Conceptions of Mathematics Enrichment
Wai Yi Feng
University of Cambridge
Faculty of Education
Paper presented at the British Educational Research Association Annual Conference, University of Warwick,6-9 September 2006
Draft: Not for Circulation
Please Do Not Cite Without Consulting Author
Despite the increasing amount of activity taking place under its label, mathematics enrichment is under-theorised and currently lacks a coherent framework of organising ideas. The term ‘enrichment’ itself is often used commonsensically, its meaning taken as shared (Feng, 2005). This paper reports on a project which is developing a framework for understanding mathematics enrichment in the UK. Beginning with a summary of the conflicting views of mathematics enrichment found in the educational literature, this paper then presents a four-strand model for organising enrichment ideas. By way of illustration, the model is then applied to map a selection of mathematics enrichment provisions currently available in the UK. This paper ends by posing questions regarding the future of enrichment and its relationship with formal education in the light of current educational concerns and anticipated curriculum development in the UK.
Conflicting Views of Mathematics Enrichment
Conceptions of ‘mathematics enrichment’ have been influenced by many of the wider debates in education ever since ‘enrichment’ first began to receive attention some eighty years ago. Such debates include ones which pertain to the relative merits of special versus inclusive education, the intrinsic value of mathematics versus its utility in the modern society, as well as discussions about who should have access to what kind of mathematics education and why. Conflicting views in these debates have given rise to a variety of enrichment discourses (Feng, 2005). Further, as conceptions of mathematics enrichment grew to accommodate and reflect the changes in educational thinking, different meanings of the term ‘enrichment’ have also been incorporated into the literature. As a result of this, existing descriptions and definitions of enrichment often lack clarity and consistency; assumptions underlying how enrichment could be accomplished are not necessarily shared.
What is Enrichment?
Considered simply, enrichment might be defined as any experience that replaces, supplements, or extends instruction beyond that normally offered by the school (Correll, 1978, as cited in Clendening & Davies, 1980). This basic definition is supported by a number of scholars (e.g. Stanley, 1979; Clendening & Davies, 1983; Eyre & Marjoram, 1990), some of whom have also elaborated on its meaning in their writing. However, since all such definitions refer to ‘normal’ practices which in fact vary between schools and classes, enrichment also becomes a relative concept. Indeed, ‘enrichment’ often acts simply as a convenient placeholder for a range of educational provisions not explicitly prescribed by the curriculum. As educational priorities changed, enrichment has also evolved with the result that different authors writing about ‘enrichment’ at different times and in different contexts may not necessarily share any interpretation of the concept.
The difficulty of defining enrichment has grave implications for its interpretation, both conceptual and practical. Questions of interest include:
Why is Enrichment Needed?
When mathematics enrichment first began to receive attention in the educational literature some eighty years ago, it was identified as a means of supporting the educational needs of mathematically-gifted students (Barbe, 1960). However, in more recent years, mathematics enrichment has increasingly been represented as an attempt to engage students in more stimulating mathematics as well as a reaction to the perceived shortfalls of the mathematics curriculum. The former has been given impetus by what has been perceived as the urgent need to reverse the trend of falling numbers recruited into post-16 mathematics and mathematics-related education, as well as by students’ disaffection with school mathematics as reported in educational research, official publications and the media (e.g. Gardiner, 2003; Nardi & Steward, 2003; Smith, 2004). The latter has been encouraged by the succession of critical evaluations of the school mathematics curriculum published in the last two decades (e.g. Cockcroft, 1982; Smith, 2004) as well as the growing debate regarding the utility of school mathematics in daily life and the type of mathematical skills that might be useful in the modern technological society (Willis, 1990; Hoyles et. al., 1999; Bramall & White, 2000).
Enrichment, then, is a means of introducing accessible aspects of mathematics not covered by the curriculum, fostering mathematical thinking, encouraging extended problem solving, presenting alternative approaches to curriculum topics, and highlighting links between aspects of mathematics presented separately in the curriculum. Through these activities, enrichment may variously aim to present students with a stimulating experience of mathematics, promote positive attitudes, raise attainment, and contribute to efforts to promote, popularise and raise public understanding of mathematics.
Who should be Enriched?
Although the enrichment literature still retains much of its original gifted focus, whereby the main concern is to support the learning of mathematically-able students whose ‘special needs’ are thought not to have been adequately met, this assumption has not gone unchallenged.
