Vol. 29 No. 1 Spring 2024 1 Editorial Team: Kirsty Behan Alan Edmiston Peter Jarrett Alison Roulstone Les Staves Nicky White Letters and other material for the attention of the Editorial Team to be sent by email to: edmiston01@btinternet.com ©The Mathematical Association The copyright for all material printed in Equals is held by the Mathematical Association Advertising enquiries: Charlotte Dyason charlotted@media-shed.co.uk D: 020 3137 9119 M: 077 1349 5481 Media Shed, The Old Courthouse, 58 High Street, Maidstone, Kent ME14 1SY Published by the The Mathematical Association, Charnwood Building, Holywell Park, Loughborough University Science and Enterprise Park, Leicestershire, LE11 3AQ Tel: 0116 221 0013 Fax: 0116 212 2835 (All publishing and subscription enquiries to be addressed here.) Designed by Nicole Lane The views expressed in this journal are not necessarily those of The Mathematical Association. The inclusion of any advertisements in this journal does not imply any endorsement by The Mathematical Association. Editors’ Page 2 Harry Hewitt Award 3 The help you need may be closer than you think 3 Alan Edmiston and ‘Darragh’ share two SEND support stories. These highlight how helpful colleagues from special schools can be in enabling mainstream schools create inclusive classrooms. How does understanding number happen? 5 In a continuation of his articles exploring of the roots of mathematics Les Staves tackles the tricky question of how the understanding of number occurs. Archive piece – Visible numbers are not only for learners with Down Syndrome! 12 Vikki Horner argues here that Number Facts to 10, 20, and beyond, are accessible by all children. In the first of two articles, she shows how Catherine Stern’s programmes for teaching maths to individuals with Down Syndrome can help. Supporting students with Dyscalculia in the maths classroom 17 Louise Langford has kindly shared her ten top tips for helping Dyscalculic students develop their mathematical thinking. Please let us know yours and which of the ideas below you use and why? The aim is to collate and then publish an extensive list of strategies for use in supporting Dyscalculic students. The catastrophic journey from SATs into NCTs 18 Mark Pepper provides us with a history lesson looking at assessment from the 90’s to the present day. Identifying, Supporting and Preventing Anxiety in the Mathematics Classroom 23 Janet Goulding shares the ideas emerging from work on anxiety she has been doing with the London SW Maths Hub for the past two years.
Vol. 29 No. 1 Spring 2024 2 Editors’ Page Since I took over editing this wonderful publication some ten years ago now there has been one guiding presence helping and supporting me. I am talking about Nicole Lane - the designer who takes the few pieces of text I send and works her magic to produce what you see before you. Without Nicole Equals would not be the publication it is – it is due to her skill and artistry that it looks the way it does. I would like to take this opportunity to thank Nicole, on behalf of all the contributors and readers, for all of her hard work, support and professionalism over the years. This editorial contains three important pieces of news: Firstly a working group has been set up by SASC (the Specific Learning Needs Assessment committee) to review the guidance and definition of Maths Difficulties and Dyscalculia from 2019. For more details see the Maths Difficulties and Dyscalculia Update (sasc.org.uk). The group consists of representatives from some of the key University departments involved in research into maths cognition/difficulties and maths anxiety; SpLD professional bodies; SpLD teacher training providers; psychologists and specialist assessors. This was partially pre-empted by the availability of recent research which suggests a rethink on the definition of Dyscalculia. The key news for us is that the first draft of the group’s findings is imminent and we will report on this and summarise the main outcomes in our next edition. Secondly we are pleased to inform you that the Second Equals Conference “Access for All’ will now be taking place during the Autumn term and to find out more information and to book please use this link: https://www.m-a.org.uk/ equals-conference The conference will be an exact duplicate of the one originally advertised. Such was the strength of speakers that we felt it important to offer you access to the same quality of support. Lastly I would like you to join me in influencing the future direction of Equals. You are invited to join the conversation about forming a special schools maths network. Preliminary meetings have been held in Bristol, Chester, London and Sunderland and the key outcome is that special schools need and would value a forum to support and facilitate collaboration and sharing. We are offering the chance to help us move into this space and to would like you to join a zoom meeting about this on the Tuesday 7th May at 4.00pm Join Zoom Meeting https://us06web.zoom.us/j/84460959061?pwd=Uz xazPNmou2cE2vPyFsbEfRfYOyVwU.1 Meeting ID: 844 6095 9061 Passcode: 996876
Vol. 29 No. 1 Spring 2024 3 The Harry Hewitt Memorial Award This prize is awarded to any pupil who has overcome barriers with mathematics and is now making progress and enjoys their mathematics. Do you have a pupil who tries hard and is growing in confidence as they engage with mathematics. Then why not celebrate their success with Equals? We are offering recognition for their efforts, a £25 book token and the opportunity to have their work published in Equals. Choose a piece of work that both you and your pupil consider successful and send it to Equals. Please try to include: • the piece of work or photograph or copy, • an explanation from the teacher about the piece of work and a description of the barriers which the pupil has overcome or is in the process of overcoming, • supportive comments from colleagues and parents, • the pupil’s age, school and the context of the class in which they learn and, if possible, some comments from the pupil about what they are pleased with about this piece of work and/or the learning it shows. The help you need may be closer than you think Alan Edmiston and ‘Darragh’ share two SEND support stories. These highlight how helpful colleagues from special schools can be in enabling mainstream schools create inclusive classrooms. This article is based upon a conversation I had with Darragh who works in an SLD (significant learning difficulties) school. Students at the school function fairly independently working at Early Years, KS1 and KS2 abilities at chronological ages of 11-19. Students at the school can also have ASC, ADHD, SEMH diagnoses of other impairments. This piece seeks to highlight the value of mainstream schools collaborating with special schools with the aim of developing
