The key focus of this research is learners¿ mathematical competence as shown through their errors in Arithmetic and Algebra. The study is a supplement to the Kassel Project. The investigation considered that error-analysis has a diagnostic, scientific and prescriptive character. The research was founded on stable ontological, epistemological, methodological, philosophical accounts of mind and on the breakthroughs of Cognitive Science to identifying the probable causes of learners¿ emergence of errors and to giving explanations. It also contributed to an understanding of the way in which mathematical knowledge should be communicated in specific contexts, natural or artificial to promote better achievement. Accordingly, appropriate teaching strategies have been specified and recommendations to textbook writers and instructional designers have been proposed. Mathematical competence results from the cyclic interrelation of representations (symbols and syntactical rules), meaning (semantics), schemata. This proposition is the thesis of the present research. It provides both a prediction of the phenomenon of the emergence of mathematical errors and a hypothesis, testable in practice.