Vol. 27 No. 1 Spring 2022 1 Editorial Team: Kirsty Behan Carol Buxton Alan Edmiston Peter Jarrett Louise Needham Nicky White Letters and other material for the attention of the Editorial Team to be sent by email to: edmiston01@btinternet.com ©The Mathematical Association The copyright for all material printed in Equals is held by the Mathematical Association Advertising enquiries: Charlotte Dyason charlotted@media-shed.co.uk D: 020 3137 9119 M: 077 1349 5481 Media Shed, The Old Courthouse, 58 High Street, Maidstone, Kent ME14 1SY Published by the Mathematical Association, 259 London Road, Leicester LE2 3BE Tel: 0116 221 0013 Fax: 0116 212 2835 (All publishing and subscription enquiries to be addressed here.) Designed by Nicole Lane The views expressed in this journal are not necessarily those of The Mathematical Association. The inclusion of any advertisements in this journal does not imply any endorsement by The Mathematical Association. Editors’ Page 2 Remembering Mundher Adhami 3 Alan Edmiston pays tribute to his friend who passed away on the 11th March 2022. Ratio for all! 4 Are difficulties with ratio due more to numeric presentations than to the concept itself? Mundher Adhami suggests that the visual handling of ratio-and-proportion problem may allow all to access the concepts intuitively before moving to addressing these number terms. Maths in Stories 8 Anne Haberfield’s school is doing some wonderful things for all of their learners. Here she writes about how they use stories to support and enhance mathematical learning. Ken is back! 11 It’s been ten years since Ken wrote about his adventures teaching maths at an academy in the north of England. Now he leads maths in a PRU and is keen to share his experiences! Tales from the Arc - Part 3 11 Sometimes in teaching, writes our special correspondent Ken, when you sit back and reflect, a tremendous sense of well being, accompanied by the thought ‘It could not get any better than that’, washes over you. What’s in a name? 15 Alan Edmiston reflects upon the term dyscalculia and what it means to him now. The University of Derby needs you 17 This appeal from Dr Tom Hunt is an opportunity for you to get involved in research into mathematical anxiety Recall of tables 18 Like many of us Mark Pepper has found himself tutoring pupils who are finding mathematics hard following lockdown. Working in such a way has afforded him the chance to reflect upon some of the barriers to progress. Book Review: ‘Care in Mathematics Education, Alternative 22 Educational Spaces and Practices’ by Anne Watson Dave Tushingham from Blaise High School has kindly sent us a review of the latest book by Anne Watson.
Vol. 27 No. 1 Spring 2022 2 Editors’ Page This edition of Equals is dedicated to the memory of Mudher Adhami who passed away on the 11th March. For those of you who do not know him, Mundher was a key member of the Equals team for many years and I pay tribute to my friend and mentor in the first article on page three. If you have any personal memories of Mundher please get in touch to share them and we will pass them onto his family and also publish them in the next edition. 2022 is a key year for Equals for it sees us host our first ever National conference. The conference will take place at Parliament Hill School, London, on the 25th November 2022. Further details will be available from May on the MA’s website. It is envisaged that this will be a very popular event and so I encourage you to book early as soon as registration is open in May. At the time of writing it is not possible to share the dates and speakers for the two Summer Seminars we are planning but they will be published on the MA’s website as soon as they become available. As Equals was going to press the Government were close to publishing a Summary of the SEND review: right support, right place, right time. We will be commenting upon this in the next edition and will include thoughts from a range of stakeholders. If you have any comments concerning this review then please send them to us for inclusion in the next edition. This is good timing for the NCETM are currently working on the SEND support that will be offered via the Maths Hubs for all schools in England. As soon as information is available we will let you know. This year some Hubs have offered two different work groups for their schools: The Characteristics of SEND and Teaching for Mastery in the context of Special Schools and Alternative provision. Advertisement Success Across all Settings SAS - Who Cares Wins! 25TH NOVEMBER 2022 LONDON
Vol. 27 No. 1 Spring 2022 3 Remembering Mundher Adhami As I write this, I am finding it hard to believe that on Friday 11th March 2022 Mundher Adhami died peacefully in Tunisia where he lived with his wife Haifa Zangana, a novelist and activist. I first met Mundher in 1996 and soon came to view him as a dear friend and trusted mentor. I became involved in Equals twenty years ago because he encouraged me to write about my lesson experiences. Mundher was the most interesting person I knew, an exiled Iraqi who had lived in Russia studying for his PhD before moving to the UK. Initially Mundher spent time at Durham University before working as a maths subject lead in London. He was part of the team led by Margaret Brown that produced the GAIM (Graded Assessment in Mathematics) materials. Following that he paired up with Michael Shayer to work on the CAME (Cognitive Acceleration through Maths) project. This collaboration saw him reside at King’s College for many years and resulted in several books, numerous publications and also the Let’s Think Maths series of activities. It was just prior to the publication of the CAME or Thinking Maths lessons that I first met him. Little did I know that meeting Mundher would change my career and see me move in a relatively short time from teaching science to someone who would spend 90% of his time teaching mathematics across all key Stages and working with maths teachers. Mundher possessed an intuitive grasp of progression within mathematics and was brilliant at devising activities that enabled children to move towards higher, abstract, levels of thinking within a conceptual strand. For me the lessons he devised will stand the test of time and in honour of him we are reproducing an article he wrote about one of the Let’s Think Maths lessons. I do not possess the words to fully pay tribute to him and to highlight the impact he had upon my life. It was a pleasure to know him and to spend time in his home and with his family. Those who knew him will miss his warmth, compassion and love deeply. Mark Pepper, who has been involved with Equals for many years, was