Approaches to analysing structures vary along a spectrum of detail, inversely with level
of abstraction4, that is, the most abstract is the least detailed, and vice versa — see Figure
2. The most detailed, and explicitly mathematical, approach is using the exact computer-
based methods (or, in a few special cases, exact algebraic solutions); the most abstract
approaches are ‘qualitative’, and there are in-between techniques that engineers call
‘rough calculations’ — for example, these might involve approximating the shape of a
wall as nearly circular because then a simple formula can be applied to give an answer
within 10% of the exact answer.
"DESIGN" "ANALYSIS"
MORE ABSTRACT MORE DETAILED
QUALITATIVE QUANTTTATTVE
Figure 2: The spectrum of abstraction and detail
Understanding comes through connecting across the levels of detail, in the simplest
instances cross-checking of, say, an exact answer against a rough calculation. Qualitative
approaches are entwined with the notion of design, in distinction with the quantitative
calculations of analysis (which are now largely in the realm of computer software).
Another term for qualitative understanding often used by engineers is ‘structural feel’,
which emphasises its (highly-prized) status as something intuitive.
The type of qualitative thinking that characterises the use of ‘feel’ in the design process is
exemplified by the concept of load path, the notion that the loads acting on a structure
have to “flow down into the ground” like a kind of fluid. It is a powerful, very physical
concept5, and extremely useful because it provides a way of thinking about a structure
before any analysis is done, allowing judgements to be made about the validity of
quantitative analysis of the structure:
A load is applied and eventually it’s got to get back into the ground. It’s so fundamental to
structural design that you have to be able to see what that route is in order to have a feeling, to
be able to calculate, what sorts of loads and forces will be apparent in any particular member.
Without a clear idea of the load path, you have nothing to judge what you’re getting from the
computer.
Formal mathematical analysis is based on the assumption of static equilibrium, which
assumes that nothing is moving in a stable structure, an assumption that appears to
conflict with the load path concept. Nevertheless, load path allows predictions of
behaviour that emerge from webbing together the actual properties of the material (e.g.
steel beams) with the (mathematically-abstracted) forces with which they are associated6.
Load path is closely-related to structural geometry: we found evidence for the idea that
engineers use mathematics to carry around in a very compact form the shapes and
4 Note: This use of the term is drawn from computer science, and it is different to the concept of abstraction
most usually discussed in mathematics education.
5 Structural engineers usually treat load path as a qualitative metaphor. The concept can be formally
described, but this is a specialist interest of structural theorists.
6 The attribution of ‘mythical’ chains of causality to formally non-causal situations has been studied by
researchers in various areas of cognitive science, although not, as yet, within the context of mathematics
education. This is a line of investigation that we have begun to pursue.
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