A novel approach is proposed to
the analysis of the expressive powers of
Concrete, Pictorial and Abstract representations of arithmetic
.
Drawing on
the theory of formal mathematical logic, it is shown that precisely defined
syntaxes and semantics can be introduced for Cuisenaire rods and
part
-
whole bar models. By interpreting sentences expressed in these
representations
as sentences in the first
-
order theory of arithmetic, it is
possible to rigorously study the potential and limits of these representations.
It is shown that different appr
oaches to bar modelling vary depending on
the semantic content given to geometric bar length, and the implications of
this observation are studied to reveal the relative power of these
representations for expression of word problems and their subsequent
so
lution within the representation.