Pre-service training of mathematics teachers with respect to conceptual understanding and constructivism

Abstract

The participants in this study were twelve undergraduate students majoring in mathematics education at the university level in Iceland. The purpose of the study was to evaluate whether their pre-service training was sufficiently oriented towards strengthening the students’ conceptual understanding in mathematics as well as their problem solving ability and analytical skills. The study was aimed at answering the question whether the students’ preparation in mathematics was solid enough to enhance their competence to work as professional schoolteachers in the spirit of constructivism. A short overview of constructivism in mathematics teaching is provided and, in order to show the constantly growing emphasis that this theory has been given in the Icelandic educational system, a brief overview is included of the curricular changes in mathematics for the past few decades in Iceland. It is pointed out that both the 1999 and the 2007 official curricula explicitly state the importance of problem solving and present a discussion on what it means to understand a mathematical concept. The growing emphasis on constructivism in mathematics teaching is also clearly witnessed in recently published mathematics textbooks for Icelandic primary schools. The paper contains a short discussion on the importance of problem solving in connection with constructivism in mathematics teaching. In the second half of the 20th century problem solving was one of the key issues of mathematics as a field of study. In the skill of problem solving as well as in the theory of constructivism the questioning mind of the student is paramount. The evaluation of the students’ preparation was based on observations of their tackling five given mathematical problems, interviews with the students about the problems and the students’ written solutions. The evaluation was based on the following research questions put forward in the study: 1) Do the students understand the problem well enough to realize how they should solve it? 2) Do the students use analytical skills in trying to solve the problem? 3) Do the students analyze the problem they are working on and show a questioning mind? Do the students try to see the problem in a wider perspective? 4) Can students be encouraged to analyze further a problem they have already solved? To shed light on why the above research questions were chosen by the researchers, there is a brief mention of two reports on competency in mathematics teaching and one research article addressing the same subject. One of these reports is an official Danish report and the other an official American report. The research article describes an Icelandic study on how successfully the pre-service education of teachers promotes their overall competency. The above mentioned five problems the students were given in this study include one ancient number theoretic problem in a typical problem solving style, a paradox of Zeno, some questions involving Fibonacci sequences, a conceptual problem concerning linear functions and approaches to a few geometric concepts through the use of graphic software. All five problems can be considered to be typical and important mathematical problems; each one is explained in the paper and a justification for their choice is given. The findings suggest that the twelve students participating in this study are lacking in some of the necessary competences for a schoolteacher intending to use constructivistic methods to teach mathematics. Comparing our conclusions with the definitions of competency in mathematics teaching, in both the above mentioned Danish report and the American report, we find that the students’ understanding of mathematical concepts as well as their ability to reason mathematically needs to be strengthened. Accordingly, we conclude that subject courses in pre-service education for mathematics teachers in Iceland should place more emphasis on those aspects of the training of mathematics teachers.