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Questions concerning origin of mathematical knowledge and roles of languageand intuition (imagery) in mathematical thoughts are long standing and widelydebated. By introspection, mathematicians usually have some beliefs regardingthese questions. But these beliefs are usually in a big contrast with therecent cognitive theoretic findings concerning mathematics. Contemporarycognitive science opens new approaches to reformulate the fundamental questionsconcerning mathematics and helps mathematicians break through the Platonicbeliefs about the essence and sources of mathematical knowledge. In thisarticle, we introduce and discuss mathematical thinking activities andfundamental processes such as symbolic/formal and visual/spatial ones. Twodifferent aspects of mathematics should be separated in mathematical cognition.One aspect considers mathematics as an explicit crystallized knowledge. Theother aspect considers mathematics as an ongoing and transient mentalprocessing. The cognitive processes and corresponding tasks involved in theseaspects are different. Ongoing mathematical activities both elementary andadvanced, demand working memory resources. Using dual-task techniques, wedesign some pilot experiments to differentiate the symbolic/formal andvisual/spatial processes. Using this memory theoretic approach, we explain thecrucial roles of language-based processes such as verbal articulation andinstructive speech and also visuo-spatial intuition such as spatial imagery andmental movement in various aspects of mathematics.

A memory theoretic approach for investigating the roles of language and intuition in mathematical thinking activities