someone who worked closely with Mundher, got in touch to share his memories of Mundher. “We were in the editorial team of Equals for a number of years. In that setting I always found Mundher to be a very good team member in which he listened closely to the views of others as well as making his own contributions. He was always constructive in his assessment of articles submitted and never expressed negative views about them. I also worked closely with Mundher in the late 1990s when we produced a series of books called Maths Direct. Mundher was the series editor whilst I was a member of the team of authors. I always found Mundher to be supportive and he provided many constructive ideas. In the course of producing the books the writing team occasionally spent weekends at hotels in various locations. These meetings were extremely productive as well as being very pleasant from a social perspective. As a person I always found Mundher to be courteous and he had a pleasing manner which helped to generate an atmosphere of relaxation.” If you have any personal memories of Mundher and his work please feel free to share them with us. Alan Edmiston
Vol. 27 No. 1 Spring 2022 4 Ratio for all In tribute to Mundher we are reprinting something he wrote for Equals in 2004 on ratio. I have chosen this as it concerns a lesson I call Jelly Babies and it is something that even today I share with almost every group of maths teachers I work with. Are difficulties with ratio due more to numeric presentations than to the concept itself? Mundher Adhami suggests that the visual handling of ratio-and-proportion problem may allow all to access the concepts intuitively before moving to addressing these number terms. Ratio for all! The reason that Number is difficult does not seem related to something inherent in the concepts themselves. Increasingly scientists are convinced that our brains are pre-wired to recognise numerosity and number relations. They see it as one of the universal human traits parallel and connected to language acquisition.1 Any engagement with the physical world would seem to assure these mental capacities in people, with or without organised schooling. After all, societies throughout human history used and recorded numbers independently from each other, as evident by the ways the ancient Egyptian, Babylonian, Chinese, Romans, Arabs and many others have used numbers. Pre-historic/ pre-settled communities, very small children, and even chimpanzees recognise quantities both in ‘counting’ terms, especially in small numbers, say up to 5, and in comparisons of size. But of course without some advanced social means of communication, and therefore social learning in general, this recognition does not go very far. In our advanced cultures many properties of Number become accessible almost to all, with or without schooling. Some other properties remain problematic. It is worth looking at examples of these two kinds, and the reasons for ease or difficulty. Of the more accessible kind are the additive properties of number, which require initially little more than being conscious of the learner’s actions on collections of objects. Ability to count itself is a way of keeping track of a series of pointing action using words so that a bowl of 7 apples comes to have a label of 7 for the whole collection, regardless of the order of counting, or differences of apple sizes. Putting on top another collection of 4 apples clearly makes the resulting collection larger while taking away Children age 7 have shown they can intuitively deal with ratio and proportion, to the extent of describing such relations in numbers, almost formally in ratio form.
Vol. 27 No. 1 Spring 2022 5 some apples makes it smaller. You can also split the 7 apples in many ways, and record all these actions on paper with comparatively little difficulty. From there on, the recording comes to have a life of its own, so doing calculations on a page can still be accessible to many, through being meaningful. Numerals are always interpretable as mere 1-to-1 labels of collections while place value conventions can be, with adequate teaching, interpreted as labels of units and collectionsof-units in agreed sizes, e.g. 10s and 100s. Beads, the abacus, and Dienes blocks, make all these ideas tactile and beyond these the number line is still visible. In relation to additive properties, or additive composition of numbers the representations, or the ‘cultural tools’ for working on Number, do seem very useful. At least they do not cause too many problems. Many problems, however, arise with numbers such as fractions and ratio, which sometimes are labelled ‘multiplicative relations’. This term seems confusing since it leads one to think of the operation of multiplication as a procedure, memorised or not. I suggest this domain is better described simply as relational, since the object of attention in each case requires handling a magnitude in relation to another, or some quantity in relation to another. It seems sometimes that we create confusion by not attending to the relational essence of these numbers. Recently in research in KS1,2 we developed a lesson that proved successful in supporting the development of such relational thinking. Children age 7 have shown they can intuitively deal with ratio and proportion, to the extent of describing such relations in numbers, almost formally in ratio form. Teachers were surprised at how an appealing context, such as making sweets in two flavours, represented as pictures of ‘jelly babies’, have kept children engaged in demanding mathematics. It is a mathematics that is often expressed in natural language, but that is easily translated into numbers and even recorded. Considering the truly huge range of ability, i.e. processing capacity, use of language, and even dexterity of children at this age - and at any age - we claim that such an activity is useful for many older children. It can also serve as a first step for more advanced cycles of work in the same contexts, as we shall see at the end of this article. Here is an account of the lesson activity. Episode 1: Comparing drawings The teacher tells a story about a factory making jelly sweets in two flavours for head and body, and asks the children to choose the two flavours. She gives children two pictures, one showing a shape in two equal size parts for head and body. The other picture has the head as only a quarter of the body. Children find names for the two types in the context of the story (e.g. ‘jelly baby’ ‘Jelly Daddy’ or Jelly man) and find ways of talking about the sizes of the head and body in each to answer the question: “What is different between them?”