Vol. 29 No. 1 Spring 2024 4 curricula that better support and include all learners. Many schools are struggling to meet the needs of their SEND learners, especially those who are unable to access specialist provision and are placed in mainstream classes. Darragh and I have recently been involved in projects that support the lowest attaining pupils in mainstream schools. In my case it was developing a new year 7 science nurture curriculum and in Darragh’s it was facilitating KS4 pupils to gain certification in Year 11. Both schools reached out for help knowing they needed to change their current practice in order to meet the needs of pupils who are operating at least five years below age expectations at Year 7. I will tell Darragh’s story first. He was approached through his trust to support a school with a very high proportion of SEND and disadvantaged pupils, in fact this school was in the top 5% with regard to the number of pupil premium in the city. The children in question were unable to access the Year 7 work at all. To cut a long story short this class was used to trial a curriculum that mirrored what Darragh put in place in his school which supported students to access a spiral curriculum in KS3 accessing KS1 and KS2 content and then accessing Entry levels 1, 2 and 3 at KS4. Darragh felt that a pathway was now in place to support these pupils to leave school with a sense of achievement and with a level of certification they could feel proud of. Momentum was building and the new curriculum was ready to take off until the school leadership changed. This story does not have a happy ending for the SEND pupils and staff that worked on this project for Darragh’s curricula amendments were dropped as the school moved to mixed ability classes and away from a distinct pathway for the lowest ability SEND students. I like teaching mixed ability but the pupils in this case were unable to access age appropriate learning and so ended up sitting GCSE in the hope of achieving a 1. I could feel the sadness in his voice as he reflected upon the damage that filing would do to the social and emotional development of this cohort of young people. My story is still being written and I hope it has a happy ending. Barry School runs two nurture classes in Year 7 who are taught by one primary specialist for their core subjects i.e. English, maths and science. These are very fragile pupils who would find life in Year 7 tough. I have worked with this school for a few years now and knew the department were unhappy with the nurture children following a curriculum that in the words of Sally, one of the nurture teachers, “Just goes over their heads”. I was in complete agreement as there is no point in expecting pupils in Year 7 who are unable to sort using two criteria to understand the particulate nature of solids, liquids and gases. Over the past year we have trialled a range of activities and amended their curriculum to ensure they have plenty opportunities for practical tasks such as measuring and consequently the more abstract aspects of Year 7 have been removed. The topics they follow are similar to the rest of year 7 but we have lowered the floor to ensure they have appropriate access to a curriculum that is more in keeping with their stage of thinking. Many schools are struggling to meet the needs of their SEND learners
Vol. 29 No. 1 Spring 2024 5 The future looks bright for the school has joined something called a SEND Enthuse Partnership run by STEM Learning and are seeking to use this project to explore KS 4 provision for these pupils. As I reflected upon our conversation it struck me that so many schools must be in the same situation i.e. they analyse their data for their year 7 pupils and realise that a significant number of the cohort (in some schools this can amount to 20 pupils or more) are working at levels far below their peers and many of these bring with them EHCP’s. I think one possible way forward can be found in the stories told here – there are colleagues close by in special schools who not only have the experience of these pupils but they also have the curriculum expertise that can be applied to ensure that life in year 7 can better support the needs of all pupils. How does understanding number happen? In a continuation of his articles exploring of the roots of mathematics Les Staves tackles the tricky question of how the understanding of number occurs. Awareness is growing in schools about the importance of developing ‘Number Sense’ and teaching the skills of subitising that develop from it. It is at last gradually reaching our curriculum documents, but what is it? Where does it come from? How does it develop in our minds? These are questions for us to understand If we are to appreciate how to teach children to improve this natural skill which underpins mathematical learning. Research is showing that students with Dyscalculia suffer from poor number sense. A sense of quantity In 1992, pioneering work on the early numerical cognition of infants, by Karen Wynn, discussed how, in the first weeks of life, infants could visually discriminate when there were quantitative differences between groups of objects. Their reactions indicated that they noticed changes, and by five months they even showed some anticipation of what outcomes of changes they were watching, should be. She dubbed this ‘Number Sense,’ but at this early level, the name is a bit of a misnomer because it is not strictly speaking ‘numeric’. Initially it is about perceiving differences of groups, not about knowing ‘numbers’, but it is the perceptual skill from which understanding Numbers grows. It was noted that the perceptual skill operated at two levels. Exact Number Sense Firstly, with small groups of things even babies have an ‘exact number sense,’ and they immediately notice differences between groups,
