Digital technologies as a means of accessing powerful mathematical ideas. A study of adults with low schooling in Mexico

This paper derives from a study which main purpose was to investigate how a group of adults with low schooling can have access to powerful mathematical ideas when working with activities that involve the use of technology resources and that take into account the adults’ previous experience with mathematics. Specifically, adults’ previous experience with area calculation was considered. Principles of the Theory of Didactical Situations (TDS) formulated by Brousseau guided the study design, and Pick’s theorem was recreated in a dynamic digital setting, with which it is possible to calculate the area of regular and irregular polygons. In this approach, intuitive notions of area and perimeter are resorted to, seeking to promote the experience with powerful ideas such as ‘the generality of a method’, ‘realizing the existence of different methods used for one and the same end’ and ‘realizing that each method possesses advantages and limitations’. Analysis of interview protocols from three noteworthy cases (which include both adults’ work in the digital setting and their discussions with the researcher) suggests the presence of powerful underlying mathematical ideas, such as the idea of generality and the power of a method and the features of the constituent elements of a geometric figure that are involved in calculating its attributes, attributes such as area.

O TEOREMA DE PICK

http://dx.doi.org/10.5902/2179460X14606This work has as a main theme the Pick’s Theorem. This theorem refers to the calculation of area of polygons with vertices at points of a flat network, such a calculation can be summarized in a simple relation between the number of points of a network, located in the interior of a polygon and the number of points of the border of the polygon that belongs to the network. Research, before, presenting your demonstration, align the central ideas of your demonstration, so that there are sufficient arguments to support it. We propose to establish a relation between this method and the well known theorem of Euler, besides presenting an arithmetical application for this theorem. The attention given to this theorem is well deserved due to its apparent simplicity and for the fact that it can awaken the interest of some students for the subject.

Pick’s Theorem in Two-Dimensional Subspace ofR3

In the Euclidean spaceR3, denote the set of all points with integer coordinate byZ3. For any two-dimensional simple lattice polygonP, we establish the following analogy version of Pick’s Theorem,kIP+1/2BP-1, whereBPis the number of lattice points on the boundary ofPinZ3,IPis the number of lattice points in the interior ofPinZ3, andkis a constant only related to the two-dimensional subspace includingP.