Very Special Pupils
Albert Einstein said:
"Since the mathematicians have invaded the theory of relativity,
I do not understand it myself any more."
Teachers working with pupils with severe or profound learning difficulties
have a very positive track record of curriculum development. From the birth
of the National Curriculum they were determined to ensure their
pupils had access. Drawing on their knowledge of language development
they have produced well-documented materials and their work has
enabled their pupils to access to the English curriculum appropriately,
to develop their skills and appreciation of communication. This
work has facilitated suitable approaches to implementation of the
literacy hour for very special pupils.
Following the introduction of the Numeracy hour schools for very
special pupils have taken the National Numeracy Strategy framework
and adapted it to meet the needs of very special pupils.
This article discusses the nature of fundamental mathematics and its
importance to pupils with severe or profound learning difficulties,
and suggests appropriate approaches to enhancing the National curriculum
to meet their needs.
These processes are entirely in accord with the QCA "Guidelines for planning,
teaching and assessing the curriculum for Pupils with learning difficulties",
as published in 2001.
Should we suffer maths anxiety?
For many people mathematics has an aura of mystery about
it, and there is a tendency to think of it in terms of an academic subject
that involves abstract thinking and complex calculations. Looked at in
such light its relevance to pupils at early stages of development seems
marginal. In practical reality we use mathematics throughout our every
day lives. Mathematics is about space and quantity, we use it to understand
and influence the world around us. Its basic concepts are so natural to
us they are part of the thinking we do even before we speak.
The part that mathematical thinking plays in our subconscious reasoning is very
important to our conscious functions and appreciation of life. Whilst
we enjoy conversation, and hearing about our world reflected through the
wonder of language, in stories poetry and song, and recognise that these
are part of the "English" curriculum. We also understand and
enjoy our world through appreciation of space, shape and form, patterns
and rhythms, but we are less inclined to recognise that these are "Mathematics".
Learning through a feeling for space and time.
It is interesting to note that Einstein, one of the greatest mathematical
minds, described his creative mathematical thinking as,
"Initially involving visual and muscular processes"
He said that words and other signs that could be used to try to describe
his feelings only came into use after the initial "associative
play." What better witness could we have to testify that mathematical
understanding has physical and sensory origins, even wider and more basic
to our life functions than knowledge of "numeracy".
Understanding starts early
In articles in Special Children Carol Aubery described
the rich background of mathematical knowledge children develop by the
time they start school.
Children as young as two develop intuitive sense of addition and subtraction.
Before they start school many children have begun to construct their mathematical
thinking, and have their own personal strategies for counting, practical
addition etc. They are able to use this knowledge in specific practical
situations, which they are familiar with, but are not always able to generalise
their knowledge to other settings. For example they may recognise number
13 if it is the number of their house, but not relate the number to a
quantity. (Vygotsky 1978). So even before formal teaching very young children
are actively learning the basis of mathematics from experience. They are
building up a fund of knowledge, when they go to school the teacher’s
job is to build on the fund, to help them to link their personal knowledge
to abstracted ideas and the formalised systems which everyone else uses.
Special children who are developmentally different may have barriers to
learning, and not be able to use their senses and social interactions
to build up this fund. When they arrive at school they may not be ready
to embark upon learning the numerical and computational aspects of mathematics
upon which the framework of the Numeracy hour is likely to focus. Nevertheless
mathematics is important and useful to them in many ways.
Mathematics is an essential part of communication
Communication requires that we are able to comprehend and express the
nature and order of things. English and maths use similar processes and
complement each other in practical communication. For example "Joint
action" and "turn taking" have patterns which facilitate
prediction. Whilst understanding "object permanence", is needed
to comprehend quantities and changes. Bringing items together, separating,
sharing etc are all experiences that relate to quantities or events. Set
in real life contexts they are situations that motivate discussion, create
curiosity, stimulate predictions, or consideration of probability, they
may promote agreement or disagreement. – In such situations communication
is promoted alongside the development of mathematical concepts.
Maths is a powerful tool for practical activity, and for learning more
Mathematical concepts are basic to skills required for
many practical purposes. For example how could we lead our everyday lives
without understanding "one to one" relations, or appreciating
increase and decrease? Skills of counting and arithmetic may develop from
such concepts but even those skills are not ends in themselves, they are
essential tools not only for daily living but also further learning.
