Mathematics

We present a new measure for evaluating focused versus overview eye movement behavior in a stimulus divided by areas of interest. The measure can be used for overall data, as well as data over time. Using data from an ongoing project with mathematical problem solving, we describe how to calculate the measure and how to carry out a statistical evaluation of the results.

The main purpose of this study is the analysis of some basic ideas of embodied cognition in Mathematics Education research. In particular, the authors are concerned with the cognitive analysis of students' activities, which are mainly based on the use of image schemas and grounding metaphors. This article describes a teaching experiment carried out with a group of students attending secondary school in Italy (9 th grade). The experiment is supported by both 'paper and pencil' environment and a technological device.

The purpose of this paper is to introduce two perspectives for observing students' activity, taking into account the main elements of their communication and thinking processes: gestures, speech, written words, mathematical signs. These perspectives give origin to two different types of analyses called parallel and serial analysis, in order to distinguish between a research attention to the contemporary production of gestures and words, or to the functions of signs, sequentially introduced during the activity.

In this paper, we consider gestures as part of the resources activated in the mathematics classroom: speech, inscriptions, artifacts, etc. As such, gestures are seen as one of the semiotic tools used by students and teacher in mathematics teaching–learning. To analyze them, we introduce a suitable model, the semiotic bundle. It allows focusing on the relationships of gestures with the other semiotic resources within a multimodal approach. It also enables framing the mediating action of the teacher in the classroom: in this respect, we introduce the notion of semiotic game where gestures are one of the major ingredients.

The paper presents an approach to modelling in secondary schools where technological instruments are used for measuring and modelling motion experiences. In all cases one or more sensors measure various quantities and are connected to a calculator. In some examples we study pupils (9-th grade) who run in the class and see the Cartesian representation of their movement produced by a sensor in real time. In others, pupils (11–13-th grade) go on switchbacks or other similar merry-go-rounds and use instruments to measure some quantities (speed, acceleration, pressure), which are recorded on graphs and tables. In both cases, pupils discuss what has happened and interpret the collected data. Within a general Vygotskian frame, the authors use different complementary tools to analyse the situations: the embodied cognition by Lakoff and Núñez, the instrumental approach by Rabardel, the definition of concept by Vergnaud. In particular the role of the perceptual-motor activity in the conceptualisation of mathematics through modelling is stressed.

This paper sits within the research on the affordances of new technologies in the mathematics classroom and focuses on a specific feature that is available in dynamic geometry environments, i.e. measuring tools, within the context of conjecturing and proving in open geometry problems. We develop a classification of different modalities of measuring, based on our previous work on dragging. The modalities are illustrated through the analysis of 15–16 year-old students’ proving processes, which focuses on how these modalities relate to the moves between the spatio-graphical field and the theoretical field and may either support or hinder the proving process. The classification of the modalities of measuring enables researchers to access students’ cognitive processes and teachers to be aware of the different possible uses and interpretation of measuring, giving them tools to support students when difficulties arise.

In this paper I present a case-study, part of a teaching experiment in the secondary school (11 th -12 th grade), focused on the construction of the concept of integral in the environment of measurements, considered by the point of view of Tall's cognitive roots. Particularly, the study refers to the students' cognitive processes with the framework of embodied cognition and instrumental approach. The aim is to analyse the passage from finite sums to infinite ones, with the mediation of the technology. O. Robutti 6 Andrea: "That it gets nearer to the average value"

In this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we r 1 3 and s = r 1 3 we obtain equations (2).

In questo lavoro viene introdotto il concetto diL-divisore di un anello. Questo concetto corrisponde esattamente a quello diL-fuzzy-ideale primo. GliL-divisori vengono studiati singolarmente, in quanto «valutazioni», e nel loro insieme, come costituenti uno «spazio strutturale». Il caso dei corpi e degli anelli booleani è trattato con completezza.

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from (0, . . . , 0, 1)) into any other impulse sequence, by two processes that we call construction and deconstruction. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that r-bonacci numbers are a Bell polynomial transform of the (r − 1)bonacci numbers.

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from $(0,...,0,1)$) into any other impulse sequence, by two processes that we call \emph{construction} and \emph{deconstruction}. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that $r$--bonacci numbers are a Bell polynomial transform of the $(r-1)$--bonacci numbers.