This assignment presents a discussion about some problems that appear in the construction of concept of distance between two points of the plan by students of elementary school. These difficulties are usually related to the difference between the subjacent concepts to the approach of scientific knowledge and the daily knowledge on this subject. In order to clarify and diagnose some of the complexities that arise in the formalization of this concept an experimental work was made with students in elementary school. In this task, the students should theoretically estimate the distance between two locations located in different types of maps and then compare the result with the official information of the real distances. In cases where the results differ from the official ones, we questioned them about why of this divergence. Here are presented and analyzed the results of this study with a view to implementing the actions requested and its consequences for the formalization of the mathematical concept and understanding of its potential for application in everyday life.