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.

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Meaningful contexts and representations

In 1992, the International Adult Literacy Survey (IALS)^{77} 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.

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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

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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?”

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Algorithms

Algorithms form part of the numeracy learning progressions, but the progressions make it clear that learners who use algorithms and calculators need to be able to determine and justify the reasonableness of the answers they obtain (by explaining or demonstrating how they know that each answer is reasonable). If learners cannot do this, they will need to develop either a better understanding of the algorithm or an alternative approach to calculating.

The traditional algorithms are methods for working out number problems that have been developed over time. They involve a sequence of steps in a procedure that can be followed to solve a problem. Each is based on performing the operation on one place value at a time with transitions to adjacent positions. Historically these transitions are referred to as renaming, trading, borrowing etc. Traditional algorithms tend to treat the problem in terms of digits rather than the composite number that the digits make up. The traditional algorithms work for all numbers but are often not the most efficient or useful method of computing. Most often, algorithms in mathematics are associated with the vertical working form traditionally used to solve operational problems.

For example:

Although it is helpful for adult learners to know written procedures for the number operations, the standard algorithms taught in school are often not the most appropriate or understandable procedures to use.^{78} Standard algorithms are accurate and efficient, but their meaning is often unclear to learners. Steps such as borrowing, carrying, moving the decimal point and shorthand notations can be confusing to learners. They can result in “buggy” procedures that the user has no way of fixing when solutions appear to be unreasonable. When adult learners try to use procedures that have “bugs”, They often become frustrated and negative attitudes towards mathematics may be reinforced.^{79}

To those who have learned an algorithm, the process of simply following that familiar algorithm may be faster and feel more comfortable than thinking about other ways to understand and solve a new problem. Learners need to know that they can continue to use this preferred method as long as they are always able to check that the answer they have obtained is reasonable and makes sense for the actual problem they are solving.

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