Some of the mathematical thinking strategies we have identified include:
• Being systematic;*
• Identifying common structures (isomorphisms);*
• Introducing variables;
• Specialising/clarifying/looking for specific examples;
• Considering a special case (the particular);
• Solving simpler related problems;
• Reflecting on experience - have you met something like this before?
• Multiple representations;
• Working backwards;
• Identifying and describing patterns;
• Representing information– diagram, table
• Testing ideas - guessing and testing (hypothesizing.
* We have begun to develop curriculum resources that illustrate and support these aspects of mathematical thinking, in the form of trails.
There is still some work to do in identifying different aspects of mathematical thinking .Not all these strategies have a similar feel to them. Currently it seems easier to implement a developmental schema for some than for others.
Problem Solving Process
There are a number of descriptions of what constitutes problem solving within the literature (Mason et al (1985), Mayer (2002), Ernest (2000), Polya (1957)). These references have many common threads and have models of the process that are broken down into a varied number of stages. The process outlined below combines a number of the features of these existing models with our own research findings.
The C.A.P.E. model
o Making sense of the problem/retelling/creating a mental image,
o Applying a model to the problem;
• Analysis and synthesis
o Identifying and accessing required pre-requisite knowledge,
o Applying facts and skills, including those listed in mathematical thinking (above),
o Conjecturing and hypothesising (what if);
• Planning and execution
o Considering novel approaches and/or solutions
o Identifying possible mathematical knowledge and skills gaps that may need addressing,
o Planning the solution/mental or diagrammatic model,
o Reflection and review of the solution,
o Self assessment about ones own learning and mathematical tools employed,
o Communicating results.
Despite its representation, this is not a simple linear model – sometimes it is necessary to revisit and review several times – one can think of the problem solving process as a spiralling inward towards a satisfactory conclusion.
Implications for teaching for enrichment
I have discussed above the curriculum content associated with mathematical enrichment in terms of the two aspects of mathematical thinking and problem solving. For this content to have meaning, the learning (and teaching) environment needs to encourage effective use of the resources so that pupils develop the necessary skills, strategies and competence to tackle problems and use underpinning thinking skills effectively. This has implications for the second thread of mathematical enrichment – that of the teaching approach adopted. There are a number of features of such a teaching approach, building on the work of Lerman (1999), Romberg (1993) and Ruthven (1989) and takes a view of pupils constructing their own learning in a social context, where communication and sharing are central to mathematical growth and understanding. Aspects of such an approach include:
• The use of problems which encourage a problem solving approach that in turn supports mathematical thinking and the contextualising of the relevance of mathematical skills and facts (known or to learn).
• Employing the use of low threshold – high ceiling tasks
• Giving pupils time to engage with the problem before moving towards a solution (exploration)
• Focus on “doing mathematics” – pupils taking responsibility for tasks and identifying possible routes to and requirements of solutions rather than being led by the teacher.
• Appropriately targeted mediation that supports entry into problems and development of solutions without leading. Building on pupil discovery and knowledge and making connections (codification)
• Transfer of knowledge which is dependent upon individuals internalising schema with the teacher identifying opportunities.
Mathematical enrichment trails
The trails are a new concept of resource management that are being developed by the NRICH team, practising teachers and mathematics educators. They aim to combine related resources (problems, activities, games, articles, other sites) into a coherent programme of activities that have problem solving at their centre and which describe a strand of an enrichment curriculum aimed at either a particular aspect of mathematical thinking, or a particular aspect of the curriculum tackled through a problem solving approach. They also reflect the view of teaching and learning mathematics outlined above and are being described in terms of:
• their mathematical content (standard curriculum facts and skills as well as mathematical thinking skills);
• a recommended pathway, or pathways, through the items
• prerequisite knowledge;
• anticipated learning outcomes;
• guidance notes for teachers which reflect the enrichment approach to teaching tha underpins our work
• guidance notes and hints for pupils;
• formative self-assessment mechanisms which will enable medium to long term planning and evaluation.
A trail, for example, might develop and support the work on number and problem solving through investigating Magic Squares. For the most able students the work might lead to investigating the idea of isomorphisms and the underlying structure of some mathematical problems (looking for pattern and familiarity in problem solving contexts – “have I seen something like this before?”). Brighter pupils may also be encouraged to consider algebraic properties and relationships in this context. A very able student may begin to generalise and look at “higher order” mathematics, looking at articles on the subject written by established mathematicians. Whilst students struggling with identifying patterns and relationships more generally may benefit from generalising their findings when working from one magic square context to another.
A trail on “being systematic” can offer opportunities in a range of mathematical contexts (number, geometry etc) to take a systematic approach to solving the problem. Whilst other proof, or algebra based methods may be just as appropriate in any particular context, the aim is to use a range of systematic strategies to access, engage in, and eventually solve, a problem. Work on the trail may extend over weeks or months or several academic years but in every case the aim is to give some structure to the development of the related skills.
The structure of a trail will enable choices concerning the routes into the resources to reflect the needs of the pupil and underlying learning theories. Trails aim to “unpick” the opportunities being offered to pupils to use and develop their problem solving and other higher order mathematical skills in terms of content, learning theories and associated teaching styles.
Implications for Implementation
Through the intertwining of the research and development of the NRICH site, and particularly the trails, the value of this curriculum innovation is being constantly assessed. All the work is grounded in appropriate theories as well as research and classroom experience that not only clarifies and informs the development itself but throws light on current views and practice with respect to the role, content and implementation of mathematics enrichment more generally. As materials are developed and tested this in turn informs our theoretical framework.
An emerging area of interest is the nature and role of mediation and how mediation can take place, or underpinning learning theory be reflected, in the materials we produce. Current small-scale research by members of the NRICH team identifies the view of problems as rivers to be crossed rather than to be studied (the process is simply about finding the answer rather that mathematical discovery). This view acts as a barrier to encouraging problem solving and mathematical thinking skills. We are currently undertaking research into the role of mediation and how we can offer relevant mediation at a distance (Back, J., et al. 2004, forthcoming).
The clarification of the terms enrichment, mathematical thinking and problem solving have all led to a clearer understanding of the potential of NRICH to support mathematical enrichment more generally, being a vehicle for the many not simply the few.
• establishing a view of enrichment/problem solving /mathematical thinking and reflecting this view within the resources we produce.
• Placing the role of factual knowledge and skills within an enrichment framework both as a precursor and a consequence
• the identification of mediation in a “remote” environment as a key area for our future research
• continuing to reflect the importance of the social role in the construction of knowledge within an online and remote resource
• that issues related to seeing the process and/or solution as the goal rather than the answer is key to our mediation and support work
• that there is a role for assessment and that self and/or peer assessment is an area we need to investigate further.
Impact on the development of the NRICH site
The NRICH had the first phase of its relaunch in January 2004. The key features of the new site that have been driven by our research findings are:
• Transparency between levels
• Range of levels and difficulty (challenge level)
• Monthly themes
• Problems also include hints and notes
• Integration of the thesaurus
• Integration of the discussion boards
• Easier access to related material within the archive.
Impact on the development of Trails
• Clear rationale for each trail
• Structure and accompanying documentation that supports learning theories and associated teaching approaches,
• Picking particular mathematical thinking and problem solving schemes as focus for each trail
• Developmental not ad-hoc organisation of resources
• Consideration of the role of mediation and developing mediation strategies.
• The choice of self-assessment as the core assessment strategy.
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