In this paper, we present a didactic analysis of the mathematical concept of implication under three points of view: sets, formal logic, deductive reasoning. For this study, our hypothesis is that most of the difficulties and mistakes, as well in the use of implication as in its understanding, are due to the lack of links in education between…

In this paper, I describe how undergraduates can develop their understanding of the concept of proof by viewing the act of proving as a procedure. Such undergraduates first understand proof as an algorithm, or a step-by-step mechanical prescription for proving certain types of statements. The students can then condense this algorithm into a…

In recent years metacognition has been receiving increased attention in mathematics education. Special attention has been focused on metacognition and its essential role in achievement settings. The basic difficulty of the study on the field of metacognition is to develop and use valid tasks in order to measure metacognitive ability especially for…

Mathematical proof is one of the most difficult topics for students to learn. Several empirical studies revealed different kinds of students' problems in this area. Our own research suggests that students' views on proofs and their abilities in proving are significantly influenced by their specific mathematics classrooms. However, the reasons for…

This paper presents key findings of my research on the approaches to justification by investigating how a sample of teachers in Hong Kong and Shanghai taught the topic Pythagoras theorem. In this study, 8 Hong Kong videos taken from TIMSS 1999 Video Study and 11 Shanghai videos videotaped by the researcher comprised the database. It was found that…

In this paper the notion of "procept" (in the sense of Gray & Tall, 1994) is extended to advanced mathematics by considering mathematical proof as "formal procept". The statement of a theorem as a symbol may theoretically evoke the proof deduction as a process that may contain sequential procedures and require the synthesis…

Having an ability to appreciate, understand, and generate proofs is crucial in being able to evaluate students' mathematical arguments and reasoning. As such, the development of this ability in perspective teachers is imperative. This study examines the work of a group of preservice elementary school teachers in their efforts to generate one-line…

The purpose of this paper is to offer a framework for categorizing and describing the different types of processes that undergraduates use to construct proofs. Based on 176 observations of undergraduates constructing proofs collected over several studies, I describe three qualitatively different ways that undergraduates use to construct proofs. In…

The equivalence between a statement and its contrapositive is so obvious for an expert that, usually, he does not need any explanation. In this paper, we shall examine the argumentations which students produce in order to justify a statement that, in their opinion, is equivalent to a given statement. We shall observe that the most common…

Researches in Mathematics Education about proof by contradiction revealed some difficulties of the students but also that this kind of argumentation comes spontaneously in certain situations. In this paper we shall show some processes that might lead the student to produce a proof by contradiction. In particular, we shall point out a deep link…

A systematic series of studies by De Bock et al. revealed a strong, deep-rooted tendency among secondary school students to apply the proportional model in non-proportional problems involving lengths, areas and volumes of similar geometrical figures. In these studies, however, items were used involving direct measures for area and volume as well…

Proof by mathematical induction is known to be conceptually difficult for high school students. This paper presents results from interviews with six experienced high school teachers, concerning the use of models in teaching mathematical induction. Along with creative and adequate use of models, we found explanations, models and examples that…

This paper explores the role of instructional scaffolding in the development of undergraduate students' understanding of mathematical proof during a one-year discrete mathematics course. We describe here the framework adapted for the analysis of whole-class discussion and examine how the teacher scaffolded students' thinking. Results suggest that…

We present results from a study which investigated the influence of dynamic geometry-based activities in the development of proving skills (in geometry) in high-school students (15-16 years of age). After a 12-week course on Cabri and the writing of conjectures and proofs, students were asked to write and prove conjectures based on their…

Two of the greatest problems of research on affective factors, and in particular, research on beliefs, is "what" and "how" we observe. The first difficulty is due to the lack of a clear terminology; but even once it has been clearly decided what to observe, it is not easy to put this into practice. This report describes from a…