Mathematical Logic in High School: Hints and Proposals

In school, students learn how to reason and argue, and logic is the art of reasoning. Aristotle, who first developed it, held it to be so, i.e., the foundation of all science. But one certainly cannot impose on girls and boys an institutional course in logic as a prerequisite to all other knowledge. Not even in high school, when the matu- ration process of the students allows the teacher some more abstraction. In truth, in the Archive of Public Education - Cultural, Educational and Professional Pro- file of High Schools [129], it is stated that the logical-argumentative area assumes a central role, because it contributes to the formation of a citizen who “supports her/his own convictions, bringing adequate examples and counterexamples and us- ing concatenations of statements; accepts to change her/his opinion by recognising the logical consequences of a correct argumentation”. But education in logic must be done prudently, in the right formative ways. Some would argue that the practice of mathematics, often based on reasoning, in particular the model of Euclidean geometry, is in itself a cue to progressively in- sinuate logical mechanisms. Unfortunately, in recent times, however, Euclidean geometry seems to be a subject in disgrace, often neglected or forgotten. Instead, there are those who emphasise, in mathematics, the importance of intuition, dis- covery, experience and error, contrasting it with the excessive rigour of too many proofs. The purpose of this thesis is to propose various ideas that, within the fun- damental programmes of high school, specifically of Italian Liceo Classico and Liceo Scientifico, attempt to insinuate logic and accustom the students to logic in a way that we hope is light, clear and pleasant. We therefore do not propose a systematic treatment. We prefer to recall basic logic and then to give scattered ideas rather than a structured and definitive theory. But, as mentioned, we are confident that these hints can best prepare students of high school for logic. Indeed we address ourselves primarily to teachers and we believe that their knowl-edge of logic is useful and indeed necessary. But through them we also wish to address students. The thesis is organised as follows. The first chapter introduces and discusses the whole topic and explains why in our opinion logic is important in high schools. We also discuss how and when to propose it to students. The next two chapters introduce basic logic to teachers and students. The second illustrates the simplest logic, the Boolean one, recapitulating its essential points and emphasising in particular the use of connectives. The third deals with first-order logic, which we may consider the most classical of logics. Here we highlight in par- ticular the function of quantifiers. The following chapters propose several topics, belonging to logic or related to logic, that seem very intriguing and could be considered in high school. First, in chapter four, we treat Aristotelian syllogistics, which, even in recent times, frequently appears in various access tests. We will present some amusing introduc- tions to it, such as those of Lewis Carroll [25] or Pagnan-Rosolini [86]. Chapter five is dedicated to proofs without words. Relying on various examples from geometry, number theory and combinatorial calculus, it illustrates how reason- ing can sometimes be successfully expressed and developed through the images and intuitions they suggest. In this chapter, we also discuss the logic of images proposed by Leonardo in the Codex Atlanticus [73] to address and solve geometric questions often linked to the Pythagorean theorem. The sixth chapter is dedicated to what Henri Poincaré called the “mathematical reasoning” par excellence, namely the principle of induction. This law governs nat- ural numbers and is often used as a powerful demonstrative tool in various exercises. However, students seldom learn it and above all understand it properly. Drawing on the history of the principle of induction, from its primitive intuitions to its for- malisation by Peano and Dedekind, we attempt to approach it in what we hope will be a pleasant and appealing way, also o ering a wide range of examples. Logical paradoxes are another logical theme that is impossible to forget: mental games that not only disorientate but also intrigue and amuse, which are the heart of the seventh chapter. To the classical logics considered at the beginning of the thesis, based on two truth values, yes or no, we then contrast multi-valued and fuzzy logics, which are better suited to analysing situations of uncertainty. We link them, in chapter eight, to the Rényi-Ulam game, which searches for truth in contexts in which the information received may be lying and deceptive. The final chapter takes up a basic topic of high school mathematics: equations. Diophantine games show us how they can be an opportunity for challenge and fun, as well as suggesting intriguing insights into fundamental themes of modern math- ematics: not only number theory and algebra, but also game theory and the theory of computability and computational complexity. For a general and in-depth overview of mathematical logic, we refer to [10] and [114]. For the theory of computability and computational complexity we refer the reader to [37] and [83], for number theory to [60]

