Geometric and algebraic approaches in the concept of complex numbers

This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from high schools in Greece (17-18 years old). Results shed light on pupils' use of two distinct approaches to solve complex number tasks: the geometric and the algebraic approach. The geometric approach was used more frequently, while the pupils used the algebraic approach more consistently and in a more persistent way. The phenomenon of compartmentalization indicating a fragmental understanding of complex numbers was revealed among pupils who implemented the geometric approach. A common phenomenon was pupils' difficulty in complex number problem solving, irrespective of their preferred type of approach. © 2006 Taylor & Francis

Geometric and algebraic approaches in the concept of "limit" and the impact of the "didactic contract"

The present study explores students' abilities in conversions between geometric and algebraic representations, in problem- solving situations involving the concept of "limit" and the interrelation of these abilities with students' constructed understanding of this concept. An attempt is also made to examine the impact of the "didactic contract" on students' performance through the processes they employ in tackling specific tasks on the concept of limit. Data were collected from 222 12th-grade high school students in Greece. The results indicated that students who had constructed a conceptual understanding of limit were the ones most probable to accomplish the conversions of limits from the algebraic to the geometric representations and the reverse. The findings revealed the compartmentalized way of students' thinking in non-routine problems by means of their performance in simpler conversion tasks. Students who did not perform under the conditions of the didactic contract were found to be more consistent in their responses for various conversion tasks and complex problems on limits, compared to students who, as a consequence of the didactic contract, used only algorithmic processes. © National Science Council, Taiwan 2009

Exploring different aspects of the understanding of function: Toward a four-facet model

Based on a synthesis of the relevant literature, this study explored students’ display of behavior in four aspects of the understanding of function: effectiveness in solving a word problem, concept definition, examples of function, recognizing functions in graphic form, and transferring function from one mode of representation to another. A main concern was to examine problem-solving in relation to the other types of displayed behavior. Data were obtained from students in grades 11 and 12. Findings indicated that students were more capable in giving examples of function rather than providing an appropriate definition of the concept. The lowest level of success was observed in problem-solving on functions. Students’ problem-solving effectiveness was found to have a predictive role in whether they would successfully employ the concept in various forms of representation, in giving a definition and examples of function. © 2008 Taylor and Francis Group, LLC