Vol. 27 No. 1 Spring 2022 6 A third picture is then introduced. “How different is this?” would lead to a discussion on the use of blocks or rods placed on the sides of the pictures. A name should be found, e.g. ‘Jelly Child’ or ‘Jelly girl’, with the children arranging the pictures in order. Attention is directed throughout to comparing the height aspect of head and the body in the pictures, rather than on their global size. The blocks and the rods on the size of the picture of the Jelly sweets point the attention of the children to the number aspect, but full use of number is not expected at this stage. Episode 2: Deciding which of other shapes is of which type. Carrying on with the story of the sweet factory the teachers tells the children that each type can come in different sizes, so the the Jelly Daddy can be smaller than the Jelly baby. ‘How can that be possible?’ Children look at nine pictures and decide for each whether it is for a jelly baby, jelly child( Jelly boy, Jelly girl) or jelly daddy. None of the pictures are of the same size as the three given earlier, nor have they the blocks or rods to help. To decide the type the pairs of children must find ways of comparing the head and the body using strips that can be folded or torn, and/or pencil marks using the head as a measure.. They must give reasons for their decisions. That may lead to dicussion of ratios, recorded in some ways, as well as of fractions. Episode 3: Enlarging the scale; A Jelly Giant The teacher tells the class the manufactures wanted to have a Jelly Giant as well. She asks, ‘What might you expect about the size of his head, compared with his body?’ For an answer of the kind: ‘LARGE body’ the teacher presses for a comparison with the the head, rather than just the size, e.g. suppose the whole sweet is of this small size, how can the body be large? Then she can direct the children to decide the size of the HEAD and Body of the Giant, measure up its sides, and then draw it. To make it more challenging the children are not to tell the scale so that others can
Vol. 27 No. 1 Spring 2022 7 guess or work it out. (all this work on comparative sizes is done on the basis of lengths.) The children either use their card strips or maybe the blue and red cubes or lines from their Daddy picture to measure the head and body, and then draw it. Then the rest of the class will have to guess what numbers they have chosen. The teacher watches them as they try to draw, and decides on two that are clear drawings but at the same time different scales. These are then pinned up to the whiteboard, and the children are asked what sizes (number of head units) the children who drew the bodies used. She asks them for reasons and uses whatever measuring instrument they suggest to check their guesses/choices. How far to go with the activity? At each stage the teacher decided, according to the range of ability in the class, and the volunteering of ideas, how far to go in formalising the writing of the ratios. The emphasis in all cases is on ways of working: How useful were the paper strips, the pictures of blocks or sticks. How useful is using numbers? With older pupils and the more able, the whole activity described above can take half the time that it took us with the Y2 class we trialled it in. Possible extensions include: • Giving pupils heads alone or bodies alone in different sizes, with the label of what they are, i.e. Jelly Baby, Jelly Daddy etc, and asking them to complete it, using rulers or strips of paper. • Suggesting that the sweet factory is thinking of adding to some jelly sweets small chocolate buttons somewhere half-way down the body. How can they find that position? What about if the button is half-way from top of head to bottom of feet? What is the difference between the two positions. • Writing the ratios for each of the type of jelly sweet, how to write these as fractions. • Then, for the still more able, how can they explain the difference between a ratio and a fraction. Hopefully some interesting spontaneous language descriptions can be aired to the effect that ratios are usually used when separate things are compared , while a fraction is part of the whole. In both cases there is a unit that is used for both parts. The potential for confusion can then be explored, since a fraction is really a special ratio kind of ratio. We ourselves have not tried this activity with children older than 7. Perhaps teachers who try can report how it goes. Kings College, London 1. Butterworth, B, (1999). The Mathematical Brain. London: Macmillan.. Also Dehaene, S, (1997) . The Number Sense. London: Allen Lane, and Dehaene, S, Dehaene-Lambertz, G., & Cohen, L..(1998). Abstract representations of numbers in the animal and human brain. Trends in Neuroscience, 21, 355-361. 2. Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus (2001/2004). Research project funded by the Economic and Social Research Council at King’s College.
