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Beyond the pop colors

Embedding the Cartesian Space Principles in a Smile, an Inclusive Learning Method for K12

Paper accepted and presented in Poster Session «MIT LINC 2019 Promoting excellence and transformation in education«, Stefano Vuga & Eleonora Vuga. *This ongoing research was suspended due to the COVID-19 emergency.

REVIEW 1, paper 118, Overall evaluation:
This work can provide useful/unique insight into specific learning obstacles and ways to address them.

REVIEW 2, paper 118, Overall evaluation:
This is a compelling illustration of a new tool that has been designed to introduce children to the Cartesian space in a playful, “Kid-friendly” way. The author describes the blocks and phases of the game, and reports promising early results. The paper would be strengthened by references to research literature on topics including children’s attitudes towards math, the use of manipulatives in elementary math, and the development of conceptual understanding.-


Image source MIT LINC 2019 Facebook page

Image source and credits: MIT LINC Conference Facebook page.


It is now well established that the negative emotions the child experiences for not understanding a mathematical topic mark their emotional memory associated with that topic. We’ve been investigating which tangible and accessible tools prevent the development of a pathological allergy to a fundamental concept as it is the Cartesian space, seeking for kid-friendly gates to the subject. When fear and pain for not understanding traces an escape pattern from this topic at a young age the child’s ability to relate to all its didactic applications can be seriously jeopardized, marking (when not identified) the school career and sub-sequent attitudes towards all the fields of theoretical and practical application of it. The elementary approach in explaining the Cartesian space principles to the children remains mainly linked to traditional visualization models of three-dimensional images on two-dimensional space, e.g., paper, blackboard, and screens.

Only recently, augmented reality has been used as a teaching aid for visualizing objects in the actual three-dimensional space. Those systems are suitable for children naturally predisposed to mathematical and/or visual-cognitive intelligence, who are not suffering from any visual impairment. This is a non-inclusive system of access to understanding such fundamental topic as the Cartesian space. Topic which is later essential to an extended comprehension of geometry, mathematics, representation of objects, and concepts. The aim of the research was to find and test a support system to complement the standard two-dimensional and visual-only approach and to guarantee a complete and consistent sensorial experience of the definition of the Cartesian space through physical, material, and modular forms. We sought to create a bond between the concept and its real representation. This system should be extended to different ages of development and types of intelligence and backgrounds, transversal to environments and contexts of usage (family/school), also for visually impaired children. The developed tools provide the child an early and positive emotional bond, prior to any traditional scholastic approach, with the fundamental principles of the Cartesian space through methods such as free play, trial and error, experimentation and share of the emotions while engaging in cooperative activities.

S. Vuga, E. Vuga

Original paper: ResearchGate I EasyChair