Leading view papers – Days 1 to 7

The ‘EIS’ approach to motivating reluctant learners

Introduction

This paper addresses the central issue of this conference: for too many learners, mathematics in school is boring and disengaging. We attempt to identify some reasons for learner boredom and alienation and suggest a possible remedy, which forms the basis of the authors’ current classroom-based research, and which we have found can motivate and inspire school-age learners of mathematics. The project, called EIS (Entering Into Symbols), brings together a range of learning styles, including kinaesthetic, use of computer-based images and traditional pencil and paper approaches to handling conventional mathematical notation.

Why are learners bored?

Cast your mind back to your school lessons - particularly mathematics lessons. If your schooling was anything like ours, these lessons were pretty boring. Yet it is a fair bet that, on average, readers of this article were the relatively high achievers in their school, for whom the rewards of being successful will have certainly sugared the learning pill.

Of course, we like to think that schools are much nicer now – learners are treated with more respect and their opinions and preferred learning styles are acknowledged and, where, possible, accommodated when difficulties are encountered. While this may have occurred to some extent, it is our experience from visiting many schools and talking to learners and parents, that school lessons are still boring for too many students, for too much of the time.

So what exactly does boring mean? At its heart, boredom involves disengagement from ongoing classroom tasks, and this can happen for many reasons. A central problem is to do with the nature and level of challenge in the tasks being offered. For some learners, the tasks are too easy. But for many underachieving students, it is their perception that the tasks are too hard, when set against their level of interest. Whether or not their perceptions are correct, these students feel that, because there are a number of significant missing links in their mathematical knowledge, further progress is simply not possible – so why bother even trying?

There are three main factors we have examined in our EIS research that impact strongly on this unsatisfactory state of affairs and these are explored below.

Learning and teaching styles

Many learners – particularly those who might be labelled ‘less academic’ – feel culturally remote from the content of what is taught, as well as the style of learning on offer. Unfortunately, mathematics doesn’t seem to reward common sense. Even successful students take the view that the closed, artificial world of mathematical problem-solving is best explored by abandoning everyday reasoning. This is very unfortunate! Although mathematics is rightly seen as an abstract subject, it doesn’t follow that mathematical learning needs to be made more difficult by being conducted in a manner that separates learners from their thinking skills and everyday experiences.

Next, mathematical ideas are too often introduced wrapped in conventional symbols and notations. As these are imperfectly understood, many learners fail to grasp the gist or overall purpose of the topic, when described in these terms. When students ask their teacher, ‘What’s the point of this?’, they are not necessarily dismissing the topic out of hand. Rather, they are seeking some sort of common sense understanding of what the topic is about and, indeed, why it is deemed by teachers and society to be sufficiently important to be worth putting effort into learning. These are real questions to which learners deserve thoughtful answers in order to prepare their minds for learning.

Third, we suggest that many learners would benefit from greater opportunities to learn about mathematical ideas in more accessible ways and, in particular, to be able to engage in kinaesthetic learning. However, merely handling physical models of mathematical concepts is not enough. Learners also need to be offered purposeful tasks where they can see more clearly the overall goal and therefore be able to know when they have succeeded in the task. While learning through physically handling concrete embodiments of mathematical concepts is still common in some primary classrooms, sadly, in many UK secondary lessons, this approach is seen by teachers and students as babyish, and so avoided. As one 11 year old remarked to us last week, while enjoying  a lesson using base 10 blocks, 'I haven’t been allowed to use apparatus like this since I was in year 3'.

The role of ICT

The learning revolution in UK schools, powered by ICT, hasn’t really happened. Although learners are clearly developing skills and confidence with using computers and calculators, many of these skills are self-taught from student experiences outside the classroom. Too often, classroom ICT-based tasks are grafted onto existing structures, based on old-style pencil and paper approaches to learning. Furthermore, ICT use tends to be offered to learners as a ‘one-off’ experience, rather than being thoughtfully placed within a wider teaching programme.

A key, potential benefit of successful ICT use is its capacity to develop, in students, their powers of mental imagery, which they can invoke in support of future learning. Imagery is particularly effective if linked to contexts, stories and prior everyday or physical experiences. However, this use of imagery in ICT use has not been much exploited with the pupils we meet.

Mathematical obstacles

We can all remember the horrible experience of sitting frozen and stuck in a mathematics lesson, being unable to progress. We suggest that when students say they find maths boring, many of them may be simply expressing dissatisfaction at the unhappiness they have experienced in mathematics lessons when they have felt stupid, or worse, publicly humiliated by their failure to understand or perform a mathematical task. It is obvious that learners will only be motivated to tackle something that is hard if they are both interested or motivated, and sufficiently confident there is a reasonable chance of success.

