“In mathematics the ability to solve problems is not just knowing some straightforward rules”
The NRICH Project (www.nrich.maths.org) has been in operation since 1996, when its original purpose was to support able young mathematicians whose access to opportunities in their local community was limited and often non-existent. Since this time the resources on the web site have grown and the project has developed a reputation for creative thinking in the area of mathematics enrichment.
The most recent work of the project has centred on making more effective use of the wealth of resources we now have available to us, both in terms of access to the enormous archive and in creating meaningful frameworks within which selections of the material can be placed (enrichment trails). As the new site and trails have developed we have questioned our understanding of mathematics enrichment and how it might be represented in classroom practice and through the NRICH site itself. The reflection and early research findings have resulted in two key outcomes that are having a fundamental impact on our work:
• the resources are not suitable solely for the most able but have something to offer pupils of nearly all abilities. This has resulted in the restructuring of the site and creation of the trails to facilitate a “free flow” of resources across age and ability boundaries.
• enrichment is not only an issue of content but a teaching approach that offers opportunities for exploration, discovery and communication,
• effective mediation offers a key with which to unlock the barriers to engagement and learning.
We are attempting to address the issues of the nature of enrichment, accessibility, mediation and the philosophies of learning and teaching that underpin our work both through the structure and content of the site, our work on enrichment trails and our face to face work with pupils and teachers.
This paper considers the key aspects of mathematics enrichment and how the content and design of trails (as well as the NRICH site itself) has been influenced by, and built upon, these philosophies.
Background and Rationale
The wider context
The United Kingdom Numeracy Framework offers guidance and exemplification of the mathematics curriculum giving content, structure and guidance on its implementation and delivery. However, although there has been an overall improvement in performance in national tests, there are areas where concerns still exist in terms of performance, teaching and attitudes to mathematics:
• Concerns exist over pupil performance in algebra, geometry and problem solving (Brown, Millett et al. 2000). These concerns have most recently resulted in changes to the national mathematics attainment tests, which will now include a problem solving section.
• Most commonly, the needs of most able pupils are met through courses of acceleration. Pupils undertaking such courses are often taught independently (and separately) from their peers, older pupils often having to go to other schools for their lessons. These models of acceleration pose medium to long term problems of sustainability and there is no evidence of long-term benefits. Ability grouping with ‘fast track’ top sets has also been shown to cause problems in the long term (Boaler, Wiliam et al. 2000).
• Fewer pupils are choosing to study mathematics and mathematics related subjects beyond the age of 16 (Nardi and Steward 2002), (Nardi and Steward 2002) (Nardi and Stewart 2003 forthcoming).
• Evidence of lack of motivation and consequent dips in performance across KS3 is available and indicates pupils are being “turned off” mathematics. (Watson, 2001). Results in 2003 show a slight decline in performance over previous years resulting in the government adjusting long term performance targets.
Enrichment can be used:
• to support the most able alongside all children in the class; often offering differentiation by outcome,
• to promote mathematical reasoning and thinking skills, preparing pupils through breadth and experience to tackle higher level mathematics with confidence and a sense of pattern and place.
Mathematics Enrichment Materials on the NRICH Website
There have been a wealth of resources that support mathematical enrichment, most notably the NRICH online mathematics project (www.nrich.maths.org). The resources on the NRICH site have been in “loose leaf” format; being stored with few pointers to their curriculum context and relevance. This has left the user with issues of access to appropriate material and knowledge of the potential of, and means by which, the material can be used to support the development of high level mathematical reasoning (and other) skills.
From these points come the foci of our recent work:
• identification of key aspects of an enrichment curriculum for mathematics that makes links between content, the national frameworks, and practice explicit;
• effective presentation and structuring of resources on the NRICH site such that they will underpin an enrichment framework by offering exemplars of content and supporting material.
It is through examining the theories underpinning the development of structured content (trails) and views of teachers as users of the trails, the nature of mathematical enrichment and how it can be represented is being implemented.
Defining a Framework
Terms such as “mathematical thinking”, “mathematical problem solving” and “enrichment” are variously described in current literature. Our work has therefore involved us in clarifying definitions of these terms. Establishing meanings has involved a literature review, interviews with colleagues and teachers and the analysis of NRICH team discussions. In addition, the process of site and trail development has involved multiple iterations which have themselves informed the definitions. These definitions are therefore constantly being reviewed and refined as we trial and test materials and build the framework within which our work is set. What is presented is our current view of these terms as they relate to our work.
In current literature, “enrichment” is almost exclusively used in the context of provision for the mathematically most able. However, there is strong evidence from the use of the NRICH site, and our own experience working with teachers and pupils, that this fails to address the value of an enrichment approach to teaching mathematics generally. Problems which offer suitable entry points can be used with pupils of a wide range of ability and therefore can be used within the “ordinary” classroom. The teacher or mentor can use such materials in flexible ways that respond to the needs (and experience) of the learner. We see enrichment as an approach to teaching and learning mathematics that is appropriate for all not simply the most able. NRICH resources therefore continue to support the most able but this is within the context of a broad interpretation and view of enrichment not within a context of provision simply targeting the most able. Good enrichment education is good education for all. Good mathematics education should incorporate an approach that is an enriching and stimulating experience for all pupils. The construction of enrichment we are adopting thus builds on two main threads:
This thread describes an enrichment curriculum, which has the following components:
• Content opportunities designed to:
o develop and use problem solving strategies,*
o encourage mathematical thinking,*
o include historical cultural contexts,
o offer opportunities for mathematical extension.
* These two strands form the focus of the content discussion in this paper
Enrichment is not simply learning facts and demonstrating skills. Mathematical skills and knowledge can be a precursors to, and also outcomes of, an enrichment curriculum (needs driven learning). The aim of an enrichment curriculum is to support:
• a problem solving approach
• improving pupil attitudes
• a growing appreciation of mathematics
• the development of conceptual structures
based on Ernest (2000)
Enrichment therefore represents an open and flexible approach to teaching mathematics which encourages experimentation and communication
This places an emphasis on teaching that reflects a constructivist view of learning and which stresses:
• non-assertive mediation,
• group work, discussion, communicating …,
• varied solutions and different approaches being valued and utilised,
• exploration, making mathematical connections, extending boundaries, celebrating ideas not simply answers, flexibility… ,
• acknowledgment that maths is hard but success is all the more enjoyable when a hurdle is overcome.
Problem Solving and Mathematical Thinking
A range of literature exists in the areas of Problem solving and Mathematical thinking. The two terms often being used synonymously or with a lack of clarity in their inter-relationship, As part of our framework for development we have been able to identify two distinct threads that appear in the use of the two terms and which are worthy of articulation and distinction. These threads pull together ideas drawn from current theory (Mayer (2002); Koshy (2001); Mason, Burton and Stacey (1985); Ernest (2000), Shoenfeld (1994), Polya (1957), Lester (1994), Cobb et al (1991), Van Zoest et al (1994), and our own work in the field.
We are taking “mathematical thinking” to mean particular mathematical strategies that are employed in solving problems of different types. Some exemplars of these strategies are given below. The aim is to identify problems where such strategies are useful and create a curriculum thread that encourages pupils to develop each strategy and identify the type of context and the ways is which such strategies can be employed.
Problem solving is reserved for the structural approach to solving problems - the overview, or steps on the journey from meeting a problem for the first time to its solution. Problem solving identifies and developments competence in utilising the stages on the route through solving a problem. Problem solving underpins the vast majority of NRICHs resources.
Thus mathematical thinking strategies are needed to tackle problems and will be used within the problem solving process.