Why the journey to mathematical excellence may be long in Scotland’s primary schools
Stenhouse (1975) noted that curriculum can be viewed as intention and as reality
The success of these is highly dependent on the learning experiences devised by teachers which in turn are informed by their own conceptual understanding of the content and processes of each subject
mathematics curriculum reform that has shifted from “an emphasis on knowing things to being able to do things
While acknowledging that the cognitive aspect of teaching is intimately related to affective factors, such as confidence in, attitudes to and beliefs about the subject, this paper addresses only the cognitive
Constructivism proposes that children construct knowledge about the world around them.
Cognitive constructivism stems largely from the work of Piaget and his followers, Bruner among them, and focuses on the individual construction of knowledge,
social constructivism derives from the work of Vygotsky and holds that the social environment plays a central role in learning
Constructivist teachers see it as their role to create learning contexts which will facilitate this and indeed they are often described as facilitators of learning.
the mathematics teaching in the former emphasised set methods and procedures, kept topics distinct from each other and closed down mathematical problems.
The approach at Phoenix Park, on the other hand, was open, gave a degree of choice to its pupils and was concerned about giving them mathematically rich experiences
Pupils in the first school had a broad understanding of facts, rules and procedures, but they found it difficult to remember these over time
while those in the second school were flexible and adaptable in their mathematics and able to use their knowledge in different situations
The Phoenix Park practices appeared to result in more secure knowledge that was transferable across a number of situations.
that attempts to transmit knowledge to students are less helpful than classrooms where children are “apprenticed into a system of knowing, thinking and doing
Curriculum advice is just that, advice
Of the seven categories of knowledge outlined by three can be thought of as specifically relating to content: content knowledge, or subject matter knowledge (SMK) had previously described it, pedagogical content knowledge (PCK) and curriculum knowledge
SMK, has been a fertile ground for researchers who in many cases found it wanting with the resulting implications for teacher effectiveness
that pupils of primary teachers with a low proportion of conceptual links in their mathematics SMK made the lowest gains in attainment,
in order to improve mathematics education the quality of teachers’ SMK needed to be improved.
Anecdotally we may know teachers who are excellent mathematicians but whose teaching skills leave a lot to be desired. So while sound content knowledge is a necessary condition for effective mathematics teaching, it is not sufficient
click to edit
Effective teachers of mathematics must be able to move back and forth between the mathematics and the pedagogy, drawing on both to meet the needs of the learner
curriculum knowledge, as the name suggests is knowledge of the curriculum and instructional resources that can be used to deliver it
Making decisions about the content to be learned and how to organise that learning in terms of materials, texts and approaches is the “stuff of the teaching profession” and the way the teacher interacts with the children through a task will shape the type of mathematical thinking that occurs
reaks down PCK further into knowledge of content and students (KCS) and knowledge of content and teaching (KCT)
In the past 30 years mathematics curriculum reform in many countries attempted to replace procedural approaches to learning the subject with inquiry based learning that promoted conceptual understanding
children working together in small groups while the teacher posed problems and then circulated to scaffold them in their progress. It was found that
increased student achievement on computational skills and higher level thinking occurred.
This knowledge is shaped, developed and refined as they interact with others and their environment.
rather than the authority within the classroom resting with the teacher and textbooks she became the initiator and guide through the students’ development of knowledge.
While the situation in Scotland provides the focus for this paper, the central message of assuming educational change can be driven by curriculum reform alone is more international.
When looking at how one teacher viewed constructivist approaches compared to the transmission model of teaching she had favoured previously reported that she changed her beliefs about the nature of mathematics from being a set of rules and procedures to meaningful activity.
pupils of constructivist teachers developed in an atmosphere of trust that resulted in pupils being enthusiastic and persistent in mathematical problem solving
the constructivist principles apparent in reform pedagogies have shifted the responsibility of how mathematics is taught to the teacher.
in the UK the constructivist approaches used in the Cognitive Acceleration in Mathematics Education (CAME) project resulted in some gains in student achievement
literature suggesting that tackling the cognitive in the first instance may in fact lead to more positive attitudes and confidence
it is clear that the transmission model of teaching may no longer result in the type of relational understanding that is now required
tended to remove content detail from teachers who may need this very guidance to prompt and augment their own subject knowledge.
addressing the subject knowledge of some primary teachers must be a key consideration
that content knowledge came top of Schulman’s categories of teacher knowledge needed to promote understanding in their pupils
identified pedagogical content knowledge (PCK) as being of special interest as it is essentially what a teacher does. It takes the content knowledge a teacher possesses and combines it with professional understanding of how children learn
it is a blending of subject knowledge and pedagogy which transforms topics from mere content to a form which appeals to the different abilities and interests of learners
This model better reflects the dynamic nature of teaching that recognises the interaction between teachers and learners, where teachers’ subject knowledge can develop as a result.
constructs do in fact exist as viable models for learning more about mathematical knowledge for teaching, it is clear that knowledge of subject content is key to both.
