U C E T
Universities Council for the Education of Teachers
Chair
Professor Michael Totterdell, BA, PGCE, MA, MID, FInstAM, FRSA
Executive Director
James Rogers, MA, FRSA
Honorary Life-President
Mary J Russell, MBE, DEd, DLitt
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REVIEW OF MATHEMATICS TEACHING IN PRIMARY SCHOOLS AND EARLY
YEARS SETTINGS
Response from UCET to Williams Review
1.What is the most effective pedagogy of maths teaching in primary schools and
early years settings? That consideration should include instructional
methodologies, teaching and learning strategies, and lesson designs that are
most effective in helping children to progress in their learning.
1. It is highly unlikely that the most effective pedagogy can be identified for such a
wide range of age, ability and learning style. It is much more probable that there
are a number of effective pedagogies and international and other comparisons
will surely reinforce the view that there is no single generic “magic bullet” that is
the teacher’s panacea. Besides, whenever we ask the question about the most
effective pedagogy, the complexity of the context means that the most valid
answer is always It depends: helping children to progress in their learning
requires that teachers and schools pay informed attention to the particular
circumstances and characteristics of individual learners.
2. That said, there are undoubtedly some approaches to teaching particular
mathematical ideas that seem overall to be more effective that others, and can be
recommended, and some broad kinds of activities that, other things being equal,
do generally work better than others. The Primary Strategy is a kind of repository
for many of these, and its continued efforts to isolate, represent, and disseminate
helpful practices is of considerable value to primary mathematics. If primary
mathematics teaching has not improved as much as might be hoped, it is not
because the Strategy is promoting the wrong teaching ideas; indeed, teaching
has been shaped (by the Strategy) to have the same characteristics as effective
teaching, the same instructional methodologies, strategies and lesson designs,
but these have been adopted superficially, without the necessary sensitivity to the
specific circumstances of individual learners. What is needed for further
improvement is not a new set of instructional methodologies and so on, but an
examination of teachers’ understandings, their beliefs and purposes, and how
these relate to the mathematical exchanges that occur in classrooms through
which mathematics is learned. An important element is clear recognition of the
significance of pedagogic knowledge and ability to apply knowledge of the subject
through this to actually teaching children.
3. Beyond that, there are three principles that we consider should govern
mathematics teaching in primary schools and early years settings. Firstly, there
is evidence that struggling learners are particularly supported by constructivist
approaches as these pedagogies are founded on principles of sense-making and
learning with understanding. Secondly, explicit instruction in taught strategies and
computational procedures may ensure that some children are competent in
number procedures. However, that approach cannot guarantee the same level of
conceptual understanding. Thirdly, prescriptive procedural instruction can
constrain investigative cultures within classrooms. Such instruction can foster
children’s beliefs that the methods that they have been explicitly taught are the
only valid ones.
2. What range of provision best supports children across the full ability range,
including the most gifted? The highest priority should be given to those who are
not progressing fast enough to reach national expectations.
1. Again, here, the notion that the specification of a range of provision can address
the issues of underachievement arising from low quality in teaching seems overoptimistic. What needs to occur is for primary teachers to be better at teaching
mathematics, to be more able to communicate mathematical ideas, and to
develop the meaningful learning of facts and skills. These aims are achieved
through more effective interaction with pupils, supported by better pedagogical
subject knowledge and its related knowledge of mathematics. Improvements of
that kind in teachers will benefit all children, but especially those who seem to
lose out most at present – higher ability pupils who are insufficiently stretched,
and children who do not respond well to the existing provision and so fall behind.
There can be no “easy fix” here. It will take time, and may have already been
begun by the work of the Strategy.
2. Problem-solving and investigation should be at the heart of all mathematics
teaching. Children should be encouraged to develop mathematical understanding
through investigation. Such understanding can occur within a wide range of
meaningful contexts including play, particularly at the earliest stages.
Investigation and problem- solving should precede and help build an
understanding of number and knowledge of number facts. Such provision should
not be used as a way of contextualising pupils’ application of abstract knowledge
of number.
3. Provision of maths education needs to support breadth and complexity, such that
children realise that the discourse concerns abstract mathematics, and not the
explicit activity in hand. If young children do not develop this basic understanding
then they are likely to experience maths teaching as a series of unconnected
activities. In their quest to find “What must I do?” they are likely to focus on the
activity rather than on the abstract mathematical topic.