Vol. 27 No. 1 Spring 2022 8 Anne Haberfield’s school is doing some wonderful things for all of their learners. Here she writes about how they use stories to support and enhance mathematical learning. Maths in Stories Everyone loves a story, but have you ever looked at the stories you read to your class through a mathematical lens? You will be amazed by how many opportunities you find. Stories can be flexible ways to introduce and simplify maths teaching, enabling you to contextualise and link learning to a central theme. You can use a story in playful and creative ways, inspiring you to find new ways to teach maths concepts to children of various abilities. At Castle School in Cambridge, we have been using this approach since 2017. Our journey began when we were exploring and developing our curriculum offer for our pupils with SEND. We initially started looking at cross-curricular themes and planning, but our deeper focus and love of maths inspired us to explore simple texts with a mathematical focus. We were stunned by how much maths we could find in each book, not just using the written words but also exploring pictures and themes. Over time this has evolved: we have discovered that it not only links well with the Maths Mastery approach, but also improves motivation, mathematical language use and engagement. The approach enables students to build on their existing knowledge and provides a ‘hook’ for future learning. How we plan We use this approach for our pre-formal sensory and semi formal learners. Pre-formal classes work on a termly text, which engages, inspires and links to the development of contingency awareness, simple maths songs, and early cause and effect. Within this framework, maths in stories gives us the freedom to explore mathematical concepts through play. Here we are engaged in an activity connected to ‘Shark in the Park’ by Nick Sharratt. We are exploring the idea of “one” through pick and put and practising our counting with our magnetic fishing game.
Vol. 27 No. 1 Spring 2022 9 Our semi-formal classes have half-termly themes, such as conservation. Each class has a storybook related to the topic, which allows two or three weeks to plan and sequence our maths teaching, learning and assessment. Each story can be used to introduce children’s mathematical thinking. Firstly, the story can be presented through an Attention Building approach, based on Gina Davies’ Attention Autism intervention. Simple maths concepts are built into the 4-stage Attention Autism structure. For example, in a Primary Class we are using Dear Greenpeace by Simon James as a counting activity: the teacher has a container of water, and the group counts different types of sea creatures. Secondly, we can create problems for children to solve whilst encouraging them to consider the different elements of the story. For example, using Michael Recycle the pupils were able to answer questions about weight, such as “What’s heavier, a pile of paper or a pile of plastic?” Thirdly, we can help students learn abstract maths concepts and ideas with concrete and pictorial representations taken from the story props, such as adding, subtracting and grouping. Fourthly, we can determine depth of understanding by encouraging children to make connections between imagined elements of the story and real life contexts. For example, in the story Noah’s Ark, children played a game to see how many animals they could count in sets of two.
Vol. 27 No. 1 Spring 2022 10 Finally, we can use the text as a bridge between developing maths and life skills, such as weighing ingredients to make a recipe linked to a text, or using money to buy items in a role play shop. This is transferred to a real-life experience. Teaching using maths in stories produces fascinating experiences for children to engage and apply their maths skills across the curriculum, that is, from book to beyond! Go on, give it a try! If you would like more information on this approach to teaching maths please look at our website https://www.castleschool.info/page/?t itle=Maths+In+Stories&pid=71 or contact Anne Haberfield Deputy Head, SEND lead Cambridge Maths Hub ahaberfield@castle.cambs.sch. uk or Davinder Mankoo – Primary Maths Lead dmankoo@castle.cambs.sch.uk
Vol. 27 No. 1 Spring 2022 11 Ken is back! It’s been ten years since Ken wrote about his adventures teaching maths at an academy in the north of England. Now he leads maths in a PRU and is keen to share his experiences! Just before Storm Arwen hit in December I received a wonderful email from a friend called ‘Ken’. Those of you with long Equals memories may recall that Ken wrote a few articles for us almost ten years ago. At the time he worked in an Academy in the North of England and took the time to reflect upon the joy he got from teaching maths to a group of SEND learners in a unit called the Arc. Ken (still not his real name) has moved on and is now the maths lead within a Pupil Referral Unit or PRU in the Midlands. Over Christmas we arranged a time to meet up, for it is six years since I last saw Ken, and as he spoke about his new role I asked if I could interview him and publish it in this edition of Equals – he readily agreed to this. Before I do that I feel it is worth reprinting his last article for us from 2013. Tales from the ARC – part 3 Sometimes in teaching, writes our special correspondent Ken, when you sit back and reflect, a tremendous sense of well being, accompanied by the thought ‘It could not get any better than that’, washes over you. To me the feeling is the ‘hit’ that provides our profession with the internal compass to demonstrate that we are meeting the needs of our learners. It has not happened so often at the Academy but my best lesson of the year occurred just after the Easter break. It may have been the sleep and renewed enthusiasm but I am not so sure – circumstances prevailed to ensure a brilliant time in the ARC. Mrs Jones, the TA, had been asking me for about a month to ‘do something’ with money as she felt her charges needed some life lessons in personal finance and I left before the holiday promising to sort something out. I had an activity, called ‘Money-go-round’, one that I first tried with a Year 1 class in mind as I felt it would give the pupils in the ARC some concrete experience of money that Mrs Jones could then develop during their ‘standard’ maths lessons. I was happy as this was the original aim of my time in the ARC – to develop a series of activities that the pupils could access which would then form the basis of a week’s worth of teaching by Kathy and her team.
