Post-14 Mathematics Inquiry 

 A NIACE Response to The Post-14 Mathematics Inquiry

Published: April 2003

NIACE, the National Institute of Adult Continuing Education, is the leading non-government organisation working with adult learners in the UK. NIACE is a registered charity and a company limited by guarantee. It is a membership organisation, with individual members and more that 440 corporate members across the full range of providers, policy makers and users of adult learning opportunities. NIACE headquarters are in Leicester, England and Cardiff, Wales. The organisation employs 180 permanent staff.

NIACE’s aim is to support an increase in the total numbers of adults engaged in formal and informal learning in England and Wales; and at the same time to take positive action to improve opportunities and widen access to learning opportunities for those communities under-represented in current provision.

2. Parameters of this response

3. Terminology

‘Numeracy’ is the term currently used by the government for mathematical activity at a basic level. However, there is no generally agreed definition of numeracy and the government and its agents use different definitions for different purposes. In the Adult Numeracy Core Curriculum, numeracy is defined in terms of mathematics, as covering ‘the ability to:

and ‘the ability to … use mathematics at a level necessary to function at work and in society generally’ (Basic Skills Agency, 2001, p 3). Although not specifically mentioned in the definition, spatial relationships and the interpretation of data are included in the curriculum. Very little geometry and no algebra are included, even though Level 2 Numeracy is claimed to be equivalent to GCSE Grades A-C.

The National Numeracy Strategy for schools, on the other hand, uses a narrower, and less functional, definition of ‘numeracy’, based on number, calculation, measurement and the interpretation of data and graphical information, including the communication of methods and reasoning, but leaving out spatial relationships (DfEE, 1998). It also does not include algebra.

In this response, ‘mathematics’ or ‘maths’ are used, except where referring to government policies and initiatives, which use the term ‘numeracy’. ‘Mathematics’ is taken to include ‘numeracy’.

NIACE is particularly concerned that the needs of adults and young people who have not had access to a full school education or have not had successful learning experiences, are fully considered. In particular, people who have experienced:

Learners may be of any age, both genders, and from any ethnic or linguistic background represented in England and Wales.

Adult learners who enrol for maths classes have very variable, not uniform, expectations. These often include:

However, many adults in the socially excluded categories above may not identify maths as something they want, or see any necessity to learn.

Others may suffer from maths anxiety, which is a major barrier to adults coming forward to learn mathematics. Adults can experience acute fear and panic symptoms, sometimes just at the thought of numbers (Buxton, 1981). In a milder form, maths anxiety affects a substantial part of the population.

If learners can identify something that they are interested in learning, they can often cope with the implicit mathematical content, where they might find formal maths learning to be too big a challenge. An example of such a project, which was funded by the Adult and Community Learning Fund (ACLF), is organising a football team, where the young learners drew up time sheets, measured and recorded their weight, and worked out their fitness programmes for the gym (Tyers and Aston, 2002). These learners are more likely to accept, and be successful at, real problem-solving in the context of their interests, than being taught abstract mathematical contexts, even if they are wrapped up in supposedly relevant ‘context’. Improving learners’ confidence and self-esteem is more important at this stage than the acquisition of specific mathematical skills, like adding and subtracting. Some, but not all, learners will then be inspired to go on to further courses.

Not all potential learners will attend courses in colleges or learning centers. Adults who have not experienced successful learning at school can find the prospect of entering a school-like building too daunting. Outreach project work to provide access to education for socially excluded groups is an important part of provision. Project funding enables innovative approaches to be trialed and evaluated (Tyers and Aston, 2002). In a field where very little research has been done, this is extremely important. NIACE and the Basic Skills Agency (BSA) are doing a great deal of work in this sphere, supported by the ACLF.

Workplace education is also important for extending provision to hard-to-reach potential learners. Creating specifically tailored provision ensures that it is relevant to learners. Providing education in or near the workplace, in work time, makes access less problematic. The fact that the management of the company or service are investing resources in them, can give learners a sense of worth.

Running family maths classes in schools, where parents and children can attend together, is another approach to introducing hard-to-reach learners to adult education. The parents’ initial motivation for attending is usually to help their children, but they can become motivated to continue learning for themselves through finding the experience of family maths learning enjoyable and interesting.

The majority of offenders in prison have not had successful school careers. Many more are dyslexic than in the general population. There are many problems associated with the provision of consistent educational experiences in prison and these are exacerbated by the increasing overcrowding. There are also problems associated with continuing education, begun in prison, on release.

NIACE believes that an important part of rehabilitation is the opportunity to have successful and interesting learning experiences, including the opportunity to learn maths, or to take part in other learning activities in which maths is embedded.

It has been estimated that 40% of learners who are diagnosed as dyslexic also experience difficulties with maths (Butterworth, 2001). Short term memory, sequencing, retrieval of basic facts and directional sense can all present difficulties. Some learners can be helped by exploring different learning styles. Some of these difficulties can be partially overcome by using calculators. When calculators are prohibited in tests and examinations, these learners need special dispensation. Other difficulties can be mitigated by giving learners extra time. Many dyslexic adults have developed their own idiosyncratic methods of doing calculations. They should be supported in using their own methods, as long as they work. Taking tests, where algorithmic answers are required, can be problematic.

