Very special maths : developing thinking and maths skills for pupils with severe or complex learning difficulties

Cover
Half Title
Title Page
Copyright Page
Table of contents
Acknowledgements
About the author
Part 1 About a special curriculum
	1 What this book is about
		Teaching very special mathematics and thinking
		A time of change
	2 About curriculum attitudes and mindsets
		About curriculum attitudes
		Mindsets
			Our mindset about maths
				Is maths an abstract subject?
			Our mindset about pupils
			The mindset of this book
		Note
	3 The place of maths in a very special curriculum
		Why mathematics is important to special pupils
			Mathematics occurs in all of these contexts:
			They all require:
		Life’s maths
			Schools are developing various curriculum models
			Special aspects of a very special mathematics curriculum
				Core learning
			Using the tools of learning
			Learning to learn
			Processes of learning
			Developing thinking
			Developing counting and the ‘big ideas’
		Notes
	4 Sensory beginnings
		Where does mathematical learning begin?
			What important mathematicians have said
		There are some mathematical senses
		Core knowledge
		The roots of a very special curriculum
		Learning to learn skills
			They are skills that all pupils require
		The roots are part of the whole curriculum
		Growing beyond sensory roots
			Personal and social mathematics
		Personal and social maths come together through communication
		Notes
	5 Introducing the parts of learning
		Introducing tools and processes of learning
			The tools children use for learning
			The processes children use to learn
			Practical experience is mathematical content
			The roots of learning mathematics in exploration and play
Part 2 Tools for learning
	6 Introducing the tools for learning
		Learning to learn
			Learning to learn develops through real experiences
			About the tools and mathematical learning
			When there are difficulties with the tools
				Motor and manipulation
				Senses and perception
				Attention difficulties
				Memory
				Communication
			Some examples of special children and using tools for learning
				Janine
				Clyde
				Jack
		Note
	7 Physical skills at the beginning of thinking
		The beginning of exploration is the beginning of thinking
		Fine motor manipulation
			Finding out – heuristic play
				Direct patterning
				Sharing looking and exploration – learning together
				Some of the fine motor activities that we can work into our shared activities
		Gross motor activity
			Movement and our sense of space
		Notes
	8 Schema – first patterns of thinking
		Major schema
		Schema and mathematical thinking
		Schema are in all our actions
			Schema work at different levels of development
			Schema are revisited as children’s abilities mature
			Schema can help our observation and planning
			Awareness of schema is a useful lens for both seeing progress and planning
			We might plan
			Or we might observe progress
			A good picture of progress requires:
		Notes
	9 Introducing the senses
		The senses
		Sensory integration
			When there are sensory difficulties
				External and internal senses
		Notes
	10 About vision
		The range of visual impairment
			Learning from social observation
			Visual memory of the environment
		Working with visual impairment
			Alternative sensory stimuli
			Guiding thinking through commentary
		Pupils with profound disabilities
			Working at different levels
				Exploration in ‘face space’
				Exploration in body space
				Extending into social space
		Neural visual difficulties
			Perceptual difficulties may be difficult to pin down
				Learning to share interest
		Notes
	11 Hearing
		The range of hearing difficulties
			Physical hearing loss
			Neural hearing difficulties
			Difficulties that arise from hearing loss
			Some thoughts about teaching related to hearing difficulties
			A whole view of sound – not just the ears
		Working with hearing impairment
			Communication
			Working on ‘multisensory hearing’
				Imitation – movement and touch
			Some ideas for promoting multisensory ‘hearing’
		Notes
	12 Touch and movement
		Haptic touch – tactile and spatial awareness
			The whole body
			A fundamental sense for mathematics
		When there are difficulties
			Some very special issues
				Working with children with profound learning difficulties
		The power of the teacher’s touch
			About providing the experience of touch and movement
		Working with children’s hands
			Hand under hand support
		Notes
	13 Attention
		The tool that makes sense of senses
		Seeking information
		Filtering and being selective
			Selective attention
		Sharing attention
			Learning from others
		When there are difficulties
			Sensory coordination
			Developing interaction skills
		Notes
	14 Perception
		Making sense of sensation
		Perception and mathematical learning
			Core knowledge
			Building on core knowledge
			Perceptual tendencies
			But these tendencies can also cause disturbances
			Developing multisensory ‘looking’
				Teaching opportunities
		Notes
Part 3 Processes of learning
	15 Introducing the processes of learning
		Developing exploration and refinement
		A cycle of learning through exploration
			The cycle and its connection to curriculum organisation
			Different levels of playing and learning
		When there are difficulties in learning to play
		Notes
	16 Learning to play