It is clear from this that as children progress through the primary school and into the early stages of secondary all is not as it should be in terms of mathematics attainment.
the gap between the lowest and highest performers in P5, measured using the inter-quartile range, has narrowed since 1995 due to an increase in the score of low performers but also a decrease in the score of high performers
Studies at the author’s institution have reported that student teachers’ mathematics subject knowledge was often lacking when assessed using an online assessment and that confidence levels could be low
The knowledge that pupils should have at each level is clearly stated and a framework for conceptual development is provided.
Curriculum for Excellence has been lauded as ‘the most radical reform of education in Scotland for a generation (Scottish Government 2009c: no page) yet in terms of mathematics how different is it? One change is that a set of numeracy outcomes exists that is distinct from those in mathematics
Numeracy, along with literacy and health and wellbeing, are the responsibility of all teachers, but as primary teachers are responsible for all aspects of the curriculum anyway this does not constitute real change at this stage.
A teacher whose own understanding of the topic is less than secure may be challenged by what is meant by ‘I can use my knowledge of rounding’ from the Second Level in table 2
As the clear progression that existed under the 5-14 curriculum will no longer be provided it is up to the teacher to fill in the missing detail and this requires a degree of understanding about and confidence in the topic that may not be present.
most mathematics lessons in Scotland still tend to feature some form of teacher-led demonstration followed by children individually practising skills and procedures from a commercially produced scheme
This would seem to suggest that attempts over more than 25 years to move to constructivist models of teaching mathematics have not been entirely successful
it becomes clear that little may change in how mathematics is taught under the new curriculum.
it has been argued that curriculum reform which attempts to increase teacher autonomy in how a subject is taught without also considering that teacher’s own subject knowledge may have little chance of success.
In order to ensure that curriculum reform in mathematics translates into practice a three-pronged approach is suggested with implications for national qualifications, initial teacher education and continuing professional development (CPD).
If teachers’ subject knowledge is not sufficient it would seem natural to call for an increase in the level of these qualifications.
If the subject knowledge of these future teachers is lacking, how likely is it that they will be able to develop into autonomous professionals who adopt constructivist models of teaching mathematics by creating motivating learning experiences through which children’s mathematical understanding will develop?
Thirty years ago Begle (1979) reported that US teachers who had taken advanced mathematics courses (calculus and beyond) had positive effects on pupils’ achievement in only 10% of cases, and more worryingly negative effects in 8%.
In England Askew et al. (1997a) found that pupils of teachers with advanced level mathematics were less likely to make positive gains in attainment than their counterparts with the lower level required for entrance to teacher education programmes.
It may seem like a contradiction to the central argument that more advanced mathematics qualifications may not be the answer to improving subject knowledge. However, a more dedicated qualification (Advanced Subsidiary (AS) level equivalent to SCQF Level 6), such as that proposed by Burghes (2009), could provide a way forward.
If, as reported by Macnab and Payne (2003), student primary teachers are more apprehensive about teaching mathematics than any other curricular area, such a qualification may deepen understanding and lead to increased subject confidence.
While a case could be made for requiring that entrants to primary education programmes have this proposed qualification, another option would be to ensure the content is covered as part of initial teacher education.
In Scotland there is no consistent approach to addressing mathematics subject knowledge during initial teacher education.
In England teacher education institutions audit students’ mathematics knowledge and address any deficiencies.
The recent review of teacher education in Scotland (Donaldson 2011) recommended that applicants to primary education programmes be assessed in literacy and numeracy at the interview stage, that any deficiencies in their knowledge be addressed during their programmes and that they demonstrate a high level of competence by the end of these
maintaining sound subject knowledge and developing deeper understanding of the subject can take many more years than those spent in initial teacher education and so consideration must also be given to the continuing professional development of teachers.
highly effective teachers were more likely to have been involved in extended mathematics CPD, with sessions of 10-20 days likely to have more impact.
found that subject-specific CPD was more effective in raising attainment than generic professional development.
Until now the only way for a Scottish teacher to develop their interest in mathematics is to undertake private study or to engage in local authority or national events, where available.