Abstract:

The present study aims to examine mathematical creativity through cognitive and perceptual approach in geometry. Mathematical creativity in school mathematics is usually connected with problem solving and the ability to solve a problem in multiple ways. Among the various areas of mathematics, geometry can be used to discover and develop different ways of thinking. Geometry provides opportunities for investigation and proving activities that resemble the work of mathematicians. Success in the solution of geometric problems depends on how the student’s understand the geometric shape. Given the need for a multidimensional approach concerning teaching and learning in geometry, here we address the following research questions:
(a) what is the influence of geometrical figure apprehension on the production of multiple solutions in geometry problems? and
(b) how does the necessity to introduce new auxiliary lines in a geometric shape influence the ability to produce multiple solution?

Data were collected from 243 tenth graders. All participants were asked to complete a two-part test. The first part that concerns geometrical figure apprehension includes six tasks developed on the basis of previous studies. The test was structured according to the Duval theory (1995) and is characterized by three category of tasks. The first category includes two tasks concerning students’ geometrical figure perceptual ability and their recognition ability, the second category includes two tasks and examine students’ operative apprehension to reconfiguration a geometrical figure and the third category includes two tasks and examine students’ operative apprehension to introduce new auxiliary lines. Regarding the students’ creativity in the second part of the test, we examine it through two multiple-solutions problem in geometry focusing on two components: fluency and flexibility. The two geometrical problems are classic with the only difference that the one is solved either by the given shape or by introduce additional auxiliary lines, while the second is solved only by introducing new auxiliary lines. The solutions provided by the students are classified on the basis of the strategy they used. In order to answer above the research questions, descriptive statistics and t-test at 95% significance level were used. Furthermore, similarity statistical analysis is conducted using statistical software C.H.I.C.

Concerning the geometrical figure apprehension, significant variances in the success rates depending on the category of tasks are indicated. Significant differences in the fluency and flexibility of student solutions were observed depending the two categories of problems. In fact, students exhibited greater fluency and flexibility in the problem in which the verbal description of the shape can give to them solutions without being necessary to construct auxiliary lines. The similarity analysis regarding the apprehension of the geometric shape and students’ fluency shows that students who “trapped” in the perceptual apprehension of the figure or resolving operative tasks in algorithmic way achieved to solve the problems with 0, 1 or 2 way at most. On the other hand, results reveal that students who solve the operative tasks with metrological way, solving simultaneously the third category of tasks with auxiliary lines, can perceive the elementary units of a scheme and understand their relationships and thus solve the problems in a 3 or 4 ways.

Keywords:

Geometry, Mathematical Creativity, problem solving.