Vol. 27 No. 1 Spring 2022 12 I resourced the lesson and that Thursday I was ready to go but sometimes it is the unforeseen circumstances that conspire to turn a good lesson into an experience that is truly memorable. That day I happened to wear a tie that had been a present from New York. All the Academy students loved it but the ARC pupils, always to be counted upon to be inquisitive and tactile, were all over it from the moment I walked into the room. In the end I had to take it off so they could all admire it. The lesson was due to begin with a time of role-play where the pupils act out buying an item from a shop and paying for it with money from a purse. As John had the tie around his neck I persuaded him to act as the customer who would like to buy it from Georgia and as with all things I have tried in the ARC they responded superbly. This then led to a discussion of where the shopkeeper puts their money at the end of the day and where shoppers get their money from. It was an ideal scenario in which to establish the three key roles necessary for the lesson: shopper, shopkeeper and banker, indeed by now they were keen to get their hands on the money they saw I had! Money-go-round is a lesson in simplicity, each pupil takes one of the three roles and counts 15 pennies into their purse. At the role of a dice they each pass money from one character to the next: the shopper buys something from the shopkeeper, the shopkeeper takes the money before putting some money in the bank and the banker then gives some money to the shopper so they can keep on purchasing. The beauty here is that after 4 rolls of the dice each player has spent money (subtraction) and received money (addition). At no point are the players allowed to see what is in their purse, they simply take out the number on the dice or take what they are given. At an agreed signal the pupils then tell you how much money is in their purse without counting. At first their answers are all over the place as they have not used any strategy to keep track of how much money is coming and going. Georgia thought she had 30p when in fact she only had 10p! What did emerge after their initial attempts to play the game was the idea of conservation i.e. the money at the end had to be the same as the money at the start as it has just been passed around the table. It struck me that this is key developmental step in the understanding of personal finance yet it is one that is missing for many people. The rise of plastic means that money is not concrete anymore – we go to the shops and pay with our cards and leave with more money (because of cash back) that we went in with. It seems to me that a better understanding of conservation (thank you Piaget!) could have gone some way to preventing the global financial crash. As they started to play the game for the second time, now with 15p instead of 10p, the pupils asked if they could write down what was happening. This proved to be a decisive step for two main reasons: 1. for the first time several of them began to link a mathematical operation with an action i.e. ‘If more and more of them were giving accurate answers when asked how much money was in their purse
Vol. 27 No. 1 Spring 2022 13 I give him two do I write down -2?’ ‘Is this an add?’ upon receiving 4 p from the shopper. 2. they began to link the giving and receiving of pennies with the fluctuating amount within their own purse. The more they played the more careful their sums became and it was encouraging to see that each time the game stopped, more and more of them were giving accurate answers when asked how much money was in their purse. After the fourth go they now wanted to talk about what they did with their own pocket money. They asked me what I did with my children and the fact that to receive money they were often given jobs to do at home. Jacob shared that he earned £30 a week which the others thought was a great deal to have, accompanied with a great deal of moaning regarding who they could borrow money from when their original allocation had been spent. With a wise nod of the head Jacob brought the lesson to a very effective close with the words ‘When my £30 is gone, its gone, no more money till payday!’ Out of the mouth of babes and ……. Ken works in an academy in the north of England Teaching Maths …… Ken is very honest in this chat and I think this is necessary if we are to be open about the challenges of supporting our colleagues in alternative provision settings. After all they are the ones that have to deal with pupils who are far too dysregulated for mainstream classroom learning. Ken will continue with us as Equals seeks to provide help for all those who struggle with mathematics. Alan – Hi Ken, could you describe for us the school that you work in please? Ken – I work in an alternative provision school that has 60 pupils on roll. It used to be called a PRU but academisation enabled us to change the name to remove some of the stigma associated with the PRU label. They are all permanently excluded pupils from KS 3 upwards. We are also developing a unit within the PRU called Respite which will support pupils who are at risk of permanent exclusion to try and support them before it’s too late. Alan – To help us understand what life is like for you could try and describe what a typical day is like in your school? Ken – There is no typical day really and it has taken me a few years to get used to the chaotic nature of work in this setting. I do plan but I have to be flexible. Today for example I had six timetabled lessons with 15 pupils spread across the day. Today, which I would say is normal, I only saw five of them! There is a high degree of absence and internal truanting which has a massive impact upon the continuity of teaching and curriculum coverage. So the notion of sequential learning goes out of the window with such random, and sparse, lesson attendance. You might have a child attend for two weeks but then not see them again for three months! In addition some pupils attend every day, some every other day and some every other week! This means that