Vol. 29 No. 1 Spring 2024 6 across a range of one to five things. It has been established that this sense is as automatic and immediate as discriminating colours. But there is a limitation, and children, or even adults, cannot easily identify randomly arranged groups of more than five or six items. We will discuss patterning and other strategies needed to identify larger groups later. Approximate Number Sense Secondly, with much larger groups we have an ‘Approximate Number Sense,’ and children can detect when there is a general difference so long as it is large. For example, infants will see a difference between 10 and 20 things, but not between 10 and 15. The ratio improves for us as adults, but never becomes perfect. This sense is obviously important for making estimations and practical choices. But one might also see that its role in making comparisons and choices is part of problem solving. It is also useful for rapid mathematical thinking, where making approximations can speed up or help check working. Whilst it deserves our consideration, I have only space in this article to focus on the roles of ‘exact number sense’ in developing fundamental understanding of number. Core perceptual knowledge With the progression of neuroscience more evidence was accrued confirming the nature of children’s perceptual experiences of quantities. Researchers like Elizabeth Spelke suggested that they are one of the aspects of our ‘core knowledge,’ that is innate biological tendencies that we have at birth to support survival and development. She has noted that it is not only a visual skill, we also discriminate between different groups and patterns of sound or touch and movement experiences. Like other animals and birds, we can identify variations in size between different groups. The difference being that humankind has extended the perceptual skill and developed ways of making representations of quantities to record, remember and communicate. Since pre- history, tokens and graphic representations were used. They developed into symbols, that in turn enabled abstract thinking and led to a system of ideas that we know as mathematics. Neurological and cultural roots Neuroscientists Stanislas Dehaene and Brian Butterworth have both researched and written about this. They have described the role that ‘number sense’ has played in the development of practices of counting, recording, calculation and mathematical thinking in all cultures over millennia. Both these eminent scientists also highlight that children’s mathematical development also mirrors the cultural processes of mathematics that started in pre-history. Just as finger representations or tokens like stones or shells and tally mark making, preceded formalised numerals in the cultural development of maths. So children start with making sense of their sensory perceptions, With the progression of neuroscience more evidence was accrued confirming the nature of children’s perceptual experiences of quantities
Vol. 29 No. 1 Spring 2024 7 working on through practical experiences and learning language. They extend thinking by using mark making for representations, which support abstract thought, and can be related to numerals they are taught. Infants perceive differences and equivalences and that gives them experience to appreciate size and sequence and order. They develop elementary ideas about quantities and numbers, that spring from and connect to the cultural models of action and language they experience with adults and peers. These perceptions of quantity are the fundamental beginnings of understanding numeracy. Without it we would not be able to lead our practical daily lives, nor would mathematics ever fly to its abstract heights. But because it develops through fundamental early experiences we often disregard it when we think about school levels of learning. Research in recent years has been uncovering more about the causes of difficulties with mathematical learning. There is growing awareness and discussion about ‘the relationships between Dyscalculia’ and difficulties with fundamental powers of visualisation, and number sense at its sensory level. Roots of number in the brain Brain scanning observations have told us that the area of the brain used for number sense is the parietal lobe, it is the area in which we process spatial information, from any of our senses. Neuroscientists have established that there is an intertwining relationship between our spatial perceptions and developing ideas of number. For many of us who work with young children this reinforces convictions about the importance of visual and manipulative equipment. Understanding number has sensory beginnings. Number sense and fingers Recent years have seen growing interest in the relationship between fine motor ability and finger gnosis in the development of mathematical thinking. Interestingly the parietal lobe is also the very part of our brain that controls the automatic actions of our hands and fingers. It is now well established that the use of fingers as representations of quantity has had a role in the development of number systems from primeval times. There is also ample evidence of connections in languages, where words for hands or fingers are commonly linguistic roots of number words. This should alert us to think again about the long-standing prejudices about finger representations that have been prevalent in school education. We should understand more about their roles, in subitising and thinking. They function as malleable manipulation equipment, connected directly to our parietal lobe. They provide not only visual, but also touch and proprioceptive information as we use them for tracking number or carrying out finger calculations. They are also temporary representations that help short term memory bridge between physical/neural There is growing awareness and discussion about ‘the relationships between Dyscalculia’ and difficulties with fundamental powers of visualisation, and number sense at its sensory level