In mathematics, there are certain particular obstacles for students to overcome. For example, there are difficulties in understanding the specialised language and notation. Second, in order to progress, most learners feel the need to grasp the overall purpose of the task and this may be hard for them to elicit. But there is a third important consideration that is explored here, which has to do with learners’ perceptions of conceptual load – that the bite-size of the skill or concept that they are trying to master may be too large to be able to comprehend quickly, as a result of which they simply give up. Ironically, the maxim that ‘it is better to have tried and failed than never to have tried at all’ simply does not hold true in a classroom mathematics lesson. Kids who are seen to have tried and failed are dismissed as dim, whereas the ones who never tried at all may yet be bright – it is just that they have never permitted the system to test them.

An underexploited role for ICT is to help reluctant learners through these obstacles, making it easier for them to achieve their goals by offering a lot of support initially before gradually weaning them off it. To borrow from the terminology of Vygotsky, we believe that learners operate most successfully when:

• the learning is contained within their ‘zone of proximal development’, that is, where there is some element of challenge, but not too much
• the ICT can be used to ‘scaffold’ the structure of their learning
• the scaffolding structures can be made to ‘fade’ so that learners can subsequently achieve independent mastery of mathematical concepts when presented in conventional forms.

The ‘EIS’ approach

The EIS research undertaken by the authors is based on an innovative approach to the teaching of mathematics topics that unsuccessful pupils tend to find difficult. We have chosen to start with subtraction, basic algebra and fractions. Our work on each topic is structured around a three-phase teaching programme, which we refer to as ‘EIS’. Based on the ideas of educational psychologist, Jerome Bruner, E, I and S represent three ways of understanding mathematical ideas that he called enactive (that is, physically handling objects that embody the concept), iconic (based on pictures of the concept) and symbolic (where the concept is represented through conventional mathematical notation). Too often in mathematics, learners are rushed into using symbols. Where this fails, and often it does, it may be because learners lack any clear foundations – for example, a helpful physical model (the ‘E’ stage) of the concept, associated mental imagery (the ‘I’ stage) and, crucially, mathematical understandings that connect with existing intuition and common sense.

We have chosen to represent the initial ‘E’ stage, with a physical or concrete embodiment of the concept. To take the example of teaching subtraction, simply handling base 10 equipment does not teach place value without learners also doing so to a purpose. To this end, we set subtraction in the context of selling sweets in a tuckshop, thereby giving learners a reason for having to ‘unpack’ 10s, 100s and 1000s. The purposefulness of this task helps learners to connect their learning with their common sense.

We have tried to exploit learners’ imagery (the ‘I’ stage) by asking them to work with software applications (in the form of specially-created Java applets). This ‘I’ stage provides opportunities for quick and efficient exploration of the imagery introduced with the preceding physical task (the ‘E’ stage) and learners can use the software either independently or as part of a class lesson.

Finally, learners move on to engage with the concept symbolically using conventional mathematical notation (the ‘S’ stage). The aim here is for them to become competent with reading, writing and performing mathematics using pencil and paper. A feature of the design of our applets has been that they contain an option (usually the final one on offer) where the supportive imagery is put to one side and, instead, users are asked to do the same mathematical tasks using conventional notation. A useful feature of the software is that they can, at any time, toggle between ‘I’ and ‘S’ and so remind themselves of the imagery that they have already acquired. This can be a valuable reminder of what the symbols represent, what the mathematical symbols are really saying and what the overall purpose of the task is.

The central questions of this conference provide educators with a strong challenge in terms of building learner confidence and enthusiasm. Our work on the ‘EIS’ project suggests that we can go some way towards achieving this goal by:

• legitimising and encouraging learners’ belief in their own common sense and intuition by providing them with purposeful and meaningful tasks
• where possible, presenting problems initially in concrete forms, so that the overall goal is clear (the ‘E’ stage)
• drawing attention to the underlying imagery, which can provide a useful learning metaphor when similar ideas are subsequently presented symbolically
• offering learners a computer micro-world within which they can learn to manipulate ideas quickly and successfully (the ‘I’ stage)
• delaying the introduction of symbolic forms of representation (even to older learners) until the purpose of the task is well established and supportive mental imagery is available to them.

Clearly, these approaches take time – time for learners to work on a task, time to think, to build on experience and to reflect on what the feedback tells them. Learners also need easy access to appropriate hardware/software (there are many practical obstacles in schools to using non-networked software). Ideally, older learners should be encouraged to use apparatus when needed, but perhaps it is easier with adolescents to persuade them to experiment with software representations of apparatus.

If you are interested to see and use our EIS software, it is freely available at: www.eisMaths.co.uk

References

Bruner, J (1966) Towards a theory of instruction. Harvard University Press, Cambridge, MA.
Duke R and Graham A (2007) Inside the letter: mathematics teaching 2000.
Vygotsky, L. See: www.west.net/~ger/vygotsky.html

ABOUT THE AUTHOR

Mr Alan Graham, Centre for Mathematics Education, The Open University, UK. Ms Sue Johnston-Wilder, Institute of Education, University of Warwick, UK. Mr Roger Duke, School of Information Technology and Electrical Engineering, The University of Queensland, Australia.

 

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