4. Breadth, compexity and problem-solving should be an integral feature of provision
for the most and the least able. It is a mistake to think that complexity can be
introduced after children have learned “simple” number facts. Where this is done,
children simply misconstrue the essence of mathematics
3. The review should specifically make recommendations to inform the
development of an early (age 5 to 7) intervention programme for children who are
failing to master the basics of numeracy ‘Every Child Counts’ - as recently
announced by the Prime Minister.
1. To refer to the need to identify children as young as age 5 to 7 as potentially
being in need of an intervention programme seems to say more about what might
be problematic in the existing curriculum than any within-child deficits that need to
be addressed. A broader view of a mathematics curriculum, one that is informed
by and respectful of children’s mathematical development, may provide more
lasting benefits than specific intervention approaches designed to help children
catch up with what may have been an inappropriate curriculum. There is also the
possibility that deficit model interventions may wash out in the long term. If
teaching is informed by children’s mathematical development and respectful of
children’s pace of learning, the need for any separate interventions may be
reduced to the most extreme cases.
2. Adequate note should be taken of the length of time that some children require to
develop basic number concepts, regardless of their abilities in other fields such as
language and literacy. This may not be apparent to teachers, who are used to
measuring what children appear to know rather than their understanding of
concepts. As a result, they often over-estimate children’s conceptual development
in the early stages of schooling. It would be best if teachers and early years
workers learned ways to assess conceptual development so that they can be
sure that their teaching is providing appropriate challenge for all children.
3. Furthermore, steps should be taken to avoid a situation where children are
labelled as “failing” before the age of 7. One of the difficulties with an arbitrary
baseline for defining “failing” performance at age 7 is that it does nothing to help
teachers ensure that appropriate challenge is present in teaching for children at
all levels of understanding, including the most able. Able children who are not
achieving their best potential are, in a very real sense, as much a failure of the
system as children who fall below an arbitrary standard.
4. When children fail to master the basics of numeracy, the most common
underlying mathematical problem rests usually with their counting skills. Most
aspects of numeracy follow on from being proficient in counting, including
knowledge of the counting string, skill in moving around it and keeping track in it,
and using it to quantify collections of objects. The Strategy recognises this aspect
of numeracy development, at least to an extent, and counting has a high priority
in interactive teaching activities promoted in the Strategy framework. The work of
the “Every Child Counts” intervention programme will presumably echo this
awareness – especially given its title – but will need to do so in a more detailed,
sustained and extensive way than is feasible in everyday classroom activity. A
detailed diagnostic assessment of counting based on the extensive work of
psychologists such as Fuson and others on the development of counting seems
vital to this initiative.
4. What conceptual and subject knowledge of mathematics should be expected of
primary school teachers and early years practitioners, and how should initial
teaching training and continuing professional development be improved to secure
that knowledge?
1. Professional development must lie at the heart of any initiatives that seek to drive
forward meaningful and authentic learning for all pupils. In a seminal paper Lee
Shulman (1986) described pedagogical content knowledge as a fundamental type
of knowledge required by teachers. Pedagogical content knowledge is a
synthesis of knowledge of subject matter, teaching approaches, the curriculum
and the learners, in terms of what they bring to the learning process. It is a
uniquely integrated form of knowledge possessed by those who teach, as
opposed to those who are experts in subject matter. Primary teachers need
pedagogical content knowledge rather than extensive knowledge of mathematics
as a subject.
2. ITE should give novice teachers the space to develop their theoretical
frameworks for early maths teaching as part of their pedagogical content
knowledge. In this way, they will have the requisite understanding of children's
mathematical development to become skilled numeracy teachers. Probationer
CPD should allow early career teachers to develop their numeracy teaching skills
through the development of their pedagogical content knowledge. Primary
teachers should be as skilled in teaching numeracy to young children as they are
in teaching literacy.
3. One very effective approach to teacher development is to have teachers in
groups developing their own ideas for teaching (from initial stimuli from a variety
of possible sources, including research papers, published teaching experiments
and the successes of others in the group) in a critical but supportive environment,
as used in the LAMP and RAMP projects in the 1980s and recently in the
Improving Attainment in Mathematics Project. Such an approach primarily
develops pedagogical subject knowledge in Shulman’s sense rather than the
knowledge of mathematics with which the Review appears to be concerned.
However, the group interaction also highlights where subject knowledge needs to
be developed in order to make the teaching ideas work, and so make teaching
effective. Arguably, this is a better way to establish what subject knowledge is
needed than the usual approach of lists based on groundless presumptions of a
relationship between particular mathematical knowledge and teaching
effectiveness. The mathematical knowledge that a teacher needs depends on
what (s)he is teaching and how, yet possession of that knowledge does nothing to
guarantee that they will teach effectively.