Since the 1960s, education in the UK has undergone momentous change, most notably in the rise of comprehensive and inclusive education, the decline of educational selection, and the growing concern for social justice within education which underpins the first two. Such wider educational concerns, when supplemented by the erosion of confidence in mainstream mathematics provisions (which resulted from the critical evaluations of the school mathematics curriculum mentioned earlier), has led to enrichment being gradually made available to a growing proportion of students. Although the explicit use of mathematics enrichment to engage disaffected or lower-attaining students (e.g. Walton, 1995) is still rare, proactive efforts are increasingly made to reach those students who have not traditionally benefited from enrichment provisions (e.g. NRICH Outreach).
Where and When should Enrichment Take Place?
When enrichment is seen as additional support for only a few mathematically-gifted students in a class, enrichment might be ‘given’ to those selected during lunch-times, in after-school clubs, or as extension work (Witty, 1960). However, as enrichment began to be seen as the entitlement of many, it has become less clear whether it should be a part of, or additional to, ‘everyday’ teaching and learning. Taken even further, if enrichment is thought of as the ultimate goal of education — e.g. if mathematics enrichment is considered to be an environment or process which encourages problem solving and mathematical thinking — then the objective might be to make all learning experience so ‘rich’ that enrichment becomes simultaneously pervasive and redundant.
Can All Learning be Enriched?
In essence, the conflicting conceptions of mathematics enrichment represent the contested views arising from the allocation of limited resources according to different perceptions of ‘interest’, ‘needs’ and ‘benefits’ in wider education. Perceptions of whether all students can benefit from enrichment and whether all topics of study are suitable for an ‘enriched’ presentation are no less a case in point.
At one extreme, if enrichment is perceived as ‘the deliberate differentiation of curriculum content and activities for the superior pupils in a heterogeneous class’ (NEA, 1950, as cited in Barbe, 1960), then only those students identified as ‘superior’ would be able to benefit from the specially-formulated learning material, which could involve more advance subject matter or require higher-level treatment of prescribed curriculum topics. At the other extreme, if mathematics enrichment involves ‘problem solving and mathematical thinking that is linked to mathematical contexts’ (Piggott, 2004a), then enrichment should underlie many, if not all, aspects of the curriculum and all students should be able to benefit from such experiences.
Despite conflicting views on the nature and practice of mathematics enrichment, four paradigmatic positions could be identified from the enrichment literature, each with its own educational standpoints and priorities. These focus on enrichment as:
In this section, a series of ‘cameos’ is presented to illustrate the four paradigmatic positions in turn.
Position 1: Enrichment as Development of Exceptional Mathematical Talent
This view sees enrichment as a means of meeting distinctive academic needs of (mathematically) gifted students. As such, the primary enrichment concerns are to identify and develop (mathematical) talent, often with a view to cultivating an elite group of (mathematically-minded) students to become leaders in civic, commercial and industrial contexts.
High (mathematical) attainment is a prerequisite to participation; it is also promoted and prized throughout enrichment. By engaging students in interesting and challenging mathematics, it is hoped that their mathematical skills could be extended, their interest in the subject could be maintained or heightened, and their awareness of the use of mathematics in daily life and as a pre-condition for future careers could be raised. Students are given considerable agency and freedom in their own work so as to help them develop the sense of personal and social responsibility necessary for leadership and independent learning. Through such efforts, mathematics enrichment aims to help mathematically-gifted students find fulfilment in mathematics, and to recruit, train, and retain these gifted students in mathematics and mathematically-related fields. Thus, this view of mathematics enrichment is selective and mathematics-centred.
In terms of enrichment content, the material covered may include additional or more difficult topics not covered by the curriculum — at least for the age group in which the students belong — or involve more advance treatment of curriculum topics. Considerable emphasis is also placed on matching enrichment material to students’ interest.
Students may be enriched during lessons (e.g. after the completion of set work) or in special extra-curricular programmes. The latter could take the form of school-run gifted programmes (e.g. lunchtime or after-school mathematics club or supported independent study) as well as externally-organised projects (e.g. mathematics competitions). Since enrichment is explicitly designated for the gifted, the involvement of other students would be considered a ‘dilution’ to the detriment of the programme. Depending on the schools’ gifted policy, the interest of the teacher (or external enrichment provider) and the availability of other resources, enrichment undertakings may vary between the frequent and the occasional.