NIACE and LSDA have recently begun a research project to investigate the most effective ways of supporting dyslexic adult learners of numeracy, literacy and ESOL (NIACE, February 2003).

Dyscalculia is not a well-defined syndrome, but symptoms can include difficulties with the idea of number size, which make it problematic to estimate or compare numbers, or navigate up and down the number sequence, or count in twos or threes (Butterworth, 2001). Dyscalculics can have problems with translating between number words and numerals or lack an understanding of the place value system. They may find it extremely difficult to memorise number facts, but also be unable to deduce one fact from another, because of their lack of understanding of the number system. Measurement, especially proportions can be difficult, as can spatial relationships. Dyscalculics may have great difficulty in understanding word problems and deciding which operation is required.

However, all these functions appear to have separate neural networks, so that one can be malfunctioning while others are functioning normally. The fact that a learner has one or more of these difficulties, does not mean that they lack competence in other mathematical areas. Therefore, care needs to be exercised in diagnosing all learners’ difficulties in maths. Simplistic testing will not necessarily reveal their real competence.

Adults may acquire dyscalculia as a result of accidents or strokes (Caporali, 2000). It has also been found to be an early symptom of dementia (Hirono, Mori, Ishii, Imamura, Shimomura, Tanimukai, Kazui, Hashimoto, Yamashita and Sasaki, 1998). Therefore teachers working with learners with disabilities or older learners need to be made aware of the possibility of their students having dyscalculic symptoms.

Until now, there has not been a diagnostic test for dyscalculia. One has just been published which can be used by teachers and has been standardised up to age 14 (Butterworth, 2003). This may be useful for adult learners. The challenge is then for teachers to develop strategies for overcoming or circumventing the problems. Most of the research on dyscalculia has been done from a neurological perspective. Educational research is also needed in this area.

A lack of recognition of dyscalculic or dyslexic symptoms, coupled with past failure in learning, can lead to maths anxiety and avoidance of maths.

The government has identified a core of mathematical knowledge and skills which they say is needed by all to function in society and be employable (Basic Skills Agency, 2001). NIACE is concerned that this assertion is not based on research, which shows that the mathematics adults use in their everyday lives is inextricably embedded in the socio-cultural context of their activities. So that the meanings they attach to calculation, or measurement, or information, is concerned with feeding the family, catching trains, carpentry, sport, etc, or work activities, not explicitly with mathematics (Lave, 1988, Lave and Wenger, 1991, Nunes, Schliemann and Carraher, 1994, Harris, 1997, Colwell, 1998, Hoyles, Noss and Pozzi, 1999, Colwell, 2000, Hoyles, Wolf, Molyneux-Hodgson, and Kent, June 2002). Moreover, much of this research indicates that most adults are very competent at their everyday activities. What some people have difficulty with is ‘school maths’: passing tests using standard written algorithms.

Rather than being the maths that people use in their everyday lives, the Adult Numeracy Core Curriculum is a reduced version of the school mathematics curriculum for Key Stages 1-4, with the requirement that the unconnected lists of elements are contextualised in learners’ lives. The model of mathematical knowledge used in the Adult Numeracy Core Curriculum, like that used in the school mathematics curriculum, is the traditional one of culture-free, neutral and value-free pure knowledge (Mackenzie, 2002). The idea that if learners acquire abstract mathematical skills and knowledge, they can then use these like a toolkit in any context, has not been borne out by research.

NIACE welcomes the injunction in the curriculum to teachers to make numeracy meaningful to learners by using their contexts. But we are concerned that a simplistic approach to this is being advocated: of wrapping up abstract mathematical procedures in artificially contrived contexts. This is an approach that learners are likely to recognise from their schooling. For example, the concept of ratios is often taught by calculating ‘best buys’: comparing the unit prices of two products, say two tins of baked beans of different weights and prices. This is not the same as investigating how adults choose the products they buy; how they may take into account taste, texture, advertising claims, shelf-life of the product, carrying weight, their storage capacity, as well as weight and price. The use of ratios may be involved, amongst many other things, including personal preferences (Lave, 1988).

NIACE believes that the latter approach is more appropriate for adult learners. It recognizes the range of ways that adults approach real problem-solving in their everyday lives and validates them, whilst allowing learners to learn from each other.

In their Adult and Community Learning Fund projects, NIACE and the Basic Skills Agency have found that hard-to-reach learners can be attracted to courses which reflect their interests (Tyers and Aston, 2002). Where learners found that they needed some basic skills support for their activities, these were provided in the contexts of those activities.

More realistic mathematics curricula, to support the mathematics adults use in their everyday lives and work, would focus on everyday and work problem-solving, not calculation, and consist of modules geared to particular situations and requirements, with the continued possibility of developing further modules. NIACE would like to see more research into the mathematics adults use in their everyday lives to inform the development of more flexible curricula.

