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Making Sense of Mathematics

An approach to learning mathematics at Key Stage 4 based on RME

Background

The Freudenthal Institute, University of Utrecht was set up in 1971 in response to a perceived need to improve the quality of mathematics teaching in Dutch schools. This led to the development of a research strategy and to a theory of mathematics pedagogy called Realistic Mathematics Education (RME). RME uses realistic contexts to help pupils develop mathematically. Pupils engage with problems using common sense/intuitions, collaboration with other pupils, well judged activities and appropriate teacher and textbook interventions. This approach to teaching is used in Dutch schools today and continues to be refined.

In 1991, the University of Wisconsin, funded by the USA’s National Science Foundation and in collaboration with the Freudenthal Institute, started to develop the Mathematics in Context (MiC) approach based on RME. The initial materials were drafted by staff from Freudenthal Institute on the basis of 20 years of experience of curriculum development. After revision by staff from the University of Wisconsin, the materials were trialled, revised and re-trialled over a period of five years.

The first version of MiC was published in 1996/7 and it has undergone several revisions since then. The teacher material, which supports the pupil books, provides a comprehensive analysis of issues pertaining to the topic and provides the teacher with insights into teaching and learning trajectories.

The research evidence from the USA suggests that this approach to teaching was particularly effective in raising standards and influencing pupils’ and teachers’ attitudes1.

In 2004, the Gatsby Charitable Foundation agreed to fund a project, run by Manchester Metropolitan University (MMU), based around trialling RME over a three year period in English secondary schools. At that time, the Economic and Social Research Council (ESRC) also agreed to fund research into how teachers’ beliefs and behaviours change as a result of engagement in the project. Essentially the project was conceived in terms of trialling the MiC materials through Key Stage 3 in a variety of schools (in total over 1500 pupils and 35 teachers were involved in the project). The project pupils fully utilised the MiC materials over a period of three years but with the expectation that they cover the National Curriculum and take the end of Key Stage examination. The aims of the project included exploring pupil development over time and generating an understanding of the needs of teachers trained in different paradigms. The outcomes from this project are summarised in Anghileri, J. (2009)2 and Hanley, U. et al (2007)3.

Moving Forward

The Making Sense of Mathematics (MSM) project began in 2007 as an extension to the work of the Gatsby funded Key Stage 3 project. Resources for the MSM project were produced as a result of collaboration between the Freudenthal Institute and MMU and currently consist of a series of 9 booklets designed to deliver the Key Stage 4 Foundation level curriculum. These booklets build upon the experiences gained from the Gatsby project and take account of difficulties highlighted by the Key Stage 3 teachers such as the need for RME based materials which feature British contexts and are more closely linked to UK assessment systems.

The MSM project has involved Foundation level classes from 6 schools in the first cohort and 10 schools in the second cohort. MMU has supplied resources to these schools and has provided ongoing support in the form of twilight training sessions and school based observations. Feedback given by the teachers has been used to revise the materials which are currently in their second version.

There have been several challenges related to working with Foundation pupils using RME based pedagogy:

(i)Foundation pupils have a repeated history of failure in mathematics; consequently they lack confidence and are reluctant to expose their ideas. Many display a range of well honed non-engagement tactics and teachers have reported that it can take several months for the pupils to start to interact in a meaningful way, which allows for the processes of listening to, discussing with, and learning from others.

(ii)Foundation pupils are well rehearsed in proceduralways of engaging with mathematics, despite their lack of success with this approach to learningand have long since lost a belief that mathematics could make sense to them. It can take time for them to recognise that less formal ways of approaching problems, more closely linked to their intuitions are indeed valid and can be very fruitful.

There were similar issues during the Key Stage 3 project in terms of recognising the need for it to be the pupils who are “doing the maths” rather than the teacher, but this has been even more striking at Key Stage 4. It highlights the need for CPD which enables teachers to create genuinely interactive classrooms even when working with less inclined pupils.

Key Findings from the MSM project to date:

Teachers

Working with MSM can influence the practice of teachers in the following ways.

(i)Over time the teachers start to recognise the need to engage pupils in genuine debate as they explain and listen to each other’s mathematical thinking. In this pursuit the teachers begin to acquire a repertoire of strategies to promote such engagement.

(ii)They realise the power of context not only as a means of engaging pupils in debate but also as a domain where pupils can make sense.

(iii)They begin to ask different types of questions which prompt the pupils to move between different mathematical representations and to visualise. Examples of such questions are ‘Where can you see the 120 squares? (showing a picture of an empty 15 by 8 rectangle), and ‘Where can you see the diameter in this? (pointing to the circumference of a circle).

(iv)They begin to question their beliefs about what is effective teaching and comment on the dilemma caused by matching this approach with the objective led climate in British classrooms.

(v)They begin to use the MSM resources and approach with some of their other classes.

Pupils

Working with MSM can influence the behaviour of pupils in the following ways.

(i)Using contexts which are familiar to them can prompt reluctant pupils to contribute and to articulate their thinking in a domain which appears less obviously mathematical and therefore, to them, less threatening.

(ii)Over time pupils begin to re-engage with mathematical thinking. They start to show more confidence in describing their own methods and a greater inclination to listen to others. In all areas of the curriculum it is possible to see evidence of mathematical development.