Expositions consistent with this position can be found in Clendening and Davies (1983, 1980), Meister and Odell (1979), Worcester (1979), and Stanley (1979, 1976).
Position 2: Enrichment as Popular Contextualisation of Mathematics
This view of enrichment is best described as mathematics- and application-centred. It sees enrichment as a means of engaging students in mathematics, which is to be valued, not just as an academic discipline, but for its applications in the modern world. In this context, enrichment is motivated by the need to raise awareness of implicit mathematics in daily life. A considerable part of enrichment therefore concerns the popularisation of mathematics through the engagement of students in meaningful and relevant mathematical practices. Through expanding students’ experience of mathematical practices, it is hoped that negative stereotypes of mathematics and mathematicians could be broken down; by deepening students’ understanding of mathematics and its applications, it is hoped that students will better appreciate the important role of mathematics in underpinning modern social and scientific processes. Thus, not only are all students able to benefit from mathematics enrichment, it is also important that as many students as possible are thus enriched.
In this conceptualisation, enrichment involves alternative presentations of curriculum topics, the integration of curriculum teaching to ease transfer, as well as the teaching and learning of accessible topics not covered by the basic curriculum. Applications of mathematical ideas in daily life and in other areas of study — notably history, science and art — and the relevance of mathematical concepts in contemporary events and issues are also emphasised in an attempt to help students uncover the ‘hidden’ mathematics underpinning the modern society.
Whole classes of students could be collectively enriched in planned lessons in this sense of popularisation. Educational fieldtrips or visits by people who are involved in the use of mathematics might be appropriate. National strategies for raising public understanding of mathematics are also relevant. The frequency of enrichment is dependent on school policy, available resources and the teachers’ interest. Where students are able to gain access to enrichment provisions independently, the frequency of enrichment is also dependent on the students’ interest and commitment.
Literary evidence supporting this interpretation of enrichment can be found in Zeeman (1990), Fox (1979), and on the Plus website.
Position 3: Enrichment as Enhancement of Mathematics Learning Processes
This view of enrichment is best described as student- and experience-centred. In this, enrichment is considered to be an informed approach to teaching and learning which should form an integral part of education for all students, whether in the regular classroom or beyond. In this respect, enrichment is an ongoing process which should pervade all aspects of teaching and learning and is not easily separated from good educational practices.
Using this interpretation of enrichment, the engagement of all students in meaningful mathematical practices is an essential and worthwhile part of education; this also forms the main goal of mathematics enrichment. Individual fulfilment is emphasised throughout. This is achieved through the extension of skills, the expansion of students’ experience of mathematics, and the development of students’ sense of responsibility for, and ownership of, their own work in the process of becoming independent learners.
In this conceptualisation, enrichment supplements and assists the delivery of the mainstream curriculum. It involves imaginative presentation of curriculum topics, integration of curriculum teaching to prevent fragmentation of knowledge and to ease transfer, as well as the teaching and learning of accessible topics not covered by the basic curriculum. It encourages the formation of links between mathematical content presented separately in the curriculum and between mathematical content and other areas of study. By presenting students with a stimulating experience of mathematics, enrichment fosters mathematics thinking and problem solving.
There is no specifically-designated enrichment material: all appropriate and available resources are brought to bear in providing students with fulfilling experiences of mathematics and of education more generally. Of course, given the diversity of the target audience, it is only to be expected that some students will require more support than others in order to make full use of the enrichment opportunities. Enrichment in this sense therefore emphasises appropriate scaffolding and the differentiation of content: enrichment tasks are often designed to be approached from a number of perspectives, each of which makes use of mathematical concepts and techniques at various levels of difficulty, and may lead to qualitatively different end-points.
Evidence supporting this view of enrichment can be found in Piggott (2004a, 2004b), Beetlestone (1998), Cook et. al. (1997), Ainley (1996), Thyer (1993), and Eyre and Marjoram (1990).
Position 4: Enrichment as Outreach to the Mathematically Underprivileged
Being aware of the important role of mathematics in modern society, educators subscribing to this view of enrichment are mainly concerned with widening students’ access to mathematics and raising levels of engagement. To them, enrichment is a means of overcoming barriers of engagement and of liberating students. As concerns about social justice and equity are also foremost on their agenda, educators subscribing to this view of enrichment not only believe that enrichment should be open to all students, they also make proactive efforts to bring mathematics enrichment to those students who have not traditionally benefited from such provisions. In so doing, it is hoped that the base from which young people are recruited into mathematics and mathematics-related education and careers can be broadened.