Accessible academic maths
Whilst the maths used in everyday life or the maths embedded in subjects that interest them are the best starting points for adult learners, they can also take pleasure in doing real mathematics: investigating such topics as the Fibonacci series and its applications in nature, magic squares, Pascal’s triangle, circle geometry, and exploring how maths is done in other cultures, simply for enjoying the beauty of mathematics. These topics do not require knowledge of fractions, decimals and percentages. If academic maths were liberated from the need to be useful, like music, poetry, drama or sport, it could concentrate on what is fascinating and beautiful about maths: exploration, finding patterns and connections. More students would enjoy it.

Effectiveness of teaching styles
NIACE strongly advocates that adult learners must be treated as adults: there should be opportunities in the learning situation for them to contribute their existing skills and knowledge, whether academic or practical. Through feeling that their knowledge is valid, learners will have the confidence to build on it. Learners must retain control of their learning and choose what they want to do.

Centralising tendencies in government policies for education are resulting in the attempt to reduce learning to one curriculum, one set of qualifications, and one set of learning materials, applicable for all students from 16 to 80+. This tendency results in learners’ voices being neglected.

Maintaining students’ interests through appropriate learning materials
The Adult Numeracy Core Curriculum requires teachers to use learners’ contexts for teaching the listed mathematical elements, in order to make learning meaningful to learners. Learning materials must feature adult concerns. In the main, learning materials developed for schoolchildren are not suitable for adult students, although they can be adapted. Similarly, materials developed for young people in colleges, which feature aspects of college life, are not necessarily suitable for adult students, especially those studying in community provision.

The idea that learning materials can be developed nationally and hallmarked by the DfES is in contradiction to the principle of using learners’ contexts for teaching. NIACE takes the view that talking with learners about their interests and developing learning materials are both part of tutors’ working practices, for which they need time and resources. But in the past, much good work developed by tutors remained in their local domains. Ways of sharing materials need to be developed, for example using the National Grid for Learning. This would give tutors the opportunity to exchange ideas while being able to adapt materials to suit their particular students’ needs.

ICT has the potential for offering learners exciting and innovative ways of developing mathematical understanding, some of which have been exploited in school maths provision: SMILE programmes, LOGO and Cabri-Geometre, for example, or the use of graphic calculators to plot real events. Spreadsheets also have a lot of potential uses. However, most of these opportunities do not appear to be being offered to adult students, although some classes are using spreadsheets for real problem-solving (Learndirect, 2002).

To develop the use of ICT with adult students, we need:

Although there is a place for drill and practice using computers, the main thrust of this development should be in the use of programmes that enable the learner:

Learning numeracy on-line
An example of the use of ICT for teaching numeracy to adults is the on-line numeracy courses Ufi Learndirect offers as part of the Skills for Life strategy (www.learndirect.co.uk). Ufi has made much of the capacity of ICT in general, and the Internet in particular, to motivate potential learners to begin to study. While this may be true, it is a pity that their courses use an impoverished version of traditional maths teaching for their subject matter in numeracy.

School teachers were recently offered computers of their own so that they could familiarise themselves with them and become confident in using them. It would be an excellent idea to extend this scheme to teachers of adults.

Adult learners, who are under no compulsion to learn, will only persist if they feel that the provision, the learning materials and the teaching are appropriate for them.

Study skills
Adult learners can benefit from explicitly exploring some of the conditions and processes which can lead to successful learning. Asking learners to keep learning diaries and share experiences of learning in a group can help them develop awareness of their own learning development. For example, they can benefit from the opportunity to explore and discuss what conditions for studying work best for them:

In a similar way, they can explicitly develop study skills:

Adult learners can also benefit from developing strategies for mathematical problem-solving which they may not have encountered before. For example:

Maths anxiety
Past experiences of maths education and family and social attitudes to learning maths can have a profoundly negative effect on learners’ feelings about maths and their perceptions of their abilities to learn it. Sharing experiences in discussions and writing can help learners understand how their attitudes to maths have been socially and politically constructed and help to reduce their anxieties about learning (Mackenzie, 2002). Investigating ways in which learners are doing quantitative and spatial problem-solving in their everyday lives and validating these activities as mathematical is another method of boosting learners’ confidence in their abilities to do maths.

Although maths is often described as a ‘universal language’ and this may be true in a broad sense, there are many differences in the way maths is expressed in different languages. Different cultures have different ways of teaching, they use different symbols, have different algorithms for calculation; and if they use mathematical problems (not all cultures do), they are written in the language that is being used for teaching.

Learners whose first language is not English may be highly educated in maths in their own language, or have had very little formal education, or be anywhere between these extremes. They need to have their mathematical competence carefully assessed. Giving them a standard test will not necessarily reveal their real level of competence: they may have difficulty in understanding the contexts of problems, as well as the language, symbols and algorithms.

ESOL students do need to learn the English form of mathematical notation (for example, the dot used for the decimal point), but it is only necessary for them to learn the standard English algorithms if they have to pass a test which requires algorithmic answers.

ESOL learners often do not want to attend special ESOL maths classes, fearing that this will be second class provision. However, maths tutors need to be more aware of potential linguistic and cultural differences, in order to teach their ESOL students effectively. This can probably be best achieved by having language teachers team-teaching with maths teachers.

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