An instrumental orchestration for online course at the university level

In this article, we present a proposal for instrumental orchestration that organizes the use of technological environments in online mathematics education, in the synchronous mode for the concepts of eigenvalue and eigenvector of a first linear algebra course with engineering students. We used the instrumental orchestration approach as a theoretical framework to plan and organize the artefacts involved in the environment (didactic configuration) and the ways in which they are implemented (exploitation modes). The activities were designed using interactive virtual didactic scenarios, in a dynamic geometry environment, guided exploration worksheets with video and audio recordings of the work of the students, individually or in pairs. The results obtained are presented and the orchestrations of a pedagogical sequence to introduce the concepts of eigenvalue and eigenvector are briefly discussed. This work allowed us to identify new instrumental orchestrations for online mathematics education.

O movimento e suas implicações na aprendizagem de Matemática: um olhar fenomenológicoMoviment and its implications in learning mathematics: a phenomenological look

ResumoEste estudo foca o movimento como fenômeno de pesquisa, explicitando-o a partir de diferentes perspectivas teóricas, quais sejam: a física, a matemática e a fenomenologia, porém, assumindo a terceira para efetuar a análise. Mediante estudo no âmbito dessas áreas e análise de uma atividade desenvolvida em ambiente de Geometria Dinâmica, o objetivo da investigação é apresentar compreensões sobre como a percepção do movimento pode direcionar o pensar e contribuir com a aprendizagem de matemática. Para tanto, realizamos um estudo qualitativo de cunho bibliográfico, que forneceu compreensões que uma vez articuladas com a análise da atividade, permitiram ao estudo o entendimento de que o movimento é correlato a um sujeito que se move, movendo, e o permite conhecer as implicações desse ato materializando-se em seu mundo circundante e em seu corpo, que é o ponto zero do movimento e das percepções que realiza. Assim, o aprender dá-se na unidade movimento-percepção-conhecimento.Palavras-chave: Movimento, Fenomenologia, Educação matemática.AbstractThis study focuses on the movement as a research phenomenon, explaining it from different theoretical perspectives, namely: physics, mathematics, and phenomenology, however, assuming the third one to carry out the analysis. Through study in the scope of these areas and analysis of an activity developed in a dynamic geometry environment, this study aims to present understandings about how the perception of movement can direct thinking and contribute to the learning of mathematics. To this end, a qualitative study of a bibliographic nature is carried out, providing understandings that, once articulated with the analysis of the activity, allowed the study to recognize that the movement is correlated to a subject who moves, moving, and allows it to know the implications of this act materialising in his surrounding world and also in their body, which is the base of the movement and the perceptions it realises. Thus, learning occurs in the movement-perception-knowledge unit.Keywords: Movement, Phenomenology, Mathematical Education.ResumenEste estudio enfoca el movimiento como fenómeno a ser investigado, explicándolo a partir de diferentes perspectivas teóricas, sean: la física, la matemática y la fenomenología, sin embargo, asumiremos la tercera para efectuar nuestro análisis. El objetivo es, mediante el estudio en el ámbito de esas areas y el análisis de una actividad desarrollada en el ambiente de la geometría dinámica, presentar cómo la comprensión de la percepción del movimiento pode direccionar el pensamiento y contribuir al aprendizaje de la matemática. Para ese objetivo, se realiza un estudio cualitativo de carácter bibliográfico, que ofreció comprensiones que una vez articuladas con el análisis de la actividad, permitieron al estudio la comprensión de que el movimiento está relacionado a un sujeto que se mueve, moviéndose, y le permite conocer las implicaciones de ese acto materializándose en su mundo circundante y también en su cuerpo, el cual es el punto cero del movimiento y de las percepciones que realiza. Así, el aprendizaje se da en la unidad movimiento-percepción-conocimiento.Palabras clave: Movimiento, Fenomenología, Educación matemática.

Enhancing the Skill of Geometric Prediction Using Dynamic Geometry

This study concerns geometric prediction, a process of anticipation that has been identified as key in mathematical reasoning, and its possible constructive relationship with explorations within a Dynamic Geometry Environment (DGE). We frame this case study within Fischbein’s Theory of Figural Concepts and, to gain insight into a solver’s conceptual control over a geometrical figure, we introduce a set of analytical tools that include: the identification of the solver’s geometric predictions, theoretical and phenomenological evidence that s/he may seek for, and the dragging modalities s/he makes use of in the DGE. We present fine-grained analysis of data collected during a clinical interview as a high school student reasons about a geometrical task, first on paper-and-pencil, and then in a DGE. The results suggest that, indeed, the DGE exploration has the potential of strengthening the solver’s conceptual control, promoting its evolution toward theoretical control.

Creativity as a function of problem-solving expertise: posing new problems through investigations

This study was inspired by the following question: how is mathematical creativity connected to different kinds of expertise in mathematics? Basing our work on arguments about the domain-specific nature of expertise and creativity, we looked at how participants from two groups with two different types of expertise performed in problem-posing-through-investigations (PPI) in a dynamic geometry environment (DGE). The first type of expertise—MO—involved being a candidate or a member of the Israeli International Mathematical Olympiad team. The second type—MM—was comprised of mathematics majors who excelled in university mathematics. We conducted individual interviews with eight MO participants who were asked to perform PPI in geometry, without previous experience in performing a task of this kind. Eleven MMs tackled the same PPI task during a mathematics test at the end of a 52-h course that integrated PPI. To characterize connections between creativity and expertise, we analyzed participants’ performance on the PPI tasks according to proof skills (i.e., auxiliary constructions, the complexity of posed tasks, and correctness of their proofs) and creativity components (i.e., fluency, flexibility and originality of the discovered properties). Our findings demonstrate significant differences between PPI by MO participants and by MM participants as reflected in the more creative performance and more successful proving processes demonstrated by MO participants. We argue that problem posing and problem solving are inseparable when MO experts are engaged in PPI.