The first goal of enrichment is to engage students in mathematical practices. To this end, students are empowered as effective agent and knowledge generators, and encouraged to take responsibility for, and make decisions about, their own work. Often, students are presented with imaginative applications of commonly-encountered curriculum material to pique their interest and to introduce them to alternative ways of thinking about familiar material. Extended investigations or problem-solving activities in which students are required to use mathematical knowledge creatively and systematically are also widely used. Through such activities and methods of working, enrichment fosters mathematical thinking, exposes students to learning processes not commonly encountered in regular mathematics lessons, and sharpens those mathematical skills already introduced in the classroom. It is hoped that in so doing, students’ interest in mathematics could be heightened and that they would be able to derive greater fulfilment from mathematical practices.
Enrichment provisions of this kind are mainly extra-curricular, although lesson time could also be set aside to enable whole classes to participate. The frequency of enrichment depends on available resources, school policy and the interest of the teacher.
Evidence of the outreach educators’ interpretation of enrichment can be found in Piggott (2004b) and on the websites detailing NRICH Outreach and the SHINE-NRICH Project in Hackney and Tower Hamlets.
Mapping Mathematics Enrichment in the UK
In the UK, mathematics enrichment has mainly taken the following forms:
These can, in turn, be mapped according to the four paradigmatic positions proposed.
By way of illustration, this section seeks to classify mathematics enrichment provisions operating nationally in the UK (i.e. the NAGTY Mathematics Summer Schools, the UKMT Mathematics Challenges, programmes run by the MMP, and the RI Mathematics Masterclasses) through documentary analysis. Supporting evidence has been taken from the websites of the respective organisations.
NAGTY Mathematics Summer Schools
Membership of the National Academy for Gifted and Talented Youth is restricted to students in the top 5% of the academic ability range (NAGTY Eligibility Criteria). Once membership has been approved, NAGTY members have access to a range of specially-organised academic programmes and on-line forums throughout the year, including Summer Schools during the summer vacation.
The NAGTY Mathematics Summer Schools focus on developing the mathematical awareness, interest and talent of NAGTY members. A rigorous application process also enables the organisers to select those most suited to participate in the courses (NAGTY Summer School Statement of Eligibility; NAGTY Summer School Application Pack (2005)). This strongly associates the Summer Schools with enrichment as development of talent (Paradigmatic Position 1).
UKMT Mathematics Challenges
The UK Mathematics Trust organises and runs a number of mathematics competitions set at various levels of difficulty throughout the year. The Junior Mathematics Challenge targets students in Year 8 or below; the Intermediate Mathematics Challenge targets students in Year 11 or below; the Senior Mathematics Challenge targets students up to Year 13. Students in Years 8 and 9 in England and Wales and Years 9 and 10 in Northern Ireland can also take part in the Team Challenge (UKMT Home Page; UKMT Team Challenge Page; UKMT Follow-On Rounds Page).
The Mathematics Challenges are ‘designed to appeal to as many students as possible in each year group’ and aim to ‘stimulate greater interest in mathematics … [encourage] participation and [reward] enthusiasm and achievement.’ (UKMT Home Page) Those performing well at the individual Challenges are invited back for follow-on rounds; mentoring can also be organised through the UKMT (UKMT Follow-On Rounds Page; UKMT Mentoring Page). Finally, students with the highest scores in successive rounds in the Senior category are selected to join the British team for the annual International Mathematics Olympiad (UKMT Follow-On Rounds Page); for the latter, further training is arranged by the UKMT.
Thus, mathematics enrichment supported by the UKMT begins with the use of interesting and challenging mathematical problems to engage students in mathematics more generally. This conceptualisation of enrichment is in keeping with popularisation (Paradigmatic Position 2). However, as students enter successive rounds, the work of the UKMT becomes increasingly focused on supporting the mathematically gifted (Paradigmatic Position 1).
