perceived objects (both specific an

perceived objects (both specific and generic), verbalizing properties and shifting from
practical mathematics to the platonic mathematics of axioms, definitions and proofs.
In the symbolic world, development begins with actions that are symbolized and
coordinated for calculation and manipulation in successively more sophisticated
contexts. The shift to the axiomatic formal world is signified by the switch from
concepts that arise from perceptions of, and actions on, objects in the physical world
to the verbalizing of axiomatic properties to define formal structures whose further
properties are deduced through mathematical proof.
Focusing on the framework appropriate to school mathematics, we find the main
structure consists of two parallel tracks, in embodiment and symbolism, each
building on previous experience (met-befores), with
embodiment developing through perception, description, construction,
definition, deduction and Euclidean proof after the broad style suggested by
van Hiele;
symbolism developing through increasingly sophisticated compression of
procedures into procepts as thinkable contexts operating in successively
broader contexts.
Figure 2: long-term developments in the three worlds