish without technology. We are in such a strong position to now harness the power of technology to support truly inclusive practice and we are excited to be at the forefront of this journey. How can cute puppies help teach Maths? Hema Tasker attended the 2022 Equals conference and was inspired to make some changes to his practice. In this piece he reflects upon his attempts to support his pupils to become more independent. I heard Steve Chin, Pete Jarret and Les Staves explain what learning Maths means for students. I had an idea about how I could help my students engage with Maths in a more positive way. The main comments I have heard over the years around Maths have been negative such as, “Maths is really tricky.” and “I can’t do maths.” It is this negative voice that stops my students trying. This negative cycle means they don’t attempt Maths questions in tests and lessons and they then use the results of the Maths test to reinforce the idea they ‘can’t do Maths’. This is echoed in Rachel Addy’s research. Although I can’t make exams easier, I can encourage the student to try and work independently. However, whatever I say the student gets more anxious… The question I attended the conference with was how do I break this negative cycle for both of us? I can’t do Maths cycle Poor test results Avoiding independent work More anxious in lessons Poor revision notes from the feedback lesson Les talked about breaking Maths down into small achievable steps- this helped get my student starting to work independently but as soon as they got stuck we were back in our loop. This is when I had the idea of using cute kittens and puppies to help. Not the actual animals, I would love that
Vol. 28 No. 2 Summer 2023 22 but I am not sure how that would help me focus when teaching the lessons. So I used the next best thing: images of cute kittens and puppies. This worked as an experiment because we had something we both liked in the Maths lesson (kittens and puppies) but the phrase “I can’t do maths!” quickly reappeared… So what next? I made a bookmark that has the following format: Front Back Sides (optional) Pictures of kittens and puppies (cute ones of course!) Images that summarise the topics we have been studying and assumed knowledge that is needed for the work to be completed independently For one of my students who is very anxious and freezes when attempting any numeracy work, I wrote of alternative words to use when describing a tricky question. Phrases that are used in my lessons to encourage problem solving skills and independent learning Did it work? Yes and no. For one student it did not work at all and this was because they ‘didn’t like cats and dogs’. So I suggest this is adapted to the interests of the child you are working with. Although all the students said they didn’t use the back of the bookmark, most of them attempted some independent work and few of them were able to reflect that the reminders ‘could be helpful’. Analysis of the answers shows that students did attempt part of some of the questions. This is progress because some of these students would not have attempted any independent work. These reminders helped them to feel confident making a guess and helped them as a prompt. They were able to start a question, linking back to the idea that they can just attempt the first step. For one student, a student who ‘freezes in lessons’ and thinks they are ‘stupid’ the words I added to the side made a positive difference. The student said, “Mrs Tasker, this question is really complicated.” while laughing. For this student we changed the language of the lesson from ‘I can’t do it’ to using different words to describe a question; testing, complicated, challenging, difficult… Any alternative to ‘impossible’. In the long term this has had the biggest impact on the student’s attitude to maths as the internal voice has changed from ‘impossible’ to ‘challenging’. This has resulted in the student achieving some
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