Fien Depaepe

The Illusion of Linearity: Expanding the Evidence towards Probabilistic Reasoning.

Previous research has shown that – due to the extensive attention spent to proportional reasoning in mathematics education – many students have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. This phenomenon is sometimes referred to as the ‘illusion of linearity’. For example, in geometry it is known that many students believe that if the sides of a figure are doubled, the area is doubled too. In this article, the empirical evidence for this phenomenon is expanded to the domain of probabilistic reasoning. First, we elaborate on the notion of chance and provide some reasons for expecting the over generalization of linear models in the domain of probability too. Afterwards, a number of well-known and less-known probabilistic misconceptions are described and analysed, showing that they have one remarkable characteristic in common: they can be interpreted in terms oft he improper application of linear relations. Finally, we report on an empirical investigation aimed at identifying the ability of 10th and12th grade students to compare the probabilities of two binomial chance situations. It appears that before instruction in probability, students have a good capability of comparing two events qualitatively, but at the same time they incorrectly quantify this qualitative insight as if the variables in the problem were linked by a linear relationship. Remarkably, these errors persist after instruction in probability. The potential of this study for improving the teaching and learning of probability, as well as suggestions for further research, are discussed.

Unraveling the Relationship Between Students’ Mathematics-Related Beliefs and the Classroom Culture

Over the past 2 decades the study of students’ (and teachers’) mathematics-related beliefs has gradually received more and more attention from researchers in the field of educational psychology as well as from scholars in the area of mathematics education. In this article positive beliefs about mathematics and mathematics learning are considered as a major component of competence in mathematics. Results of empirical studies are presented showing that primary school students often have negative and/or naive beliefs about mathematics learning, focused on the phenomenon of “suspension of sense-making” in mathematical problem solving. A design experiment is then described in which a learning environment was developed and implemented, which was intended to improve students’ performance in problem solving as well as their mathematics-related beliefs. This and related work support the hypothesis that changes in the classroom culture and practices can foster students’ mathematical thinking and learning as well as...

Developing Pedagogical Content Knowledge: Lessons Learned from Intervention Studies

Pedagogical content knowledge (PCK) is generally accepted as positively impacting teaching quality and student learning. Therefore, research on PCK development in (prospective) teachers is highly relevant. Based on a search in three databases (ERIC, PsycInfo, and Web of Science), a systematic review is conducted on intervention studies aiming at PCK development. The research questions are threefold: (1) How are the studies designed? (2) How are the interventions designed? and (3) What elements of interventions contribute to PCK development? The results show that most intervention studies are conducted in math and science education and use a qualitative methodology. Reflection, PCK courses, contact with other teachers, and experiences in educational practice are typically part of effective interventions. The review enables the identification of clear guidelines that may strengthen future research on stimulating PCK.

Students’ Non-realistic Mathematical Modeling as a Drawback of Teachers’ Beliefs About and Approaches to Word Problem Solving

Over the past decades numerous scholars have become aware of many compelling observations of students in mathematics classes abandoning their sense-making capabilities when doing word problems, and, in particular, carrying out arithmetic calculations that do not make sense in relation to the situations described. This led us, together with several other scholars, to embark upon an extended investigation of the phenomenon, the results of which are reported, among others, in two books (Verschaffel L, Greer B, De Corte E, Making sense of word problems. Swets & Zeitlinger, Lisse, 2000; Verschaffel L, Greer B, Van Dooren W, Mukhopadhyay S, Words and worlds: modelling verbal descriptions of situations. Sense Publishers, Rotterdam, 2009). The goal of the present chapter is to bring together and critically review the theoretical analyses and empirical studies that have focused on major aspects of teachers’ instructional practices that affect – directly or indirectly – students’ non-realistic approaches to and beliefs about word problem solving. Special attention will be given to the problems that appear in students’ mathematical textbooks as well as to teachers’ beliefs about word problems and what are appropriate ways to solve them, and to their instructional behavior, and how these factors affect students’ beliefs about and approaches to word problems. While the focus is on research that has been done in our own center, we also integrate relevant studies by others.

