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This article presents a study of configural reasoning and written discourse developed by students of the National Polytechnic School of Ecuador when performing geometrical exercises of proving.

The study of geometry at all levels of education is very important because it “can be seen as a reflective tool that allows human beings to solve various problems and understand a world that offers a wide range of various geometric forms, each of the scenarios that comprise it, whether they be natural or artificial” [

According to McClure [

Jones et al. [

A traditional approach to defining mathematical proof states that: “a mathematical proof is a formal, logical line of reasoning that begins with a set of axioms and progresses through logical steps towards a conclusion,” Griffiths [

Harel and Sowder [

Alcolea [

From the psychological point of view, vision and perception are associated two functions: the first called epistemological and related function with direct access any physical object and the second synoptic function, interpreted as the simultaneous apprehension of several objects or specific complete field. Visualization is the vision manifested in synoptic function [

Visual perception (epistemological function) requires physical examination, it cannot grasp the object at once, as a whole. On the contrary, visualization can have a complete “snapshot” apprehension of any organization of relations.

When solving geometry problems we are constantly interacting with figures and for these to constitute a mathematical object they must be a combination of several related gestalts that determine what we are observing (configuration) and be linked to a (mathematical) proposition that sets some properties represented by the gestalt (hypotheses). Based on Duval two ways of apprehending a figure emerge from these conditions: perceptual apprehension and discursive apprehension.

Perceptual apprehension is characterized as simply identifying a configuration and discursive as cognitive action produces an association of mathematical statements identified with given configuration. In the process of solving problems we interact with geometric figures, making changes to the original settings. These changes are manifestations of operational apprehension, which can be of two types: when new geometric elements (operational figural apprehension of change) are added when the components and subassemblies are manipulated like pieces of a puzzle (operational apprehension of reconfiguration). After each change, new properties can be made visible and they can be associated with definitions, axioms, theorems (discursive apprehension); after this analysis further changes may be made to the settings previously obtained (operational apprehension) repeating the discursive/operational apprehension cycle in a coordinated manner until the solution is reached or the strategy being followed is abandoned.

The configural reasoning should be understood as the development of coordinated action of discursive/opera- tional apprehension that the students made when solving a geometric problems; generating an interaction between the initial configuration and any changes in this with the appropriate mathematical statements [

Torregrosa and Quesada [

The coordination provides solution to distinguishing two types of processes: a) Truncation: capturing the idea that allows deductive problem solving and b) Conjecture without proof: the problem is resolved assuming as valid certain assumptions that arise from the mere perception.

Coordination fails to solve the problem and a phenomenon called loop occurs: it is up to a blocking situation where you cannot move towards the solution, causing a blockage of reasoning.

The text produced by students when communicating the resolution of a problem may reflect their cognitive styles and the development of these types of processes, as some people reason better with words and others reason better with figures [

The interaction between configuration representations and discourse when students are solving geometric exercises of proving provide information on the geometric reasoning of students, as both written discourse and verbal expressions or gestures can be considered semiotic resources used by students when they are engaged in problem solving and in communicating those resolutions [

Clemente and Linares work [

The study involved 46 students of the preparatory course of technologists who take a course of geometry. These students answered a questionnaire that included two proving problems (P2 and P4) as part of the course evaluation. For resolution the students should develop operative and discursive apprehensions identifying sub configurations to allow them to recognize some geometric objects to generate a proof,

In this section we show the numerical results of configural reasoning in relation to the identification of subassemblies and the association of knowledge. The style of the written discourse of students is also shown,

P2 | P4 | |||
---|---|---|---|---|

Given the figure, prove that the QT segment is equal to KN. | Prove that the area of a square on the diagonal of another square has twice of his area. | |||

Sub configurations | ||||

P2 | P2.1 | P2.2 | P2.3 | |

P4 | P4.1 | P4.2 | ||

Sub configurations | Discourse style | Truncation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

P2 | P1.1 | P1.2 | P1.3 | G 2 | T 1 | G/T 38 | 29 | |||||

Identified | Associated knowledge | Identified | Associated knowledge | Iidentified | Associated knowledge | |||||||

25 | 25 | 30 | 8 | 33 | 25 | |||||||

P4 | P2.1 | P2.2 | Others | 5 | 2 | 28 | 22 | |||||

Constructed | Associated knowledge | Constructed | Associated knowledge | Constructed | Associated knowledge | |||||||

22 | 19 | 19 | 11 | 23 | 23 | |||||||

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The analysis of the table shows that the 29 (63.04%) students in P2 and 22 (47.82%) in P4 achieved truncation. In the P2problem the sub configuration P2.2 is associated the less number of knowledge despite having been identified 30 (61.25%) times. In the P2 problem we note that a large number of new sub configurations 23 (50%) have been created with 100% of associated knowledge.

Written discourse analysis shows a strong tendency to mixed type G/T which perhaps is due to the nature of the proposed problems.

Ruth Cueva, (2016) Study of Configural Reasoning and Written Discourse in Geometric Exercises of Proving. Open Journal of Social Sciences,04,33-37. doi: 10.4236/jss.2016.42007