Lily Allen’s recent tweet to Schools’ Minister Nick Gibb – “I left school 15 years ago and I've not used Pythagoras's Theorem once” – generated a wave of responses. Some agreed with her sentiment, citing the irrelevance of their school mathematics experiences; some justified the place of school mathematics by the critical role mathematics plays in today’s ubiquitous technology:
Many of us in the classroom have to field questions along the lines of “But when am I ever going to use this piece of algebra / trigonometry / geometry / nonsense outside school?” One response to the challenge of making mathematics in the classroom relevant to all learners is Realistic Mathematics Education (RME), initiated in the Netherlands and subsequently explored in projects in the US and the UK.
The starting point for RME is that “students should develop their mathematical understanding by working from contexts that make sense to them” (Dickinson and Hough, 2012). Learners use their intuition to analyse a meaningful situation, whether from everyday life or even something purely mathematical. The key is that the context is “realistic”, in the sense of something learners can “realise”, from the Dutch verb “zich realiseren”, meaning “to imagine”. From their intuitive models, through guided instruction, learners are led to build more formal models, moving from “models for the situation” to “models of the situation” (van den Heuvel-Panhuizen, 2003, pp.14-15). An initial model may be as simple as a picture, but then this model is gradually refined and made more abstract, so that the model can become a tool for solving other problems. In this way models bridge the gap between the informal and the formal – but the route back to the context is always reviewed, so that the connection with the abstract is maintained. The development of more formal models is described as “progressive mathematisation” (Hough and Gough, 2007). Treffers distinguished between two types of mathematisation: “horizontal mathematisation”, the process of using “mathematical tools to organize and solve problems situated in real-life situations” and “vertical mathematisation” which “concerns moving within the abstract world of symbols”, “using connections between concepts and strategies” (van den Heuvel-Panhuizen and Drijvers, 2014, p. 522). The two modes of mathematising are considered of equal value and may overlap.
Dickinson and Eade (2005) give an example of the “progressive formalisation” of models in learning about fractions when investigating a problem concerning cutting up submarine sandwiches (figure 1). Models may start as pictures and then progress through levels of abstraction. The "term ‘model’ is not taken in a very literal way. Materials, visual sketches, paradigmatic situations, schemes, diagrams, and even symbols can serve as models” (van den Heuvel-Panhuizen, 2003, p.13):
Figure 1 (Dickinson and Eade 2005)
According to Dickinson and Hough (2012), the distinctiveness of RME lies in:
- “Use of realistic situations” to allow students to develop their mathematics “as opposed to using contexts” for applying learnt mathematics.
- “Less emphasis on algorithms and more on making sense and gradual refinement of informal procedures.”
- “Emphasis on refining and systemising understanding.”
- “Less emphasis on linking single lessons to direct content acquisition and more on gradual development over a longer period of time. Students stay with a topic for long periods of time, remaining in context throughout.”
- “Discussion and reflection play a significant part in supporting student development.”
In an evaluation of two projects run in the UK, targeted at low prior attaining Key Stage 3 and Key Stage 4 pupils, the Centre for Evaluation & Monitoring at Durham University (Searle and Barmby, 2012) found a number of benefits as well as issues. The teachers “reported that the contexts and related activities interest the pupils and so engage them in the lesson … The starting contexts are rich and sometimes [the] pupils do not realise they are doing maths; this can only be a good thing for pupils who have so little confidence in the subject” (Dickinson and Hough, 2012).” It was found that several lessons might be needed to internalise the models, but once achieved the pupils “can understand how these models can be applied in a variety of contexts” (Searle and Barmby, 2012). “Exposing students to new, rich contexts and at the same time highlighting the ‘mathematical’ elements of these situations, allows children to learn maths that they see as ‘relevant’ but that also contains all the ‘abstract’ content that they would learn in a more ‘traditional’ classroom setting.” (Dickinson and Hough, 2012). Using assessment data from Year 7 pupils it was found that “those pupils who had experienced RME were not only more likely to solve a problem correctly, but showed considerably more understanding through their ability to explain their strategy” (Searle and Barmby, 2012).
Key issues that emerged from the evaluation included concern from parents and school management that little was written in the pupils’ exercise books, and that there was a lack of formal assessment. At the time of the evaluation there was also a perceived incompatibility with GCSE, although the problem solving approach may now be more in sympathy with the reformed examinations and the new (from September 2014) National Curriculum. The two other key issues identified by the evaluation related to pupils experiencing a mix of approaches, some teachers using RME, others not, and the need for “a support network of teachers” for “initial training and ongoing professional development” (Searle and Barmby, 2012). Teachers emphasised the need to understand the philosophy and to be trained in the use of the materials: “you can’t just pick up the books and use them; it will not be effective” (Dickinson et al, 2011).
Perhaps this does not directly address Lily Allen’s concern that the mathematics we teach is not explicitly relevant to most peoples’ everyday lives. But if we believe a mathematical education is of value, does RME provide a route for making “mathematising” relevant and engaging? How do you respond to pupils’ groans that maths is not relevant – or do you have an approach that overcomes their concerns from the outset? We would like to know your experiences.
MEI has a webpage devoted to RME including links to videos demonstrating use of RME resources.
The Freudenthal Institute has a number of applets to support “Maths in context”.
Two of the references below, Hough & Gough and van den Heuvel-Panhuizen, contain examples of RME models and how they can be used.
The American Mathematical Society has produced Mathematical Moments, a series of posters and associated podcasts on a wide range of applications of mathematics, from thwarting poaching of rhinos, to designing rollercoasters, and improving understanding of the dynamics of cities.
Dickinson, P., & Eade, F. (2005). Trialling realistic mathematics education (RME) in English secondary schools. Proceedings of the British Society for Research into Learning Mathematics, 25(3).
Dickinson, P., & Hough, S. (2012). Using realistic mathematics education in UK classrooms. Centre for Mathematics Education, Manchester Metropolitan University, Manchester, UK.
Dickinson, P., Hough, S., Searle, J., & Barmby, P. (2011). Evaluating the impact of a Realistic Mathematics Education project in secondary schools. Proceedings of the British Society for Research into Learning Mathematics 31 (3).
Hough, S., & Gough, S. (2007). Realistic Mathematics Education. Mathematics Teaching Incorporating Micromath, 203, 34-38.
Searle, J., & Barmby, P. (2012). Evaluation report on the realistic mathematics education pilot project at Manchester Metropolitan University. Durham: Durham University.
van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational studies in Mathematics, 54(1), 9-35.
van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In Encyclopedia of mathematics education (pp. 521-525). Springer Netherlands.