A Calculus of Number Based on Spatial Forms - University of ...

3
Table 1.1: Simple Calculations in Standard and Boundary Forms.
Standard Boundary
1+1=2 =
21 + 11 = 32 ([b][])([b][])=([b][])
2 3=6 ([][])=
3 2 =33 (([[]][]))=([][])
x +0=x x=x
x1=x ([x][])=x
x 0=0 ([x][])=
x 1 = x (([[x]][]))=x
x 0 =1 (([[x]][]))=
1=x = x ,1 ()=(([[x]][]))
x y x = x y+1 ([(([[x]][y]))][x])=(([[x]][y]))
objects which carry the mathematical constraints explicitly, such as counting blocks.
The counting blocks serve as computational objects upon which calculation can be
directly performed. Because standard notation does not embody the transformation
constraints, surrogate representations are necessary for machines and children, both
<strong>of</strong> which require explicit declarations.
Standard notation conceals mathematical behavior in ways besides hiding digit
magnitudes. The equations on the left side <strong>of</strong> Table 1.1 collectively reect mathematical
knowledge which, when known, make the equalities obvious. Visually, they are
not obvious: the features <strong>of</strong> the forms provide no clues as to their manipulative constraints.
The forms carry none <strong>of</strong> this knowledge, a failure which places unnecessary
demands on users <strong>of</strong> mathematics.
The boundary calculus shifts the focus <strong>of</strong> representation from the conceptual
objects <strong>of</strong> numbers and functions to the computational objects where the dynamics
<strong>of</strong> the forms can be directly captured. It makes a clear separation between the
fundamental constraints on number manipulation and the concepts that direct this
manipulation. The equations on the right side <strong>of</strong> Table 1.1 are the same equalities
translated to the boundary calculus. Here, b represents the base <strong>of</strong> the place-value system,
i.e. b = : : The expressions are longer because they are constructed