Pedagogical Content Knowledge in Teacher Education

Over the past 25 years, PCK has become widely accepted as a useful construct in understanding the unique nature and development of teachers’ knowledge, and PCK research in the field of teacher education has rapidly accumulated. Yet while there is consensus in the literature about the importance of the construct in bringing together specific types of knowledge that teachers require to support effective student learning of particular subject matter, questions still remain about how PCK can be understood, developed and organised in the context of initial teacher education. The development of PCK has been the subject of studies in different programme formats of teacher education, at different levels (primary, secondary), in different subjects, and in various countries. From these studies, it is apparent that whatever the organizational format of the programme, the development of PCK should be related to pre-service teachers’ subject matter knowledge on the one hand, and their teaching experiences, notably during practicum, on the other. Equally, developing PCK requires pre-service teachers’ ability and willingness to view subject matter from a learners’ perspective. Depending on the level of subject matter knowledge of pre-service teachers at the start of their programme, the development of PCK may benefit from a design which integrates institute based activities with authentic classroom teaching of subject matter.

Who is granted authority in the mathematics classroom? An analysis of the observed and perceived distribution of authority

The article deals with the way in which authority was established and interpreted by teachers and students in two Flemish sixth-grade mathematics classrooms. Problem-solving lessons during a seven-month observation period were analysed regarding three aspects of teacher–student interactions that explicitly or implicitly reflect who bears mathematical authority: (1) to whom were students allowed to ask for help; (2) who was allowed to answer students’ mathematics-related questions; and (3) who was allowed to evaluate students’ responses. For each of these aspects, we were interested whether authority was ascribed only to the teacher or whether authority was distributed more broadly, with students being considered accountable for their own and other’s learning process. Interviews with teachers and students were conducted to investigate the meaning that participants gave to the practices in which they were involved.

Perceptions of instructional quality: impact on acceptance and use of an online learning environment

ABSTRACT Given the growing use of online learning environments in higher education, it is important to further unravel how students’ use is influenced by their perceptions towards these learning environments. This study includes the perceived quality of the instructional design based on the First Principles of Instruction of Merrill and students’ acceptance based on the constructs perceived usefulness and perceived ease of use of the technology acceptance model (TAM). The aim of this study is twofold: a first aim is to investigate the influence of the perceived instructional quality on students’ acceptance and the second aim is to investigate the impact of students’ acceptance and the perceived instructional quality on the quantity (i.e. course activity) and quality (i.e. course performance) of use. In this study, a Moodle-based online learning environment for learning French as a foreign language was studied. Participants were 161 university students. Structural equation modeling (SEM) indicates that the perceived instructional quality has a significant positive influence on students’ acceptance. Furthermore, students’ perceived instructional quality has a positive influence on the quality, but not on the quantity of use, whereas students’ acceptance of the online learning environment has no impact on the use of the learning environment.

Pedagogical content knowledge of French as a foreign language: differences between pre-service and in-service teachers

Abstract Pedagogical content knowledge (PCK) is an important predictor of educational quality. PCK develops from various sources, such as teaching experience. Studies investigating the effects of teaching experience on PCK development mainly focus on the question whether experienced teachers differ from novices in the amount of PCK. In a mixed methods study, we study both quantitative and qualitative differences between pre-service and in-service teachers’ PCK. Total scores on a paper-and-pencil instrument are used as a measure of French PCK. To analyse the participants’ PCK in a qualitative way, answers are coded in four categories and the prevalence of answers in each category is compared between groups. Furthermore, the nature of answers is studied. No differences on our quantitative PCK measure are observed. The qualitative analysis shows that in-service teachers have more difficulties with conceptual knowledge, while showing more practical knowledge of students’ understanding than pre-service teachers.

Flemish Students' Historical Reference Knowledge and Narratives of the Belgian National Past at the End of Secondary Education.

Since the early nineteenth century, western governments have expected history education to play a vital role in the formation of a national identity and the pursuit of national cohesion, by fostering shared knowledge and a shared (master)narrative of the national past. This article reports on a qualitative study that examines which narratives young adults construct about their national past, to what extent those narratives are underpinned by existing narrative templates, whether they reflect on the fact that the national past can be narrated in different ways, and to what extent they share a common reference knowledge. The study addresses the Flemish region of Belgium, a case characterized by a specific context of a nation state in decline, wherein diverse and often conflicting historical narratives coexist in popular historical culture and where the national past is almost absent from history education. A total of 107 first-year undergraduate history students were asked to write an essay on how they saw the national past. The influence of both history education and popular historical culture was reflected in the reference knowledge as well as in the (absence of) templates that students used to build their essays. Templates were not critically deconstructed, although some students nevertheless were able to discern and criticize existing ‘myths’ in the national past.