Technical Debt Prioritization: State of the Art. A Systematic Literature Review

Background. Software companies need to manage and refactor Technical Debt issues. Therefore, it is necessary to understand if and when refactoring Technical Debt should be prioritized with respect to developing features or fixing bugs. Objective. The goal of this study is to investigate the existing body of knowledge in software engineering to understand what Technical Debt prioritization approaches have been proposed in research and industry. Method. We conducted a Systematic Literature Review among 384 unique papers published until 2018, following a consolidated methodology applied in Software Engineering. We included 38 primary studies. Results. Different approaches have been proposed for Technical Debt prioritization, all having different goals and optimizing on different criteria. The proposed measures capture only a small part of the plethora of factors used to prioritize Technical Debt qualitatively in practice. We report an impact map of such factors. However, there is a lack of empirical and validated set of tools. Conclusion. We observed that technical Debt prioritization research is preliminary and there is no consensus on what are the important factors and how to measure them. Consequently, we cannot consider current research conclusive and in this paper, we outline different directions for necessary future investigations

Lode a Maria SS. Addolorata

Italian hymn to Our Lady of Sorrows, composed by Carlo Gerbaldi with lyrics by Father Antonio Fontana. This sheet music was published by the Church of San Carlo in Torino.https://ecommons.udayton.edu/imri_sheetmusic/1058/thumbnail.jp

Efficacy of spa-therapy, mud-pack therapy, balneotherapy and mud-bath therapy in the management of knee osteoarthritis. A systematic review

Background: Osteoarthritis (OA) is the most common musculoskeletal disease in the world. OA is the result of an inflammatory and degenerative process affecting the entire joint. Osteoarthritis, especially involving the knee, has a relevant socio-economic impact in terms of drugs, hospital admissions, work absences and temporary or permanent invalidity. Therapy of knee osteoarthritis is based on pharmacological and non-pharmacological measures. Methods: We conducted a systematic review of the studies published between 2002 and 2017 on spa-therapy, mud-pack therapy, balneotherapy and mud-bath therapy in the treatment of knee osteoarthritis in order to investigate the evidence of the efficacy of such treatment on pain, functional limitation, drug use and quality of life. Overall, 35 studies were examined among which 12 were selected and included in the review if trial comparative. Each report was reviewed to identify the criteria used for study enrolment and for assignment to experimental vs control groups, sample size, type and characteristics of treatment, features of mineral water, control intervention, assessment point, endpoints, outcome measures, tests used for statistical analysis of the results. We have been able to illustrate the main results obtained in the individual studies and to elaborate these results in order to allow as much a unitary presentation as possible, and hence an overall judgment. Results: Because the studies we reviewed differed markedly from one another in terms of the methods used, we were unable to conduct a quantitative analysis (meta-analysis) of pooled data from the 12 studies. For the purposes of the present review, we re-evaluated the results of the different studies using the same statistical method, the Student’s t test, which is used to compare the means of two frequency distributions. Among all the studies, the most relevant indexes used to measure effectiveness of spa therapy were improved including VAS, Lequesne’s and WOMAC Score. Conclusions: The mud-pack therapy, balneotherapy, mud-bath therapy and spa therapy has proved to be effective in the treatment and in the secondary prevention of knee osteoarthritis, by reducing pain, non-steroidal antiinflammatory drug consumption, functional limitation and improving quality of life of affected patients. Is a noninvasive, complication-free, and cost-effective alternative modality for the conservative treatment of knee osteoarthritis

Suitability of ground-based SfM-MVS for monitoring glacial and periglacial processes