Vol. 27 No. 1 Spring 2022 14 teaching a topic to a class is impossible and as a consequence we personalise everything and teach to meet the needs of the child in front of us on any one day. Alan – Given this, what do you find works best with these pupils? Ken – Relationships are the key. I have seen some teachers who are wonderful and many times I wish I had been going into their rooms to be taught but because they have not been able to establish relationships the children hate them and being in their lessons. I have seen lessons that I would call really boring but the children love going in because that member of staff has a relationship with them. Without this you are not going to get anywhere at all and you can predict that disaster, with the accompanying profanities and expletives, will soon follow. Alan – What have been some of your mathematical highs this year Ken? Ken – This year we have managed to get all Year 10’s through their entry level qualification which was a significant success for us. We use this as a contingency as some children attend but then disappear so our philosophy is that every student will leave with a qualification. We always try to bank that qualification and then move on to provide the challenge that they need. Once you know the child and their triggers you can think about helping them, for when I hear words such as appropriate challenge it concerns me for when ‘challenged’ these pupils pull the shutters down and that’s when you will have chairs flying across the classroom. It’s a very delicate balancing act. Alan – What barriers to maths are you coming across in your pupils and are there any issues that stand out for you with? Ken – For us we have post-code wars as we take pupils from all across the city. A gang mentality and mindset is very problematic for us in the PRU. The pupils from each compass point, shall we say, are very territorial and there often a number of issues that occur before they get into my classroom and they are often not addressed before they come to me. This means I am having to diffuse situations and put out fires even before I start teaching. Their retention of knowledge is very poor and significantly reduced compared to mainstream pupils and much of this is associated with drug, cannabis use. Some pupils, during a time when their brains have not fully grown, are smoking cannabis every single day and so the impact of this use is significant. It’s not a good mix for their long-term memory and other aspects of their cognitive development. In addition to this they don’t have role models that show them anything other than chaotic life-style choices. Such lifestyles become more appealing to them once they realise how far they need to go and how much work it will take to get there. Something needs to be done in society to break these negative feedback loops that my pupils are stuck in. Alan – Just to finish, could you describe how you approach a lesson in the PRU as opposed to a mainstream lesson. Ken – Some would be expecting me to talk about how I plan experiences that are as creative as possible for the pupils such as you would see on those ‘Welcome to teaching’ adverts that are shown on the television. I find you need to go in completely the opposite direction! My lessons
Vol. 27 No. 1 Spring 2022 15 need to be highly structured and boring almost as this is what the pupils need - they need a rigid structure. For example in my early days here I planned a wonderful lesson on area that involved buying new turf for a football pitch. I had it all ready to go and we viewed it all from Google Earth and were just about to go out with trundle wheels to measure the dimensions of the pitch but what I can only describe as ‘Trundle Wheel Sword Fight War’ ensued! If you try to be creative it all goes to pot. For me I would measure success in the following terms: small steps. By this I mean the following: 1. Has a pupil come in and left knowing something that they did not know before. 2. They know how to get some marks in an exam. If this interview has resonated with you then please get in touch to share your thoughts, insights and experiences. Also if you have any questions for Ken then please get in touch and I will forward them to him. What’s in a name? Alan Edmiston reflects upon the term dyscalculia and what it means to him now. Recently I have been thinking a lot about the difficulties that pupils face within mathematics and time and time again I keep coming back to the word dyscalculia. In order to clarify my own understanding, I feel I need to put pen to paper and I have recently been helped in this matter by listening to two people share their understanding: Daniel Ansari and Janet Goring. Daniel was kind enough to take a seminar on the matter for Equals and Janet gave up her time to talk to one of my SEND groups for the Origin Maths Hub. I will therefore divide this piece into two reflections and a summary. In January’s seminar Daniel Ansari kindly shared with us his work concerning dyscalculia. Daniel is Professor and Canada Research Chair in Developmental Cognitive Neuroscience and Learning in the Department of Psychology and the Faculty of Education at Western University in Canada, where he heads the Numerical Cognition Laboratory (www. numericalcognition. org). Ansari and his team explore the developmental trajectory underlying both the typical and atypical development of numerical and mathematical skills, using both behavioural and neuroimaging methods. He began by acknowledging that dyscalculia exists and that it represents a significant challenge for many pupils and their teachers.
Vol. 27 No. 1 Spring 2022 16 This session proved to be something of a cautionary tale for me and I feel it is worth sharing some of the key messages that came from Professor Ansari. At Equals we feel it is vitally important to help teachers understand the specific mathematical difficulties that the pupils in their classrooms may exhibit. The reality of our knowledge about developmental dyscalculia means that there is not as much significant help for teachers who wish to know more about this condition as we would like. Daniel’s seminar was thoughtful and honest and he began by acknowledging that dyscalculia exists and that it represents a significant challenge for many pupils and their teachers. As he spoke it became clear that we know some things about dyscalculia but there is much we do not know about it. Helpfully he shared that we should be wary of those who say they have the answers to help dyscalculia for even its diagnosis is not consistent, in the way that it is with diabetes for example. The word itself may be educationally more familiar and in common usage in SEND circles but we do not yet have a shared understanding of dyscalculia. Due to the many gaps in our knowledge, so well articulated by Daniel, I now wonder what indeed this understanding is actually based upon. What emerged for me was the knowledge that much of the research into dyscalculia comes from studies with small sample sizes and varied inclusion criteria and so we are unclear as to whether like is being compared with like. It seems that when it comes to dyscalculia we are not there yet and that there is so much more to learn. It was refreshing to hear someone speak so honestly about the fact that many studies are arbitrary and that inclusion criteria are not consistently applied. For example, there is no clear evidence for a common cause and a simple genetic basis for dyscalculia and we do not even know how early we can identify it. There are screening tests but these are not diagnostic. Daniel’s lab uses the following screening test: https:// www.numeracyscreener.org/home.html with students from the age of six. Many of the studies investigating dyscalculia focus upon the difference in performance of children in symbolic (the spatial arrangements of dots) verses non-symbolic (i.e. numerals) tasks and