By ANA CLÁUDIA HENRIQUES of UNIVERSITY OF LISBON
Mathematical reasoning constitutes a powerful personal and social tool. Today’s society, characterized by high demands, competition and constant change related to new scientific and technological developments, requires individuals who, in addition to knowledge, have the ability to solve the challenging problems they face in their lives. Many of those challenges are based in mathematics. It is the role of educators to instill mathematical reasoning abilities, thereby providing students with mathematical tools and processes required to solve everyday problems at home, during leisure and in various fields of work.
The development of a person’s mathematical reasoning depends mainly on the experiences that they encounter in life, particularly while a student. Students do not develop all of their mathematical abilities by simply memorizing concepts and carrying out routine procedures. Whatever topics are taught, students need to learn them in a way that deepens their mathematical reasoning. An important question for educators is how teachers stimulate students to become thoughtful problem solvers. My recent research into that question analysed ways by which mathematical reasoning can be developed in mathematics courses. The study goals were to identify the mathematical processes used by university students in their classroom work and to describe how these processes contribute to the development of mathematical reasoning.
In that study, activities that require high levels of mathematical reasoning were designed and then used in a class of 36 second-year university students enrolled in a numerical analysis course. The study activities were designed to offer the students a non-traditional approach to numerical analysis. The classroom work included participation in four mathematical investigation tasks on different numerical analysis topics. The activities were undertaken in three phases. Initially, students were presented with a task for which they had not been provided with background theory or routine processes. The task was designed to challenge the students to pose their own questions about the mathematical problem and to then develop and defend their own strategies to solve it. Following task introduction, the students worked autonomously in self-selected pairs or small groups. Upon finishing task exploration, the students presented their work orally to the class. These presentations and class discussions were important learning opportunities as, when prompted by classmates’ and teacher’s questioning, they used their reasoning abilities to justify and discuss aspects of a problem that they had not considered.
Working on mathematical investigation in such a manner and on a regular basis provides students with the opportunity to develop, through individual or group-based efforts, their mathematical reasoning abilities. Such abilities, including looking for regularities/irregularities, posing questions, formulating and testing theories, proving results and generalizing to other problems, may then be applied during mathematical problem solving in other areas. In that sense, the ability to reason mathematically is at least as important as the gathering and mastery of specific knowledge of mathematic facts. The study also showed that students, when given such activities, were able to build mathematics concepts and procedures by using individual or group reasoning. The strong commitment and effort exhibited by the students in this study contributed to the development of a thorough understanding of the issues with which they worked. Thus, restructuring the activity structure of daily classroom practice can enhance the development of students’ mathematical reasoning abilities, increase their mathematical knowledge and improve their mathematical process skills.
To establish new teaching tools and goals and to improve classroom approaches to the development of students’ learning of mathematical processes are issues that require consideration by the education system. Regardless, it is and will continue to be the role of the teacher to design and implement teaching strategies that promote students’ mathematical reasoning. The results of this study indicate the potential of using mathematical investigation in classroom settings to allow students to develop such reasoning abilities. Such a teaching approach should be integrated into future mathematics classrooms.
Ana Cláudia Henriques
University of Lisbon
www.atomiumculture.eu
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