التربية والتعليم

Mental strategies used by school students in solving problems involving operations with whole numbers have been documented for some time. There is, however, little information about how students solve fraction, decimal, and percent - or part-whole - number problems mentally. This is despite the fact that there is an increased emphasis on facility with these types of numbers as students move from primary to secondary school. In this study 24 students from grades 3 to 10 participated in individual interviews involving mental computation. This paper will document mental strategies used in operations involving part-whole numbers and relationships. Of particular interest are the links to many of the mental strategies that have been observed with whole numbers. As well, responses will be analysed with respect to their conceptual or instrumental usage. Since many students appear to find part-whole number relationships conceptually challenging, understanding the strategies children use when solving part-whole problems mentally is valuable. This knowledge may help teachers in strengthening students’ understanding about intuitive ideas for part-whole numbers, in providing transitions to written algorithms, and in reinforcing estimation practices that will assist in checking answers obtained in other ways, including by calculator

In this article, data from a study of the mental computation competence of students in grades 3 to 10 are presented. Students responded to mental computation items, presented orally, that included operations applied to fractions, decimals and percents. The data were analysed using Rasch modelling techniques, and a six-level hierarchy of part-whole computation was identified. This hierarchy is described in terms of the three different representations of part-whole reasoning – fraction, decimal, and percent – and is elaborated by a consideration of the likely cognitive demands of the items. Discussion includes reasons for the relative difficulties of the items, performance across grades and directions for future research.

This paper reports on part of a larger study that investigated this question with students in the Perth Metropolitan area who were given the same mental computation test twice – once with the items in context and once without. Money was used as the context and students’ experience with money was also measured. Despite many curriculum documents espousing the importance of teaching and learning mathematics in context, this study found that neither experience with nor use of the context of money had any effect at all, except in Year 3. Keywords: context, mental computation, number sense, primary school, secondary school Teaching and learning mathematics in context is often considered to be a significant goal in school mathematics, as confirmed in various curriculum documents (Australian Education Council, 1991; Curriculum Council, 1998; NCTM, 2000). However, it is generally accepted that this goal is given much less prominence in Asian countries. The landmark study by Carraher, Carraher and Schliemann (1985) clearly demonstrated students’ higher performance on market-place mathematics – a context within their experience – compared to the same items presented in a ‘school mathematics’ format. Maier (1980) used the term ‘folk mathematics’ to define mental computation and estimation that is developed by individuals through self-discovery – as distinct from school mathematics which is often unrelated to real life. The nature of folk mathematics implies that it is embedded in context. A number of researchers have used money as a context – for example, Guberman (1992); Nunes, Schliemann, and Carraher (1993); and Irwin (2001) – but these studies have been somewhat limited in scope. Current mathematics curricula are devoting greater attention to mental computation. This is stressed by leading researchers in the area of number sense who see mental computation as the way forward in the development of number sense, alongside the need to reduce traditional written computation due to the widespread use of calculators (McIntosh & Sparrow, 2004). In a study in Taiwan, Yang (1995) found that while students performed well in written computation items, they struggled when the same items were presented as requiring an estimate – thus the need to use mental computation and number sense.