Learning algebra in a computer algebra environment : design research on the understanding of the concept of parameter

It is well known that algebra is a difficult topic in the school mathematics curriculum, and is often experienced as a stumbling-block. One of the directions in which solutions to the problems with the learning of algebra can be sought is the integration of information technology (IT) into mathematics education. Although originally not developed for educational purposes, a computer algebra system is an IT tool that seems promising because of its algebraic power. The basic aim of this study, therefore, is to investigate whether computer algebra use can contribute to the understanding of algebra. This leads to the following main research question: How can the use of computer algebra promote the understanding of algebraic concepts and operations? Chapter 1 contains the research questions and explains the aims and backgrounds of the study. In Chapter 2 the research design and methodology are described. Key words are design research and hypothetical learning trajectory. Chapters 1 and 2 together indicate what the research is about and how it is conducted. Chapters 3, 4 and 5 form the theoretical part of the thesis. They treat the main themes of the study: algebra in general, the concept of parameter in particular and the possible roles of computer algebra. Chapter 3 concerns algebra in general. It sketches different views on algebra and describes the standpoint of this study. The theoretical issues of symbol sense, symbolizing, the process-object duality and Realistic Mathematics Education are addressed. In Chapter 4, we zoom in on the concept of parameter. After a brief historical perspective, a conceptual analysis of the parameter is given. Then we describe what we consider a higher level understanding of the concept of parameter. This is connected to the theoretical notions from Chapter 3. Chapter 5 deals with the tool that students use in this research project: computer algebra. Besides an overview of previous research in this domain, it contains a description of the theory of instrumentation that will be used in Chapter 10 in particular. Chapters 6 - 10 form the empirical part of the dissertation. Chapters 6, 7 and 8 describe the development of the hypothetical learning trajectory and the classroom experiences during the three subsequent research cycles. Chapter 9 concerns the contribution of computer algebra use to the understanding of the concept of parameter. In Chapter 10, the results concerning the instrumentation of computer algebra are presented. Chapter 11, finally, answers the main research question. After that, we look back on the study and discuss the results and the methodology. Also, the relevance of the theoretical framework and the generalizability of the findings are evaluated. The chapter ends with recommendations for teaching, for software design and for further researc

Новий навчальний посібник “Україна в міжнародних організаціях”

Рецензія на посібник: Макар Ю. І. Україна в міжнародних організаціях : навч. посібник / Ю. І. Макар, Б. П. Гдичинський, В. Ю. Макар, С. Д. Попик, Н. Ю. Ротар ; за ред. Ю. І. Макара. – Чернівці : Прут, 2009. – 880 с

Combined inner and outer loop feedback in an intelligent tutoring system for statistics in higher education

Intelligent tutoring systems (ITSs) can provide inner loop feedback about steps within tasks, and outer loop feedback about performance on multiple tasks.While research typically addresses these feedback types separately, many ITSs offer them simultaneously. This study evaluates the effects of providing combined inner and outer loop feedback on social sciences students' learning process and performance in a first-year university statistics course. In a 2 x 2 factorial design (elaborate inner loop vs. minimal inner loop and outer loop vs. no outer loop feedback) with 521 participants, the effects of both feedback types and their combination were assessed through multiple linear regression models. Results showed mixed effects, depending on students' prior knowledge and experience, and no overall effects on course performance. Students tended to use outer loop feedback less when also receiving elaborate inner loop feedback. We therefore recommend introducing feedback types one by one and offering them for substantial periods of time

Die Schweizerdeutschen dialecte

В статье рассматриваются диалекты Швейцарии, а также, основные грамматические, лексические и словообразовательные особенности швейцарского варианта немецкого языка

Embodied collaboration to foster instrumental genesis in mathematics

As cognitive science reports joint action requiring tight intercorporeal coordination between two partners, we aim to evaluate the role of this coordination in computer-supported instrumental genesis for mathematics. In our dual eye-tracking design study we developed an embodied activity that potentially contributes to technologically extended problem solving in trigonometry. We tested three versions of the design: (a) individual sensorimotor enactment only, (b) individual and then collaborative enactments, and (c) individual enactment and then collaborative description followed by enactment. As our first case showed, the required sensorimotor coordination was developed but never used in the following problem solving when a student worked alone. In contrast, in both collaborative cases the relevant sensorimotor coordination became a part of instrumented action scheme. Future research is needed to investigate if intercorporeal coordination with the other is crucial for the transfer of sensorimotor coordination from their original source to instrumental activity in mathematics

Promoting insight into algebraic formulas through graphing by hand

Student insight into algebraic formulas, including the ability to identify the structure of a formula and its components and to reason with and about formulas, is an issue in mathematics education. In this study, we investigated how 16- and 17-year-old pre-university students’ insight into algebraic formulas can be promoted through graphing formulas by hand. In an intervention of five 90-min lessons, 21 grade 11 students were taught to graph formulas by hand. The intervention’s design was based on experts’ strategies in graphing formulas, that is, using a combination of recognition and qualitative reasoning, and on principles of teaching complex skills. To assess the effect of this intervention, pre-, post-, and retention tests were administered, as well as a post-intervention questionnaire. Six students were asked to think aloud during the pre- and posttests. The results show that all students improved their abilities to graph formulas by hand. The think-aloud data suggest that the students improved both on recognition and reasoning, and give a detailed picture of how students used recognition and qualitative reasoning in combination. We conclude that graphing formulas by hand, based on the interplay of recognition and qualitative reasoning, might be a means to promote students’ insight into algebraic formulas

Context, abstractie en vaardigheid in schoolalgebra

Van allerlei kanten wordt er geklaagd over het rendement van het wiskundeonderwijs in de tweede fase van het vwo. Studenten voelen zich slecht voorbereid op de universiteit en hebben de ‘Lieve Maria’ actie op touw gezet. Op universiteiten zijn zogenaamde deficiëntiecursussen gemeengoed. De opvattingen over benodigde maatregelen zijn verdeeld. Stemmen gaan op om minder tijd aan contexten te besteden en meer algebraïsche basisvaardigheden te oefenen. Een aantal medewerkers van het Freudenthal Instituut heeft in een recente bundel ‘Wat a is, dat kun je niet weten’ [1] de standpunten over algebraonderwijs opnieuw bepaald. Paul Drijvers, redacteur van deze publicatie, stelt dat de gesignaleerde problematiek serieus genomen moet worden, maar dat de voorgestelde oplossingen hun doel voorbij schieten

Tools and taxonomies: a response to Hoyles

In this response paper to Hoyles’ contribution “Transforming the mathematical practices of learners and teachers through digital technology” (Hoyles, 2018) focuses on three points. First, more knowledge is needed on why teaching and learning practices should transform, into what will they transform, and by what or by whom will they be transformed. Second, a suggestion is made for a more specific taxonomy on the didactical functionality of digital tools in mathematics education. Third, a plea is made for a future research agenda that addresses the ways in which activities with digital tools mediate the learning of mathematics in a fruitful way. This includes the interpretation and grading of online student work through intelligent mathematical software, and the notion of embodiment, as to do justice to the bodily experiences in which mathematical experiences are rooted

Wat bedoelen ze toch met... symbol sense?

In elke aflevering van de rubriek Wat bedoelen ze toch met... staat een spraakmakend begrip uit de wiskundedidactiek of de onderwijskunde centraal, waarover veel is geschreven, maar waarvan toepassing in de wiskundeles niet altijd meteen duidelijk is. Wat wordt met het wetenschappelijk jargon bedoeld en hoe vertaalt zich dit naar de onderwijspraktijk