		From sensory exploration to social play
		Good contexts for scaffolding learning
			Learning from reality
			Learning in games
				Devising games
				Levels of participation and learning
			Potential in technology
		About modelling
			The importance of mutual interactions
				Germinating play is not a process of instruction – it requires participation
				Mirroring
			The importance of our playfulness
		Notes
	17 There are many ways of playing
		Phases of development from roots of play
			All phases are important to very special pupils
		Phases of play
			Attunement play
				Serve and return
				Serve and return – mutual experiences that promote attunement
			Reflexes
				Physical problems
				Neural problems
			Sensory play
				Sensory play is practical
			Solitary play
				Sliding in mirroring and modelling
				Solitary play also has lifelong value
		Becoming more social
			Onlooker play
				With special children
			Parallel play
				With special children
			Associative play
				Barriers for special children
			Cooperative play
				Creative elements
				Imaginative play – special children and play partners
		There are many ways of playing
		Notes
Part 4 Thinking about thinking
	18 About the development of thinking
		From biological drives to communication and abstract thinking
		Sensory and physical actions are thinking processes
			Multisensory mental images
			Schema again
		Phases of thinking
			Understanding is a process of making connections between different kinds of thought
		Notes
	19 Thinking with objects and fingers
		Enactive thinking with objects and actions
			Manipulatives – using things to think with
				Commercial manipulatives
				Virtual manipulatives and apps
					Starting to game
				Apps for number activities
		A special place for fingers in mathematical thinking
			Fingers and their connection to core knowledge
			The perspective on fingers and counting
			About fingers and special children
		Notes
	20 Visualisation – using all senses
		Mental images
		Developing visualisation
			The very beginning
			Peek-a-boo, memory and anticipation
			Object permanence
		Visualisation is a form of thinking
		Visualisation and number
		Notes
	21 Thinking using marks and graphic representations
		Understanding representation role of pictures
		Making marks
			Making marks, making meaning
				Sensory mark making
				Schematic level
				Alternatives to drawing
				Pictorial level
				Narrative level
			Using tokens and tally marks
				As icons to represent groups
			Dice, playing cards and spot cards
		Notes
	22 Using numerals – a medium for abstract thinking
		Having the sense of size
		Ideas that are needed
		About numerals and numbers
			Are numerals numbers?
		When children have very special needs
			Sensory learners
			Concrete learners
				Concrete learners with some graphic awareness
			Pupils with numeric performance
		Possible confusions with numbers and numerals
		Perceptual difficulties
		Writing numerals
		Notes
	23 Language, thinking and memory
		The way we use language matters
		Creating problems to think about, talk about and solve
		Thinking aloud
			Giving ourselves instructions
		Enhancing memory skills
			Rehearsal and repetition
			Recall
		Thinking about thinking
		Using a narrative is a thinking process
		Using ‘narrative’ to provide context and structure for special children
			The value of using narratives to teach begins before numerate levels
		The narratives of practical activities
			Recording narratives
		Notes
Part 5 Developing mathematical ideas
	24 Introducing aspects of mathematical thinking
		There are some very fundamental elements
		Developing mathematical thinking
			Sensory learning – about objects and space
				Fundamental awareness of objects
				Awareness of groups
				Understanding sequences
				Understanding sizes
			The beginnings of numeric learning
				Number sense – perceptions of quantity
					Subitising
				Fundamental ideas about quantity
				Counting
			Some ‘big ideas’
		Notes
	25 Some sensory beginnings of number
		Awareness of objects, groups and sequences
			Object awareness
			Object permanence
				The natural development of object permanence
			Awareness of movement, spatial arrangement and groups
				Movement and spatial distribution
				Perception of groups
		Notes
	26 Comprehending space, shape and measures
		Understanding the physical world
		Shape and space – the geometry of life
			Sensory levels
			Concrete levels
			Abstract thinkers
		Measures for living
			Sensory levels
			Concrete levels
			Abstract thinkers
		Connections with number
		Note
	27 Learning about size and comparisons
		Making comparisons
			Thinking about physical comparisons
				At sensory levels
				Concrete and social levels
				Beginning to use practical reasoning
			Processes of observation
			Processes of discrimination – pairing, matching, sorting and ordering
		The language of size and comparisons
			Absorbing general words about size
				Relating the language of size to groups
				Beginning to think about comparisons
		But mathematical language can be confusing
		Notes
	28 About number sense
		Seeing number is an essential skill of life
			A human adaptation of a general biological skill
				Number senses and special children
		First perceptions
		Neuroscience has observed number sense
		What is our intuitive number sense like?