Vol. 29 No. 1 Spring 2024 8 perceptions to abstract representations of numerals. Pre numeric perceptual beginnings Though the psychologists who researched the ability to notice differences called it ‘number sense,’ the awareness of quantities and comparisons is not initially ‘numeric,’ in a symbolic sense. The perceptions are not related to number words, or symbols. Infants cannot initially name them, those connections have to be learned and refined. Whilst it is true that most children start making these connections with small numbers, through everyday interactions even before school age, it is worthwhile for teachers to know of these processes and understand how to promote and reinforce them. In this article I will start by focussing on the ‘exact number sense’ of small quantities, because that is what leads children towards realising what number names ARE, it is the background from which ideas about numbers grows. There are two perceptual processes that very young children use at this level 1. Looking at whole sets - When whole groups of up to four or five things, are different (This is called analogue magnitude). It is the root of understanding about group quantity, and is modelled for children as adults or peers name the groups. 2. Tracking individual items - Following or scanning and noticing the extent of the sequence of individual items, (which is called Parallel individuation). This is at the roots of understanding sequential quantity. As with whole sets, its extent is limited, and if there are more than four or five objects, or sounds in a sequence, children lose track. There is a strong case for modelling and practising perceptual processes of whole group recognition; sequential scanning; and itemisation. Many opportunities can be created either playing games, or in social and practical activities. It is interesting to note that research by Wang and Feigenson at John Hopkins University, suggests that playing number sense games – making rapid comparisons of dot arrays and simply calling out more or less, without even naming the groups, primes children to perform better in formal maths tests later. Knowing without counting - levels of Subitising Young children use and develop their perceptual skills toward becoming numeric even before they can count. They are often able to identify and name small quantities that they look at without counting. How does this happen? They use the perceptual skills as exact number sense to immediately discriminate and generalise different configurations of small groups. Meanwhile, adults or peers will name groups or use number words to track a sequence. Infants soak in this modelling and associate the image generalisations and the language. Eventually they can immediately Young children use and develop their perceptual skills toward becoming numeric even before they can count
Vol. 29 No. 1 Spring 2024 9 recognise and name the small quantities. The skill is called ‘subitising’ which is a word that comes from the Latin for ‘sudden.’ Subitising is an integration of spatial recognition skills with language drawn from general social interaction. It becomes useful practical knowledge for quickly naming and comparing small groups of all kinds of things- objects- pictures- marks etc. It is a powerful naturally acquired tool. But traditionally maths teaching has not valued subitising, which has been looked upon as guessing. Counting is stressed so much as being the accurate logical way to find value, that a natural mathematical tool has not only been unappreciated but actively discouraged. Perceptual subitising We might like to make an extra distinction, of calling the identification of small numbers ‘Perceptual Subitising’, because it involves naming from immediate perception. In later paragraphs we will discuss more complex processes, related to quantities larger than five. Subitising is a process of rapidly looking, knowing and naming. It’s a quick-fire skill, and it is worth practicing and refining. Not only because it’s a useful practical skill, but also because it has vital effects and roles in learning about number. Children can usually subitise before they can use meaningful counting. In fact, though adults might not even notice it, children often subitise as they are practising emerging counting skills. Number sense and counting Perceptual number sense and exact subitising of small groups exists before counting. It actually plays a role in children developing understanding about counting. When children are learning counting skills, their instincts of quantity- ‘Exact number sense’- are inevitably at play. So, without consciously expressing it they may instinctively subitise before they count things. Then if the outcome of the count matches their underlying expectations, there can be both a confidence and a reinforcement. If that process happens over and over with small quantities it helps form some simple but important ideas in children’s minds, such as: - Numbers are names for quantities - Quantities each have their own consistent name - Counting is not just a recitation it is to find group size - Different arrangements can be the same numeric quantity - Different arrangements can also be moved into a number line. There are more ideas, often so fundamental that most of us take them for granted, but they are important background knowledge to both understanding what counting is about, and to being able to think quickly and flexibly. Perceptual number sense is the bedrock of building understanding about number. If it is harnessed to support emergent counting it can Perceptual number sense is the bedrock of building understanding about number
Vol. 29 No. 1 Spring 2024 10 be a powerful reinforcement. If we teach children to trust it (and the related tools of approximation) they will have mental tools for flexibility. If children do not have fluency, or if they have mismatches and misconceptions then we need to work on it. Working on exact number sense is not complex it is natural, accentuating and reinforcing everyday social learning. It involves teachers recognising that the beginnings of mathematical thinking lay in connecting perceptual and concrete experience to developing language. We can use apparatus, but we can also find opportunities in everyday experiences or games, where we can accentuate processes of observing and naming different configurations of small groups. We can comment on the connections of quantity, along with making representations in various ways, tokens, marks, pictures, finger shapes, as well as numerals. Conceptual subitising We need to remember that the perceptual subitising I have been discussing above has limitations. Even adults cannot immediately perceive random groups larger than five, so to identify larger groups they must use other strategies, which takes fractionally longer to do. The additional processes might include either: - Counting