Maths helps us appreciate relationships
The language of mathematics describes space, quantities
and events, with it we can compare things and describe changes. Moreover
the various parts of maths are themselves interrelated and its language
describes relations. For example sorting is related to discrimination;
volume to size; subtraction to addition; algebraic pattern to multiplication.
In practice no one idea stands alone and learning one set of ideas will
also help us understand related concepts. For example observing a pattern
of multiplication will also help us understand division.
Maths helps us be systematic
Using its structures and language we can record and bring
order to our observations. We can put experiences into memorable form.
Looking for and describing patterns we can move forward from trial and
error, and improve our powers of estimation and prediction. Developing
systematic approaches helps make learning clear, and we can remember things
better the next time we encounter a similar situation. This applies as
much to concepts of quantity,order or space and time, as it does to remembering
number bonds and tables.
Mathematics is a tool of the imagination
The structures of mathematics use systematic rules but
they are not necessarily restrictive, rules can help put imaginative responses
into order. Learning benefits from a two-way relationship between systematic
thinking and imagination. When mathematical concepts are brought to bear
on an experience new knowledge may dawn. This may be through gradual development
or sudden revelation. It is true for children or for great thinkers. Archimedes
struggled with how to measure the volume of an irregular object until
he got in the bath. Newton was hit on the head by inspiration.
Maths is fascinating
Even though many people are anxious about mathematical
language and processes they are often fascinated by patterns, changes
and comparisons, they are also interested in outcomes and predicting them.
These fascinations are powerful tools to motivate communication and learning.
- Important to communication.
- Important in our practical lives.
- Important in helping us understand relationships.
- Important because it helps us be systematic.
- Important because it helps us harness our imaginations.
- Important because it fascinates us.
Mathematics is in the course of life
Mathematics is so beguiling that it even creeps into our
enjoyment. Mathematical patterns and emotion are connected. Play rock
and roll, or play a waltz, and the counted rhythms are quite different,
but each of the patterns have undeniable charm in their power to move
us. Across the catalogue of music there is an amazing range of variations
on this theme, and the connections between music, movement, emotion, and
thinking suggest that those patterns have important influences on our
physical and mental development. Professor Ian Stewart described the connections
between, mathematics, bodily rhythms and movement in the Royal Institute
Christmas Lecture in 1997. He also related geometry to our responses to
The influence of pattern is wider than music, speak in nursery rhyme or
through a Shakespearean stanza and the magic of pattern will be there.
Even changes of intonation in music or voice have mathematical dimensions,
and these are important in even the earliest of children's communications.
As Ian Stewart emphasised,
"The mathematical mind is rooted in the human visual tactile and
motor systems. Counting is based on touch and movement, geometry is visual".
Let my hand trace a sphere, or grasp a banana I will sense differences,
my sense of enquiry will be roused in contrasting ways, I will want to
tell you different things. We are constantly exposed to experiences, both
of practical necessity and of pleasure which have mathematical aspects,
and which stimulate our natural desires to communicate and know more.
Such experiences provide opportunities to expand children’s knowledge,
and these areas of connection where mathematics and early cognitive development
are intertwined are important for children who are at early stages of
However the Programmes of Study for Mathematics in the National
Curriculum assume that before they start school children have developed
many skills that support mathematical understanding, and therefore they
skim over the relevance of underlying experiences upon which mathematics
is built. They assume that before children start school they have already
developed the basic intuitive understandings, which are the basis of mathematical
Consequently the Programmes of Study make little reference to
early cognitive activities that are the basis of everyone’s mathematical
knowledge. In these respects they are not detailed enough to help us recognise
or organise mathematical learning for pupils with very special needs and
they need to be enhanced. Similarly the framework of the National Numeracy
Strategy needs to be adapted to relate to patterns of learning suitable
for such students.
Enhancing the curriculum
Taking account of patterns of learning
Learning is driven by curiosity it begins as children explore with their
senses. They realise significant things about their observations and develop
ideas through which they begin to understand about things and relationships.
