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

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Encircling Boundaries
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Distinction Network
Figure 8.2: Visual Interpretations <strong>of</strong> Boundary <strong>Number</strong>s.
visually simple. No part <strong>of</strong> the calculus involves dissociated textual code; the picture
itself completely and unambiguously describes a mathematical expression.
The calculus also performs computation in-place, a valuable feature <strong>of</strong> a visual
programming language [18]. Expressions change themselves into new forms, exactly
where they are, in many cases simply by fading in or out parts <strong>of</strong> the expression.
This computation is naturally parallel and concurrent, without explicit control points:
computation occurs where and when it can.
The boundary calculus can improve the mathematical interface by making it visual:
expressions are visually specied and computation occurs visually on the same
forms.
8.5 Education
The standard algebraic notation|made <strong>of</strong> operators and equations|dominates mathematical
experience. As the primary means <strong>of</strong> communicating algebraic concepts, it
is used in textbooks, on ash cards and posters, on tests and homework. The notation
holds an exclusive role: it is the only common thread through all mathematical
experience and so is commonly confused as being mathematics itself.
The notation not only serves to implement mathematical concepts, it also serves
as an authority onhow these concepts are made to work. The notation links ideas
into common abstractions that serve as the identifying references for the ideas. All
mathematical explanations are ultimately tied back to these representations; they are
relied upon pedagogically as fundamental to mathematics.
Standard notation conceals knowledge <strong>of</strong> the relationships between functions. For-