The study was primarily undertaken to describe the nature of the students’ conceptual understanding of the definite integral and related topics. Specifically, to be able to contribute into the pedagogic expertise of mathematics teachers by helping them in describing the nature of the students’ conceptual understanding on the mathematical concept and guide teachers explore and develop indicators for the four APOS - based levels (Action – Process – Object – Schema) of understanding that will describe the mathematical concepts that students should mentally construct. The study made use of eight randomly selected subjects from one intact mathematics class. Using the theory’s perspective as a guide in investigation, the sampled students’ responses to a talk-aloud test were audio – and video - taped to record their actions and written words. The audiotapes were transcribed to written form and were analyzed based on issues related to the research problems. The talk-aloud test used in the study is designed to reveal students’ mental constructions and measure the extent of their conceptual understanding. The students’ APOS based – levels of understanding are determined quantitatively and qualitatively. Results showed that the levels of understanding of students belonging to two different levels of mathematical performance (High Group and Low Group) for the definite integral and other aforementioned concepts do not vary for the lowest – based cognitive demand (Action). However, as the APOS – based cognitive demand for a problem-solving task is increased, the levels of understanding attained by the two groups differ, in favor of the High Group.
Edgardo Santos, Bulacan State University, Philippines