Xenia Vamvakoussi

University of Ioannina | UOI · Early Childhood Education

Rational number density has been investigated through open-ended question tasks and multiple-choice tasks or by asking to interpolate a number between two numbers. However, students' responses to these three types of tasks were not directly compared. The objective is to look for relationships between the three types of tasks in order to identify di...

In this study, we investigated how secondary students interpret algebraic expressions that contain literal symbols to stand for variables. We hypothesized that the natural number bias (i.e., the tendency to over-rely on knowledge and experiences based on natural numbers) would affect students to think that the literal symbols stand for natural numb...

We present part of an ongoing topic-specific design research aiming at enhancing preprimary students’ multiplicative reasoning. We focus on an activity introducing vocabulary for multiples and submultiples with a view to enable children to express multiplicative relations verbally. We present the rationale and the design of the activity and finding...

We report a case study intervention pilot-testing a program of activities aiming at enhancing pre-primary children's multiplicative reasoning competences. The program treated discrete and continuous quantities in a unified manner; provided learning experiences pertaining to three fundamental multiplicative operations, namely iteration of a quantity...

Over the last years, there is a growing interest in studying students’ difficulties with rational numbers from a cognitive/developmental perspective, focusing on the role of prior knowledge in students’ understanding of rational numbers. The present study tests the effect of the whole or natural number bias (i.e., the tendency to count on natural n...

In the present study, we tested the hypotheses that: a) there are individual differences in secondary students' conceptual and procedural fraction knowledge, and b) these differences are predicted by students' approach (deep vs. surface) to mathematics learning. We used two instruments developed and evaluated for the purposes of the study which wer...

Στην παρούσα μελέτη, ελέγξαμε τις υποθέσεις ότι α) υπάρχουν ατομικές διαφορές στην εννοιολογική και διαδικαστική γνώση για τα κλάσματα και οι διαφορές αυτές παραμένουν ακόμη και στις τελευταίες τάξεις του Γυμνασίου, β) οι διαφορές αυτές εξηγούνται από διαφορές στη μάθηση και τη μελέτη των μαθηματικών (επιφανειακή/βαθιά προσέγγιση). Οι συμμετέχοντες...

In this article we present an overview of four studies investigating Greek secondary students’ conceptual and procedural knowledge of fractions. We discuss the problem of defining conceptual and procedural knowledge, and the implications of adopting one particular definition over others. We draw on the studies and their results to discuss the probl...

Είναι ευρέως αποδεκτό ότι η βαθιά προσέγγιση στη µάθηση των µαθηµατικών συνοδεύεται ποιοτικά από υψηλού επιπέδου µαθησιακά αποτελέσµατα. Σκοπός της εργασίας αυτής είναι να διερευνήσει τις συνιστώσες της βαθιάς προσέγγισης στη µάθηση των µαθηµατικών. Πραγµατοποιήθηκε µελέτη περίπτωσης δύο µαθητών µε εξαιρετική επίδοση και βαθιά εννοιολογική γνώση στ...

We constructed and calibrated an instrument targeting conceptual and procedural fraction knowledge. We used this instrument in a quantitative study with 126 secondary students (7 th and 9 th graders), testing the hypothesis that there are individual differences in the way students combine the two types of knowledge. Cluster analysis revealed four d...

In this paper we focus on the development of rational number knowledge and present three research programs that illustrate the possibility of bridging research between the fields of cognitive developmental psychology and mathematics education. The first is a research program theoretically grounded in the framework theory approach to conceptual chan...

Σε μια ποσοτική μελέτη, ελέγξαμε την υπόθεση ότι υπάρχουν ατομικές διαφορές στον τρόπο που οι μαθητές συνδυάζουν την εννοιολογική και διαδικαστική γνώση για τα κλάσματα που παραμένουν σημαντικές ακόμα και στο Γυμνάσιο. Για το σκοπό αυτό κατασκευάσαμε και αξιολογήσαμε ένα ερωτηματολόγιο που επιδόθηκε σε 138 μαθητές Α΄ και Γ΄ Γυμνασίου. Σε συμφωνία μ...

The problem of adverse effects of prior knowledge in mathematics learning has been amply documented and theorized by mathematics educators as well as cognitive/developmental psychologists. This problem emerges when students’ prior knowledge about a mathematical notion comes in contrast with new information coming from instruction, giving rise to sy...

It is widely acknowledged that there are individual differences in the way students approach the learning process, and that these are reflected in the learning outcomes. Little research has been done from the learning approaches perspective regarding mathematics learning. We report an exploratory study investigating the features of the deep approac...

The development of rational number knowledge has been studied extensively by mathematics education researchers, cognitive-developmental psychologists and, more recently, by neuroscientists as well. Building on a rich body of prior research, and some exciting new ideas, the target articles re-visit several topics, with a view to refine and deepen ou...

We present the results of an in-depth qualitative study that examined ninth graders' conceptual and procedural knowledge of fractions as well as their approach to mathematics learning, in particular fraction learning. We traced individual differences, even extreme, in the way that students combine the two kinds of knowledge. We also provide prelimi...

This study tested the hypothesis that intuitions about the effect of operations, e.g., “addition makes bigger” and “division makes smaller”, are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated adu...

In two experiments we explored the instructional value of a cross‐domain mapping between “number” and “line” in secondary school students' understanding of density. The first experiment investigated the hypothesis that density would be more accessible to students in a geometrical context (infinitely many points on a straight line segment) compared...

A major source of errors in rational number tasks is the inappropriate application of natural number rules. We hypothesized that this is an instance of intuitive reasoning and thus can persist in adults, even when they respond correctly. This was tested by means of a reaction time method, relying on a dual process perspective that differentiates be...

In the present study we examined ninth graders' procedural and conceptual strategies in rational number tasks and their relationship to students' approach (deep/superficial) to the study of mathematics. We found individual differences-sometimes extreme-in the way that students combine procedural and conceptual knowledge for rational numbers. Moreov...

It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number bas...

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students’ understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The...

The framework theory approach to conceptual change hypothesizes that in the shift from natural to rational numbers, overcoming the idea that numbers are discrete is gradual and that, in the process, certain intermediate states of understanding will appear in students. This paper presents a study in two different groups of 9 th graders, namely Greek...

Fifty-six teachers, from four European countries, were interviewed to ascertain their attitudes to and beliefs about the Collaborative Learning Environments (CLEs) which were designed under the Innovative Technologies for Collaborative Learning Project. Their responses were analysed using categories based on a model from cultural-historical activit...

The term conceptual change is used to characterize the kind of learning required when the new information to be learned comes in conflict with the learners’ prior knowledge. The conceptual change approach has been applied extensively to explain students’ difficulties in science learning. In this paper, we argue that the conceptual change approach c...

This paper reports the conceptions of teachers from four European countries of the Innovative Technologies for Collaborative Learning project tools for collaborative learning. Fifty six teachers were interviewed about different aspects of the CLE (Webbased Collaborative Learning Environment) implementations and about their own evaluations of the CL...

In the present article, we argue that the conceptual change approach to learning can apply in the case of mathematics, taking into consideration the particular nature of mathematical knowledge and the neurobiological bases of mathematical cognition. In the empirical study that is reported in this article, we investigated ninth graders’ understandin...

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Prior knowledge of natural numbers often stands in the way of children's understanding rational numbers and their properties. In particular, the idea of discreteness is a fundamental presupposition of children's' initial theories about numbers. Understanding of the dense structure of rational numbers requires radical reorganization of children's pr...