all - Detecting a small group and then counting on - Recognising two small groups and using addition knowledge. Because these processes entail using extra ideas, it is called ‘Conceptual Subitising’. The strategies required in addition to perceptual subitising include, being able to count on, or appreciate that combining numbers can make larger numbers. This suggests that any child who cannot count beyond five will be unable to identify any larger groups we show them. Perhaps this accounts for the wild guesses some pupils make in our classrooms. It drives home our need to work on extending exact number sense including: developing powers of observing patterns and extensions; combining groups or understanding counting on. Many of the manipulatives used in schools such as Numicon or Rekenrek, support these processes, as does the use of ten frames, which brings in more flexible imaging for number bonds to ten. An ever-increasing battery of programmes for computer and tablets can provide greater variety of arrangements and context. The diagram that shows phases through which subitising develops from initial sensory perception towards Conceptual Subitising Staff must be aware of the reasons for teaching number sense, and aware of the nuances. The recent encouragement to use Rekenrek in schools is a case in point, it’s important for classroom staff to realise that the point of Rekenrek, is not that it’s
Vol. 29 No. 1 Spring 2024 11 a frame for counting on. Primarily its line of beads are coloured into groups to support subitising first. Using the one touch approach is important for promoting confident number sense and number bond knowledge. Growing awareness Recent research into mathematical development has suggested, that teaching which encourages children to ignore their instincts, and rely only on counting, undermines their confidence in their number sense. Later mathematical abilities are affected as it slows down processes and inhibits intuitive judgements. Disregarding, or having poor number sense ossifies flexibility of thinking, and the ability to predict or check quickly to have confidence in your result. The research at John Hopkins university found that practicing dot estimation games improves children’s capacities in later formal maths tests. It may not explain why number sense exercise helps children to improve, or how long improvements last. But it adds to evidence that has been growing over many years from neuroscientists, and from educationists like Steve Chinn, that there is something fundamental about visualisation in mathematical learning. But one other lesson we may need to absorb from neuroscience and psychological research is that ‘visualisation’ is not restricted to vision! Children learn about early number through hearing patterns, or touching and moving things, number sense is a multi-sensory phenomenon. Most early years teachers instinctively understand this. But the application of this knowledge is restricted with the advent of curriculum models that put regard for subject structure above awareness of how people learn. The role of number sense in school learning is increasingly being substantiated, and awareness of Dyscalculia is accelerating. For some children, perhaps for many, we are being shown the need for curriculum that takes account of HOW learning happens in children’s minds and bodies. Can we have curriculum delivery which incorporate practical attention to sensory and perceptual aspects of learning, and can we enhance our teaching by using them. This article has referred to Atsushi Asakawa. Shinichiro Sugimura. (2022) . Acta Psychol (Amst) 2022Nov. Mediating processes between fine motor skills, finger gnosis, and calculation abilities in preschool children. Butterworth, B. (1999) What Counts. New York..The Free Press. Dehaene S. (1997) The number Sense. New York: Oxford University Press, 1997; Cambridge (UK): Penguin press, 1997. ISBN 0-19-511004-8 Lakoff,G. Nunez,R.E. “2000. Where mathematics come from: How the embodied mind brings mathematics into being. Basic Books. New York. Spelke, E.S. and Kinzler, K.D. (2007), Core knowledge. Developmental Science, 10: 89-96. https://doi.org/10.1111/ j.1467-7687.2007.00569.x Wang,J. Dark,O. Halberda, J. Feigenson, (2016) L. Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance. Journal of Experimental Child Psychology 147 (2016) 82–99 Wynn, K (1992). “Addition and subtraction by human infants”. Nature. 358 (6389): 749–750.
Vol. 29 No. 1 Spring 2024 12 Archive piece – Visible numbers Several special schools I have visited this academic year have told me they are considering taking part in the KS 1 Mastering Number training on offer with their local Maths Hub from September 2024. With this in mind please read the following article from by Vikki Horner that was written in 2007 and let us know what you think about the ideas contained in her lovely reflections on making numbers visible. Visible numbers are not only for learners with Down Syndrome! Vikki Horner argues here that Number Facts to 10, 20, and beyond, are accessible by all children. In the first of two articles, she shows how Catherine Stern’s programmes for teaching maths to individuals with Down Syndrome can help. I have personally worked with structured materials for six years and with Stern apparatus for the past three years, mainly with children with Down Syndrome. In this first article I describe the materials and programmes while the second gives accounts of four cases of Down Syndrome learners ages from 3 to 36. Although the two articles focus on individuals with Down Syndrome, please be aware that Stern was designed to teach ALL children number and arithmetic from aged 3 to age 11, and for children with learning difficulties/disabilities, these materials are of special importance. They may have difficulty in discrimination, memory, inter-sensory organisation, perceptual processing (both visual and auditory) and the ability to sustain their concentration. They must learn how to receive and integrate information from as many different senses as possible in order to form concepts. Making the number system visible Structural Arithmetic was invented by mathematician Dr Catherine Stern. Her rationale for this teaching approach was that learning should not be based on rote memory, but on visualisation of the structural characteristics of the concept, thus giving pupils insight into the relationships that are to be grasped. It was
