Watson, J.M., Kelly, M.N., & Callingham, R.A. (2004, December). Number sense
and errors on mental computation tasks. Refereed paper presented at the annual
conference of the Australian Association for Research in Education, Melbourne,
December, 2004.
The analysis of errors while completing mental computation tasks reported in
this paper represents the first stage of analysis of 5535 test responses from
students in Grades 3 to 10 over a period of three years to various subsets of
374 items. Following a previous analysis that suggested the items represented
eight increasing levels of difficulty covering nine sub-domains of basic number
skills, this report focuses on responses to items at Level 6. Of particular
interest are the performances across the grades, the types of errors diagnosed,
and the relationship of errors from different types of operations. In analysing
errors a developmental approach is adopted, suggesting that more than the
“right-wrong” nature of responses is involved. Some errors appear to demonstrate
a “partial number sense” that could be used to help construct more complete
understanding. Suggestions for future research and classroom practice are made.
Callingham, R.A., & Watson, J.M. (2004). A developmental scale of mental
computation with part-whole numbers. Mathematics Education Research Journal,
16(2), 69-86.
This report presents data from a study of the mental computation competence of
students in Grades 3 to 10. 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.
Caney, A., & Watson, J.M. (2003). Mental computation for part-whole number
operations. In Educational Research, Risks, & Dilemmas: Proceedings of the joint
conferences of the New Zealand Association for Research in Education and the
Australian Association for Research in Education [CD-ROM]. Auckland, New
Zealand, December, 2003. (Refereed paper)
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 twenty-four
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 a
structural model of cognitive development. 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.
Faculty of Education
University of Tasmania
Private Bag 66 Hobart Tasmania Australia 7001
Phone: 61-3-6226-2570; Fax: 61-3-6226-2569
Jane.Watson@utas.edu.au