Abstract—When proposing structural changes in an educational system three elements are required: a set of oppositional principles to evaluate alternatives; a formal analysis of the status quo; and a comprehensive description and evaluation of proposed spaces. Using this three-phase approach, this paper proposes a set of alternative spaces of mathematics for K-12 education in which the skills demanded by today's society are taken into consideration. In addition, these spaces of mathematics supplement the traditional and limited fields of arithmetic, Euclidean and Cartesian geometries, algebra, and basic trigonometry. The cognitive capabilities and limitations of the human mind that in the last two decades have been identified are described and illustrated as guide for the evaluation of oppositional principles. This paper formally evaluates both the cognitive demands and the potentials for growth and development of areas in mathematics in current and alternative educational spaces, and uses specific educational examples to illustrate the mathematical, pedagogical and cognitive potentials and limitations of current and alternative mathematical educational systems.