As with our discussion about adult numeracy theories, adult numeracy theories are simply ideas about how adults learn maths and numerical concepts. To some extent, these mirror some of the content we’ve already covered regarding adult education in general and adult literacy in particular. Three key theories of adult numeracy include:

**Behaviourist**: For behaviourists, if you can observe a change in your learner’s behaviour due to some stimulus that you have provided as educator. The behaviourist approach promotes the idea that the teacher should provide mathematical knowledge perhaps in the form of a problem (the stimulus), which the learner absorbs and then produces a solution for (the response). Following the rules correctly results in the correct answer which educators can measure easily with tests.**Constructivist**: As with ideas about literacy, numeracy has shifted towards a constructivist approach in recent years. In the constructivist point of view, learners actively construct mathematical knowledge as they bring what they already know together with new information and experiences. Constructivist theories of adult numeracy usually follow either Piaget, who emphasised ways in which individual learners make sense of mathematics, particularly through the importance of developmental stages, or Vygotsky who or saw learning as a social activity where teachers provided “scaffolding” to help learners move to higher levels of development.**Sociocultural**: Sociocultural theories of adult numeracy build on Vygotsky’s approach. As with the sociocultural approach to literacy, the basic idea is that numeracy learning and teaching is influenced by different social and cultural factors. A good illustration of this is the kind of “hands on” numeracy knowledge that a trades person would use to estimate or measure area in a real life work situation as compared to a textbook area calculation that you would find in a school setting. Textbook mathematics problems do not usually prepare apprentice farmers, gardeners, or horticulturalists for the kinds of issues associated with area, ratios, and measurement that they experience in the reality of their daily work.

## Professional Reading

Review the questions below then read the article

- What theories underpin the approach to numeracy taken by the adult numeracy progressions that we use in Aotearoa New Zealand?

## What is numeracy?

The term numeracy is relatively new. It was first used in 1959 in the UK Crowther Report, where it was characterised as the mirror image of literacy. Since then, numeracy has been interpreted in different ways internationally, mostly because of the very different needs of the users of the term. The view of numeracy that underpins the numeracy learning progressions is about knowing and understanding: it is therefore both broad and contextualised. The following definitions most closely represent the view taken here.

To be numerate is to have the ability and inclination to use mathematics effectively in our lives – at home, at work and in the community.

*Ministry of Education, 2001, page 1*

To be numerate means to be competent, confident and comfortable with one’s judgements on whether to use mathematics in a particular situation and if so, what mathematics to use, how to do it, what degree of accuracy is appropriate and what the answer means in relation to the context.

*Coben, 2000, cited in Coben, 2003, page 10*

We believe that numeracy is about making meaning in mathematics and being critical about maths. This view of numeracy is very different from numeracy just being about numbers and it is a big step from numeracy or everyday maths that meant doing some functional maths. It is about using mathematics in all its guises – space and shape, measurement, data and statistics, algebra and of course, number – to make sense of the real world and using maths critically and being critical of maths itself. It acknowledges that numeracy is a social activity.

*Tout, 1997, cited in Coben, 2003, page 11*

The view of numeracy that underpins the numeracy learning progressions places an emphasis on the need for learners to gain:

- a conceptual understanding of mathematical knowledge, and
- the ability to use mathematical knowledge to meet the varied demands of their personal, study and work lives.

The numeracy learning progressions are based on the belief that in order to meet the demands of being a worker, a learner and a family and community member, adults need to use mathematics to solve problems.

### key concepts

Several key concepts can be identified as central to the understandings about numeracy and about adult learners that have informed the development of the numeracy learning progressions. These concepts are covered below, under the following headings:

- Meaningful contexts and representations
- Understanding and reasoning
- Degree of precision
- Algorithms.

### Meaningful contexts and representations

In 1992, the International Adult Literacy Survey (IALS) was redesigned to include a numeracy survey that assessed the distribution of basic numeracy skills in adult populations. The concepts underlying the assessment included the recognition that mathematical ideas are embedded within meaningful contexts and may be represented in a range of ways, for example, by objects and pictures, numbers and symbols, formulas, diagrams and maps, graphs and tables, and texts. The importance of teaching mathematics in meaningful contexts was also emphasised in the SCANS report (1991) and is an integral part of national adult education standards in Australia and the United Kingdom.

