Abstract:

The aim of this study is to explore how problem posing affects high school students’ mathematical creativity (fluency, flexibility, originality) and how their mathematical creative ability is related to their geometrical figure apprehension.

For the purpose of the study, we address the following research questions:
(a) How an integrated apprehension of a geometric figure facilitates the problem posing ability?
(b) How does the necessity to introduce new auxiliary lines in a geometric shape promote the problem posing ability? and
(c) How do mathematical creativity components (fluency-flexibility-originality) relate to the two issues above?

In the field of mathematics education, creativity defined by three dimensions: fluency, flexibility and originality. Fluency in problem posing refers to the ability of students to generating as many problems as possible, flexibility refers to posing problems that can be solved in various ways by applying the What-if-not? approach, and originality refers to posing a different problem after examining several posed ones. The problem-posing situational model is comprised of three aspects:
(a) free-originate problems based on a given real-life scenario,
(b) semi-structured—complete an open scenario by applying previous mathematical knowledge and experiences, and
(c) structured—create problems similar to given problems provided with a very specific scenario.

The present study uses Duval's theory of how geometrical figure are apprehended by students. Duval distinguishes four apprehensions for a geometrical figure: perceptual, sequential, discursive and operative. Perceptual apprehension refers to the recognition of a shape, sequential apprehension is required whenever one must construct a figure or describe its construction, discursive apprehension is related to the fact that mathematical properties represented in a drawing cannot be determined through perceptual apprehension and operative apprehension that we can get an insight into a problem solution when looking at a figure.

Data were collected from 243 tenth graders (15 and 16 years old) from five secondary schools in Greece. All participants were asked to complete a two-part test, a problem posing test and a geometrical figure apprehension test. The results indicate the multidimensional character of the relations between creativity, problem posing and geometrical figure apprehension.

Keywords:

Geometrical Figure Apprehension, Geometry, Mathematical Creativity, Problem Posing.