evidence seems to highlight that those who may have dyscalculia perform more poorly on tasks involving symbolic reasoning. Dyscalculia is a word I have been hearing for many years and looking back to find the article by Mundher Adhami for this issue I noticed that Jane Gabb mentioned dyscalculia in 2004. The second half of this piece is more practical in nature for it comes from listening to someone who works on a daily basis helping pupils with specific maths difficulties. Janet Goring works in Wandsworth, heading up a team who support pupils who have difficulties in both literacy and numeracy. Janet is also one of the leads for the RIWG Characteristics of SEND for the maths Hubs. I found it helpful that Janet chose to title her presentation ‘Number Sense’ and not dyscalculia for she came at it from a very practical angle and that in itself really helped me to understand where to place dyscalculia within my understanding of SEND within mathematics. Janet helpfully defined dyscalculia as a problem with number sense some studies show that 1-2% of the population are effected although most studies suggest 3-8% suffer but this is dependent on the cohort. It seems to affect the ability of children to retrieve number facts and is associated
Vol. 27 No. 1 Spring 2022 17 with a difficulty in understanding the magnitude of numbers and also an ability to subitise. There is a connection with dyslexia and this is linked to a co-occurring problem with working memory. There are signs that we can look for as children who may be dyscalculic have a weak understanding of prepositions, are unable to sort based upon similarities and differences and have a limited grasp of place value. As a consequence, their ability to calculate fluently is limited. A key feature of dyscalculics is an inability to know which is more when presented with two groups of dots representing 6 and 4 say. This is described as the cardinal property of number i.e. the size of the collection. Janet was careful to point out that dyscalculia could only be assessed by a specialist and that screening alone does not provide the full picture. It was the latter part of her talk that I found the most helpful for it focused upon what we can do to help pupils with dyscalculia or a poor number sense. This condition is not linked to ability and so pupils may have real strengths that we can work with to support their learning. The development of metacognitive strategies (we will focus upon this in the next edition of Equals) and the use of clear and simple instructions can bring real change. Repetition and revision through low stakes quizzes really help alongside enabling support from an adult. The use of what I would see as good early years practise i.e. a multisensory approach using visual and concrete resources alongside songs, the physical acting out of word problems really resonated with me as a parent and a teacher. Enactive teaching during which maths involves an interaction with the world around us and the slow move to more abstract ideas are vital in the development of mathematical thinking. It is thanks to Daniel and Janet that my thinking is becoming clearer regarding how I both view and place dyscalculia into my growing understanding of the difficulties that pupils face within mathematics. It also reinforced my belief, which was triggered by Mundher Adhami, that we all have mathematical brains hard wired to interact with and make sense of the world around us and it is part of this interaction and thinking that we call mathematics. Given the misunderstanding that surrounds dyscalculia we at Equals will endeavour to provide the best support we can for teachers and their pupils. If you have any questions or need any support and advice then please get in touch. The University of Derby needs your help! • Do you want to help children who are suffering from maths anxiety? • Are you interested in finding out how we can make the maths classroom a better place for all learners? If the answer to either of those questions is yes then please click on the following link: Get involved in research – Mathematics Anxiety Research Group (derby.ac.uk) This page highlights the areas in which you can get involved with research into the causes, and solutions, of anxiety in mathematics.
Vol. 27 No. 1 Spring 2022 18 Recall of tables Like many of us Mark Pepper has found himself tutoring pupils who are finding mathematics hard following lockdown. Working in such a way has afforded him the chance to reflect upon some of the barriers to progress. Students unable to rapidly recall multiplication tables are at a considerable disadvantage in exams and in everyday life Part 1 I recently commenced working on a 1-to-1 basis with a Year 9 student whom I will call Jim. In order to make an initial assessment on his mathematical progress I sat next to him while he worked on a non-calculator GCSE Foundation paper. I was impressed with his positive attitude, his logical reasoning and his general competence in answering the questions. I was, however, struck by the fact that he was unable to recall multiplication tables. An example of this occurred in a question in which he was required to calculate the area of a rectangle which had a width of 6 cms and length of 8 cms. He knew that he had to multiply length by width, (8 x 6) sq cms. He hesitated for a considerable amount of time whilst he mentally tried to continuously add 6 on 8 occasions. Eventually he lost track and then he made use of pen and paper. He wrote 6+6+6+6+6+6+6+6 in a column and then proceeded to add the numbers. Eventually he tentatively produced an answer of 48 and I confirmed that this was correct. This brought back to me memories of the many occasions in the past in which students of various ages and abilities had used the same strategy when required to multiply 2 numbers without the use of a calculator. This included students in the further education sector. It seems to be probable that poor recall of multiplication tables continues to be a fairly widespread occurrence within secondary education and beyond. There are two distinctly different issues involved in order to remedy this situation: 1. The need for effective teaching strategies to enable pupils in KS1 & 2 to become proficient in the recall of multiplication tables. 2. The immediate need to implement teaching strategies to enable students in Year 7 upwards who are having difficulties with the recall of multiplication tables to become more proficient in this. The need for effective teaching strategies to enable pupils in KS1&2 to become proficient in the recall of multiplication tables. Within the structure of maths lessons I have found the regular use of a mental maths component to be extremely beneficial. During the relatively short period that the National Numeracy Strategy (NNS) was in force a mental maths starter was an It seems to be probable that poor recall of multiplication tables continues to be a fairly widespread occurrence within secondary education and beyond.
