TEACHING MENTAL CALCULATION STRATEGIES

Mental Calculation Strategies



Mental Calculation Strategies

Task 1

When simple arithmetic problems are calculated mentally, the approach used is commonly called a 'strategy. When attention is focused on "efficiency" and "effectiveness" (Askew, 1998) in mental calculation, however, some of the approaches used by children are not considered to be 'strategies', in particular those involving counting, and those where a visualised 'sum' in the mind is made in which the figures are manipulated mentally in ways corresponding to the standard paper and pencil algorithm. In each of these "a set procedure is followed irrespective of the numbers or the order of difficulty" (QCA, 1999, p. 14). The element of flexibility, where "calculations which may appear similar can be amenable to the use of different strategies depending on the numbers involved" (op. cit.), which is crucial to efficient and effective calculation, is missing (Koshy Murray 2002).

The sorts of mental calculation 'strategies' which are needed for flexibility are those in which the operands of the number problem are transformed in some way to make them easier to deal with, perhaps being broken up into constituent parts, or treated in terms of nearby numbers. Such inference-based approaches to mental calculation have been sorted and labelled by different writers in different ways and at different levels of generality.  for example classified these strategies into two broad categories: 

1. accumulation or iterative strategies, in which "a transformation on one of the operands is followed by a compensating transformation on the answer" (p. 301), for example 83 -27 is approached as 80 - 20 [60] then - 7 [53] then +3 [56]; and

2. replacement strategies, in which "transformations are made on both operands in the original problem before any attempt at computation, resulting in an equivalent numerical expression which is easier to solve" (op. cit.), for example 83 - 27 is first 'transformed' into 86 - 30.

Beishuizen and Anghileri (1998) describe two very common kinds of strategies that seem to arise differentially as a result of different emphases in the mathematics programmes of different countries:

1. The 'N10' strategies of "jumping by tens (up or down) e.g. 48 + 26 via 48 + 10 (= 58) + 10 = 68, 68 + 6 = 74" (p. 523) which develops in Dutch Realistic Mathematics Education contexts, probably through the extensive use of the 'empty number line' (op. cit.); and

2. The ' 1010' strategies in which "tens and units are split off and handled separately" (p. 523), e.g. 48 + 26 via 40 + 20 = 60 and 8 + 6 = 14, answer 60 + 14 = 74, which is widely used in the USA and UK, probably as a result of an emphasis on the formal 'place value' structure of 'Hundreds, Tens and Units'. More specifically Thompson (1997a and 1997b) identifies, for addition (for example):

Derived fact strategies, where something that is known is altered to find an unknown, e.g. 46 + 57 as 50 + 50 (=loo) + 7 - 4 (=103)

Cumulative sum strategies, where one number is broken into parts and added bit by bit to the other number, ...