Vol. 29 No. 1 Spring 2024 13 the founder of Gestalt Psychology – Dr Max Wertheimer who named her approach Structural Arithmetic. The apparatus was designed to make the number system visible. Learning takes place at the child’s pace, and involves all aspects of natural development at this age: sensori-motor, psychological and social. Children work through a series of experiments and games using concrete apparatus. It includes two different sets of representations of the numbers 1 to 10 in the form of 2cm graded Number Blocks – size being relevant to a child’s sensorymotor development – and pattern boards 1 to 10. The structure of the patterns provides unforgettable imagery and promotes cognitive growth. Children gain solid understanding of concepts through five important areas of development: visualisation language; receptive and auditory memory; action and reversibility. Number names are not used until the second stage, and written work is not introduced until later, so that children with delayed motor skills are not held back. Stern materials are self-checking devices, specifically designed to encourage children to think for themselves and to ensure learning is successful. Through their experimentation, self-correcting, and increased visualisation ability, children develop mathematical reasoning. For example, if a block is too small or too big for an empty groove, they can see and feel in what way it does not fit and try others until they are successful. One of the most important principles built into Stern materials is in their arrangement that keeps many different relationships in view and in the child’s consciousness at any given time. The apparatus offers excellent diagnostic tools where assessments can take place simply by watching and listening to what pupils are doing/ saying. Counting Board 10-box Addition and Subtraction Facts to 10 Experimenting with Numbers – the first of five programmes – is a multi-sensory approach where children learn the basic addition and subtraction facts with numbers up to and including 10. Taught in small-steps, through three stages, each level
Vol. 29 No. 1 Spring 2024 14 broadens previous learning. From the beginning, (level1) simple experiments take place in the Counting Board, 10-Box, and Pattern Boards. Learners discover ‘size’ relationships, position and sequencing. They fill the 10-Box with pairs of blocks: (later to become the bonds to 10) – the teachers places a random block in the 10-Box; a child finds the block that fits. They find two pairs of like blocks, (preparation for the commutativity of addition). They recognise and match patterns, construct patterns, sequence from the smallest to the biggest, to know where each pattern lives in the sequence. Pattern Boards In terms of cognitive growth these experiences can be seen in the following: • Hand-eye coordination improves as children practice over and over again to fit blocks into matching grooves. • The ability to scan develops as children search for one block among many scattered blocks. • The ability to judge sizes is developed when children constantly compare blocks with empty grooves to find a matching combination. • Left/right directionality and one-to-one correspondence is practised as children fit cubes into the empty pattern boards. • Spatial awareness is increased. KIT A After the ‘puzzle’ stage of experiments, level 2 broadens and builds further understanding. Children are keen to talk about what has been discovered so will learn the name of each Block and Pattern Board; learn to count; know which block is meant when described with the words ‘one bigger’ ‘one smaller’ ‘after’ ‘before’ ‘between’ and ‘equal to’; will add ‘0’ to any number and know that it results in the same number; zero – will know that 10 and nothing makes 10; will add 1 and see that it is the next higher number; add 2 to an even number and learn that it gives the next higher even number; add 2 to an odd number gives the next highest odd number. When subtracting 1 from any number, the apparatus provides unforgettable imagery showing that this results in the next lower number and subtracting 2 from and even/odd number results in the next lower even/odd number. At this stage in the 10-box, with the number names in place, children can now name the familiar combinations of blocks that go together to fill the box and express them orally for example: 8 and 2 makes 10, 10 and nothing makes 10.
Vol. 29 No. 1 Spring 2024 15 They will become aware that the two like pairs of blocks can be placed in any order and it makes no difference to the sum. “8 and 2 makes 10, 2 and 8 makes 10.” Here also, they will begin to understand the concept and language of the ‘missing’ addend, “8 and what makes 10?” Hide two blocks behind your back and say “I have 10 altogether. In one hand I have 8, (show it) what is in the other hand?” Pupils begin to see the relationship between addition and subtraction that of ‘doing’ and ‘undoing’. When your child gets to be the ‘teacher’ this is a wonderful way of assessing how many facts are known. With this knowledge in place it is time to move to level 3 which introduces the numerals 1 to 10 and links them to the named number blocks and pattern boards. Equation work begins. To do this the plus sign and equal sign are introduced. With the pattern boards children are able to act out a subtraction word problem; able to read and understand an equation and use the plus or minus sign in an equation. Back to the 10-box in level 3 children begin to record from memory addition facts with the sum of 10 and record the related subtraction facts. Using the wooden number markers, children will record an equation from hearing an addition or a subtraction story. Number Bonds to 20 The above levels give the reader an outline of how pupils will develop their numerical and operational understanding with numbers to 10. Once this foundation is in place, the bonds to 20 are tackled using the 20-tray. This is a marvellous piece of apparatus, specifically designed to show relational understanding of what was learned in the 10-box and the combinations to 10. With the size of this piece of apparatus, children can clearly see that the same ordered blocks 1 to 10, are now sitting on top of a base of ten 10-blocks. They ‘see’ that the same facts to 10 hold true in the structure of the teen numbers.