They respond, communicating both about the ideas they have developed and
the questions that their ideas and curiosity raise. The processes of these
stages spiral forward like waves. There is repeated feedback and progression,
as personal exploration results in gaining knowledge, and developing ideas,
which encourage responses and so promote communication. Learning thus
progresses from self-centred exploration towards social communication,
and with feedback progresses again in a smooth flow.
Personal and Social Maths Lead Towards Numeracy
The processes I have described make up the earliest stages
of our pupils’ mathematics.
- Our teaching should begin with their personal exploration, using
their senses to begin to explore the world about them and developing
their perceptual and attention skills to help them appreciate and extend
their understanding and order their knowledge - we might describe these
processes as personal exploration of space, shape, time, quantity etcetera
--- personal maths.
- As children develop people contrbute commentary
to the childs experiences and their own observations encourage them to
communicate, to share, request and question. We might describe the processes
that occur during communication, which relate to understanding and describing
quantities, space, time and changes as social maths.
In practical reality personal and social spects of learning
are often intertwined.
- Once the children have begun grasp ideas which
are the personal and social background to maths they can begin
realise how to use the ideas of maths for practical purposes. It is then
that they are ready to appreciate and learn the beginning
skills of numeracy, first learning to itemise groups, name quantities
and numbers then to count and describe numeric events
Personal and social mathematics lead towards numeracy
- Personal Maths - Children's
self-centred exploration, beginning with body awareness and consciousness
of self, moving towards awareness of self and
others and the environment.
- Social Maths - Communicating and
learning about self others and the environment.
- Numeracy - Learning about numbers, counting
Processes of interactive teaching and learning
This sequence of learning points us towards using processes
of interactive teaching and learning as means of offering our special
pupils access to the National Curriculum. Through interactive teaching
bulding upon sensory exploration, developing perceptual and attention
skills we can encourage biological, cognitive and social experiences that
promote bedrock mathematical understanding.
Our approach should take advantage
of cross-curricular opportunities, but mathematical detail does need to
be described if it is to adequately guide our teaching, and provide for
As we delve more deeply into what is "Personal and Social Maths"
it will be evident that much of what will be described is already present
in good practice working with very special pupils. However it is important
to outline detail of activities and processes because we need to understand
how the jig-saw pieces of our teaching fit together.
A wealth of "Personal
maths" occurs in sensory sessions, and "Social maths"
occurs whilst communication is being promoted. Knowledge of detail will
equip us to exploit and emphasise the mathematical aspects of those experiences.
It will also make us aware of pupils’ progress and more able to
credit their achievements.
|The next article entitled "Magnifying
Maths" will relate theory to practice and offer some content
for the structure described in this article.
It will consider:-
Personal and Social Maths For Very Special Pupils
The development of mathematical ideas:-
- Through personal exploration.
- In parallel with perceptual and thinking skills
- Alongside communication skills
- Through social development.
- In cross curricular contexts.
Beginning Numeracy For Special Pupils
- Developing the language of counting.
- Relating quantities and numbers.
- Learning the principles of counting – what counting means.
- The development of counting and activites with numbers.
- AUBERY, C. (1998) Mathematics and the SENCO Special Children 108. Feb 1998
- EINSTEIN, A. In A. Somerfelt "To Albert Einstein’s
Seventieth Birthday" in Paul A Schlipp (ed) "Albert Einstein,
Philosopher – Scientist" Evanston 1949.
- EINSTEIN, A. Letter to J Hadamard.
In "The Psychology of Invention in the Mathematical Field". J Hadamard. Princeton
University Press Princeton.
- STEWART, I. (1997) Life really is a tuneful little number. The Sunday Times 28.12.1997.
- STEWART, I. (1997) The Magical Maze - The Natural
world and the Mathematical Mind. Royal Institute Christmas Lecture BBC
Videos for Education and Training. London.
- VIGOTSKY, LS. (1978) Mind in Society.
The Development of Higher Psychological Processes. Translated by M Cole. Cambridge
MA Harvard University Press.
- This article by Les Staves was first published in Issue 117 of
under the title "Painting By Numbers".
Retired as the head teacher of Turnshaws Special School in Kirklees following
an outstanding Ofsted report.
He has thirty years teaching experience in mainstream and special education.
He now works as a freelance trainer and consultant.