An Overview of Theoretical Frameworks and Contemporary Approaches for Facilitating Conceptual and Procedural Knowledge in Mathematics

Abstract

Mathematics is one of the key educational areas and the acquisition of mathematical competence has far-reaching effects on individuals' academic and professional development. The basic aspects of mathematical competence are conceptual knowledge, which represents the understanding of concepts, and procedural knowledge, which refers to the application of procedures in order to solve the tasks. Both types of knowledge are important for the development of adaptive expertise and success in mathematics. In order to encourage the acquisition of conceptual and procedural knowledge during education, it is useful to adjust the teaching methods in accordance with the approaches that have shown to be effective through research and practice. Therefore, one aim of this paper would be to present the basic theoretical frameworks of teaching mathematics during the 20th century, as well as two recent teaching approaches, which are rooted in the currently dominant theoretical framework, constructivism: using comparison and productive failure. While using the comparison, students compare one procedure of task solving with the new procedure, and through this comparison, they can detect key features of procedures that also differentiate them. In relation to the sequential procedures presentation, the comparison has been shown to be more effective on the measures of procedural knowledge and flexibility, and conceptual knowledge. Productive failure is based on the integration of guided discovery and direct instruction, and is implemented in two phases: 1) students individually or in the group discover the solutions of tasks; 2) during the instruction provided by teachers, students compare their solutions with the correct solutions, which leads to the discovery and correction of negative knowledge, and the understanding of the basic concepts in the lecture. Despite its positive effects, this approach has been criticized, what is also described in this paper. Based on the presented information, the second aim of the paper would be to suggest guidelines for the practice and future research, in order to increase the number of students who have high levels of mathematical competences, which are important for students' involvement in increasingly demanding STEM professions.