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# Key concepts

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Last updated 26 October 2012 15:28 by NZTecAdmin

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 examples
• Understanding and reasoning
• Degree of precision
• Algorithms.

## Meaningful contexts and examples

By using familiar, real-life contexts and examples, 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 levels, all mathematics needs to be learned with understanding so the learner can apply what they learn in a variety of situations.

Knowing some mathematical facts or routines is not enough to enable learners to use that knowledge flexibly in a wide range of situations. Being able to do mathematics does not necessarily mean being able to use mathematics well for solving real-life problems. A learner who counts decimal places to decide 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. If the learner does not understand the mathematical processes involved, they will have no way of knowing why some of their answers are correct and others incorrect.

## Degree of precision

When adults need to use mathematics to solve reallife problems, there is generally a certain amount of flexibility around the degree of precision necessary. In order to choose the best approach to solving a
problem, an adult needs to begin by deciding how precise the solution needs to be. For example, a practical problem may involve working out how much carpet is needed to cover
the floor of a room. As a real problem for an adult, solving this problem may involve first asking and answering some 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 adult decides which measurement units and tools to use, based on how accurate the measurement must be. For example:

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

## Algorithms

Traditional algorithms are methods for working out answers to 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. Algorithms form part of the numeracy learning progressions, but the progressions make it clear that learners who use algorithms and calculators also need to be able to decide if the answers they obtain are 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 steps used in algorithms are known as renaming, trading, borrowing, carrying over and other similar terms. 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:

 345 x 41,380

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.

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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