General principles of mental computation Facts about mental computation Many children can solve computation problems mentally BEFORE they learn the relevant formal written algorithms at school. Most mental methods are not taught. People work them out from their good understanding of place value, their number sense and their understanding of the meaning of the arithmetic operation and its properties. People good at mental computation use a wide variety of methods. The methods used for mental computation are often quite different to the paper-and-pencil algorithms taught at school. In general, less competent students use less efficient strategies (such as counting on by ones rather than by tens) and they use them for longer. Focussed teaching is needed to help them move on. People good at mental computation select strategies which do not make high demands on short-term memory. This is the reason why short term memory does not correlate highly with proficiency in mental computation. A good knowledge of number facts is essential for mental computation. It reduces the demands on short-term memory. Teaching rules such as "add a zero to multiply by ten" without understanding is dangerous because they are misused by all but the best students. Some mental strategies are cognitively easier than others to understand and to create. For example, breaking one number into constituent parts as in decomposition subtraction is cognitively easier than changing two numbers to an equivalent calculation (compare the principles behind decomposition and equal additions subtraction). Characteristics of mental methods Mental methods are often varied to take advantage of known properties of the actual numbers in the problem. For example, mental methods use facts such as 8 is close to 10, 25 is one quarter of 100 or 6 and 4 add to 10. Favourite number combinations are often used as a basis of computation. Many mental methods follow unconventional patterns like subtracting or multiplying from left to right so that the big quantities are dealt with first (e.g. hundreds before ones). This is advantageous when an estimate, rather than a precise answer, is enough. In real life, estimation is as important a skill as exact calculation. It is a skill essential to complement calculator use. It is common in mental computation to modify the question and then compensate later (eg. by rounding, doubling, halving etc). Mental methods are often based on using round numbers (e.g. 600, 1400, 30). In contrast, some formal written algorithms are hard to carry out with round numbers (think about 1000 - 657 done by a formal subtraction algorithm). Children make many mistakes dealing with zero in formal written algorithms. Mental computation is often step-by-step, rather than dealing with all the relationships in the problem simultaneously. Mental computation sometimes uses a primitive version of an operation. For example, addition may be done by counting on, multiplication may be done by repeated addition. For many people, the types of numbers that can be dealt with by mental computation is limited. For example, many people can calculate with 1/2 but not with other fractions.