Excursions to and from semantic oblivion
This report offers some recent experiences from the author's teaching that highlight the fragility of language and mathematical formalism in communicating ideas and concepts. Syntactic reasoning is at the superficial end of the spectrum, "skating on the surface", and involves formal manipulation of symbols, simple rules and substitutions. Semantic reasoning is deeper, "diving down towards the seabed", drawing conclusions from underlying meanings and heuristics. The author argues that illuminating the tension and interplay between these complementary modes of reasoning, and creating heightened awareness, may enhance approaches to successful learning, improve morale and attitudes, and lead to more robust outcomes and willingness to engage in challenging mathematical activities.Keywords: syntax, semantics, reasoning, mathematics education.
AMS Subject Classification: Primary 97C30.
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