Vol. 27 No. 1 Spring 2022 19 obligatory opening of all maths lessons. Whilst this no longer became an official requirement with the abandonment of the NNS, I continued to use it throughout my teaching career. Content of the mental maths starter. Whilst in general a teaching approach that helps to develop autonomy, logical thought and problem-solving abilities provides an effective means of teaching maths, there are some instances in which rote learning can be helpful. This applies in the learning of multiplication tables and in memorising number bonds up to 10 initially and then up to 20. As children of a young age usually have very good memories then it seems reasonable to ask them to memorise multiplication tables. Various activities can then be used to reinforce this knowledge of number facts. One of these is a fun activity called Down the Pit. I have made reference to this in the past and am repeating brief details of it here as I have found it to be a really effective resource. Down the Pit The game commences with the teacher writing the numbers 1 to 20 on the whiteboard and then drawing a ring round three of the numbers which would have been randomly selected. The teacher sits opposite the group or class and holds a jumbo calculator with the display panel facing the class. The learners then have to pick a low number that they think will miss the ringed numbers when they are continuously added. The teacher would then make use of the constant button and continuously add the suggested number. E.g. If the pupil chose to add in 4s then the numbers obtained would be 4,8,12,16 and 20. If the ringed numbers are avoided then the player does not fall in the pit, whereas if a ringed number is accessed then the player would fall into the pit. Hence if one of the ringed numbers had been 16 then the player would fall into the pit. A record could then be kept of the outcome of each game by placing a tick in the appropriate space of a grid consisting of two columns, one headed “does work” with the other column headed “does not work”. This helps the class to recognise number patterns such as that 2 should not be chosen if any of the ringed numbers are even. It also helps the class to become familiar with the multiples of numbers up to 10. It is also beneficial for pupils to develop rapid responses to questions initially involving number bonds up to 10 and then up to 20. Activity aimed at reinforcement of number bonds up to 10 and later up to 20. This is an activity that can take place in an individual, small group or whole class setting. The teacher holds up a jumbo calculator with the display panel facing the class. A low number is entered e.g. 4. The class then has to respond by rapidly naming the number that when added to 4 would total 10. This process can be repeated a number of times. When this has been done over the course of several lessons, the usual outcome is that the class would then provide instant accurate answers. Once this position has been reached, then the same process can be used for number bonds to 20. There are some instances in which rote learning can be helpful.
Vol. 27 No. 1 Spring 2022 20 Students with poor multiplication tables recall in KS 3&4. The teacher with a new class needs at an early stage to identify any students who have poor number fact recall. Whilst it seems probable that many of these would be lower attainers in maths, it can also include students who are otherwise quite able mathematicians. There may well be a belief among some teachers that whilst this is unfortunate there is little that can be done about it, especially if the students involved refuse to try to memorise the tables. Whilst such an attitude is to some extent understandable it can be demonstrated that it is ill advised as the consequences of poor number fact recall are extremely detrimental to the students both in the context of day to day life and in taking the GCSE maths paper. Reasons why it is essential to have good multiplication table recall. Within the context of day-to-day life it is true that aids such as mobile phones or watches would normally be available and that their use would enable the student to rapidly obtain answers to multiplication questions. There are, however, times when a more immediate response is required. An example of this could occur with shopping in which a person wanted to check that the correct amount had been charged and that appropriate change had been given. Within the context of GCSE maths exams the disadvantages associated with poor multiplication table recall are significant. This particularly applies within word problems such as the calculation of income from wages, short and long multiplication questions and finding the LCM or HCF of a number and to a lesser extent in questions relating to area of 2D shapes and the volume of 3D shapes. The students who respond to such questions with the use of repeated addition pay a heavy price in terms of the possibility of making an error in their calculations and in the inefficient use of time. This could result in them using up the allotted time for the paper before they have had time to attempt some of the questions. Strategies that can be used in order to minimise the adverse effects of poor number table recall. Once the students have been identified then a strategy needs to be developed to address their difficulties. This can involve encouragement to memorise multiplication tables up to x10. Some students may well be reluctant to do this. In such cases it is helpful to give them one number fact to memorise such as that 7 x 7 = 49. Then if they need to know the value of 7 x 8 they can simply add 7 to 49 and they can use a similar method for other numbers around the middle of the 7 times table. A similar process can be used for the 5,6 and 8 tables. Another strategy involves making use of the 10 times table to find multiples of 9. An example of this would be in response to 9 x 8, they could do (10 x 8) – 8. Tables 2, 3 and 4 involve such low numbers that there should be little difficulty in counting on. It would also be helpful for students to develop efficient methods of identifying factors. This could be done with the whole class in quick ways to determine if a number less than 10 is a factor of a specified larger number or not. Each of the numbers could be considered as follows: The consequences of poor number fact recall are extremely detrimental to the students.
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