The Millennium Mathematics Project
The Millennium Mathematics Project was launched in 1999 as a mathematics education initiative for those aged 5 to 19 as well as the general public. Based at the University of Cambridge but active internationally, the broader goal of the MMP is to ‘help people of all ages and abilities share in the excitement of mathematics and understand the enormous range and importance of its applications and its vital contribution to shaping the everyday world.’ (MMP Home Page) This conception of mathematics enrichment is rooted in the popular contextualisation of mathematics (Paradigmatic Position 2).
A number of enrichment schemes fall under the umbrella of the MMP. Notably:
Royal Institution Mathematics Masterclasses
The Royal Institution of Great Britain has a long and illustrious history of supporting popular seminars and talks given by eminent mathematicians and scientists for the general public. More recently, a series of regional networks has been formed to provide mathematics masterclasses for secondary students as an extension to the RI’s commitment to ‘facilitate dialogue between scientists and the public’ (RI Events Page). Since each regional base enjoys considerable autonomy, the resulting series of masterclasses present a differing range of subject matter and take a variety of formats. However, some common goals are shared by all the masterclasses as each aims to engage students in mathematics, pique interest, expand students’ experience, and raise awareness of mathematics.
Masterclass participants are usually selected by their own schools. Since places are limited, it is often the gifted who are chosen to take part. The masterclasses, therefore, also serve to support the development of mathematical talent.
In sum, enrichment supported by the RI stems from a conceptualisation that is primarily based on the popular contextualisation of mathematics (Paradigmatic Position 2). However, this can be — and in some cases, has been — influenced by pragmatic considerations, which has the effect of pushing enrichment providers to adopt a stance in gifted education (Paradigmatic Position 1).
Summary and Discussion
Although all bona fide mathematics enrichment is fundamentally motivated by commitments to the provision of high-quality educational experiences in mathematics, different perceptions of how this could best be achieved and whom might benefit ‘most’ have given rise to opposing views about effective enrichment organisation and suitable audience. Four paradigmatic positions regarding mathematics enrichment have been described in this paper. Each position is underpinned by a different set of enrichment motivations and gives rise to distinct enrichment practices. These focus on mathematics enrichment as:
The enrichment characteristics of the four positions are summarised in Table 1. Looking down the columns, the Table highlights the many methods for enriching mathematics learning, the many reasons for doing so, and the many occasions in which enrichment may be appropriate or desirable. These views are organised to form four distinct but individually coherent conceptions of mathematics enrichment, running along the rows of the Table and corresponding to the four paradigmatic positions listed above. Theoretically, it is possible to map enrichment practices according to the framework suggested by the four conceptualisations. In practice, it is also possible to retrace the rationale underlying many mathematics enrichment schemes back to at least one of these interpretations of enrichment. Conceptions of enrichment may, of course, stem from a combination of two or more paradigmatic positions or straddle more than one position. Equally, pragmatic considerations can also influence enrichment practices and modify conceptualisation.
Since all four conceptualisations of mathematics enrichment are well-supported, our preference for, or choice of, enrichment rationale and practices may betray more about our own educational standpoint. Indeed, as educational priorities have changed over the years, different forms of enrichment have come into focus. In effect, enrichment is itself an educational curriculum, albeit in miniature and mostly independently run. An awareness of this is important, not only for appreciating the influences that have shaped the development of mathematics enrichment to date, but also for understanding the relationship between enrichment and formal mathematics education in the light of current educational concerns and anticipated curriculum developments.
For example, as a result of recent policy recommendations (Smith, 2004; Clarke, 2004), there is currently much interest in the creation of a double award for mathematics GCSE and in the introduction of an extension curriculum for able students at Key Stage 3 and 4. Some of the functions expected of the proposed curriculum — e.g. challenging mathematically-able students and enhancing mathematical thinking and problem solving — are currently served informally by mathematics enrichment outside of mainstream education. This raises a number of questions regarding the future of mathematics enrichment:
Such questions lie at the heart of research into mathematics enrichment; our responses to them can, in turn, inform mathematics education more generally. By understanding the educational ideals and social influences which shaped enrichment, the mechanisms through which this was done, and their relationship with the resulting enrichment provisions and educational outcomes, we can also begin to understand mainstream education and the changes within it.
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The author would like to thank Professor Kenneth Ruthven for his guidance in preparing this paper as well as the ESRC, the Isaac Newton Trust and Newnham College Cambridge for their generous sponsorship.
This document was added to the Education-Line database on 28 June 2007