Vol. 29 No. 1 Spring 2024 16 Transferring Basic Facts to Higher Decades Learning about place-value using the Dual board really makes this concept visible and enables the transfer of basic facts to higher decades up to 100. It is a wonderful way to show how working with ‘ones’ or ‘units’ transfer to working with ‘tens’. For example, 3 ones and 3 ones make 6 ones, to 3 tens and 3 tens make 6 tens. It is also an excellent way to demonstrate the concept of regrouping; adding two numbers that make more than 10. By filling the ones column with 10 single cubes it is easy to see that these 10 cubes are now as big as one 10- block. This is then exchanged for one whole ten and is moved into the tens compartment. Dual Board The number track provides opportunities to see the same topic, taught in the dual board, in a different light, thus aiding the transfer of conceptual understanding. This apparatus is part of the Stern Kit B, and this level of teaching can be found in Teachers Manual Book 3. KIT B Vikki has actively contributed to her daughter’s development, especially for numeracy. She is passionate about helping children develop maths skills which include learning to tell the time and handling money. Further teaching includes multiplication and division, long division, the structure of numbers to 1,000,000; rounding numbers, fractions, decimal notation, ratio and proportion, the use of percentages and problem solving with these concepts. Books 4 and 5 (Availability 2007). Vikki currently advises and provides training using Stern’s Structural Arithmetic. She can be contacted on: 01747 861 503 vikki.horner@ mathsextra.com
Vol. 29 No. 1 Spring 2024 17 Supporting students with Dyscalculia in the maths classroom Louise Langford has kindly shared her ten top tips for helping Dyscalculic students develop their mathematical thinking. Please let us know yours and which of the ideas below you use and why? The aim is to collate and then publish an extensive list of strategies for use in supporting Dyscalculic students. I happened to be collaborating on something with Louise Langford (Dyscalculia Assessor and Teacher) and in the process, she kindly shared some tips she uses to help teachers support dyscalculic students. I am including her ideas below in the hope that you can add to the list, and we can begin to create a toolkit of approaches to supporting such students in the classroom. 1. Use concrete resources to introduce all new learning, modelling, and encouraging visual representation, as well as being aware that such pupils will need longer to connect to abstract symbols (Concrete, Pictorial and Abstract approach-CPA). 2. Specifically develop pictorial representations to support the expression of problems visually, rather than in purely symbolic form. 3. Build mathematical vocabulary, by explicitly teaching the language and symbols of mathematics alongside the CPA approach (connections/connected approach), creating linked key word and symbol cards or a maths wall for them to refer to. 4. Recognise that these pupils may need to be given the ‘tools to communicate’ and develop their mathematical thinking, alongside scaffolded language. 5. Develop and promote meta-cognition by allowing them to ‘think aloud’ when solving problems, building on their meta-cognitive strategies, evaluating their methods, and encouraging them to prove their answer in another way. 6. Model how you think and encourage students to jot down their thinking to keep track when dealing with two, or more, step problems. 7. Help them to estimate what their answer is likely to be and to look at their answer once they have completed the calculation to see if it is reasonable. 8. Teach and develop strategies to recall number facts, using what you know to work out what you don’t know i.e. generalising. Representations of ten are a key part of this and provide an effective way to transfer knowledge.
Vol. 29 No. 1 Spring 2024 18 9. Times table grids are helpful when teaching a strategy but input still needs to be given to teach these facts and develop strategies for recall. 10. Use games to make practice and consolidation of key number facts and concepts enjoyable and purposeful. …The use of worked examples, verbal/visual instructions and flow charts/diagrams are other strategies that will support students struggling in maths. The catastrophic journey from SATs into NCTs Mark Pepper provides us with a history lesson looking at assessment from the 90’s to the present day. There have been hugely significant changes to the content of the statutory maths tests taken at the conclusion of KS2 in the course of their journey from their inception in 1991 to the present time. In this period the title of the tests have changed from Standard Assessment Tests (SATS) to National Tests (NTs) to National Curriculum Tests (NCTs). An incidental curiosity regarding the changes to the title of the tests consists of the fact that despite the acronym SATS becoming obsolete nearly two decades ago, the term is still in widespread use amongst teachers, parents, pupils and even some educational commentators. Furthermore the title National Curriculum Tests is prominently displayed as a heading on all of the statutory test papers. The misuse of the term SATs is reinforced by the media who frequently use it in headlines of articles concerning education. A headline in The Guardian dated 12 May 2023 read: Headteachers express concern over SATs amid claims a paper left “pupils in tears.” It is also used by some official bodies including the Government. An example of this is contained in a House of Commons briefing paper dated 26th February 2016 entitled The school curriculum and SATs in England Reforms since 2010. The term SATs is also regularly used by companies that supply educational resources. A recent on-line post stated: 2023 SATs now available to download. It may be felt that to suggest that the current title of the tests, National Curriculum Tests (NCTs), should be used is being pedantic as everyone knows what the acronym SATs mean so there is no need to stop using it. If all of the acronyms SATs (in both representations of tasks and tests),NTs and NCTs were synonyms that could be used interchangeably then this could be considered a valid point. The difficulty with this interpretation is,
