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New Approaches to Basic Calculus: An Experimentation via Numerical Computation

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Book cover Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11973))

Abstract

The introduction of the first elements of calculus both in the first university year and in the last class of high schools, presents many problems both in Italy and abroad. Emblematic are the (numerous) cases in which students decide to change their course of study or give it up completely cause the difficulties with the first exam of mathematics, which usually deals with basic calculus. This work concerns an educational experimentation involving (with differentiated methods) about 170 students, part at the IPS “F. Besta” in Treviso (IT) with main focus on two 5th classes where the students’ age is about 19 years old, and part at the Liceo Classico Scientifico “XXV Aprile” in Pontedera, prov. of Pisa (IT). The experimental project aims to explore the teaching potential offered by non-classical approaches to calculus jointly with the so-called “unimaginable numbers”. In particular, we employed the computational method recently proposed by Y.D. Sergeyev and widely used both in mathematics, in applied sciences and, recently, also for educational purposes. In the paper will be illustrated tools, investigation methodologies, collected data (before and after the teaching unit), and the results of various class tests.

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Notes

  1. 1.

    The paper [4] concerns a rather complex two-year experimental project conducted at the University of Calabria within the Master’s Degree Program in Electronic Engineering. There, once two groups were formed each year, an experimental one equipped with computer-based tools and a control group associated with more traditional teaching methods, the aim was to analyze a series of student performances. We inform the reader that in the experimentations described in the present paper, we will find some slight traces of part of the methods used in [4]; the most visible is the employ of the computational system called Infinity Computer which will be used by a (very small) part of the sample, creating a hint of parallelism with the role of the software used in [4].

  2. 2.

    The mathematical intuition will be important also for us, when our students will work with the infinite and, in particular, when they will approach the new concept of grossone (see also [44]).

  3. 3.

    See, for instance, the manual [22] adopted in many academic courses and now in its third edition. See also [17] for an interesting comparison between Leibniz’ and Robinson’s systems and “Sarah”s conceptions”.

  4. 4.

    See, for instance, [16, page 2] and the proceedings [7] of the national Italian conference “Analisi nonstandard per le scuole superiori. VII Giornata di studio”. This conference takes place every year and has now reached its ninth edition, Verona, October 5, 2019.

  5. 5.

    The numbers used here to enumerate questions are different from those in the students’ test (cf. [21, Sect. 3]). We moreover precise that in some classes in Trento we prepared two or four test versions changing for the order of questions and answers, to prevent, together with other appropriate measures, any kind of influence among students.

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Aknowledgments

This work is partially supported by the research projects “IoT&B, Internet of Things and Blockchain”, CUP J48C17000230006, POR Calabria FESR-FSE 2014-2020.

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Authors and Affiliations

  1. Deparment of Mathematics and Computer Science, Università della Calabria, Cubo 31/A, 87036, Arcavacata di Rende, CS, Italy

    Luigi Antoniotti, Fabio Caldarola, Gianfranco d’Atri & Marco Pellegrini

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  1. Luigi Antoniotti
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  2. Fabio Caldarola
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Correspondence to Fabio Caldarola .

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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Antoniotti, L., Caldarola, F., d’Atri, G., Pellegrini, M. (2020). New Approaches to Basic Calculus: An Experimentation via Numerical Computation. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11973. Springer, Cham. https://doi.org/10.1007/978-3-030-39081-5_29

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