5. What is the most effective design and sequencing of the mathematics
curriculum? Recommendations in this area should inform a future review of the
primary curriculum as a whole.
1. Too often the maths curriculum is delivered through a series of activities instead
of through the teacher's sense of how children need to develop their concepts.
Mathematics instruction should be based on sense-making models where
children make meaning by engaging with mathematical ideas that are challenging
and problematic. The mathematics curriculum should encourage mathematics
learning as a skill for life. Sense-making approaches and learning that is
authentically engaging and contextualised supports this aim. The most effective
design will be one that helps teachers see the mathematical thinking in the
children’s activities.
2. There is a need to reconsider the stages of introduction of some important
mathematical concepts. For example, there is evidence that multiplication and
division can be introduced earlier, provided that the relation with addition and
subtraction is properly understood. There is also evidence that the introduction of
the principles of measurement division can aid an understanding of place value.
Multiplication and division should be taught alongside addition and subtraction.
These operations should not be seen as simple and complex versions of each
other respectively, but as similar points across the complexity of developing
concepts.
3. The curriculum design should help teachers to see children's mathematical
development as a broadly based continuum rather than as a series of short-term
goals. Teachers are sometimes overly concerned with what children need to
know, or be able to do, for the next step in the system rather than for the longterm development of their understanding. It would help if the curriculum
encouraged them to take a broader and more long-term view of how children are
using their mathematical concepts both inside and outside school. In this way the
development of early concepts may be judged in terms of their usefulness for
later (three to seven years later) mathematical learning rather than as an end in
itself.
4. At the same time teachers need to be on their guard with regard to sequencing in
the mathematics curriculum. However, such sequencing is rarely sustained, partly
because there are always different ways to learn ideas, and the
interconnectedness between mathematical ideas has too many links. Indeed,
some of the attempts to structure the curriculum on that basis (set theory before
counting; base 3 before base 10) have not been entirely successful..
5. Again one must wonder why it is supposed that the current design and sequence
of the mathematics curriculum is not acceptable. It may be arbitrary in places, but
perhaps that is unavoidable. Perhaps it is better to work with what is there than to
change it into something different with different arbitrary elements. Upheaval can
sometimes give the impression of progress, especially when it has been skilfully
justified, but the challenge here is in achieving progress when there are no
structural solutions.
6. How should parents and families best be helped to support young children’s
mathematical development?
1. Parents come with their own school and life experiences of mathematics, and
teachers need to value these positively. There is a need to develop parents’
understanding of the purpose and value of mathematics for young children as a
skill for life. Parents and families need to be made aware of developments in what
is known about children’s mathematical thinking and the implications that this has
for teaching in general and their expectations in particular. This can be done
through school-based workshops as well as through the media, including online
communication.
2. Parents play two roles in influencing children's experiences of maths. Firstly, they
provide immediate learning environments for children at home and, secondly,
they exert a powerful indirect influence on schools and teachers through their
expectations of formal learning. Teachers can be inhibited in using novel or
innovative approaches through their expectations that parents will not accept
different kinds of evidence that their children are learning mathematics. If
teachers are confident in what they are doing, then their communication with
parents can be used to manage parental expectations.
3. Support for parents and families needs to be routed indirectly through teachers,
and teachers need to be supported in involving parents in their children's maths
learning. At the level of formal schooling, Martin Hughes' projects on homework
and home-school links provide excellent guidance on positive directions that
teachers can take, for instance, to inform parents' expectations that there is a
single “best” algorithm for any numerical operation, and to help them realise that
authentic learning in maths often produces quite subtle evidence. Projects that
follow the lead of the Hughes Bristol Project will also help teachers to understand
the complex mathematical learning that often takes place out of school, and that
can supercede the limited views on maths learning imposed by the school
curriculum.
4. One of the most important contributions that parents and families make to
children’s mathematical development is in nurturing their emotional development.
Enough mathematics occurs naturally in children’s lives both in and out of school
for them to learn all that is needed, but many children are unable to engage
appropriately because of difficulties in their emotional development. Attitudes to
mathematics and to learning are crucially important and helping families in their
general interaction with children – albeit that this may go beyond the terms of
reference of the committee – is critical.
Gordon Kirk,
Academic Secretary
15 November, 2007