		We need a bigger picture
		We have two systems of number sense
		Levels of number sense
			From perceptual beginnings to developing verbal and symbolic coding
		Notes
	29 An exact number sense for small quantities
		Noticing numerosity
			Dyscalculia
			Exercising number sense
		Noticing differences and reacting to changes
			Noticing differences
				Seeking and finding
				Expressing their understanding
		Intuitive number sense is multisensory
			Sound and rhythm
			Movement and touch
		Spatial perception and groups
		Fingers and number sense
			Fingers are our natural link between number sense and representation
				The origins of representation
				Neurological links
				Fingers – a link from intuitive to conceptual number sense
		Notes
	30 An approximate number sense
		Larger quantities – approximate number sense
			A sense of proportions
			Approximation
				The distance effect
			Why approximating and comparing quantity is important to developing number
				Flexible thinking
			Improving approximate number sense
			Ideas about maths develop even before numbers
			Some experiences for exercising approximation and comparisons
		Notes
	31 Understanding comparative value – including exchange and money
		Pre-monetary starting points
			Awareness of exchange
			Awareness of reward and payment
			Paying to receive
			Awareness of value
		Thinking about money
			Realistic use of money
				Comparing coinage
				Larger values
		Notes
	32 Number is like space
		The ideas of number and space are linked
		The number line
			Can you see it?
				Developing ideas about lines
			The connection to fingers
		Pattern seeking and number naming
			Practicing group perception
				The interplay of perception and language
			Everyday experience
			When the number gets bigger
		Notes
	33 Subitising – connecting perception with number
		Stepping from intuition to ideas
			Subitising – immediate naming of number
			Subitising starts before counting
			Early ideas of comparison stem from subitising
				The cardinal number is the number that represents the size of the group
			Subitising plays a role in developing counting
		The place of subitising in the curriculum
		The place of subitising in life
			Don’t miss opportunities
		Levels of subitising
			Perceptual subitising
			Conceptual subitising
				When numbers get a little larger
				Conceptual subitising and special children
					About teaching conceptual subitising
			Symbolic subitising
				The meaning of numerals
				Connecting images and symbols
		To summarise
		Notes
	34 Counting
		Is counting a simple skill?
			Coordinating actions
			Understanding language
			Awareness of the purposes of counting
		Five counting principles
			Principles related to how to count
			Principles related to meaningfulness
			The principles help our teaching
		Teaching the parts of counting
			Principles related to how to count
				The one-to-one principle
					Problems and teaching related to the one-to-one principle
					Focusing on one-to-one itemisation leads to enumeration
					Some ideas about teaching related to the one-to-one principle
			The stable order principle
				Numbers are sequential names used in fixed order
					Problems and teaching related to the stable order principle
						Working memory
						The familiar song of counting
					Some ideas about working together on the stable order principle
		Principles related to the meaningfulness of counting
			The cardinal principle
				Cardinal numbers are used at an earlier level than counting
				Cardinality – connecting naming and awareness of value
				Confusion about purpose
				Hierarchical inclusion
				Counting for cardinality
				Some ideas about teaching related to the cardinal principle
					Counting to find out
			The abstraction principle
				Understanding that counting can be applied to any collection – real or imagined
				Some ideas about teaching related to the abstraction principle
			The order irrelevance principle
				Some ideas about teaching related to the order irrelevance principle
		Notes
	35 Calculation and big ideas
		Informal calculations
			Ideas about changes
		Some more big ideas
			Some big ideas that are important in the development of practical or numeric maths for special children
			1. Understanding there are things
			2. Understanding there are numbers
			3. Understanding there are patterns of change
			4. Understanding there are regular rules
		Addition and subtraction
			Foundations of ideas
			Practical foundations of addition
				Linear addition
				Change by joining
				Readiness for numeric addition
				Phases of learning the process of addition
				Hands and calculation
		Foundations of subtraction
			Forms of subtraction
				Taking away
				Other, more complex forms
				Comparisons
					The inverse of addition
					Complement of a set
		A note about multiplication and division – fractions
		Notes
Index