Photo-based surface reconstruction is rapidly emerging as an alternative survey technique to lidar (light detection and ranging) in many fields of geoscience fostered by the recent development of computer vision algorithms such as structure from motion (SfM) and dense image matching such as multi-view stereo (MVS). The objectives of this work are to test the suitability of the ground-based SfM-MVS approach for calculating the geodetic mass balance of a 2.1km2 glacier and for detecting the surface displacement of a neighbouring active rock glacier located in the eastern Italian Alps. The photos were acquired in 2013 and 2014 using a digital consumer-grade camera during single-day field surveys. Airborne laser scanning (ALS, otherwise known as airborne lidar) data were used as benchmarks to estimate the accuracy of the photogrammetric digital elevation models (DEMs) and the reliability of the method. The SfM-MVS approach enabled the reconstruction of high-quality DEMs, which provided estimates of glacial and periglacial processes similar to those achievable using ALS. In stable bedrock areas outside the glacier, the mean and the standard deviation of the elevation difference between the SfM-MVS DEM and the ALS DEM was-0.42 \ub1 1.72 and 0.03 \ub1 0.74 m in 2013 and 2014, respectively. The overall pattern of elevation loss and gain on the glacier were similar with both methods, ranging between-5.53 and + 3.48 m. In the rock glacier area, the elevation difference between the SfM-MVS DEM and the ALS DEM was 0.02 \ub1 0.17 m. The SfM-MVS was able to reproduce the patterns and the magnitudes of displacement of the rock glacier observed by the ALS, ranging between 0.00 and 0.48 m per year. The use of natural targets as ground control points, the occurrence of shadowed and low-contrast areas, and in particular the suboptimal camera network geometry imposed by the morphology of the study area were the main factors affecting the accuracy of photogrammetric DEMs negatively. Technical improvements such as using an aerial platform and/or placing artificial targets could significantly improve the results but run the risk of being more demanding in terms of costs and logistics

A Proof of Descartes’ Rule of Signs

In 1637 Descartes, in his famous Géométrie, gave the rule of the signs without a proof. Later many different proofs appeared of algebraic and analytic nature. Among them in 1828 the algebraic proof of Gauss. In this note we present a proof of Descartes’ rule of signs that uses the roots of the first derivative of a polynomial and that can be presented to the students of the last year of a secondary school

Smooth Lasso Estimator for the Function-on-Function Linear Regression Model

A new estimator, named as S-LASSO, is proposed for the coefficient function of a functional linear regression model where values of the response function, at a given domain point, depends on the full trajectory of the covariate function. The S-LASSO estimator is shown to be able to increase the interpretability of the model, by better locating regions where the coefficient function is zero, and to smoothly estimate non-zero values of the coefficient function. The sparsity of the estimator is ensured by a functional LASSO penalty whereas the smoothness is provided by two roughness penalties. The resulting estimator is proved to be estimation and pointwise sign consistent. Via an extensive Monte Carlo simulation study, the estimation and predictive performance of the S-LASSO estimator are shown to be better than (or at worst comparable with) competing estimators already presented in the literature before. Practical advantages of the S-LASSO estimator are illustrated through the analysis of the well known \textit{Canadian weather} and \textit{Swedish mortality dat

La evaluación en la educación para la sostenibilidad desde el paradigma de la complejidad

La inclusión de la sostenibilidad en el curriculum implica la asunción de unos presupuestos metodológicos coherentes con los principios que la inspiran. Los sistemas de evaluación juegan un papel crucial a la hora de apostar por dichos presupuestos ligados a la acción docente. Es intención de la investigación que se presenta recorrer los caminos hacia la evaluación, tierra de nadie, paso obligado por todos, aproximándonos a qué entendemos por un sistema de evaluación coherente con la sostenibilidad curricular bajo el paradigma de la complejidad. Se trata de elaborar un instrumento que permita analizar la coherencia de la inclusión de la sostenibilidad en las aulas universitarias, con los sistemas de evaluación de los procesos de enseñanza aprendizaje desarrollados en dichas aulas. Para ello hemos iniciado un estudio de caso