(iii)The early stages of analysis of Problem Solving data for MSM pupils from Cohort 1 has revealed considerable progress over control pupils in their approach to a fractions question and to an area question. MSM pupils tended to access these questions by drawing pictures, by making sense, as opposed to performing some routine on the numbers.

(iv)Pupils recognise that this approach is different, they comment ‘It doesn’t feel like you are doing maths’, ‘It’s like you are doing geography….lots of subjects’. Likewise some pupils who enjoy working procedurally through exercises find it difficult to adjust to a more exposed way of learning.

Implications for developing MSM at Higher Level

In designing and writing the materials for Higher level pupils, the following issues will be explored in detail.

(i)One of the concerns raised about the RME approach is the facility it has for enabling the learner to ‘vertically mathematise’ (to have the ability to move from informal or contextualised mathematics to more formal or abstract mathematics). This is pertinent to achieving success at higher level where pupils are required to work effectively and efficiently with formal mathematics across a variety of representations. In the previous projects we saw evidence that pupils were vertically mathematising . For example many pupils started to recognise the power of the ratio table model for solving problems from a wide range of mathematical topics. What models and learning trajectories are helpful to enable vertical mathematisation at Higher level?

(ii)The use of context: - What contexts are helpful at Higher level to enable pupils to make sense? Is it necessary to return to real life contexts or can these contexts come from the world of mathematics? How do we design materials which prompt pupils to shift between the levels of abstraction and to recognise the power of returning to a previous level as a means of making sense of the formal?

(iii)The use of models: - Some extremely powerful models emerged at Foundation level including the number bar, the ratio table and the use of squares for area. What are the significant models at Higher level, those that will be effective in promoting higher order thinking?

(iv)Higher level pupils have usually experienced a good deal of success through learning mathematics in a procedural way; many will not see the need to change their approach. What strategies can the teacher employ to promote real engagement with mathematical thinking and re-focus pupils on exploring underlying structures as well as learning how to ‘do’?

(v)An emerging thread from the MSM project is the use of questions which

prompt pupils to look for one mathematical representation within another. What strategies will be effective at Higher level as a means of enabling pupils to make links across the mathematical domains?

In order to address these issues there is a need for:

(i)Consultation with the Freudenthal Institute. In conjunction with MMU they devised the outline for the Foundation scheme and wrote the first few booklets. They have vast experience writing across all levels including higher ability.

(ii)A field study which closely examines the implementation of some sample material at Higher level prior to writing the whole scheme. This will take place in 09/10 working closely with 4 schools that have previous experience using RME. The field study is to include:

(a)Consultation with experienced RME teachers;

(b)Observation in RME classrooms;

(c)Discussion with a range of higher level pupils.

Implications for adopting MSM across the UK

The long term vision is that courses like MSM will be adopted across the UK throughout the various Key Stages. The following are some of the issues that would need to be considered before this could happen.

(i)The need for ongoing CPD over a substantial period of time was highlighted in both the Key Stage 3 and MSM projects. This is a radically different approach and teachers need guidance about how to act and what to expect both in terms of their own development and the development of their pupils. Given the current ‘rarely cover’ situation in schools CPD would need to take the form of twilight sessions and/or paid holiday INSET days.

(ii)It should also be noted that time is needed for the practitioners to acquire the appropriate skills and to develop a level of understanding of RME which enables them to make their own choices about how to act.

(iii)Online support is needed as a means of updating schools, supplementing resources and providing a helpline service to address individual queries. Several teachers in the Key Stage 3 project experienced doubt around the November of the first term. Support from the Curriculum developers and from experienced RME practitioners is particularly useful at this time.

(iv)An emerging factor from the MSM project is the recognition that teachers need to develop a repertoire of interactive teaching skills alongside the RME pedagogy. The training programme must address this need.

(v)The use of classroom video has been extremely productive as a means of enabling teachers to analyse their practice and to share effective strategies. This would need development into a video package which includes lesson excerpts with commentary.

(vi)The RME approach provides a strikingly different route through the mathematical content of UK national schemes. In recognition of this it will be necessary to outline trajectories through the mathematics curriculum which take as much account of the journey as the outcomes. This would help to relieve teachers from the pressure they feel to teach to a formal end-point even when this is not appropriate.

(vii)Further classroom based research would be needed to consider the possible adoption of this approach at Key Stages 1 and 2.

Conclusion

There are many lessons to be learnt from other countries where RME has been introduced successfully.

These include the need to develop and trial scaffolding materials which take account of research in RME and teachers’ needs, and to provide high quality CPD to ensure that teachers have a deep understanding of the pedagogy they are using. Meeting these requirements forms a major part of this project.

There are, however, other issues that are specific to the UK, particularly those associated with the assessment regime and the culture within which education is taking place. It would be quite unrealistic to expect to achieve success simply by importing a package from another country. It is essential for RME to be presented in a form that is fit for purpose in the UK. This, too, is being addressed in this project.

References

1.Romberg, T. A. (2001). Designing middle-school mathematics materials using problems set in context to help students progress from informal to formal mathematical reasoning. Madison: University of Wisconsin-Madison.

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2.Anghileri, J (2009) Gatsby Project: Mathematics in Context: Summary Evaluation Report. (Provided as a separate document)

3.Hanley, U. and Darby, S. (2007) Working with curriculum innovation: teacher identity and the development of viable practice, Research in Mathematics Education, vol 8, pages 53-66.

Making sense of MathematicsJuly 2009