When adult learners need to know and use mathematics, the need always arises within a particular context. Numeracy is the bridge between mathematics and the diverse contexts that exist in the real world.

In this sense … [there] is no particular ‘level’ of Mathematics associated with it: it is as important for an engineer to be numerate as it is for a primary school child, a parent, a car driver or a gardener. The different contexts will require different Mathematics to be activated and engaged in.

*Johnston, 1995, page 54*

Many adults are unaware of the ways in which they use mathematics in the course of their everyday lives. For example, measurement is used in a great many routine activities.

All in all, measurement is revealed as a complex and somewhat contradictory area for teaching and learning: at once at the heart of mathematics and surprisingly absent, for some people, from activities which are commonly assumed to involve a lot of measurement, such as cooking, shopping and merchant banking.

*Baxter et al., 2006, page 52*

By grounding learning within authentic contexts, the numeracy learning progressions can raise learners’ awareness of the mathematics all around them – and of the mathematical knowledge, skills and strategies they already possess.

### Understanding and reasoning

The demands for adult numeracy arise from three main sources: community and family, the workplace and further learning. While each of these sources is likely to require different mathematical skills at varying achievement levels, all mathematics needs to be learnt with understanding so that it can be generalised and adapted by the learner for a variety of situations.

Knowing certain mathematical facts or routines is not enough to enable learners to use that knowledge flexibly in a wide range of contexts. Being able to do mathematics does not necessarily mean being able to use mathematics in effective ways. Knowledge of procedural operations and facts is essential to reasoned mathematical activity, but is of little value in itself. A learner who counts decimal places to determine the number of decimal places in an answer without understanding the number operation involved may get 0.7 x 0.5 correct, but 0.7 + 0.5 incorrect. The learner’s lack of understanding of the mathematical process means that they have no way of knowing why some of their answers are correct and others incorrect, because they are unable to use reasoning.

… the notion of understanding mathematics is meaningless without a serious emphasis on reasoning.

*Ball and Bass, 2003, page 28*

### Degree of precision

In real-life problems that require adults to use mathematics for a solution, there is generally a certain amount of flexibility around the degree of precision necessary. When students in schools solve mathematics problems, the problems are often purely theoretical, but adult learners need to make decisions about how to manage problems in real-life situations. In order to choose the best approach to solving a problem, an adult needs to begin by making a decision about the degree of precision required. For example, a practical problem may involve working out how much carpet is needed to cover the floor of a room. As a classroom exercise in school, the purpose of setting the problem may be to have the students learn and practise measuring skills. The task would probably involve scaled drawings with precise measurements. The students might be expected to use calculators or to apply what they have learnt about formulas and multiplying numbers to arrive at a solution. As a real problem for an adult, solving this problem may involve first asking and answering practical questions, for example:

- “How accurate do I need to be?”
- “What tools (such as a calculator, a measuring tape, or pen and paper) should I use?”

Depending on their specific purpose in this situation, the adult judges the degree of precision that would be reasonable. This could vary from very precise (for ordering and cutting the carpet) to a rough estimate (for thinking about whether or not to re-carpet). The degree of precision required dictates the measurement units and tools to be used, for example:

- “Will I use hand spans, strides, or a tape?”
- “Should I measure in metres, centimetres, or millimetres?”

### Journal Task

Write at least 250 words on how adult numeracy theories underpin your approach to teaching and training.

If you need to, use the prompt to get yourself started:

*How much of your approach to teaching adult numeracy is underpinned by behaviourism?*

- How much of your approach to teaching adult numeracy is supported by a constructivism?

- What about sociocultural theory? Do you think your approach is underpinned by any aspects of this approach to teaching adult numeracy?

Really great stuff Graeme. This course looks fantastic. This is a far better fit for literacy tutors and enthusiasts than the NCALNE. I can imagine the conversations now, Freire versus Piaget, should numeracy include language, should we teach phonics or jump to whole word recognition??!!! It’s gonna be great!

Thanks Sittingbull…! We just need a way to make those conversations happen between people in a way that leverages the technology but is still human friendly

The trick will be to make it relevant to our graduates from trades and vocational training, i.e. the NCALNE (Voc) audience who want to go further… has to retain practical nature as well… that’s my current conundrum… hopefully the outcome of the TROQ review will clear this up for me