Vol. 29 No. 1 Spring 2024 19 however, that they are not synonymous. A brief history of the statutory tests will demonstrate that the content of the tests are very different. A brief history of the statutory maths tests at the conclusion of KS2 In 1987, Paul Black (Kings College London) and the Task Group of Assessment (TGAT) commenced on the task of producing tests directly related to the content of the National Curriculum (NC). They were entitled Standard Assessment Tasks (SATs) and they were first introduced for 7 year olds in 1991. As the name implies they were physical tasks and not written tests. Shortly after this there was a significant change as although the acronym SATs was retained this now stood for Standard Assessment Tests. Hence the whole nature of the assessment changed from the requirement to complete physical tasks to one of taking written exams. Shortly after this the Government was obliged to drop the acronym SATs as this was already in force in America where it represented Scholastic Assessment Tests. As a consequence of this the tests were then renamed National Tests (NTs). It should be stressed that at the point of the introduction of this change it simply involved a change of title with the content of the tests remaining unchanged. The title was amended again in 2016 when the tests were renamed National Curriculum Tests (NCTs). This time there were significant changes to the content of these tests. These changes will be considered later. The purpose of the statutory tests has been to assess the attainment in maths of the pupils at the conclusion of KS2. Hence there has consistently been a direct link between the content of the tests and the changing content of the various iterations of the maths National Curriculum of 1989, 1991, 2005 and 2013. In the 1989 and 1991 versions, there weren’t any dramatic differences in the content related to number facts and maths skills. There was a change of format in the 1991 version as the number of Attainment Targets (ATs) were reduced from 14 to 5 but this was achieved by condensing similar content into an abridged format. The only significant change in the 1991 and 1995 versions involved the Using and Applying AT. The crucial significance of the Using and Applying AT The Using and Applying AT was unique in that, in contrast to all of the other Ats, it encouraged a learning environment in which knowledge of number facts and maths skills could be applied to practical tasks and problems encountered in day-to-day life. Hence the Using and Applying AT made it a statutory requirement for teachers to produce a balanced curriculum designed to develop problem solving skills, creativity and increased levels of autonomy. How the Using and Applying AT was amended in the various versions of the National Curriculum The original maths NC (1989) consisted of 14 ATs. Two of these were devoted to Using and Applying mathematics. AT 1 has a sub-heading: Pupils should use number, algebra and measures in practical tasks, in real-life problems, and to investigate within mathematics itself. AT 9 provided identical instructions within the context of shape and space and of handling data.
Vol. 29 No. 1 Spring 2024 20 The truncated 1991 NC consisted of a single generic Using and Applying AT, named AT 1, which specified that: Pupils should choose and make use of knowledge, skills and understanding outlined in the programmes of study in practical tasks, in real-life problems and to investigate within mathematics itself. Pupils would be expected to use with confidence the appropriate mathematical content specified in the programmes of study relating to the other attainment targets. Using and Applying continued to be a prominent component of the 1995 NC- though it appeared in a much changed format. The Using and Applying element was not included as an individual entity but was subsumed into the other Programmes of Study. The 2013 NC omitted any reference to Using and Applying! This drastically transformed the balance of the mathematics NC such that it now became entirely devoted to the rote learning of facts and the application of taught algorithms. It can be concluded that the level of prominence of Using and Applying has consisted of a downward spiral culminating in its total demise in 2013. The National Curriculum 2013 In addition to the removal of any requirement for the use of an Using and Applying element, there were further number facts to be memorised. The multiplication tables to be memorised were increased from × 10 to × 12. Additionally there was a requirement for specified facts regarding Roman Numerals to be memorised. The crucial significance of the change of name from National Tests to National Curriculum Tests in 2016 The change of the name of the statutory maths tests at KS2 from NTs to NCTs had huge significance as it made clear that, unlike the previous tests, these would be directly related to the 2013 version of the NC that was then in force. When the first NCTs were introduced in 2016 the content was wholly based on the content of the 2013 NC. The most devastating aspect of this was that for the first time the statutory test omitted any reference to Using and Applying. National Curriculum Tests 2016 As there has consistently been a high correlation between the content of the statutory tests and the contemporary National Curriculum, it is not surprising that the NCTs provided no opportunities for the application of Using and Applying. This changed the whole balance of the assessments from a blend of learning number facts, developing maths skills with encouragement for the development of creativity and increased levels of autonomy, to one entirely devoted to rote learning in the form of memorised number facts and the application of taught algorithms. NCTs introduced a major restructuring of the format of the statutory maths tests The NCTs introduced major changes to the structure of the tests which included the abolition of an audio mental maths test, the withdrawal of permission to use a calculator for any of the exam papers and an increase in the number of papers from 2 to 3. One of these was entitled
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