This research project investigates the difficulties students encounter when solving physics problems involving the integral and the area under the curve concepts and the strategies to facilitate students learning to solve those types of problems. The research contexts of this project are calculus-based physics courses covering mechanics and electromagnetism.

In phase I of the project, individual teaching/learning interviews were conducted with 20 students in mechanics and 15 students from the same cohort in electromagnetism. The students were asked to solve problems on several topics of mechanics and electromagnetism. These problems involved calculating physical quantities (e.g. velocity, acceleration, work, electric field, electric resistance, electric current) by integrating or finding the area under the curve of functions of related quantities (e.g. position, velocity, force, charge density, resistivity, current density). Verbal hints were provided when students made an error or were unable to proceed. A total number of 140 one-hour interviews were conducted in this phase, which provided insights into students’ difficulties when solving the problems involving the integral and the area under the curve concepts and the hints to help students overcome those difficulties.

In phase II of the project, tutorials were created to facilitate students’ learning to solve physics problems involving the integral and the area under the curve concepts. Each tutorial consisted of a set of exercises and a protocol that incorporated the helpful hints to target the difficulties that students expressed in phase I of the project. Focus group learning interviews were conducted to test the effectiveness of the tutorials in comparison with standard learning materials (i.e. textbook problems and solutions). Overall results indicated that students learning with our tutorials outperformed students learning with standard materials in applying the integral and the area under the curve concepts to physics problems.

The results of this project provide broader and deeper insights into students’ problem solving with the integral and the area under the curve concepts and suggest strategies to facilitate students’ learning to apply these concepts to physics problems. This study also has significant implications for further research, curriculum development and instruction.