Conceptions about mathematics, its teaching and learning in Compendio Mathematico (1707) written by the Spanish Thomas Vicente Tosca (1651-1723)

The preface of a book is one of the main examples of paratexts, defined by Gérard Genette as those devices and conventions that enable a text to become a book. It can provide information about aspects such as the author''s motivation and intention, the origin of the presented ideas, the potential readers, etc. In the particular case of a mathematical text devoted to some extent to teaching, the preface can provide information about the author''s conceptions and beliefs about mathematics, its teaching and learning. In this work, we analyze the preface of Compendio Mathematico written by Thomas Vicente Tosca in 1707. This approach will allow us to have a view of how the teaching and learning of mathematics as well as mathematics itself were conceived during Spanish pre-enlightenment. O prólogo de um livro é um dos múltiplos exemplos de paratextos, definidos por Gérard Genette como esses elementos e convenções que fazem um texto tornar-se um livro. Isso pode fornecer informações sobre aspectos como a motivação e a intenção do autor, a origem das ideias apresentadas, potenciais leitores etc. No caso particular de textos matemáticos dedicados em certa medida ao ensino, o prólogo pode informar as concepções e crenças do autor sobre a matemática, seu ensino e sua aprendizagem. Neste trabalho, analisamos o prólogo do Compendio Mathematico, escrito por Thomas Vicente Tosca em 1707. Essa abordagem nos permitirá obter uma visão sobre como a matemática, seu ensino e sua aprendizagem foi concebida durante a pré-ilustração na Espanha

On the diameter of the commuting graph of the full matrix ring over the real numbers

In a recent paper C. Miguel proved that the diameter of the commuting graph of the matrix ring Mn(R) is equal to 4 if either n = 3 or n = 5. But the case n = 4 remained open, since the diameter could be 4 or 5. In this work we close the problem showing that also in this case the diameter is 4

Characterization of behavior of correctors when grading mathematics tests

In this work, we present some results obtained from the analysis of the behavior of 91 mathematics teachers (prospective, secondary education and university) when they grade three different types of correct answers to a classical high school problem through a questionnaire. In addition to a descriptive analysis that studies the variability and the interrater reliability, we analyze the role of experience and training as well as the influence of the different solving methods. Furthermore, we try to identify profiles of correctors among secondary education teachers using both quantitative (cluster analysis) and qualitative (content analysis) methods. In particular, we observe a great variability on the assigned grades as well as a low interrater reliability. The belonging to a particular group has impact over the assigned rates while experience has no significant influence. The grades are higher when methods closer to the corrector are used. Finally, we have been able to identify three different clusters, which are determined by the comments and actions regarding three aspects of the students’ answers: argumentation, correctness and method

Problemas de relojes. Ejemplos históricos y consideraciones didácticas

Abordamos un problema matemático clásico: aquel en el que se trata de calcular el tiempo que debe transcurrir, a partir de una hora determinada, para que las manecillas de un reloj ocupen una posición concreta. En particular, nos centramos en el caso en que la disposición requerida es que las agujas estén superpuestas. En este artículo, presentamos diversos ejemplos extraídos de textos clásicos y del siglo XIX que ilustran distintos contextos en los que se presenta el problema, así como diferentes métodos de resolución. Además, como consecuencia de dicho análisis, presentamos algunas consideraciones didácticas que pueden motivar el trabajo de estos materiales con profesorado en formación. We approach a classical Mathematical problem: that of computing the time passed, from a given moment, until the hands of a clock reach certain position. In particular, we focus on the case when the required position is the superposition of both hands. In this paper, we present some examples from classic and nineteenth century texts presenting different contexts where the problem arises as well as different solving methods. In addition, and as a consequence of this analysis, we present some didactical considerations that motivate the use of these resources with prospective teachers

Diseño e implementación de tareas de alta demanda cognitiva basadas en la sucesión look and say

Pese a que su tratamiento escolar usual se centra en aspectos principalmente de cálculo, las sucesiones son un tópico matemático con el potencial para desarrollar en los alumnos aspectos del razonamiento matemático. En este trabajo se diseña una secuencia de tareas de alta demanda cognitiva basadas en la sucesión ‘look and say’ y se implementa en un grupo de secundaria con especial interés por las matemáticas durante una sesión del Taller de Talento Matemático en la Universidad de Zaragoza. La metodología es exploratoria y descriptiva con análisis mixto de datos cualitativos. Los alumnos participantes resolvieron las tareas con un alto grado de éxito y surgieron bastantes respuestas de gran riqueza conceptual. Estas tareas pueden ser útiles para trabajar aspectos transversales del currículo e identificar alumnos con altas capacidades matemáticas. Even though their school treatment is mainly based on calculations, numerical sequences are a mathematical topic with the potential to develop aspects of mathematical reasoning amongststudents.In this work, we design a sequence of tasksofcognitive high-demandbased on the ‘look and say’se-quence andimplement them with a group of secondary schoolstudents particularly interested in math-ematics during a session of the Workshop of Mathematical Talent at the University of Zaragoza.The methodology is exploratory and descriptive with mixedanalysis of qualitative data.Participants solved the taskswith a high rate of success and several answers were ofhigh conceptual richness. Thesetasksmight be useful to work transversal curricular aspectsand to identifythosemathematically gifted

Fast computation of the number of solutions to x12+ ··· + xk2 = ¿ (mod n)

In this paper we study the multiplicative function ¿k,¿(n) that counts the number of solutions of the equation x1 2+...+xk 2=¿(modn) in (Z/nZ)k. In particular we give closed explicit formulas for ¿k,¿(ps). This leads to an algorithm with an arithmetic complexity of constant order that improves previous work by Tóth [10] and completes the quadratic case considered by Li and Ouyang in [8]

Strategies used by students from different years when they face compound proportion problems

En este trabajo se analizan las actuaciones de alumnos desde 6.º de Educación Primaria hasta 2.º de Educación Secundaria Obligatoria (11-14 años) con distintos grados de instrucción en proporcionalidad al resolver ciertos problemas de proporcionalidad compuesta. En particular, observamos la tasa de éxito y las distintas estrategias empleadas, tanto correctas como incorrectas. Los resultados muestran un aumento progresivo de la tasa de éxito y una evolución en las estrategias utilizadas desde aquellas que hacen referencia a aspectos de razonamiento proporcional, hacia aquellas en las que prima el componente algorítmico.In this work, we analyze the behavior of students from 6th to 8th grade (age 11-14) with different degrees of instruction regarding proportionality, when they solve certain compound proportionality problems. In particular, we focus on the success rate and on their strategies, both correct and incorrect. Results show a gradual increasing in the success rate and an evolution in the strategies from those involving proportional reasoning to those giving priority to algorithmic procedures.Este artículo surge del trabajo desarrollado por el grupo de investigación "S119-Investigación en Educación Matemática" financiado por el Gobierno de Aragón y el Fondo Social Europeo