This book offers a theory for the analysis of how children learn and are
taught about whole numbers. Two meanings of numbers are distinguished -
the analytical meaning, defined by the number system, and the
representational meaning, identified by the use of numbers as
conventional signs that stand for quantities. This framework makes it
possible to compare different approaches to making numbers meaningful in
the classroom and contrast the outcomes of these diverse aspects of
teaching. The book identifies themes and trends in empirical research on
the teaching and learning of whole numbers since the launch of the major
journals in mathematics education research in the 1970s. It documents a
shift in focus in the teaching of arithmetic from research about
teaching written algorithms to teaching arithmetic in ways that result
in flexible approaches to calculation. The analysis of studies on
quantitative reasoning reveals classifications of problem types that are
related to different cognitive demands and rates of success in both
additive and multiplicative reasoning. Three different approaches to
quantitative reasoning education illustrate current thinking on teaching
problem solving: teaching reasoning before arithmetic, schema-based
instruction, and the use of pre-designed diagrams. The book also
includes a summary of contemporary approaches to the description of the
knowledge of numbers and arithmetic that teachers need to be effective
teachers of these aspects of mathematics in primary school. The
concluding section includes a brief summary of the major themes
addressed and the challenges for the future.
The new theoretical framework presented offers researchers in
mathematics education novel insights into the differences between
empirical studies in this domain. At the same time the description of
the two meanings of numbers helps teachers distinguish between the
different aims of teaching about numbers supported by diverse methods
used in primary school. The framework is a valuable tool for comparing
the different methods and identifying the various assumptions about
teaching and learning.
